id	sid	tid	token	lemma	pos
cana-1630	1	1	communications	communication	NOUN
cana-1630	1	2	on	on	ADP
cana-1630	1	3	applied	apply	VERB
cana-1630	1	4	nonlinear	nonlinear	ADJ
cana-1630	1	5	analysis	analysis	NOUN
cana-1630	1	6	issn	issn	NOUN
cana-1630	1	7	:	:	PUNCT
cana-1630	1	8	1074	1074	NUM
cana-1630	1	9	-	-	PUNCT
cana-1630	1	10	133x	133x	NUM
cana-1630	1	11	vol	vol	NOUN
cana-1630	1	12	32	32	NUM
cana-1630	1	13	no	no	NOUN
cana-1630	1	14	.	.	NOUN
cana-1630	1	15	1	1	NUM
cana-1630	1	16	(	(	PUNCT
cana-1630	1	17	2025	2025	NUM
cana-1630	1	18	)	)	PUNCT
cana-1630	1	19	184	184	NUM
cana-1630	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	1	21	stability	stability	NOUN
cana-1630	1	22	and	and	CCONJ
cana-1630	1	23	preservation	preservation	NOUN
cana-1630	1	24	of	of	ADP
cana-1630	1	25	fuzzy	fuzzy	ADJ
cana-1630	1	26	membership	membership	NOUN
cana-1630	1	27	functions	function	NOUN
cana-1630	1	28	in	in	ADP
cana-1630	1	29	adaptive	adaptive	ADJ
cana-1630	1	30	gaussian	gaussian	ADJ
cana-1630	1	31	derivative	derivative	ADJ
cana-1630	1	32	filters	filter	NOUN
cana-1630	1	33	aarthi	aarthi	VERB
cana-1630	1	34	d1	d1	PROPN
cana-1630	1	35	,	,	PUNCT
cana-1630	1	36	panimalar	panimalar	ADJ
cana-1630	1	37	a2	a2	PROPN
cana-1630	1	38	,	,	PUNCT
cana-1630	1	39	santhosh	santhosh	PROPN
cana-1630	1	40	kumar	kumar	PROPN
cana-1630	1	41	s3	s3	PROPN
cana-1630	1	42	,	,	PUNCT
cana-1630	1	43	anitha	anitha	NOUN
cana-1630	1	44	k4	k4	PROPN
cana-1630	1	45	1,3	1,3	NUM
cana-1630	1	46	sri	sri	PROPN
cana-1630	1	47	ramakrishna	ramakrishna	PROPN
cana-1630	1	48	mission	mission	PROPN
cana-1630	1	49	vidyalaya	vidyalaya	PROPN
cana-1630	1	50	college	college	PROPN
cana-1630	1	51	of	of	ADP
cana-1630	1	52	arts	art	NOUN
cana-1630	1	53	and	and	CCONJ
cana-1630	1	54	science	science	NOUN
cana-1630	1	55	,	,	PUNCT
cana-1630	1	56	coimbatore	coimbatore	PROPN
cana-1630	1	57	,	,	PUNCT
cana-1630	1	58	india	india	PROPN
cana-1630	1	59	2kgisl	2kgisl	PROPN
cana-1630	1	60	institute	institute	PROPN
cana-1630	1	61	of	of	ADP
cana-1630	1	62	technology	technology	PROPN
cana-1630	1	63	,	,	PUNCT
cana-1630	1	64	coimbatore	coimbatore	PROPN
cana-1630	1	65	,	,	PUNCT
cana-1630	1	66	india	india	PROPN
cana-1630	1	67	4amrita	4amrita	NUM
cana-1630	1	68	school	school	NOUN
cana-1630	1	69	of	of	ADP
cana-1630	1	70	engineering	engineering	PROPN
cana-1630	1	71	,	,	PUNCT
cana-1630	1	72	amrita	amrita	PROPN
cana-1630	1	73	vishwa	vishwa	PROPN
cana-1630	1	74	vidyapeetham	vidyapeetham	PROPN
cana-1630	1	75	chennai	chennai	PROPN
cana-1630	1	76	,	,	PUNCT
cana-1630	1	77	india	india	PROPN
cana-1630	1	78	e	e	PROPN
cana-1630	1	79	-	-	NOUN
cana-1630	1	80	mail	mail	NOUN
cana-1630	1	81	fuzzysansrmvcas@gmail.com	fuzzysansrmvcas@gmail.com	NOUN
cana-1630	1	82	article	article	NOUN
cana-1630	1	83	history	history	NOUN
cana-1630	1	84	:	:	PUNCT
cana-1630	1	85	received	receive	VERB
cana-1630	1	86	:	:	PUNCT
cana-1630	1	87	10	10	NUM
cana-1630	1	88	-	-	SYM
cana-1630	1	89	07	07	NUM
cana-1630	1	90	-	-	PUNCT
cana-1630	1	91	2024	2024	NUM
cana-1630	1	92	revised	revise	VERB
cana-1630	1	93	:	:	PUNCT
cana-1630	1	94	26	26	NUM
cana-1630	1	95	-	-	SYM
cana-1630	1	96	08	08	NUM
cana-1630	1	97	-	-	PUNCT
cana-1630	1	98	2024	2024	NUM
cana-1630	1	99	accepted	accept	VERB
cana-1630	1	100	:	:	PUNCT
cana-1630	1	101	07	07	NUM
cana-1630	1	102	-	-	PUNCT
cana-1630	1	103	09	09	NUM
cana-1630	1	104	-	-	PUNCT
cana-1630	1	105	2024	2024	NUM
cana-1630	1	106	abstract	abstract	NOUN
cana-1630	1	107	:	:	PUNCT
cana-1630	1	108	this	this	DET
cana-1630	1	109	paper	paper	NOUN
cana-1630	1	110	examines	examine	VERB
cana-1630	1	111	the	the	DET
cana-1630	1	112	stability	stability	NOUN
cana-1630	1	113	and	and	CCONJ
cana-1630	1	114	preservation	preservation	NOUN
cana-1630	1	115	of	of	ADP
cana-1630	1	116	fuzzy	fuzzy	ADJ
cana-1630	1	117	membership	membership	NOUN
cana-1630	1	118	functions	function	NOUN
cana-1630	1	119	in	in	ADP
cana-1630	1	120	gaussian	gaussian	ADJ
cana-1630	1	121	filters	filter	NOUN
cana-1630	1	122	,	,	PUNCT
cana-1630	1	123	particularly	particularly	ADV
cana-1630	1	124	focusing	focus	VERB
cana-1630	1	125	on	on	ADP
cana-1630	1	126	the	the	DET
cana-1630	1	127	adaptive	adaptive	ADJ
cana-1630	1	128	gaussian	gaussian	NOUN
cana-1630	1	129	derivative	derivative	NOUN
cana-1630	1	130	(	(	PUNCT
cana-1630	1	131	agd	agd	PROPN
cana-1630	1	132	)	)	PUNCT
cana-1630	1	133	filter	filter	NOUN
cana-1630	1	134	.	.	PUNCT
cana-1630	2	1	gaussian	gaussian	ADJ
cana-1630	2	2	filters	filter	NOUN
cana-1630	2	3	are	be	AUX
cana-1630	2	4	essential	essential	ADJ
cana-1630	2	5	for	for	ADP
cana-1630	2	6	smoothing	smoothing	NOUN
cana-1630	2	7	and	and	CCONJ
cana-1630	2	8	noise	noise	NOUN
cana-1630	2	9	reduction	reduction	NOUN
cana-1630	2	10	in	in	ADP
cana-1630	2	11	image	image	NOUN
cana-1630	2	12	processing	processing	NOUN
cana-1630	2	13	.	.	PUNCT
cana-1630	3	1	the	the	DET
cana-1630	3	2	agd	agd	PROPN
cana-1630	3	3	filter	filter	NOUN
cana-1630	3	4	,	,	PUNCT
cana-1630	3	5	which	which	PRON
cana-1630	3	6	adapts	adapt	VERB
cana-1630	3	7	based	base	VERB
cana-1630	3	8	on	on	ADP
cana-1630	3	9	local	local	ADJ
cana-1630	3	10	image	image	NOUN
cana-1630	3	11	statistics	statistic	NOUN
cana-1630	3	12	,	,	PUNCT
cana-1630	3	13	outperforms	outperform	VERB
cana-1630	3	14	traditional	traditional	ADJ
cana-1630	3	15	methods	method	NOUN
cana-1630	3	16	.	.	PUNCT
cana-1630	4	1	key	key	ADJ
cana-1630	4	2	concepts	concept	NOUN
cana-1630	4	3	like	like	ADP
cana-1630	4	4	fuzzy	fuzzy	ADJ
cana-1630	4	5	sets	set	NOUN
cana-1630	4	6	,	,	PUNCT
cana-1630	4	7	boundary	boundary	ADJ
cana-1630	4	8	input	input	NOUN
cana-1630	4	9	boundary	boundary	ADJ
cana-1630	4	10	output	output	NOUN
cana-1630	4	11	(	(	PUNCT
cana-1630	4	12	bibo	bibo	ADJ
cana-1630	4	13	)	)	PUNCT
cana-1630	4	14	stability	stability	NOUN
cana-1630	4	15	,	,	PUNCT
cana-1630	4	16	and	and	CCONJ
cana-1630	4	17	convolution	convolution	NOUN
cana-1630	4	18	are	be	AUX
cana-1630	4	19	defined	define	VERB
cana-1630	4	20	,	,	PUNCT
cana-1630	4	21	and	and	CCONJ
cana-1630	4	22	the	the	DET
cana-1630	4	23	mathematical	mathematical	ADJ
cana-1630	4	24	formulation	formulation	NOUN
cana-1630	4	25	of	of	ADP
cana-1630	4	26	the	the	DET
cana-1630	4	27	gaussian	gaussian	NOUN
cana-1630	4	28	and	and	CCONJ
cana-1630	4	29	its	its	PRON
cana-1630	4	30	derivative	derivative	NOUN
cana-1630	4	31	is	be	AUX
cana-1630	4	32	presented	present	VERB
cana-1630	4	33	.	.	PUNCT
cana-1630	5	1	the	the	DET
cana-1630	5	2	agd	agd	PROPN
cana-1630	5	3	filter	filter	NOUN
cana-1630	5	4	's	's	PART
cana-1630	5	5	bibo	bibo	ADJ
cana-1630	5	6	stability	stability	NOUN
cana-1630	5	7	ensures	ensure	VERB
cana-1630	5	8	bounded	bound	VERB
cana-1630	5	9	outputs	output	NOUN
cana-1630	5	10	for	for	ADP
cana-1630	5	11	bounded	bounded	ADJ
cana-1630	5	12	inputs	input	NOUN
cana-1630	5	13	,	,	PUNCT
cana-1630	5	14	guaranteeing	guarantee	VERB
cana-1630	5	15	consistent	consistent	ADJ
cana-1630	5	16	behavior	behavior	NOUN
cana-1630	5	17	.	.	PUNCT
cana-1630	6	1	it	it	PRON
cana-1630	6	2	also	also	ADV
cana-1630	6	3	preserves	preserve	VERB
cana-1630	6	4	fuzzy	fuzzy	ADJ
cana-1630	6	5	membership	membership	NOUN
cana-1630	6	6	function	function	NOUN
cana-1630	6	7	properties	property	NOUN
cana-1630	6	8	,	,	PUNCT
cana-1630	6	9	maintaining	maintain	VERB
cana-1630	6	10	convexity	convexity	NOUN
cana-1630	6	11	and	and	CCONJ
cana-1630	6	12	boundedness	boundedness	NOUN
cana-1630	6	13	through	through	ADP
cana-1630	6	14	linearity	linearity	NOUN
cana-1630	6	15	and	and	CCONJ
cana-1630	6	16	continuity	continuity	NOUN
cana-1630	6	17	.	.	PUNCT
cana-1630	7	1	frequency	frequency	NOUN
cana-1630	7	2	response	response	NOUN
cana-1630	7	3	analysis	analysis	NOUN
cana-1630	7	4	using	use	VERB
cana-1630	7	5	fourier	fourier	NOUN
cana-1630	7	6	transform	transform	NOUN
cana-1630	7	7	confirms	confirm	VERB
cana-1630	7	8	the	the	DET
cana-1630	7	9	agd	agd	PROPN
cana-1630	7	10	filter	filter	NOUN
cana-1630	7	11	retains	retain	VERB
cana-1630	7	12	the	the	DET
cana-1630	7	13	gaussian	gaussian	ADJ
cana-1630	7	14	shape	shape	NOUN
cana-1630	7	15	in	in	ADP
cana-1630	7	16	the	the	DET
cana-1630	7	17	frequency	frequency	NOUN
cana-1630	7	18	domain	domain	NOUN
cana-1630	7	19	,	,	PUNCT
cana-1630	7	20	preserving	preserve	VERB
cana-1630	7	21	image	image	NOUN
cana-1630	7	22	smoothness	smoothness	ADJ
cana-1630	7	23	.	.	PUNCT
cana-1630	8	1	theorems	theorem	NOUN
cana-1630	8	2	and	and	CCONJ
cana-1630	8	3	proofs	proof	NOUN
cana-1630	8	4	validate	validate	VERB
cana-1630	8	5	the	the	DET
cana-1630	8	6	agd	agd	PROPN
cana-1630	8	7	filter	filter	NOUN
cana-1630	8	8	's	's	PART
cana-1630	8	9	stability	stability	NOUN
cana-1630	8	10	and	and	CCONJ
cana-1630	8	11	its	its	PRON
cana-1630	8	12	capability	capability	NOUN
cana-1630	8	13	to	to	PART
cana-1630	8	14	preserve	preserve	VERB
cana-1630	8	15	fuzzy	fuzzy	ADJ
cana-1630	8	16	membership	membership	NOUN
cana-1630	8	17	functions	function	NOUN
cana-1630	8	18	,	,	PUNCT
cana-1630	8	19	ensuring	ensure	VERB
cana-1630	8	20	reliable	reliable	ADJ
cana-1630	8	21	processing	processing	NOUN
cana-1630	8	22	in	in	ADP
cana-1630	8	23	applications	application	NOUN
cana-1630	8	24	such	such	ADJ
cana-1630	8	25	as	as	ADP
cana-1630	8	26	medical	medical	ADJ
cana-1630	8	27	image	image	NOUN
cana-1630	8	28	analysis	analysis	NOUN
cana-1630	8	29	.	.	PUNCT
cana-1630	9	1	these	these	DET
cana-1630	9	2	properties	property	NOUN
cana-1630	9	3	make	make	VERB
cana-1630	9	4	the	the	DET
cana-1630	9	5	agd	agd	PROPN
cana-1630	9	6	filter	filter	NOUN
cana-1630	9	7	a	a	DET
cana-1630	9	8	robust	robust	ADJ
cana-1630	9	9	tool	tool	NOUN
cana-1630	9	10	for	for	ADP
cana-1630	9	11	advanced	advanced	ADJ
cana-1630	9	12	image	image	NOUN
cana-1630	9	13	processing	processing	NOUN
cana-1630	9	14	tasks	task	NOUN
cana-1630	9	15	.	.	PUNCT
cana-1630	10	1	keywords	keyword	NOUN
cana-1630	10	2	:	:	PUNCT
cana-1630	10	3	agd	agd	PROPN
cana-1630	10	4	filter	filter	PROPN
cana-1630	10	5	,	,	PUNCT
cana-1630	10	6	fuzzy	fuzzy	ADJ
cana-1630	10	7	,	,	PUNCT
cana-1630	10	8	image	image	NOUN
cana-1630	10	9	,	,	PUNCT
cana-1630	10	10	gaussian	gaussian	NOUN
cana-1630	10	11	.	.	PUNCT
cana-1630	11	1	1	1	X
cana-1630	11	2	.	.	X
cana-1630	11	3	introduction	introduction	NOUN
cana-1630	11	4	fuzzy	fuzzy	ADJ
cana-1630	11	5	logic	logic	NOUN
cana-1630	11	6	systems	system	NOUN
cana-1630	11	7	have	have	AUX
cana-1630	11	8	garnered	garner	VERB
cana-1630	11	9	significant	significant	ADJ
cana-1630	11	10	attention	attention	NOUN
cana-1630	11	11	for	for	ADP
cana-1630	11	12	their	their	PRON
cana-1630	11	13	capability	capability	NOUN
cana-1630	11	14	to	to	PART
cana-1630	11	15	handle	handle	VERB
cana-1630	11	16	uncertainty	uncertainty	NOUN
cana-1630	11	17	and	and	CCONJ
cana-1630	11	18	imprecision	imprecision	NOUN
cana-1630	11	19	in	in	ADP
cana-1630	11	20	various	various	ADJ
cana-1630	11	21	computational	computational	ADJ
cana-1630	11	22	tasks	task	NOUN
cana-1630	11	23	.	.	PUNCT
cana-1630	12	1	these	these	DET
cana-1630	12	2	systems	system	NOUN
cana-1630	12	3	utilize	utilize	VERB
cana-1630	12	4	fuzzy	fuzzy	ADJ
cana-1630	12	5	membership	membership	NOUN
cana-1630	12	6	functions	function	NOUN
cana-1630	12	7	to	to	PART
cana-1630	12	8	map	map	VERB
cana-1630	12	9	inputs	input	NOUN
cana-1630	12	10	to	to	ADP
cana-1630	12	11	a	a	DET
cana-1630	12	12	degree	degree	NOUN
cana-1630	12	13	of	of	ADP
cana-1630	12	14	membership	membership	NOUN
cana-1630	12	15	,	,	PUNCT
cana-1630	12	16	which	which	PRON
cana-1630	12	17	is	be	AUX
cana-1630	12	18	crucial	crucial	ADJ
cana-1630	12	19	for	for	ADP
cana-1630	12	20	decision	decision	NOUN
cana-1630	12	21	-	-	PUNCT
cana-1630	12	22	making	make	VERB
cana-1630	12	23	processes	process	NOUN
cana-1630	12	24	.	.	PUNCT
cana-1630	13	1	however	however	ADV
cana-1630	13	2	,	,	PUNCT
cana-1630	13	3	the	the	DET
cana-1630	13	4	stability	stability	NOUN
cana-1630	13	5	and	and	CCONJ
cana-1630	13	6	preservation	preservation	NOUN
cana-1630	13	7	of	of	ADP
cana-1630	13	8	these	these	DET
cana-1630	13	9	membership	membership	NOUN
cana-1630	13	10	functions	function	NOUN
cana-1630	13	11	are	be	AUX
cana-1630	13	12	paramount	paramount	ADJ
cana-1630	13	13	for	for	ADP
cana-1630	13	14	maintaining	maintain	VERB
cana-1630	13	15	the	the	DET
cana-1630	13	16	accuracy	accuracy	NOUN
cana-1630	13	17	and	and	CCONJ
cana-1630	13	18	reliability	reliability	NOUN
cana-1630	13	19	of	of	ADP
cana-1630	13	20	fuzzy	fuzzy	ADJ
cana-1630	13	21	logic	logic	NOUN
cana-1630	13	22	systems	system	NOUN
cana-1630	13	23	[	[	X
cana-1630	13	24	1	1	NUM
cana-1630	13	25	]	]	PUNCT
cana-1630	13	26	.	.	PUNCT
cana-1630	14	1	in	in	ADP
cana-1630	14	2	scenarios	scenario	NOUN
cana-1630	14	3	where	where	SCONJ
cana-1630	14	4	these	these	DET
cana-1630	14	5	systems	system	NOUN
cana-1630	14	6	are	be	AUX
cana-1630	14	7	applied	apply	VERB
cana-1630	14	8	,	,	PUNCT
cana-1630	14	9	such	such	ADJ
cana-1630	14	10	as	as	ADP
cana-1630	14	11	in	in	ADP
cana-1630	14	12	image	image	NOUN
cana-1630	14	13	processing	processing	NOUN
cana-1630	14	14	and	and	CCONJ
cana-1630	14	15	pattern	pattern	NOUN
cana-1630	14	16	recognition	recognition	NOUN
cana-1630	14	17	,	,	PUNCT
cana-1630	14	18	ensuring	ensure	VERB
cana-1630	14	19	the	the	DET
cana-1630	14	20	robustness	robustness	NOUN
cana-1630	14	21	of	of	ADP
cana-1630	14	22	membership	membership	NOUN
cana-1630	14	23	functions	function	NOUN
cana-1630	14	24	is	be	AUX
cana-1630	14	25	essential	essential	ADJ
cana-1630	14	26	for	for	ADP
cana-1630	14	27	consistent	consistent	ADJ
cana-1630	14	28	performance	performance	NOUN
cana-1630	14	29	.	.	PUNCT
cana-1630	15	1	the	the	DET
cana-1630	15	2	introduction	introduction	NOUN
cana-1630	15	3	of	of	ADP
cana-1630	15	4	gaussian	gaussian	ADJ
cana-1630	15	5	derivative	derivative	ADJ
cana-1630	15	6	filtering	filtering	NOUN
cana-1630	15	7	[	[	X
cana-1630	15	8	9	9	NUM
cana-1630	15	9	]	]	PUNCT
cana-1630	15	10	offers	offer	VERB
cana-1630	15	11	a	a	DET
cana-1630	15	12	promising	promising	ADJ
cana-1630	15	13	approach	approach	NOUN
cana-1630	15	14	to	to	PART
cana-1630	15	15	enhance	enhance	VERB
cana-1630	15	16	the	the	DET
cana-1630	15	17	stability	stability	NOUN
cana-1630	15	18	and	and	CCONJ
cana-1630	15	19	preservation	preservation	NOUN
cana-1630	15	20	of	of	ADP
cana-1630	15	21	fuzzy	fuzzy	ADJ
cana-1630	15	22	membership	membership	NOUN
cana-1630	15	23	functions	function	NOUN
cana-1630	15	24	.	.	PUNCT
cana-1630	16	1	gaussian	gaussian	ADJ
cana-1630	16	2	derivatives	derivative	NOUN
cana-1630	16	3	are	be	AUX
cana-1630	16	4	widely	widely	ADV
cana-1630	16	5	recognized	recognize	VERB
cana-1630	16	6	for	for	ADP
cana-1630	16	7	their	their	PRON
cana-1630	16	8	ability	ability	NOUN
cana-1630	16	9	to	to	PART
cana-1630	16	10	capture	capture	VERB
cana-1630	16	11	fine	fine	ADJ
cana-1630	16	12	details	detail	NOUN
cana-1630	16	13	and	and	CCONJ
cana-1630	16	14	subtle	subtle	ADJ
cana-1630	16	15	variations	variation	NOUN
cana-1630	16	16	in	in	ADP
cana-1630	16	17	data	datum	NOUN
cana-1630	16	18	.	.	PUNCT
cana-1630	17	1	by	by	ADP
cana-1630	17	2	incorporating	incorporate	VERB
cana-1630	17	3	these	these	DET
cana-1630	17	4	derivatives	derivative	NOUN
cana-1630	17	5	into	into	ADP
cana-1630	17	6	the	the	DET
cana-1630	17	7	filtering	filtering	NOUN
cana-1630	17	8	process	process	NOUN
cana-1630	17	9	,	,	PUNCT
cana-1630	17	10	it	it	PRON
cana-1630	17	11	is	be	AUX
cana-1630	17	12	possible	possible	ADJ
cana-1630	17	13	to	to	PART
cana-1630	17	14	maintain	maintain	VERB
cana-1630	17	15	the	the	DET
cana-1630	17	16	integrity	integrity	NOUN
cana-1630	17	17	of	of	ADP
cana-1630	17	18	the	the	DET
cana-1630	17	19	membership	membership	NOUN
cana-1630	17	20	functions	function	NOUN
cana-1630	17	21	while	while	SCONJ
cana-1630	17	22	also	also	ADV
cana-1630	17	23	improving	improve	VERB
cana-1630	17	24	their	their	PRON
cana-1630	17	25	resilience	resilience	NOUN
cana-1630	17	26	to	to	PART
cana-1630	17	27	noise	noise	NOUN
cana-1630	17	28	and	and	CCONJ
cana-1630	17	29	distortions	distortion	NOUN
cana-1630	17	30	.	.	PUNCT
cana-1630	18	1	this	this	DET
cana-1630	18	2	integration	integration	NOUN
cana-1630	18	3	can	can	AUX
cana-1630	18	4	lead	lead	VERB
cana-1630	18	5	to	to	ADP
cana-1630	18	6	more	more	ADV
cana-1630	18	7	accurate	accurate	ADJ
cana-1630	18	8	and	and	CCONJ
cana-1630	18	9	reliable	reliable	ADJ
cana-1630	18	10	outputs	output	NOUN
cana-1630	18	11	in	in	ADP
cana-1630	18	12	applications	application	NOUN
cana-1630	18	13	that	that	PRON
cana-1630	18	14	rely	rely	VERB
cana-1630	18	15	on	on	ADP
cana-1630	18	16	fuzzy	fuzzy	ADJ
cana-1630	18	17	logic	logic	NOUN
cana-1630	18	18	.	.	PUNCT
cana-1630	19	1	communications	communication	NOUN
cana-1630	19	2	on	on	ADP
cana-1630	19	3	applied	apply	VERB
cana-1630	19	4	nonlinear	nonlinear	ADJ
cana-1630	19	5	analysis	analysis	NOUN
cana-1630	19	6	issn	issn	NOUN
cana-1630	19	7	:	:	PUNCT
cana-1630	19	8	1074	1074	NUM
cana-1630	19	9	-	-	PUNCT
cana-1630	19	10	133x	133x	NUM
cana-1630	19	11	vol	vol	NOUN
cana-1630	19	12	32	32	NUM
cana-1630	19	13	no	no	NOUN
cana-1630	19	14	.	.	NOUN
cana-1630	19	15	1	1	NUM
cana-1630	19	16	(	(	PUNCT
cana-1630	19	17	2025	2025	NUM
cana-1630	19	18	)	)	PUNCT
cana-1630	19	19	185	185	NUM
cana-1630	19	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	19	21	adaptive	adaptive	ADJ
cana-1630	19	22	filtering	filtering	NOUN
cana-1630	19	23	techniques	technique	NOUN
cana-1630	20	1	[	[	X
cana-1630	20	2	3	3	NUM
cana-1630	20	3	]	]	PUNCT
cana-1630	20	4	further	far	ADV
cana-1630	20	5	enhance	enhance	VERB
cana-1630	20	6	the	the	DET
cana-1630	20	7	potential	potential	NOUN
cana-1630	20	8	of	of	ADP
cana-1630	20	9	gaussian	gaussian	ADJ
cana-1630	20	10	derivative	derivative	ADJ
cana-1630	20	11	filtering	filtering	NOUN
cana-1630	20	12	.	.	PUNCT
cana-1630	21	1	by	by	ADP
cana-1630	21	2	adjusting	adjust	VERB
cana-1630	21	3	the	the	DET
cana-1630	21	4	filter	filter	NOUN
cana-1630	21	5	parameters	parameter	NOUN
cana-1630	21	6	dynamically	dynamically	ADV
cana-1630	21	7	based	base	VERB
cana-1630	21	8	on	on	ADP
cana-1630	21	9	the	the	DET
cana-1630	21	10	characteristics	characteristic	NOUN
cana-1630	21	11	of	of	ADP
cana-1630	21	12	the	the	DET
cana-1630	21	13	input	input	NOUN
cana-1630	21	14	data	datum	NOUN
cana-1630	21	15	,	,	PUNCT
cana-1630	21	16	adaptive	adaptive	ADJ
cana-1630	21	17	filters	filter	NOUN
cana-1630	21	18	can	can	AUX
cana-1630	21	19	provide	provide	VERB
cana-1630	21	20	a	a	DET
cana-1630	21	21	more	more	ADV
cana-1630	21	22	tailored	tailored	ADJ
cana-1630	21	23	and	and	CCONJ
cana-1630	21	24	effective	effective	ADJ
cana-1630	21	25	approach	approach	NOUN
cana-1630	21	26	to	to	ADP
cana-1630	21	27	preserving	preserve	VERB
cana-1630	21	28	membership	membership	NOUN
cana-1630	21	29	functions	function	NOUN
cana-1630	21	30	.	.	PUNCT
cana-1630	22	1	this	this	DET
cana-1630	22	2	adaptability	adaptability	NOUN
cana-1630	22	3	is	be	AUX
cana-1630	22	4	particularly	particularly	ADV
cana-1630	22	5	beneficial	beneficial	ADJ
cana-1630	22	6	in	in	ADP
cana-1630	22	7	complex	complex	ADJ
cana-1630	22	8	and	and	CCONJ
cana-1630	22	9	varying	vary	VERB
cana-1630	22	10	environments	environment	NOUN
cana-1630	22	11	where	where	SCONJ
cana-1630	22	12	static	static	ADJ
cana-1630	22	13	filter	filter	NOUN
cana-1630	22	14	parameters	parameter	NOUN
cana-1630	22	15	may	may	AUX
cana-1630	22	16	not	not	PART
cana-1630	22	17	suffice	suffice	VERB
cana-1630	22	18	.	.	PUNCT
cana-1630	23	1	the	the	DET
cana-1630	23	2	ability	ability	NOUN
cana-1630	23	3	to	to	PART
cana-1630	23	4	adapt	adapt	VERB
cana-1630	23	5	in	in	ADP
cana-1630	23	6	real	real	ADJ
cana-1630	23	7	-	-	PUNCT
cana-1630	23	8	time	time	NOUN
cana-1630	23	9	ensures	ensure	VERB
cana-1630	23	10	that	that	SCONJ
cana-1630	23	11	the	the	DET
cana-1630	23	12	membership	membership	NOUN
cana-1630	23	13	functions	function	NOUN
cana-1630	23	14	remain	remain	VERB
cana-1630	23	15	stable	stable	ADJ
cana-1630	23	16	and	and	CCONJ
cana-1630	23	17	accurate	accurate	ADJ
cana-1630	23	18	under	under	ADP
cana-1630	23	19	different	different	ADJ
cana-1630	23	20	conditions	condition	NOUN
cana-1630	23	21	.	.	PUNCT
cana-1630	24	1	implementing	implement	VERB
cana-1630	24	2	adaptive	adaptive	ADJ
cana-1630	24	3	gaussian	gaussian	ADJ
cana-1630	24	4	derivative	derivative	ADJ
cana-1630	24	5	filtering	filtering	NOUN
cana-1630	24	6	requires	require	VERB
cana-1630	24	7	careful	careful	ADJ
cana-1630	24	8	consideration	consideration	NOUN
cana-1630	24	9	of	of	ADP
cana-1630	24	10	various	various	ADJ
cana-1630	24	11	factors	factor	NOUN
cana-1630	24	12	,	,	PUNCT
cana-1630	24	13	including	include	VERB
cana-1630	24	14	the	the	DET
cana-1630	24	15	selection	selection	NOUN
cana-1630	24	16	of	of	ADP
cana-1630	24	17	appropriate	appropriate	ADJ
cana-1630	24	18	derivative	derivative	ADJ
cana-1630	24	19	[	[	X
cana-1630	24	20	5,7	5,7	NUM
cana-1630	24	21	]	]	PUNCT
cana-1630	24	22	orders	order	NOUN
cana-1630	24	23	and	and	CCONJ
cana-1630	24	24	the	the	DET
cana-1630	24	25	design	design	NOUN
cana-1630	24	26	of	of	ADP
cana-1630	24	27	adaptive	adaptive	ADJ
cana-1630	24	28	mechanisms	mechanism	NOUN
cana-1630	24	29	.	.	PUNCT
cana-1630	25	1	the	the	DET
cana-1630	25	2	interplay	interplay	NOUN
cana-1630	25	3	between	between	ADP
cana-1630	25	4	these	these	DET
cana-1630	25	5	elements	element	NOUN
cana-1630	25	6	determines	determine	VERB
cana-1630	25	7	the	the	DET
cana-1630	25	8	overall	overall	ADJ
cana-1630	25	9	effectiveness	effectiveness	NOUN
cana-1630	25	10	of	of	ADP
cana-1630	25	11	the	the	DET
cana-1630	25	12	filtering	filtering	NOUN
cana-1630	25	13	process	process	NOUN
cana-1630	25	14	.	.	PUNCT
cana-1630	26	1	a	a	DET
cana-1630	26	2	thorough	thorough	ADJ
cana-1630	26	3	analysis	analysis	NOUN
cana-1630	26	4	and	and	CCONJ
cana-1630	26	5	optimization	optimization	NOUN
cana-1630	26	6	of	of	ADP
cana-1630	26	7	these	these	DET
cana-1630	26	8	parameters	parameter	NOUN
cana-1630	26	9	can	can	AUX
cana-1630	26	10	lead	lead	VERB
cana-1630	26	11	to	to	ADP
cana-1630	26	12	significant	significant	ADJ
cana-1630	26	13	improvements	improvement	NOUN
cana-1630	26	14	in	in	ADP
cana-1630	26	15	the	the	DET
cana-1630	26	16	stability	stability	NOUN
cana-1630	26	17	[	[	X
cana-1630	26	18	10,11	10,11	NOUN
cana-1630	26	19	]	]	X
cana-1630	26	20	and	and	CCONJ
cana-1630	26	21	preservation	preservation	NOUN
cana-1630	26	22	of	of	ADP
cana-1630	26	23	fuzzy	fuzzy	ADJ
cana-1630	26	24	membership	membership	NOUN
cana-1630	26	25	functions	function	NOUN
cana-1630	26	26	,	,	PUNCT
cana-1630	26	27	thereby	thereby	ADV
cana-1630	26	28	enhancing	enhance	VERB
cana-1630	26	29	the	the	DET
cana-1630	26	30	performance	performance	NOUN
cana-1630	26	31	of	of	ADP
cana-1630	26	32	the	the	DET
cana-1630	26	33	associated	associated	ADJ
cana-1630	26	34	fuzzy	fuzzy	ADJ
cana-1630	26	35	logic	logic	NOUN
cana-1630	26	36	systems	system	NOUN
cana-1630	26	37	.	.	PUNCT
cana-1630	27	1	this	this	DET
cana-1630	27	2	study	study	NOUN
cana-1630	27	3	explores	explore	VERB
cana-1630	27	4	the	the	DET
cana-1630	27	5	development	development	NOUN
cana-1630	27	6	and	and	CCONJ
cana-1630	27	7	application	application	NOUN
cana-1630	27	8	of	of	ADP
cana-1630	27	9	adaptive	adaptive	ADJ
cana-1630	27	10	gaussian	gaussian	ADJ
cana-1630	27	11	derivative	derivative	ADJ
cana-1630	27	12	filtering	filtering	NOUN
cana-1630	27	13	for	for	ADP
cana-1630	27	14	the	the	DET
cana-1630	27	15	stability	stability	NOUN
cana-1630	27	16	and	and	CCONJ
cana-1630	27	17	preservation	preservation	NOUN
cana-1630	27	18	of	of	ADP
cana-1630	27	19	fuzzy	fuzzy	ADJ
cana-1630	27	20	membership	membership	NOUN
cana-1630	27	21	functions	function	NOUN
cana-1630	27	22	.	.	PUNCT
cana-1630	28	1	by	by	ADP
cana-1630	28	2	evaluating	evaluate	VERB
cana-1630	28	3	different	different	ADJ
cana-1630	28	4	derivative	derivative	ADJ
cana-1630	28	5	orders	order	NOUN
cana-1630	28	6	and	and	CCONJ
cana-1630	28	7	adaptive	adaptive	ADJ
cana-1630	28	8	strategies	strategy	NOUN
cana-1630	28	9	,	,	PUNCT
cana-1630	28	10	the	the	DET
cana-1630	28	11	research	research	NOUN
cana-1630	28	12	aims	aim	VERB
cana-1630	28	13	to	to	PART
cana-1630	28	14	identify	identify	VERB
cana-1630	28	15	the	the	DET
cana-1630	28	16	most	most	ADV
cana-1630	28	17	effective	effective	ADJ
cana-1630	28	18	configurations	configuration	NOUN
cana-1630	28	19	for	for	ADP
cana-1630	28	20	maintaining	maintain	VERB
cana-1630	28	21	the	the	DET
cana-1630	28	22	integrity	integrity	NOUN
cana-1630	28	23	of	of	ADP
cana-1630	28	24	membership	membership	NOUN
cana-1630	28	25	functions	function	NOUN
cana-1630	28	26	.	.	PUNCT
cana-1630	29	1	the	the	DET
cana-1630	29	2	findings	finding	NOUN
cana-1630	29	3	of	of	ADP
cana-1630	29	4	this	this	DET
cana-1630	29	5	study	study	NOUN
cana-1630	29	6	hold	hold	VERB
cana-1630	29	7	promise	promise	NOUN
cana-1630	29	8	for	for	ADP
cana-1630	29	9	advancing	advance	VERB
cana-1630	29	10	the	the	DET
cana-1630	29	11	robustness	robustness	NOUN
cana-1630	29	12	and	and	CCONJ
cana-1630	29	13	reliability	reliability	NOUN
cana-1630	29	14	of	of	ADP
cana-1630	29	15	fuzzy	fuzzy	ADJ
cana-1630	29	16	logic	logic	NOUN
cana-1630	29	17	systems	system	NOUN
cana-1630	29	18	,	,	PUNCT
cana-1630	29	19	with	with	ADP
cana-1630	29	20	potential	potential	ADJ
cana-1630	29	21	implications	implication	NOUN
cana-1630	29	22	for	for	ADP
cana-1630	29	23	various	various	ADJ
cana-1630	29	24	fields	field	NOUN
cana-1630	29	25	where	where	SCONJ
cana-1630	29	26	these	these	DET
cana-1630	29	27	systems	system	NOUN
cana-1630	29	28	are	be	AUX
cana-1630	29	29	employed	employ	VERB
cana-1630	29	30	.	.	PUNCT
cana-1630	30	1	1	1	X
cana-1630	30	2	.	.	X
cana-1630	30	3	preliminaries	preliminary	NOUN
cana-1630	30	4	2.1	2.1	NUM
cana-1630	30	5	fuzzy	fuzzy	ADJ
cana-1630	30	6	set	set	NOUN
cana-1630	30	7	:	:	PUNCT
cana-1630	30	8	a	a	DET
cana-1630	30	9	fuzzy	fuzzy	ADJ
cana-1630	30	10	set	set	VERB
cana-1630	30	11	a	a	PRON
cana-1630	30	12	in	in	ADP
cana-1630	30	13	a	a	DET
cana-1630	30	14	universe	universe	NOUN
cana-1630	30	15	of	of	ADP
cana-1630	30	16	discourse	discourse	NOUN
cana-1630	30	17	x	x	SYM
cana-1630	30	18	is	be	AUX
cana-1630	30	19	characterized	characterize	VERB
cana-1630	30	20	by	by	ADP
cana-1630	30	21	a	a	DET
cana-1630	30	22	membership	membership	NOUN
cana-1630	30	23	function	function	NOUN
cana-1630	30	24	𝜇𝐴	𝜇𝐴	ADP
cana-1630	30	25	:	:	PUNCT
cana-1630	30	26	𝑋	𝑋	PROPN
cana-1630	30	27	⟶	⟶	NOUN
cana-1630	30	28	[	[	X
cana-1630	30	29	0,1	0,1	NUM
cana-1630	30	30	]	]	PUNCT
cana-1630	30	31	.	.	PUNCT
cana-1630	31	1	for	for	ADP
cana-1630	31	2	each	each	DET
cana-1630	31	3	element	element	NOUN
cana-1630	31	4	𝑥	𝑥	PRON
cana-1630	31	5	∈	∈	PROPN
cana-1630	31	6	𝑋	𝑋	PROPN
cana-1630	31	7	,	,	PUNCT
cana-1630	31	8	the	the	DET
cana-1630	31	9	membership	membership	NOUN
cana-1630	31	10	function	function	VERB
cana-1630	31	11	𝜇𝐴(𝑥	𝜇𝐴(𝑥	NUM
cana-1630	31	12	)	)	PUNCT
cana-1630	31	13	indicates	indicate	VERB
cana-1630	31	14	the	the	DET
cana-1630	31	15	degree	degree	NOUN
cana-1630	31	16	of	of	ADP
cana-1630	31	17	membership	membership	NOUN
cana-1630	31	18	of	of	ADP
cana-1630	31	19	x	x	PUNCT
cana-1630	31	20	in	in	ADP
cana-1630	31	21	the	the	DET
cana-1630	31	22	fuzzy	fuzzy	ADJ
cana-1630	31	23	set	set	VERB
cana-1630	31	24	a.	a.	NOUN
cana-1630	31	25	2.2	2.2	NUM
cana-1630	31	26	convolution	convolution	NOUN
cana-1630	31	27	:	:	PUNCT
cana-1630	31	28	it	it	PRON
cana-1630	31	29	is	be	AUX
cana-1630	31	30	mathematical	mathematical	ADJ
cana-1630	31	31	operation	operation	NOUN
cana-1630	31	32	that	that	PRON
cana-1630	31	33	combines	combine	VERB
cana-1630	31	34	two	two	NUM
cana-1630	31	35	functions	function	NOUN
cana-1630	31	36	to	to	PART
cana-1630	31	37	produce	produce	VERB
cana-1630	31	38	a	a	DET
cana-1630	31	39	third	third	ADJ
cana-1630	31	40	function	function	NOUN
cana-1630	31	41	.	.	PUNCT
cana-1630	32	1	in	in	ADP
cana-1630	32	2	the	the	DET
cana-1630	32	3	context	context	NOUN
cana-1630	32	4	of	of	ADP
cana-1630	32	5	image	image	NOUN
cana-1630	32	6	processing	process	VERB
cana-1630	32	7	the	the	DET
cana-1630	32	8	convolution	convolution	NOUN
cana-1630	32	9	of	of	ADP
cana-1630	32	10	an	an	DET
cana-1630	32	11	input	input	NOUN
cana-1630	32	12	image	image	NOUN
cana-1630	32	13	i	i	PRON
cana-1630	32	14	with	with	ADP
cana-1630	32	15	a	a	DET
cana-1630	32	16	filter	filter	NOUN
cana-1630	32	17	g	g	NOUN
cana-1630	32	18	is	be	AUX
cana-1630	32	19	given	give	VERB
cana-1630	32	20	by	by	ADP
cana-1630	32	21	(	(	PUNCT
cana-1630	32	22	𝐼	𝐼	PROPN
cana-1630	32	23	∗	∗	X
cana-1630	32	24	𝐺)(𝑥	𝐺)(𝑥	PROPN
cana-1630	32	25	,	,	PUNCT
cana-1630	32	26	𝑦	𝑦	NOUN
cana-1630	32	27	)	)	PUNCT
cana-1630	32	28	=	=	PUNCT
cana-1630	33	1	∑	∑	PUNCT
cana-1630	33	2	∑	∑	PUNCT
cana-1630	33	3	𝐼(𝑥	𝐼(𝑥	PROPN
cana-1630	33	4	−	−	PROPN
cana-1630	33	5	𝑖	𝑖	SYM
cana-1630	33	6	,	,	PUNCT
cana-1630	33	7	𝑦	𝑦	NOUN
cana-1630	33	8	−	−	PROPN
cana-1630	33	9	𝑗	𝑗	NOUN
cana-1630	33	10	)	)	PUNCT
cana-1630	33	11	∙	∙	PROPN
cana-1630	33	12	𝐺(𝑖	𝐺(𝑖	PROPN
cana-1630	33	13	,	,	PUNCT
cana-1630	33	14	𝑗)𝑘	𝑗)𝑘	PUNCT
cana-1630	34	1	𝑗=−𝑘	𝑗=−𝑘	INTJ
cana-1630	34	2	𝑘	𝑘	INTJ
cana-1630	35	1	𝑖=−𝑘	𝑖=−𝑘	INTJ
cana-1630	35	2	where	where	SCONJ
cana-1630	35	3	k	k	PROPN
cana-1630	35	4	determines	determine	VERB
cana-1630	35	5	the	the	DET
cana-1630	35	6	size	size	NOUN
cana-1630	35	7	of	of	ADP
cana-1630	35	8	the	the	DET
cana-1630	35	9	filter	filter	NOUN
cana-1630	35	10	window	window	NOUN
cana-1630	35	11	.	.	PUNCT
cana-1630	36	1	2.3	2.3	NUM
cana-1630	36	2	bounded	bound	VERB
cana-1630	36	3	input	input	NOUN
cana-1630	36	4	bounded	bound	VERB
cana-1630	36	5	output	output	NOUN
cana-1630	36	6	(	(	PUNCT
cana-1630	36	7	bibo	bibo	ADJ
cana-1630	36	8	)	)	PUNCT
cana-1630	36	9	stability	stability	NOUN
cana-1630	36	10	:	:	PUNCT
cana-1630	36	11	a	a	DET
cana-1630	36	12	system	system	NOUN
cana-1630	36	13	is	be	AUX
cana-1630	36	14	bibo	bibo	ADJ
cana-1630	36	15	stable	stable	ADJ
cana-1630	36	16	[	[	X
cana-1630	36	17	12,13	12,13	NUM
cana-1630	36	18	]	]	PUNCT
cana-1630	36	19	if	if	SCONJ
cana-1630	36	20	every	every	DET
cana-1630	36	21	bounded	bounded	ADJ
cana-1630	36	22	input	input	NOUN
cana-1630	36	23	produces	produce	VERB
cana-1630	36	24	a	a	DET
cana-1630	36	25	bounded	bounded	ADJ
cana-1630	36	26	output	output	NOUN
cana-1630	36	27	.	.	PUNCT
cana-1630	37	1	if	if	SCONJ
cana-1630	37	2	the	the	DET
cana-1630	37	3	input	input	NOUN
cana-1630	37	4	image	image	NOUN
cana-1630	37	5	pixel	pixel	NOUN
cana-1630	37	6	intensities	intensity	NOUN
cana-1630	37	7	(	(	PUNCT
cana-1630	37	8	or	or	CCONJ
cana-1630	37	9	membership	membership	NOUN
cana-1630	37	10	values	value	NOUN
cana-1630	37	11	in	in	ADP
cana-1630	37	12	the	the	DET
cana-1630	37	13	case	case	NOUN
cana-1630	37	14	of	of	ADP
cana-1630	37	15	fuzzy	fuzzy	ADJ
cana-1630	37	16	images	image	NOUN
cana-1630	37	17	)	)	PUNCT
cana-1630	37	18	are	be	AUX
cana-1630	37	19	bounded	bound	VERB
cana-1630	37	20	,	,	PUNCT
cana-1630	37	21	the	the	DET
cana-1630	37	22	output	output	NOUN
cana-1630	37	23	image	image	NOUN
cana-1630	37	24	pixel	pixel	PROPN
cana-1630	37	25	intensities	intensity	NOUN
cana-1630	37	26	(	(	PUNCT
cana-1630	37	27	or	or	CCONJ
cana-1630	37	28	membership	membership	NOUN
cana-1630	37	29	values	value	NOUN
cana-1630	37	30	)	)	PUNCT
cana-1630	37	31	will	will	AUX
cana-1630	37	32	also	also	ADV
cana-1630	37	33	be	be	AUX
cana-1630	37	34	bounded	bound	VERB
cana-1630	37	35	after	after	ADP
cana-1630	37	36	processing	process	VERB
cana-1630	37	37	with	with	ADP
cana-1630	37	38	a	a	DET
cana-1630	37	39	filter	filter	NOUN
cana-1630	37	40	.	.	PUNCT
cana-1630	38	1	2.4	2.4	NUM
cana-1630	38	2	membership	membership	NOUN
cana-1630	38	3	function	function	NOUN
cana-1630	38	4	:	:	PUNCT
cana-1630	38	5	in	in	ADP
cana-1630	38	6	a	a	DET
cana-1630	38	7	fuzzy	fuzzy	ADJ
cana-1630	38	8	set	set	NOUN
cana-1630	38	9	,	,	PUNCT
cana-1630	38	10	the	the	DET
cana-1630	38	11	membership	membership	NOUN
cana-1630	38	12	function	function	VERB
cana-1630	38	13	𝜇𝐴(𝑥	𝜇𝐴(𝑥	NUM
cana-1630	38	14	)	)	PUNCT
cana-1630	38	15	represent	represent	VERB
cana-1630	38	16	the	the	DET
cana-1630	38	17	degree	degree	NOUN
cana-1630	38	18	of	of	ADP
cana-1630	38	19	membership	membership	NOUN
cana-1630	38	20	of	of	ADP
cana-1630	38	21	an	an	DET
cana-1630	38	22	element	element	NOUN
cana-1630	38	23	x	x	PUNCT
cana-1630	38	24	in	in	ADP
cana-1630	38	25	the	the	DET
cana-1630	38	26	fuzzy	fuzzy	ADJ
cana-1630	38	27	set	set	NOUN
cana-1630	38	28	.	.	PUNCT
cana-1630	39	1	it	it	PRON
cana-1630	39	2	assigns	assign	VERB
cana-1630	39	3	a	a	DET
cana-1630	39	4	value	value	NOUN
cana-1630	39	5	between	between	ADP
cana-1630	39	6	0	0	NUM
cana-1630	39	7	and	and	CCONJ
cana-1630	39	8	1	1	NUM
cana-1630	39	9	to	to	ADP
cana-1630	39	10	each	each	DET
cana-1630	39	11	element	element	NOUN
cana-1630	39	12	x	x	PUNCT
cana-1630	39	13	,	,	PUNCT
cana-1630	39	14	where	where	SCONJ
cana-1630	39	15	0	0	NUM
cana-1630	39	16	means	mean	VERB
cana-1630	39	17	no	no	DET
cana-1630	39	18	membership	membership	NOUN
cana-1630	39	19	and	and	CCONJ
cana-1630	39	20	1	1	NUM
cana-1630	39	21	means	mean	VERB
cana-1630	39	22	full	full	ADJ
cana-1630	39	23	membership	membership	NOUN
cana-1630	39	24	.	.	PUNCT
cana-1630	40	1	2.5	2.5	NUM
cana-1630	40	2	bounded	bounded	ADJ
cana-1630	40	3	function	function	NOUN
cana-1630	40	4	:	:	PUNCT
cana-1630	40	5	a	a	DET
cana-1630	40	6	function	function	NOUN
cana-1630	40	7	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1630	40	8	)	)	PUNCT
cana-1630	40	9	is	be	AUX
cana-1630	40	10	bounded	bound	VERB
cana-1630	40	11	if	if	SCONJ
cana-1630	40	12	there	there	PRON
cana-1630	40	13	exist	exist	VERB
cana-1630	40	14	a	a	DET
cana-1630	40	15	real	real	ADJ
cana-1630	40	16	number	number	NOUN
cana-1630	40	17	m	m	NOUN
cana-1630	40	18	such	such	ADJ
cana-1630	40	19	that	that	SCONJ
cana-1630	40	20	|𝑓(𝑥)|	|𝑓(𝑥)|	ADP
cana-1630	40	21	≤	≤	ADJ
cana-1630	40	22	𝑀	𝑀	PROPN
cana-1630	40	23	for	for	ADP
cana-1630	40	24	all	all	DET
cana-1630	40	25	x	x	NOUN
cana-1630	40	26	in	in	ADP
cana-1630	40	27	the	the	DET
cana-1630	40	28	domain	domain	NOUN
cana-1630	40	29	of	of	ADP
cana-1630	40	30	𝑓.	𝑓.	NOUN
cana-1630	40	31	2.6	2.6	NUM
cana-1630	40	32	smooth	smooth	ADJ
cana-1630	40	33	function	function	NOUN
cana-1630	40	34	:	:	PUNCT
cana-1630	40	35	a	a	DET
cana-1630	40	36	function	function	NOUN
cana-1630	40	37	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1630	40	38	)	)	PUNCT
cana-1630	40	39	is	be	AUX
cana-1630	40	40	smooth	smooth	ADJ
cana-1630	40	41	if	if	SCONJ
cana-1630	40	42	it	it	PRON
cana-1630	40	43	is	be	AUX
cana-1630	40	44	continuously	continuously	ADV
cana-1630	40	45	differentiable	differentiable	ADJ
cana-1630	40	46	to	to	ADP
cana-1630	40	47	a	a	DET
cana-1630	40	48	desired	desire	VERB
cana-1630	40	49	degree	degree	NOUN
cana-1630	40	50	.	.	PUNCT
cana-1630	41	1	in	in	ADP
cana-1630	41	2	image	image	NOUN
cana-1630	41	3	processing	processing	NOUN
cana-1630	41	4	,	,	PUNCT
cana-1630	41	5	a	a	DET
cana-1630	41	6	smooth	smooth	ADJ
cana-1630	41	7	function	function	NOUN
cana-1630	41	8	typically	typically	ADV
cana-1630	41	9	refers	refer	VERB
cana-1630	41	10	to	to	ADP
cana-1630	41	11	one	one	NUM
cana-1630	41	12	that	that	PRON
cana-1630	41	13	does	do	AUX
cana-1630	41	14	not	not	PART
cana-1630	41	15	have	have	VERB
cana-1630	41	16	abrupt	abrupt	ADJ
cana-1630	41	17	changes	change	NOUN
cana-1630	41	18	in	in	ADP
cana-1630	41	19	value	value	NOUN
cana-1630	41	20	.	.	PUNCT
cana-1630	42	1	communications	communication	NOUN
cana-1630	42	2	on	on	ADP
cana-1630	42	3	applied	apply	VERB
cana-1630	42	4	nonlinear	nonlinear	ADJ
cana-1630	42	5	analysis	analysis	NOUN
cana-1630	42	6	issn	issn	NOUN
cana-1630	42	7	:	:	PUNCT
cana-1630	42	8	1074	1074	NUM
cana-1630	42	9	-	-	PUNCT
cana-1630	42	10	133x	133x	NUM
cana-1630	42	11	vol	vol	NOUN
cana-1630	42	12	32	32	NUM
cana-1630	42	13	no	no	NOUN
cana-1630	42	14	.	.	NOUN
cana-1630	42	15	1	1	NUM
cana-1630	42	16	(	(	PUNCT
cana-1630	42	17	2025	2025	NUM
cana-1630	42	18	)	)	PUNCT
cana-1630	42	19	186	186	NUM
cana-1630	42	20	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1630	42	21	2.7	2.7	NUM
cana-1630	42	22	fourier	fourier	NOUN
cana-1630	42	23	transform	transform	NOUN
cana-1630	42	24	(	(	PUNCT
cana-1630	42	25	ft	ft	NOUN
cana-1630	42	26	):	):	PUNCT
cana-1630	42	27	the	the	DET
cana-1630	42	28	fourier	fourier	NOUN
cana-1630	42	29	transform	transform	NOUN
cana-1630	42	30	of	of	ADP
cana-1630	42	31	a	a	DET
cana-1630	42	32	function	function	NOUN
cana-1630	42	33	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1630	42	34	,	,	PUNCT
cana-1630	42	35	𝑦	𝑦	NOUN
cana-1630	42	36	)	)	PUNCT
cana-1630	42	37	is	be	AUX
cana-1630	42	38	a	a	DET
cana-1630	42	39	mathematically	mathematically	ADV
cana-1630	42	40	transformation	transformation	NOUN
cana-1630	42	41	used	use	VERB
cana-1630	42	42	to	to	PART
cana-1630	42	43	analyze	analyze	VERB
cana-1630	42	44	the	the	DET
cana-1630	42	45	frequencies	frequency	NOUN
cana-1630	42	46	present	present	ADJ
cana-1630	42	47	in	in	ADP
cana-1630	42	48	the	the	DET
cana-1630	42	49	function	function	NOUN
cana-1630	42	50	.	.	PUNCT
cana-1630	43	1	it	it	PRON
cana-1630	43	2	is	be	AUX
cana-1630	43	3	defined	define	VERB
cana-1630	43	4	as	as	ADP
cana-1630	43	5	:	:	PUNCT
cana-1630	43	6	𝐹(𝑢	𝐹(𝑢	NUM
cana-1630	43	7	,	,	PUNCT
cana-1630	43	8	𝑣	𝑣	NOUN
cana-1630	43	9	)	)	PUNCT
cana-1630	43	10	=	=	SYM
cana-1630	43	11	∫	∫	PROPN
cana-1630	43	12	∫	∫	PROPN
cana-1630	43	13	𝑓(𝑥	𝑓(𝑥	NOUN
cana-1630	43	14	,	,	PUNCT
cana-1630	43	15	𝑦)𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥	𝑦)𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥	NOUN
cana-1630	43	16	𝑑𝑦	𝑑𝑦	ADP
cana-1630	43	17	∞	∞	PROPN
cana-1630	43	18	−∞	−∞	ADP
cana-1630	43	19	∞	∞	PROPN
cana-1630	43	20	−∞	−∞	ADP
cana-1630	43	21	2.8	2.8	NUM
cana-1630	43	22	frequency	frequency	NOUN
cana-1630	43	23	response	response	NOUN
cana-1630	43	24	:	:	PUNCT
cana-1630	43	25	the	the	DET
cana-1630	43	26	frequency	frequency	NOUN
cana-1630	43	27	response	response	NOUN
cana-1630	43	28	of	of	ADP
cana-1630	43	29	a	a	DET
cana-1630	43	30	filter	filter	NOUN
cana-1630	43	31	describes	describe	VERB
cana-1630	43	32	how	how	SCONJ
cana-1630	43	33	the	the	DET
cana-1630	43	34	filter	filter	NOUN
cana-1630	43	35	affects	affect	VERB
cana-1630	43	36	the	the	DET
cana-1630	43	37	amplitude	amplitude	NOUN
cana-1630	43	38	and	and	CCONJ
cana-1630	43	39	phase	phase	NOUN
cana-1630	43	40	of	of	ADP
cana-1630	43	41	the	the	DET
cana-1630	43	42	input	input	NOUN
cana-1630	43	43	signals	signal	NOUN
cana-1630	43	44	frequency	frequency	NOUN
cana-1630	43	45	components	component	NOUN
cana-1630	43	46	.	.	PUNCT
cana-1630	44	1	for	for	ADP
cana-1630	44	2	a	a	DET
cana-1630	44	3	gaussian	gaussian	ADJ
cana-1630	44	4	filter	filter	NOUN
cana-1630	44	5	[	[	X
cana-1630	44	6	14,15	14,15	NUM
cana-1630	44	7	]	]	X
cana-1630	44	8	the	the	DET
cana-1630	44	9	frequency	frequency	NOUN
cana-1630	44	10	response	response	NOUN
cana-1630	44	11	𝐻(𝑢	𝐻(𝑢	ADP
cana-1630	44	12	,	,	PUNCT
cana-1630	44	13	𝑣	𝑣	NOUN
cana-1630	44	14	)	)	PUNCT
cana-1630	44	15	is	be	AUX
cana-1630	44	16	given	give	VERB
cana-1630	44	17	by	by	ADP
cana-1630	44	18	:	:	PUNCT
cana-1630	44	19	𝐻(𝑢	𝐻(𝑢	ADP
cana-1630	44	20	,	,	PUNCT
cana-1630	44	21	𝑣	𝑣	NOUN
cana-1630	44	22	)	)	PUNCT
cana-1630	44	23	=	=	SYM
cana-1630	44	24	𝑒−2𝜋2𝜎2(𝑢2+𝑣2	𝑒−2𝜋2𝜎2(𝑢2+𝑣2	PROPN
cana-1630	44	25	)	)	PUNCT
cana-1630	44	26	2.9	2.9	NUM
cana-1630	44	27	high	high	ADJ
cana-1630	44	28	frequency	frequency	NOUN
cana-1630	44	29	components	component	NOUN
cana-1630	44	30	:	:	PUNCT
cana-1630	44	31	it	it	PRON
cana-1630	44	32	corresponds	correspond	VERB
cana-1630	44	33	to	to	ADP
cana-1630	44	34	rapid	rapid	ADJ
cana-1630	44	35	changes	change	NOUN
cana-1630	44	36	in	in	ADP
cana-1630	44	37	intensity	intensity	NOUN
cana-1630	44	38	values	value	NOUN
cana-1630	44	39	,	,	PUNCT
cana-1630	44	40	such	such	ADJ
cana-1630	44	41	as	as	ADP
cana-1630	44	42	edges	edge	NOUN
cana-1630	44	43	and	and	CCONJ
cana-1630	44	44	noise	noise	NOUN
cana-1630	44	45	.	.	PUNCT
cana-1630	45	1	they	they	PRON
cana-1630	45	2	are	be	AUX
cana-1630	45	3	characterized	characterize	VERB
cana-1630	45	4	by	by	ADP
cana-1630	45	5	large	large	ADJ
cana-1630	45	6	values	value	NOUN
cana-1630	45	7	of	of	ADP
cana-1630	45	8	u	u	NOUN
cana-1630	45	9	and	and	CCONJ
cana-1630	45	10	v	v	NOUN
cana-1630	45	11	in	in	ADP
cana-1630	45	12	the	the	DET
cana-1630	45	13	frequency	frequency	NOUN
cana-1630	45	14	domain	domain	NOUN
cana-1630	45	15	.	.	PUNCT
cana-1630	46	1	2.10	2.10	NUM
cana-1630	46	2	high	high	ADJ
cana-1630	46	3	frequency	frequency	NOUN
cana-1630	46	4	components	component	NOUN
cana-1630	46	5	:	:	PUNCT
cana-1630	46	6	it	it	PRON
cana-1630	46	7	corresponds	correspond	VERB
cana-1630	46	8	to	to	ADP
cana-1630	46	9	changes	change	NOUN
cana-1630	46	10	in	in	ADP
cana-1630	46	11	intensity	intensity	NOUN
cana-1630	46	12	values	value	NOUN
cana-1630	46	13	,	,	PUNCT
cana-1630	46	14	such	such	ADJ
cana-1630	46	15	as	as	ADP
cana-1630	46	16	smooth	smooth	ADJ
cana-1630	46	17	regions	region	NOUN
cana-1630	46	18	.	.	PUNCT
cana-1630	47	1	these	these	PRON
cana-1630	47	2	are	be	AUX
cana-1630	47	3	characterized	characterize	VERB
cana-1630	47	4	by	by	ADP
cana-1630	47	5	small	small	ADJ
cana-1630	47	6	values	value	NOUN
cana-1630	47	7	of	of	ADP
cana-1630	47	8	u	u	NOUN
cana-1630	47	9	and	and	CCONJ
cana-1630	47	10	v	v	NOUN
cana-1630	47	11	in	in	ADP
cana-1630	47	12	the	the	DET
cana-1630	47	13	frequency	frequency	NOUN
cana-1630	47	14	domain	domain	NOUN
cana-1630	47	15	.	.	PUNCT
cana-1630	48	1	2.11	2.11	NUM
cana-1630	48	2	gaussian	gaussian	ADJ
cana-1630	48	3	filter	filter	NOUN
cana-1630	48	4	:	:	PUNCT
cana-1630	48	5	a	a	DET
cana-1630	48	6	gaussian	gaussian	ADJ
cana-1630	48	7	filter	filter	NOUN
cana-1630	48	8	[	[	X
cana-1630	48	9	2	2	NUM
cana-1630	48	10	]	]	PUNCT
cana-1630	48	11	is	be	AUX
cana-1630	48	12	a	a	DET
cana-1630	48	13	linear	linear	ADJ
cana-1630	48	14	filter	filter	NOUN
cana-1630	48	15	used	use	VERB
cana-1630	48	16	in	in	ADP
cana-1630	48	17	image	image	NOUN
cana-1630	48	18	processing	processing	NOUN
cana-1630	48	19	to	to	PART
cana-1630	48	20	smooth	smooth	ADJ
cana-1630	48	21	images	image	NOUN
cana-1630	48	22	and	and	CCONJ
cana-1630	48	23	reduce	reduce	VERB
cana-1630	48	24	noise	noise	NOUN
cana-1630	48	25	.	.	PUNCT
cana-1630	49	1	it	it	PRON
cana-1630	49	2	is	be	AUX
cana-1630	49	3	characterized	characterize	VERB
cana-1630	49	4	by	by	ADP
cana-1630	49	5	a	a	DET
cana-1630	49	6	gaussian	gaussian	ADJ
cana-1630	49	7	function	function	NOUN
cana-1630	49	8	𝐺𝜎(𝑥	𝐺𝜎(𝑥	PROPN
cana-1630	49	9	,	,	PUNCT
cana-1630	49	10	𝑦	𝑦	NOUN
cana-1630	49	11	)	)	PUNCT
cana-1630	49	12	which	which	PRON
cana-1630	49	13	is	be	AUX
cana-1630	49	14	defined	define	VERB
cana-1630	49	15	as	as	ADP
cana-1630	49	16	:	:	PUNCT
cana-1630	49	17	𝐺𝜎(𝑥	𝐺𝜎(𝑥	PROPN
cana-1630	49	18	,	,	PUNCT
cana-1630	49	19	𝑦	𝑦	NOUN
cana-1630	49	20	)	)	PUNCT
cana-1630	49	21	=	=	SYM
cana-1630	50	1	1	1	NUM
cana-1630	50	2	2𝜋𝜎2	2𝜋𝜎2	NUM
cana-1630	50	3	𝑒	𝑒	PRON
cana-1630	50	4	−	−	PROPN
cana-1630	50	5	𝑥2+𝑦2	𝑥2+𝑦2	PROPN
cana-1630	50	6	2𝜎2	2𝜎2	NUM
cana-1630	50	7	where	where	SCONJ
cana-1630	50	8	𝜎	𝜎	PROPN
cana-1630	50	9	is	be	AUX
cana-1630	50	10	the	the	DET
cana-1630	50	11	standard	standard	ADJ
cana-1630	50	12	deviation	deviation	NOUN
cana-1630	50	13	of	of	ADP
cana-1630	50	14	the	the	DET
cana-1630	50	15	gaussian	gaussian	ADJ
cana-1630	50	16	distribution	distribution	NOUN
cana-1630	50	17	,	,	PUNCT
cana-1630	50	18	controlling	control	VERB
cana-1630	50	19	the	the	DET
cana-1630	50	20	width	width	NOUN
cana-1630	50	21	of	of	ADP
cana-1630	50	22	the	the	DET
cana-1630	50	23	filter	filter	NOUN
cana-1630	50	24	.	.	PUNCT
cana-1630	51	1	2.12	2.12	NUM
cana-1630	51	2	gaussian	gaussian	ADJ
cana-1630	51	3	2d	2d	NUM
cana-1630	51	4	function	function	NOUN
cana-1630	51	5	:	:	PUNCT
cana-1630	51	6	a	a	DET
cana-1630	51	7	two	two	NUM
cana-1630	51	8	-	-	PUNCT
cana-1630	51	9	dimensional	dimensional	ADJ
cana-1630	51	10	gaussian	gaussian	ADJ
cana-1630	51	11	filter	filter	NOUN
cana-1630	51	12	[	[	X
cana-1630	51	13	4	4	NUM
cana-1630	51	14	]	]	PUNCT
cana-1630	51	15	,	,	PUNCT
cana-1630	51	16	the	the	DET
cana-1630	51	17	formula	formula	NOUN
cana-1630	51	18	is	be	AUX
cana-1630	51	19	the	the	DET
cana-1630	51	20	product	product	NOUN
cana-1630	51	21	of	of	ADP
cana-1630	51	22	two	two	NUM
cana-1630	51	23	one	one	NUM
cana-1630	51	24	-	-	PUNCT
cana-1630	51	25	dimensional	dimensional	ADJ
cana-1630	51	26	gaussians	gaussian	NOUN
cana-1630	51	27	along	along	ADP
cana-1630	51	28	the	the	DET
cana-1630	51	29	rows	row	NOUN
cana-1630	51	30	and	and	CCONJ
cana-1630	51	31	columns	column	NOUN
cana-1630	51	32	,	,	PUNCT
cana-1630	51	33	forming	form	VERB
cana-1630	51	34	a	a	DET
cana-1630	51	35	2d	2d	NUM
cana-1630	51	36	kernel	kernel	NOUN
cana-1630	51	37	.	.	PUNCT
cana-1630	52	1	the	the	DET
cana-1630	52	2	filter	filter	NOUN
cana-1630	52	3	effectively	effectively	ADV
cana-1630	52	4	reduces	reduce	VERB
cana-1630	52	5	high	high	ADJ
cana-1630	52	6	-	-	PUNCT
cana-1630	52	7	frequency	frequency	NOUN
cana-1630	52	8	noise	noise	NOUN
cana-1630	52	9	in	in	ADP
cana-1630	52	10	the	the	DET
cana-1630	52	11	image	image	NOUN
cana-1630	52	12	while	while	SCONJ
cana-1630	52	13	preserving	preserve	VERB
cana-1630	52	14	its	its	PRON
cana-1630	52	15	overall	overall	ADJ
cana-1630	52	16	structure	structure	NOUN
cana-1630	52	17	.	.	PUNCT
cana-1630	53	1	it	it	PRON
cana-1630	53	2	is	be	AUX
cana-1630	53	3	based	base	VERB
cana-1630	53	4	on	on	ADP
cana-1630	53	5	the	the	DET
cana-1630	53	6	mathematical	mathematical	ADJ
cana-1630	53	7	formulation	formulation	NOUN
cana-1630	53	8	of	of	ADP
cana-1630	53	9	a	a	DET
cana-1630	53	10	two	two	NUM
cana-1630	53	11	-	-	PUNCT
cana-1630	53	12	dimensional	dimensional	ADJ
cana-1630	53	13	gaussian	gaussian	ADJ
cana-1630	53	14	distribution	distribution	NOUN
cana-1630	53	15	and	and	CCONJ
cana-1630	53	16	operates	operate	VERB
cana-1630	53	17	by	by	ADP
cana-1630	53	18	convolving	convolve	VERB
cana-1630	53	19	the	the	DET
cana-1630	53	20	image	image	NOUN
cana-1630	53	21	with	with	ADP
cana-1630	53	22	a	a	DET
cana-1630	53	23	gaussian	gaussian	ADJ
cana-1630	53	24	kernel	kernel	NOUN
cana-1630	53	25	.	.	PUNCT
cana-1630	54	1	the	the	DET
cana-1630	54	2	mathematical	mathematical	ADJ
cana-1630	54	3	formula	formula	NOUN
cana-1630	54	4	for	for	ADP
cana-1630	54	5	a	a	DET
cana-1630	54	6	two	two	NUM
cana-1630	54	7	-	-	PUNCT
cana-1630	54	8	dimensional	dimensional	ADJ
cana-1630	54	9	gaussian	gaussian	ADJ
cana-1630	54	10	function	function	NOUN
cana-1630	54	11	is	be	AUX
cana-1630	54	12	given	give	VERB
cana-1630	54	13	by	by	ADP
cana-1630	54	14	:	:	PUNCT
cana-1630	54	15	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	54	16	,	,	PUNCT
cana-1630	54	17	𝑦	𝑦	NOUN
cana-1630	54	18	,	,	PUNCT
cana-1630	54	19	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	54	20	,	,	PUNCT
cana-1630	54	21	𝜎𝑦	𝜎𝑦	PROPN
cana-1630	54	22	)	)	PUNCT
cana-1630	54	23	=	=	SYM
cana-1630	54	24	1	1	NUM
cana-1630	54	25	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	54	26	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	54	27	(	(	PUNCT
cana-1630	54	28	−	−	PROPN
cana-1630	54	29	𝑥2	𝑥2	NOUN
cana-1630	54	30	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	54	31	2	2	NUM
cana-1630	54	32	−	−	NOUN
cana-1630	54	33	𝑦2	𝑦2	NOUN
cana-1630	54	34	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	54	35	2	2	NUM
cana-1630	54	36	)	)	PUNCT
cana-1630	54	37	where	where	SCONJ
cana-1630	54	38	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	54	39	,	,	PUNCT
cana-1630	54	40	𝑦	𝑦	NOUN
cana-1630	54	41	,	,	PUNCT
cana-1630	54	42	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	54	43	,	,	PUNCT
cana-1630	54	44	𝜎𝑦	𝜎𝑦	X
cana-1630	54	45	)	)	PUNCT
cana-1630	54	46	is	be	AUX
cana-1630	54	47	the	the	DET
cana-1630	54	48	value	value	NOUN
cana-1630	54	49	of	of	ADP
cana-1630	54	50	the	the	DET
cana-1630	54	51	two	two	NUM
cana-1630	54	52	-	-	PUNCT
cana-1630	54	53	dimensional	dimensional	ADJ
cana-1630	54	54	gaussian	gaussian	ADJ
cana-1630	54	55	function	function	NOUN
cana-1630	54	56	at	at	ADP
cana-1630	54	57	positions	position	NOUN
cana-1630	54	58	x	x	PUNCT
cana-1630	54	59	and	and	CCONJ
cana-1630	54	60	y	y	PROPN
cana-1630	54	61	,	,	PUNCT
cana-1630	54	62	𝜎	𝜎	PROPN
cana-1630	54	63	is	be	AUX
cana-1630	54	64	the	the	DET
cana-1630	54	65	standard	standard	ADJ
cana-1630	54	66	deviation	deviation	NOUN
cana-1630	54	67	determining	determine	VERB
cana-1630	54	68	the	the	DET
cana-1630	54	69	width	width	NOUN
cana-1630	54	70	of	of	ADP
cana-1630	54	71	the	the	DET
cana-1630	54	72	gaussian	gaussian	ADJ
cana-1630	54	73	distribution	distribution	NOUN
cana-1630	54	74	.	.	PUNCT
cana-1630	55	1	2.13	2.13	NUM
cana-1630	55	2	fuzzy	fuzzy	ADJ
cana-1630	55	3	gaussian	gaussian	NOUN
cana-1630	55	4	1	1	NUM
cana-1630	55	5	d	d	NOUN
cana-1630	56	1	and	and	CCONJ
cana-1630	56	2	it	it	PRON
cana-1630	56	3	’s	’	VERB
cana-1630	56	4	derivative	derivative	ADJ
cana-1630	56	5	:	:	PUNCT
cana-1630	56	6	let	let	VERB
cana-1630	56	7	the	the	DET
cana-1630	56	8	function	function	NOUN
cana-1630	56	9	be	be	AUX
cana-1630	56	10	represented	represent	VERB
cana-1630	56	11	as	as	ADP
cana-1630	56	12	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	56	13	,	,	PUNCT
cana-1630	56	14	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	56	15	)	)	PUNCT
cana-1630	56	16	=	=	SYM
cana-1630	57	1	1	1	NUM
cana-1630	57	2	√2𝜋𝜎	√2𝜋𝜎	INTJ
cana-1630	57	3	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	57	4	(	(	PUNCT
cana-1630	57	5	−	−	PROPN
cana-1630	57	6	(	(	PUNCT
cana-1630	57	7	𝑥	𝑥	PROPN
cana-1630	57	8	−	−	PROPN
cana-1630	57	9	𝑚)2	𝑚)2	PROPN
cana-1630	57	10	2𝜎2	2𝜎2	NUM
cana-1630	57	11	)	)	PUNCT
cana-1630	57	12	where	where	SCONJ
cana-1630	57	13	𝐹(𝑥	𝐹(𝑥	X
cana-1630	57	14	,	,	PUNCT
cana-1630	57	15	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	57	16	)	)	PUNCT
cana-1630	57	17	is	be	AUX
cana-1630	57	18	fuzzy	fuzzy	ADJ
cana-1630	57	19	gaussian	gaussian	NOUN
cana-1630	57	20	1d	1d	NOUN
cana-1630	57	21	,	,	PUNCT
cana-1630	57	22	x	x	X
cana-1630	57	23	is	be	AUX
cana-1630	57	24	a	a	DET
cana-1630	57	25	variable	variable	NOUN
cana-1630	57	26	,	,	PUNCT
cana-1630	57	27	𝜎	𝜎	PROPN
cana-1630	57	28	is	be	AUX
cana-1630	57	29	the	the	DET
cana-1630	57	30	standard	standard	ADJ
cana-1630	57	31	deviation	deviation	NOUN
cana-1630	57	32	and	and	CCONJ
cana-1630	57	33	m	m	NOUN
cana-1630	57	34	is	be	AUX
cana-1630	57	35	a	a	DET
cana-1630	57	36	fuzziness	fuzziness	NOUN
cana-1630	57	37	parameter	parameter	NOUN
cana-1630	57	38	.	.	PUNCT
cana-1630	58	1	also	also	ADV
cana-1630	58	2	,	,	PUNCT
cana-1630	58	3	the	the	DET
cana-1630	58	4	derivative	derivative	NOUN
cana-1630	58	5	of	of	ADP
cana-1630	58	6	(	(	PUNCT
cana-1630	58	7	11	11	NUM
cana-1630	58	8	)	)	PUNCT
cana-1630	58	9	(	(	PUNCT
cana-1630	58	10	i.e	i.e	X
cana-1630	58	11	)	)	PUNCT
cana-1630	58	12	fuzzy	fuzzy	ADJ
cana-1630	58	13	gaussian	gaussian	ADJ
cana-1630	58	14	function	function	NOUN
cana-1630	58	15	with	with	ADP
cana-1630	58	16	respect	respect	NOUN
cana-1630	58	17	to	to	ADP
cana-1630	58	18	x.	x.	NOUN
cana-1630	58	19	the	the	DET
cana-1630	58	20	derivative	derivative	NOUN
cana-1630	58	21	is	be	AUX
cana-1630	58	22	given	give	VERB
cana-1630	58	23	as	as	ADP
cana-1630	58	24	:	:	PUNCT
cana-1630	58	25	𝐹′(𝑥	𝐹′(𝑥	NUM
cana-1630	58	26	,	,	PUNCT
cana-1630	58	27	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	58	28	)	)	PUNCT
cana-1630	58	29	=	=	SYM
cana-1630	58	30	−	−	PROPN
cana-1630	58	31	(	(	PUNCT
cana-1630	58	32	𝑥−𝑚	𝑥−𝑚	NOUN
cana-1630	58	33	𝜎2	𝜎2	NOUN
cana-1630	58	34	)	)	PUNCT
cana-1630	58	35	1	1	NUM
cana-1630	59	1	√2𝜋𝜎	√2𝜋𝜎	INTJ
cana-1630	59	2	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	59	3	(	(	PUNCT
cana-1630	59	4	−	−	PROPN
cana-1630	59	5	(	(	PUNCT
cana-1630	59	6	𝑥−𝑚)2	𝑥−𝑚)2	NOUN
cana-1630	59	7	2𝜎2	2𝜎2	NUM
cana-1630	59	8	)	)	PUNCT
cana-1630	59	9	2.14	2.14	NUM
cana-1630	59	10	fuzzy	fuzzy	ADJ
cana-1630	59	11	gaussian	gaussian	NOUN
cana-1630	59	12	2	2	NUM
cana-1630	59	13	d	d	NOUN
cana-1630	59	14	and	and	CCONJ
cana-1630	59	15	derivative	derivative	NOUN
cana-1630	59	16	:	:	PUNCT
cana-1630	59	17	consider	consider	VERB
cana-1630	59	18	a	a	DET
cana-1630	59	19	2d	2d	NUM
cana-1630	59	20	fuzzy	fuzzy	ADJ
cana-1630	59	21	gaussian	gaussian	ADJ
cana-1630	59	22	function	function	NOUN
cana-1630	59	23	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	59	24	,	,	PUNCT
cana-1630	59	25	𝑦	𝑦	NOUN
cana-1630	59	26	,	,	PUNCT
cana-1630	59	27	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	59	28	,	,	PUNCT
cana-1630	59	29	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	59	30	,	,	PUNCT
cana-1630	59	31	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	59	32	,	,	PUNCT
cana-1630	59	33	𝑚𝑦	𝑚𝑦	ADP
cana-1630	59	34	)	)	PUNCT
cana-1630	59	35	where	where	SCONJ
cana-1630	59	36	x	x	PRON
cana-1630	59	37	and	and	CCONJ
cana-1630	59	38	y	y	PROPN
cana-1630	59	39	are	be	AUX
cana-1630	59	40	the	the	DET
cana-1630	59	41	variables	variable	NOUN
cana-1630	59	42	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	59	43	,	,	PUNCT
cana-1630	59	44	𝜎𝑦	𝜎𝑦	DET
cana-1630	59	45	,	,	PUNCT
cana-1630	59	46	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	59	47	,	,	PUNCT
cana-1630	59	48	𝑚𝑦	𝑚𝑦	ADP
cana-1630	59	49	are	be	AUX
cana-1630	59	50	the	the	DET
cana-1630	59	51	standard	standard	ADJ
cana-1630	59	52	deviation	deviation	NOUN
cana-1630	59	53	and	and	CCONJ
cana-1630	59	54	mean	mean	VERB
cana-1630	59	55	along	along	ADP
cana-1630	59	56	the	the	DET
cana-1630	59	57	respective	respective	ADJ
cana-1630	59	58	axis	axis	NOUN
cana-1630	60	1	[	[	X
cana-1630	60	2	6	6	NUM
cana-1630	60	3	]	]	PUNCT
cana-1630	60	4	.	.	PUNCT
cana-1630	61	1	the	the	DET
cana-1630	61	2	function	function	NOUN
cana-1630	61	3	is	be	AUX
cana-1630	61	4	given	give	VERB
cana-1630	61	5	by	by	ADP
cana-1630	61	6	:	:	PUNCT
cana-1630	61	7	communications	communication	NOUN
cana-1630	61	8	on	on	ADP
cana-1630	61	9	applied	apply	VERB
cana-1630	61	10	nonlinear	nonlinear	ADJ
cana-1630	61	11	analysis	analysis	NOUN
cana-1630	61	12	issn	issn	NOUN
cana-1630	61	13	:	:	PUNCT
cana-1630	61	14	1074	1074	NUM
cana-1630	61	15	-	-	PUNCT
cana-1630	61	16	133x	133x	NUM
cana-1630	61	17	vol	vol	NOUN
cana-1630	61	18	32	32	NUM
cana-1630	61	19	no	no	NOUN
cana-1630	61	20	.	.	NOUN
cana-1630	61	21	1	1	NUM
cana-1630	61	22	(	(	PUNCT
cana-1630	61	23	2025	2025	NUM
cana-1630	61	24	)	)	PUNCT
cana-1630	61	25	187	187	NUM
cana-1630	61	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	61	27	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	61	28	,	,	PUNCT
cana-1630	61	29	𝑦	𝑦	NOUN
cana-1630	61	30	,	,	PUNCT
cana-1630	61	31	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	61	32	,	,	PUNCT
cana-1630	61	33	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	61	34	,	,	PUNCT
cana-1630	61	35	𝑚𝑥,𝑚𝑦	𝑚𝑥,𝑚𝑦	NOUN
cana-1630	61	36	)	)	PUNCT
cana-1630	61	37	=	=	SYM
cana-1630	62	1	1	1	NUM
cana-1630	62	2	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	62	3	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	63	1	[	[	X
cana-1630	63	2	−	−	X
cana-1630	63	3	(	(	PUNCT
cana-1630	63	4	𝑥−𝑚𝑥)2	𝑥−𝑚𝑥)2	NUM
cana-1630	63	5	2𝜎𝑥	2𝜎𝑥	ADJ
cana-1630	63	6	2	2	NUM
cana-1630	63	7	−	−	PROPN
cana-1630	63	8	(	(	PUNCT
cana-1630	63	9	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	63	10	)	)	PUNCT
cana-1630	63	11	2	2	NUM
cana-1630	63	12	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	63	13	2	2	NUM
cana-1630	63	14	]	]	PUNCT
cana-1630	63	15	the	the	DET
cana-1630	63	16	fuzzy	fuzzy	ADJ
cana-1630	63	17	partial	partial	ADJ
cana-1630	63	18	derivative	derivative	NOUN
cana-1630	63	19	of	of	ADP
cana-1630	63	20	this	this	DET
cana-1630	63	21	2d	2d	NUM
cana-1630	63	22	fuzzy	fuzzy	ADJ
cana-1630	63	23	gaussian	gaussian	ADJ
cana-1630	63	24	function	function	NOUN
cana-1630	63	25	with	with	ADP
cana-1630	63	26	respect	respect	NOUN
cana-1630	63	27	to	to	ADP
cana-1630	63	28	x	x	PUNCT
cana-1630	63	29	and	and	CCONJ
cana-1630	63	30	y.	y.	NOUN
cana-1630	63	31	the	the	DET
cana-1630	63	32	partial	partial	ADJ
cana-1630	63	33	derivatives	derivative	NOUN
cana-1630	63	34	are	be	AUX
cana-1630	63	35	given	give	VERB
cana-1630	63	36	by	by	ADP
cana-1630	63	37	:	:	PUNCT
cana-1630	63	38	𝐹𝑥(𝑥	𝐹𝑥(𝑥	ADJ
cana-1630	63	39	,	,	PUNCT
cana-1630	63	40	𝑦	𝑦	NOUN
cana-1630	63	41	,	,	PUNCT
cana-1630	63	42	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	63	43	,	,	PUNCT
cana-1630	63	44	𝜎𝑦	𝜎𝑦	DET
cana-1630	63	45	,	,	PUNCT
cana-1630	63	46	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	63	47	,	,	PUNCT
cana-1630	63	48	𝑚𝑦	𝑚𝑦	ADP
cana-1630	63	49	)	)	PUNCT
cana-1630	63	50	=	=	SYM
cana-1630	64	1	−	−	PROPN
cana-1630	64	2	(	(	PUNCT
cana-1630	64	3	𝑥−𝑚𝑥	𝑥−𝑚𝑥	NOUN
cana-1630	64	4	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	64	5	2	2	NUM
cana-1630	64	6	)	)	PUNCT
cana-1630	64	7	1	1	NUM
cana-1630	64	8	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	64	9	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	65	1	[	[	X
cana-1630	65	2	−	−	X
cana-1630	65	3	(	(	PUNCT
cana-1630	65	4	𝑥−𝑚𝑥)2	𝑥−𝑚𝑥)2	NUM
cana-1630	65	5	2𝜎𝑥	2𝜎𝑥	ADJ
cana-1630	65	6	2	2	NUM
cana-1630	65	7	−	−	PROPN
cana-1630	65	8	(	(	PUNCT
cana-1630	65	9	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	65	10	)	)	PUNCT
cana-1630	65	11	2	2	NUM
cana-1630	65	12	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	65	13	2	2	NUM
cana-1630	65	14	]	]	PUNCT
cana-1630	65	15	𝐹𝑦(𝑥	𝐹𝑦(𝑥	NUM
cana-1630	65	16	,	,	PUNCT
cana-1630	65	17	𝑦	𝑦	NOUN
cana-1630	65	18	,	,	PUNCT
cana-1630	65	19	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	65	20	,	,	PUNCT
cana-1630	65	21	𝜎𝑦	𝜎𝑦	DET
cana-1630	65	22	,	,	PUNCT
cana-1630	65	23	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	65	24	,	,	PUNCT
cana-1630	65	25	𝑚𝑦	𝑚𝑦	ADP
cana-1630	65	26	)	)	PUNCT
cana-1630	65	27	=	=	SYM
cana-1630	66	1	−	−	PROPN
cana-1630	66	2	(	(	PUNCT
cana-1630	66	3	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	66	4	𝜎𝑦	𝜎𝑦	PRON
cana-1630	66	5	2	2	NUM
cana-1630	66	6	)	)	PUNCT
cana-1630	66	7	1	1	NUM
cana-1630	66	8	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	66	9	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	67	1	[	[	X
cana-1630	67	2	−	−	X
cana-1630	67	3	(	(	PUNCT
cana-1630	67	4	𝑥−𝑚𝑥)2	𝑥−𝑚𝑥)2	NUM
cana-1630	67	5	2𝜎𝑥	2𝜎𝑥	ADJ
cana-1630	67	6	2	2	NUM
cana-1630	67	7	−	−	PROPN
cana-1630	67	8	(	(	PUNCT
cana-1630	67	9	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	67	10	)	)	PUNCT
cana-1630	67	11	2	2	NUM
cana-1630	67	12	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	67	13	2	2	NUM
cana-1630	67	14	]	]	SYM
cana-1630	67	15	2	2	NUM
cana-1630	67	16	.	.	X
cana-1630	67	17	main	main	ADJ
cana-1630	67	18	results	result	NOUN
cana-1630	67	19	combining	combine	VERB
cana-1630	67	20	gaussian	gaussian	ADJ
cana-1630	67	21	filtering	filtering	NOUN
cana-1630	67	22	with	with	ADP
cana-1630	67	23	derivatives	derivative	NOUN
cana-1630	67	24	to	to	PART
cana-1630	67	25	create	create	VERB
cana-1630	67	26	an	an	DET
cana-1630	67	27	adaptive	adaptive	ADJ
cana-1630	67	28	gaussian	gaussian	ADJ
cana-1630	67	29	filter	filter	NOUN
cana-1630	67	30	offers	offer	VERB
cana-1630	67	31	significant	significant	ADJ
cana-1630	67	32	advantages	advantage	NOUN
cana-1630	67	33	by	by	ADP
cana-1630	67	34	enhancing	enhance	VERB
cana-1630	67	35	edge	edge	NOUN
cana-1630	67	36	detection	detection	NOUN
cana-1630	67	37	and	and	CCONJ
cana-1630	67	38	noise	noise	NOUN
cana-1630	67	39	reduction	reduction	NOUN
cana-1630	67	40	capabilities	capability	NOUN
cana-1630	67	41	.	.	PUNCT
cana-1630	68	1	this	this	DET
cana-1630	68	2	combination	combination	NOUN
cana-1630	68	3	is	be	AUX
cana-1630	68	4	particularly	particularly	ADV
cana-1630	68	5	effective	effective	ADJ
cana-1630	68	6	for	for	ADP
cana-1630	68	7	identifying	identify	VERB
cana-1630	68	8	subtle	subtle	ADJ
cana-1630	68	9	changes	change	NOUN
cana-1630	68	10	and	and	CCONJ
cana-1630	68	11	edges	edge	NOUN
cana-1630	68	12	,	,	PUNCT
cana-1630	68	13	crucial	crucial	ADJ
cana-1630	68	14	for	for	ADP
cana-1630	68	15	accurate	accurate	ADJ
cana-1630	68	16	feature	feature	NOUN
cana-1630	68	17	extraction	extraction	NOUN
cana-1630	68	18	in	in	ADP
cana-1630	68	19	image	image	NOUN
cana-1630	68	20	processing	processing	NOUN
cana-1630	68	21	.	.	PUNCT
cana-1630	69	1	gaussian	gaussian	ADJ
cana-1630	69	2	filters	filter	NOUN
cana-1630	69	3	are	be	AUX
cana-1630	69	4	renowned	renowned	ADJ
cana-1630	69	5	for	for	ADP
cana-1630	69	6	their	their	PRON
cana-1630	69	7	ability	ability	NOUN
cana-1630	69	8	to	to	PART
cana-1630	69	9	smooth	smooth	VERB
cana-1630	69	10	out	out	ADP
cana-1630	69	11	noise	noise	NOUN
cana-1630	69	12	while	while	SCONJ
cana-1630	69	13	preserving	preserve	VERB
cana-1630	69	14	essential	essential	ADJ
cana-1630	69	15	signal	signal	NOUN
cana-1630	69	16	characteristics	characteristic	NOUN
cana-1630	69	17	,	,	PUNCT
cana-1630	69	18	and	and	CCONJ
cana-1630	69	19	when	when	SCONJ
cana-1630	69	20	paired	pair	VERB
cana-1630	69	21	with	with	ADP
cana-1630	69	22	derivatives	derivative	NOUN
cana-1630	69	23	,	,	PUNCT
cana-1630	69	24	they	they	PRON
cana-1630	69	25	further	far	ADV
cana-1630	69	26	improve	improve	VERB
cana-1630	69	27	noise	noise	NOUN
cana-1630	69	28	reduction	reduction	NOUN
cana-1630	69	29	while	while	SCONJ
cana-1630	69	30	maintaining	maintain	VERB
cana-1630	69	31	critical	critical	ADJ
cana-1630	69	32	details	detail	NOUN
cana-1630	69	33	.	.	PUNCT
cana-1630	70	1	the	the	DET
cana-1630	70	2	adaptive	adaptive	ADJ
cana-1630	70	3	nature	nature	NOUN
cana-1630	70	4	of	of	ADP
cana-1630	70	5	the	the	DET
cana-1630	70	6	filter	filter	NOUN
cana-1630	70	7	allows	allow	VERB
cana-1630	70	8	it	it	PRON
cana-1630	70	9	to	to	PART
cana-1630	70	10	dynamically	dynamically	ADV
cana-1630	70	11	adjust	adjust	VERB
cana-1630	70	12	its	its	PRON
cana-1630	70	13	parameters	parameter	NOUN
cana-1630	70	14	based	base	VERB
cana-1630	70	15	on	on	ADP
cana-1630	70	16	the	the	DET
cana-1630	70	17	specific	specific	ADJ
cana-1630	70	18	characteristics	characteristic	NOUN
cana-1630	70	19	of	of	ADP
cana-1630	70	20	the	the	DET
cana-1630	70	21	input	input	NOUN
cana-1630	70	22	data	datum	NOUN
cana-1630	70	23	,	,	PUNCT
cana-1630	70	24	ensuring	ensure	VERB
cana-1630	70	25	effectiveness	effectiveness	NOUN
cana-1630	70	26	across	across	ADP
cana-1630	70	27	diverse	diverse	ADJ
cana-1630	70	28	conditions	condition	NOUN
cana-1630	70	29	and	and	CCONJ
cana-1630	70	30	applications	application	NOUN
cana-1630	70	31	.	.	PUNCT
cana-1630	71	1	this	this	DET
cana-1630	71	2	approach	approach	NOUN
cana-1630	71	3	optimizes	optimize	VERB
cana-1630	71	4	[	[	X
cana-1630	71	5	8	8	NUM
cana-1630	71	6	]	]	PUNCT
cana-1630	71	7	performance	performance	NOUN
cana-1630	71	8	by	by	ADP
cana-1630	71	9	balancing	balance	VERB
cana-1630	71	10	noise	noise	NOUN
cana-1630	71	11	reduction	reduction	NOUN
cana-1630	71	12	and	and	CCONJ
cana-1630	71	13	detail	detail	NOUN
cana-1630	71	14	preservation	preservation	NOUN
cana-1630	71	15	,	,	PUNCT
cana-1630	71	16	making	make	VERB
cana-1630	71	17	it	it	PRON
cana-1630	71	18	a	a	DET
cana-1630	71	19	versatile	versatile	ADJ
cana-1630	71	20	and	and	CCONJ
cana-1630	71	21	powerful	powerful	ADJ
cana-1630	71	22	tool	tool	NOUN
cana-1630	71	23	for	for	ADP
cana-1630	71	24	various	various	ADJ
cana-1630	71	25	data	datum	NOUN
cana-1630	71	26	processing	processing	NOUN
cana-1630	71	27	and	and	CCONJ
cana-1630	71	28	analysis	analysis	NOUN
cana-1630	71	29	tasks	task	NOUN
cana-1630	71	30	.	.	PUNCT
cana-1630	72	1	gaussian	gaussian	ADJ
cana-1630	72	2	filters	filter	NOUN
cana-1630	72	3	,	,	PUNCT
cana-1630	72	4	both	both	CCONJ
cana-1630	72	5	in	in	ADP
cana-1630	72	6	1d	1d	NUM
cana-1630	72	7	and	and	CCONJ
cana-1630	72	8	2d	2d	NUM
cana-1630	72	9	forms	form	NOUN
cana-1630	72	10	,	,	PUNCT
cana-1630	72	11	are	be	AUX
cana-1630	72	12	crucial	crucial	ADJ
cana-1630	72	13	for	for	ADP
cana-1630	72	14	medical	medical	ADJ
cana-1630	72	15	image	image	NOUN
cana-1630	72	16	processing	processing	NOUN
cana-1630	72	17	,	,	PUNCT
cana-1630	72	18	especially	especially	ADV
cana-1630	72	19	for	for	ADP
cana-1630	72	20	mri	mri	NOUN
cana-1630	72	21	scans	scan	NOUN
cana-1630	72	22	,	,	PUNCT
cana-1630	72	23	as	as	SCONJ
cana-1630	72	24	they	they	PRON
cana-1630	72	25	effectively	effectively	ADV
cana-1630	72	26	reduce	reduce	VERB
cana-1630	72	27	noise	noise	NOUN
cana-1630	72	28	while	while	SCONJ
cana-1630	72	29	preserving	preserve	VERB
cana-1630	72	30	key	key	ADJ
cana-1630	72	31	image	image	NOUN
cana-1630	72	32	details	detail	NOUN
cana-1630	72	33	.	.	PUNCT
cana-1630	73	1	by	by	ADP
cana-1630	73	2	incorporating	incorporate	VERB
cana-1630	73	3	fuzzy	fuzzy	ADJ
cana-1630	73	4	logic	logic	NOUN
cana-1630	73	5	to	to	PART
cana-1630	73	6	handle	handle	VERB
cana-1630	73	7	uncertainties	uncertainty	NOUN
cana-1630	73	8	in	in	ADP
cana-1630	73	9	image	image	NOUN
cana-1630	73	10	data	datum	NOUN
cana-1630	73	11	,	,	PUNCT
cana-1630	73	12	fuzzy	fuzzy	ADJ
cana-1630	73	13	gaussian	gaussian	ADJ
cana-1630	73	14	filters	filter	NOUN
cana-1630	73	15	further	far	ADV
cana-1630	73	16	enhance	enhance	VERB
cana-1630	73	17	this	this	DET
cana-1630	73	18	capability	capability	NOUN
cana-1630	73	19	.	.	PUNCT
cana-1630	74	1	both	both	PRON
cana-1630	74	2	standard	standard	ADJ
cana-1630	74	3	and	and	CCONJ
cana-1630	74	4	fuzzy	fuzzy	ADJ
cana-1630	74	5	gaussian	gaussian	ADJ
cana-1630	74	6	filters	filter	NOUN
cana-1630	74	7	smooth	smooth	ADJ
cana-1630	74	8	images	image	NOUN
cana-1630	74	9	while	while	SCONJ
cana-1630	74	10	maintaining	maintain	VERB
cana-1630	74	11	edges	edge	NOUN
cana-1630	74	12	,	,	PUNCT
cana-1630	74	13	striking	strike	VERB
cana-1630	74	14	a	a	DET
cana-1630	74	15	balance	balance	NOUN
cana-1630	74	16	in	in	ADP
cana-1630	74	17	noise	noise	NOUN
cana-1630	74	18	reduction	reduction	NOUN
cana-1630	74	19	.	.	PUNCT
cana-1630	75	1	adaptive	adaptive	ADJ
cana-1630	75	2	filters	filter	NOUN
cana-1630	75	3	excel	excel	VERB
cana-1630	75	4	in	in	ADP
cana-1630	75	5	context	context	NOUN
cana-1630	75	6	-	-	PUNCT
cana-1630	75	7	aware	aware	ADJ
cana-1630	75	8	smoothing	smoothing	NOUN
cana-1630	75	9	by	by	ADP
cana-1630	75	10	distinguishing	distinguish	VERB
cana-1630	75	11	between	between	ADP
cana-1630	75	12	different	different	ADJ
cana-1630	75	13	image	image	NOUN
cana-1630	75	14	regions	region	NOUN
cana-1630	75	15	and	and	CCONJ
cana-1630	75	16	applying	apply	VERB
cana-1630	75	17	the	the	DET
cana-1630	75	18	appropriate	appropriate	ADJ
cana-1630	75	19	level	level	NOUN
cana-1630	75	20	of	of	ADP
cana-1630	75	21	smoothing	smooth	VERB
cana-1630	75	22	to	to	PART
cana-1630	75	23	enhance	enhance	VERB
cana-1630	75	24	image	image	NOUN
cana-1630	75	25	quality	quality	NOUN
cana-1630	75	26	and	and	CCONJ
cana-1630	75	27	reduce	reduce	VERB
cana-1630	75	28	abnormalities	abnormality	NOUN
cana-1630	75	29	.	.	PUNCT
cana-1630	76	1	adaptive	adaptive	ADJ
cana-1630	76	2	gaussian	gaussian	ADJ
cana-1630	76	3	filters	filter	NOUN
cana-1630	76	4	are	be	AUX
cana-1630	76	5	particularly	particularly	ADV
cana-1630	76	6	beneficial	beneficial	ADJ
cana-1630	76	7	in	in	ADP
cana-1630	76	8	scenarios	scenario	NOUN
cana-1630	76	9	where	where	SCONJ
cana-1630	76	10	preserving	preserve	VERB
cana-1630	76	11	fine	fine	ADJ
cana-1630	76	12	structural	structural	ADJ
cana-1630	76	13	details	detail	NOUN
cana-1630	76	14	is	be	AUX
cana-1630	76	15	essential	essential	ADJ
cana-1630	76	16	,	,	PUNCT
cana-1630	76	17	such	such	ADJ
cana-1630	76	18	as	as	ADP
cana-1630	76	19	in	in	ADP
cana-1630	76	20	medical	medical	ADJ
cana-1630	76	21	imaging	imaging	NOUN
cana-1630	76	22	,	,	PUNCT
cana-1630	76	23	which	which	PRON
cana-1630	76	24	is	be	AUX
cana-1630	76	25	vital	vital	ADJ
cana-1630	76	26	for	for	ADP
cana-1630	76	27	accurate	accurate	ADJ
cana-1630	76	28	diagnosis	diagnosis	NOUN
cana-1630	76	29	and	and	CCONJ
cana-1630	76	30	analysis	analysis	NOUN
cana-1630	76	31	.	.	PUNCT
cana-1630	77	1	their	their	PRON
cana-1630	77	2	adaptability	adaptability	NOUN
cana-1630	77	3	and	and	CCONJ
cana-1630	77	4	effectiveness	effectiveness	NOUN
cana-1630	77	5	make	make	VERB
cana-1630	77	6	them	they	PRON
cana-1630	77	7	highly	highly	ADV
cana-1630	77	8	suitable	suitable	ADJ
cana-1630	77	9	for	for	ADP
cana-1630	77	10	these	these	DET
cana-1630	77	11	critical	critical	ADJ
cana-1630	77	12	applications	application	NOUN
cana-1630	77	13	.	.	PUNCT
cana-1630	78	1	the	the	DET
cana-1630	78	2	stability	stability	NOUN
cana-1630	78	3	analysis	analysis	NOUN
cana-1630	78	4	of	of	ADP
cana-1630	78	5	adaptive	adaptive	ADJ
cana-1630	78	6	filter	filter	NOUN
cana-1630	78	7	is	be	AUX
cana-1630	78	8	performed	perform	VERB
cana-1630	78	9	in	in	ADP
cana-1630	78	10	this	this	DET
cana-1630	78	11	section	section	NOUN
cana-1630	78	12	.	.	PUNCT
cana-1630	79	1	3.1	3.1	NUM
cana-1630	79	2	adaptive	adaptive	ADJ
cana-1630	79	3	gaussian	gaussian	NOUN
cana-1630	79	4	filter1d	filter1d	PROPN
cana-1630	79	5	and	and	CCONJ
cana-1630	79	6	derivative	derivative	NOUN
cana-1630	79	7	of	of	ADP
cana-1630	79	8	gaussian	gaussian	ADJ
cana-1630	79	9	1d	1d	NUM
cana-1630	79	10	(	(	PUNCT
cana-1630	79	11	agd-1d	agd-1d	PROPN
cana-1630	79	12	)	)	PUNCT
cana-1630	79	13	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	79	14	,	,	PUNCT
cana-1630	79	15	𝜎	𝜎	X
cana-1630	79	16	)	)	PUNCT
cana-1630	79	17	=	=	SYM
cana-1630	79	18	1	1	NUM
cana-1630	79	19	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	79	20	[	[	PUNCT
cana-1630	79	21	1	1	NUM
cana-1630	79	22	−	−	PROPN
cana-1630	79	23	𝛼.	𝛼.	NOUN
cana-1630	79	24	𝑥	𝑥	PROPN
cana-1630	80	1	𝜎2	𝜎2	NOUN
cana-1630	80	2	]	]	PUNCT
cana-1630	80	3	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	80	4	(	(	PUNCT
cana-1630	80	5	−𝑥2	−𝑥2	PROPN
cana-1630	80	6	2𝜎2	2𝜎2	NUM
cana-1630	80	7	)	)	PUNCT
cana-1630	80	8	(	(	PUNCT
cana-1630	80	9	1	1	X
cana-1630	80	10	)	)	PUNCT
cana-1630	80	11	where	where	SCONJ
cana-1630	80	12	𝜎	𝜎	NOUN
cana-1630	80	13	=	=	NOUN
cana-1630	80	14	√	√	ADP
cana-1630	80	15	1	1	NUM
cana-1630	80	16	𝑚𝑛	𝑚𝑛	NOUN
cana-1630	80	17	∑	∑	PROPN
cana-1630	80	18	∑	∑	PROPN
cana-1630	80	19	(	(	PUNCT
cana-1630	80	20	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	80	21	,	,	PUNCT
cana-1630	80	22	𝑗	𝑗	NOUN
cana-1630	80	23	)	)	PUNCT
cana-1630	80	24	−	−	NOUN
cana-1630	80	25	𝑚)2𝑛	𝑚)2𝑛	NOUN
cana-1630	80	26	𝑗=1	𝑗=1	PROPN
cana-1630	80	27	𝑚	𝑚	X
cana-1630	80	28	𝑖=1	𝑖=1	PROPN
cana-1630	80	29	;	;	PUNCT
cana-1630	80	30	∝=	∝=	NOUN
cana-1630	80	31	𝐴	𝐴	PROPN
cana-1630	80	32	max	max	PROPN
cana-1630	80	33	(	(	PUNCT
cana-1630	80	34	|𝐴|	|𝐴|	NOUN
cana-1630	80	35	)	)	PUNCT
cana-1630	80	36	theorem	theorem	ADJ
cana-1630	80	37	1	1	NUM
cana-1630	80	38	:	:	PUNCT
cana-1630	80	39	boundary	boundary	ADJ
cana-1630	80	40	input	input	NOUN
cana-1630	80	41	boundary	boundary	ADJ
cana-1630	80	42	output	output	NOUN
cana-1630	80	43	(	(	PUNCT
cana-1630	80	44	bibo	bibo	ADJ
cana-1630	80	45	)	)	PUNCT
cana-1630	80	46	of	of	ADP
cana-1630	80	47	the	the	DET
cana-1630	80	48	filter	filter	NOUN
cana-1630	80	49	a	a	DET
cana-1630	80	50	filter	filter	NOUN
cana-1630	80	51	𝐹(𝑥	𝐹(𝑥	NOUN
cana-1630	80	52	,	,	PUNCT
cana-1630	80	53	𝜎	𝜎	X
cana-1630	80	54	)	)	PUNCT
cana-1630	80	55	is	be	AUX
cana-1630	80	56	bibo	bibo	VERB
cana-1630	80	57	stable	stable	ADJ
cana-1630	80	58	if	if	SCONJ
cana-1630	80	59	for	for	ADP
cana-1630	80	60	every	every	DET
cana-1630	80	61	bounded	bounded	ADJ
cana-1630	80	62	input	input	NOUN
cana-1630	80	63	x	x	NOUN
cana-1630	80	64	,	,	PUNCT
cana-1630	80	65	the	the	DET
cana-1630	80	66	output	output	NOUN
cana-1630	80	67	𝑦	𝑦	NOUN
cana-1630	80	68	=	=	SYM
cana-1630	80	69	𝐹(𝑥	𝐹(𝑥	PROPN
cana-1630	80	70	,	,	PUNCT
cana-1630	80	71	𝜎	𝜎	X
cana-1630	80	72	)	)	PUNCT
cana-1630	80	73	is	be	AUX
cana-1630	80	74	also	also	ADV
cana-1630	80	75	bounded	bound	VERB
cana-1630	80	76	.	.	PUNCT
cana-1630	81	1	proof	proof	NOUN
cana-1630	81	2	:	:	PUNCT
cana-1630	81	3	suppose	suppose	VERB
cana-1630	81	4	x	x	PRON
cana-1630	81	5	is	be	AUX
cana-1630	81	6	bounded	bound	VERB
cana-1630	81	7	then	then	ADV
cana-1630	81	8	there	there	PRON
cana-1630	81	9	exist	exist	VERB
cana-1630	81	10	a	a	DET
cana-1630	81	11	constant	constant	ADJ
cana-1630	81	12	m	m	NOUN
cana-1630	81	13	such	such	ADJ
cana-1630	81	14	that	that	DET
cana-1630	81	15	|𝑥|	|𝑥|	PROPN
cana-1630	81	16	≤	≤	ADJ
cana-1630	81	17	𝑀.	𝑀.	PROPN
cana-1630	81	18	consider	consider	VERB
cana-1630	81	19	communications	communication	NOUN
cana-1630	81	20	on	on	ADP
cana-1630	81	21	applied	apply	VERB
cana-1630	81	22	nonlinear	nonlinear	ADJ
cana-1630	81	23	analysis	analysis	NOUN
cana-1630	81	24	issn	issn	NOUN
cana-1630	81	25	:	:	PUNCT
cana-1630	81	26	1074	1074	NUM
cana-1630	81	27	-	-	PUNCT
cana-1630	81	28	133x	133x	NUM
cana-1630	81	29	vol	vol	NOUN
cana-1630	81	30	32	32	NUM
cana-1630	81	31	no	no	NOUN
cana-1630	81	32	.	.	NOUN
cana-1630	81	33	1	1	NUM
cana-1630	81	34	(	(	PUNCT
cana-1630	81	35	2025	2025	NUM
cana-1630	81	36	)	)	PUNCT
cana-1630	81	37	188	188	NUM
cana-1630	81	38	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	81	39	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	81	40	,	,	PUNCT
cana-1630	81	41	𝜎	𝜎	NOUN
cana-1630	81	42	)	)	PUNCT
cana-1630	81	43	=	=	SYM
cana-1630	82	1	1	1	NUM
cana-1630	82	2	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	82	3	[	[	PUNCT
cana-1630	82	4	1	1	NUM
cana-1630	82	5	−	−	PROPN
cana-1630	82	6	𝛼.	𝛼.	NOUN
cana-1630	82	7	𝑥	𝑥	PROPN
cana-1630	83	1	𝜎2	𝜎2	NOUN
cana-1630	83	2	]	]	PUNCT
cana-1630	83	3	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	83	4	(	(	PUNCT
cana-1630	83	5	−𝑥2	−𝑥2	PROPN
cana-1630	83	6	2𝜎2	2𝜎2	NUM
cana-1630	83	7	)	)	PUNCT
cana-1630	83	8	where	where	SCONJ
cana-1630	83	9	the	the	DET
cana-1630	83	10	term	term	NOUN
cana-1630	83	11	1	1	NUM
cana-1630	83	12	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	83	13	is	be	AUX
cana-1630	83	14	a	a	DET
cana-1630	83	15	constant	constant	ADJ
cana-1630	83	16	for	for	ADP
cana-1630	83	17	a	a	DET
cana-1630	83	18	given	give	VERB
cana-1630	83	19	𝜎.	𝜎.	NOUN
cana-1630	83	20	also	also	ADV
cana-1630	83	21	,	,	PUNCT
cana-1630	83	22	[	[	X
cana-1630	83	23	1	1	NUM
cana-1630	83	24	−	−	PROPN
cana-1630	83	25	𝛼.	𝛼.	NOUN
cana-1630	83	26	𝑥	𝑥	PROPN
cana-1630	84	1	𝜎2	𝜎2	PROPN
cana-1630	84	2	]	]	PUNCT
cana-1630	84	3	is	be	AUX
cana-1630	84	4	bounded	bound	VERB
cana-1630	84	5	since	since	SCONJ
cana-1630	84	6	x	x	PRON
cana-1630	84	7	is	be	AUX
cana-1630	84	8	bounded	bound	VERB
cana-1630	84	9	,	,	PUNCT
cana-1630	84	10	|1	|1	PRON
cana-1630	85	1	−	−	PROPN
cana-1630	85	2	𝛼.	𝛼.	NOUN
cana-1630	85	3	𝑥	𝑥	PROPN
cana-1630	85	4	𝜎2	𝜎2	NOUN
cana-1630	85	5	|	|	ADV
cana-1630	85	6	≤	≤	NUM
cana-1630	85	7	1	1	NUM
cana-1630	86	1	+	+	NUM
cana-1630	86	2	|𝛼|	|𝛼|	PROPN
cana-1630	86	3	𝑀	𝑀	PROPN
cana-1630	86	4	𝜎2	𝜎2	NOUN
cana-1630	86	5	also	also	ADV
cana-1630	86	6	,	,	PUNCT
cana-1630	86	7	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	86	8	(	(	PUNCT
cana-1630	86	9	−𝑥2	−𝑥2	PROPN
cana-1630	86	10	2𝜎2	2𝜎2	NUM
cana-1630	86	11	)	)	PUNCT
cana-1630	86	12	is	be	AUX
cana-1630	86	13	bounded	bound	VERB
cana-1630	86	14	for	for	ADP
cana-1630	86	15	all	all	DET
cana-1630	86	16	x	x	PRON
cana-1630	86	17	⇒	⇒	PROPN
cana-1630	86	18	|𝐹(𝑥	|𝐹(𝑥	NOUN
cana-1630	86	19	,	,	PUNCT
cana-1630	86	20	𝜎)|	𝜎)|	ADJ
cana-1630	86	21	≤	≤	NUM
cana-1630	86	22	1	1	NUM
cana-1630	86	23	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	86	24	(	(	PUNCT
cana-1630	86	25	1	1	NUM
cana-1630	86	26	+	+	NUM
cana-1630	86	27	|𝛼|	|𝛼|	PROPN
cana-1630	86	28	𝑀	𝑀	PROPN
cana-1630	86	29	𝜎2	𝜎2	PROPN
cana-1630	86	30	)	)	PUNCT
cana-1630	86	31	since	since	SCONJ
cana-1630	86	32	all	all	DET
cana-1630	86	33	the	the	DET
cana-1630	86	34	components	component	NOUN
cana-1630	86	35	are	be	AUX
cana-1630	86	36	bounded	bound	VERB
cana-1630	86	37	,	,	PUNCT
cana-1630	86	38	then	then	ADV
cana-1630	86	39	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	86	40	,	,	PUNCT
cana-1630	86	41	𝜎	𝜎	X
cana-1630	86	42	)	)	PUNCT
cana-1630	86	43	is	be	AUX
cana-1630	86	44	also	also	ADV
cana-1630	86	45	bounded	bound	VERB
cana-1630	86	46	.	.	PUNCT
cana-1630	87	1	hence	hence	ADV
cana-1630	87	2	,	,	PUNCT
cana-1630	87	3	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	87	4	,	,	PUNCT
cana-1630	87	5	𝜎	𝜎	X
cana-1630	87	6	)	)	PUNCT
cana-1630	87	7	is	be	AUX
cana-1630	87	8	bibo	bibo	ADJ
cana-1630	87	9	stable	stable	ADJ
cana-1630	87	10	.	.	PUNCT
cana-1630	88	1	hence	hence	ADV
cana-1630	88	2	the	the	DET
cana-1630	88	3	proof	proof	NOUN
cana-1630	88	4	.	.	PUNCT
cana-1630	89	1	theorem	theorem	VERB
cana-1630	89	2	2	2	NUM
cana-1630	89	3	:	:	PUNCT
cana-1630	89	4	preservation	preservation	NOUN
cana-1630	89	5	of	of	ADP
cana-1630	89	6	fuzzy	fuzzy	ADJ
cana-1630	89	7	membership	membership	NOUN
cana-1630	89	8	distribution	distribution	NOUN
cana-1630	89	9	theorem	theorem	VERB
cana-1630	89	10	if	if	SCONJ
cana-1630	89	11	a	a	DET
cana-1630	89	12	fuzzy	fuzzy	ADJ
cana-1630	89	13	membership	membership	NOUN
cana-1630	89	14	function	function	NOUN
cana-1630	89	15	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	89	16	)	)	PUNCT
cana-1630	89	17	is	be	AUX
cana-1630	89	18	processed	process	VERB
cana-1630	89	19	through	through	ADP
cana-1630	89	20	a	a	DET
cana-1630	89	21	filter	filter	NOUN
cana-1630	89	22	𝐹(𝑥	𝐹(𝑥	NOUN
cana-1630	89	23	,	,	PUNCT
cana-1630	89	24	𝜎	𝜎	NOUN
cana-1630	89	25	)	)	PUNCT
cana-1630	89	26	,	,	PUNCT
cana-1630	89	27	the	the	DET
cana-1630	89	28	resulting	result	VERB
cana-1630	89	29	function	function	NOUN
cana-1630	89	30	𝜇′(𝑥	𝜇′(𝑥	PUNCT
cana-1630	89	31	)	)	PUNCT
cana-1630	89	32	=	=	SYM
cana-1630	89	33	𝐹(𝜇(𝑥	𝐹(𝜇(𝑥	NOUN
cana-1630	89	34	)	)	PUNCT
cana-1630	89	35	,	,	PUNCT
cana-1630	89	36	𝜎	𝜎	X
cana-1630	89	37	)	)	PUNCT
cana-1630	89	38	preserves	preserve	VERB
cana-1630	89	39	the	the	DET
cana-1630	89	40	properties	property	NOUN
cana-1630	89	41	of	of	ADP
cana-1630	89	42	a	a	DET
cana-1630	89	43	fuzzy	fuzzy	ADJ
cana-1630	89	44	membership	membership	NOUN
cana-1630	89	45	function	function	NOUN
cana-1630	89	46	.	.	PUNCT
cana-1630	90	1	proof	proof	NOUN
cana-1630	90	2	:	:	PUNCT
cana-1630	90	3	a	a	DET
cana-1630	90	4	fuzzy	fuzzy	ADJ
cana-1630	90	5	membership	membership	NOUN
cana-1630	90	6	function	function	NOUN
cana-1630	90	7	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	90	8	)	)	PUNCT
cana-1630	90	9	is	be	AUX
cana-1630	90	10	normalized	normalize	VERB
cana-1630	90	11	such	such	ADJ
cana-1630	90	12	that	that	DET
cana-1630	90	13	sup	sup	NOUN
cana-1630	90	14	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	90	15	)	)	PUNCT
cana-1630	90	16	=	=	SYM
cana-1630	91	1	1	1	X
cana-1630	91	2	.	.	X
cana-1630	91	3	for	for	ADP
cana-1630	91	4	any	any	DET
cana-1630	91	5	𝑥1	𝑥1	NOUN
cana-1630	91	6	,	,	PUNCT
cana-1630	91	7	𝑥2𝜖ℝ	𝑥2𝜖ℝ	PROPN
cana-1630	91	8	and	and	CCONJ
cana-1630	91	9	𝜆	𝜆	PRON
cana-1630	91	10	∈	∈	PROPN
cana-1630	91	11	[	[	X
cana-1630	91	12	0,1	0,1	NUM
cana-1630	91	13	]	]	PUNCT
cana-1630	91	14	then	then	ADV
cana-1630	91	15	𝜇(𝜆𝑥1	𝜇(𝜆𝑥1	ADJ
cana-1630	91	16	+	+	SYM
cana-1630	91	17	(	(	PUNCT
cana-1630	91	18	1	1	NUM
cana-1630	91	19	−	−	PROPN
cana-1630	91	20	𝜆)𝑥2	𝜆)𝑥2	PROPN
cana-1630	91	21	)	)	PUNCT
cana-1630	91	22	≥	≥	NOUN
cana-1630	91	23	min(𝜇(𝑥1	min(𝜇(𝑥1	PROPN
cana-1630	91	24	)	)	PUNCT
cana-1630	91	25	,	,	PUNCT
cana-1630	91	26	𝜇(𝑥2	𝜇(𝑥2	NOUN
cana-1630	91	27	)	)	PUNCT
cana-1630	91	28	)	)	PUNCT
cana-1630	91	29	which	which	PRON
cana-1630	91	30	is	be	AUX
cana-1630	91	31	bounded	bound	VERB
cana-1630	91	32	that	that	PRON
cana-1630	91	33	is	be	AUX
cana-1630	91	34	0	0	NUM
cana-1630	91	35	≤	≤	NUM
cana-1630	91	36	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	91	37	)	)	PUNCT
cana-1630	91	38	≤	≤	NUM
cana-1630	91	39	1	1	NUM
cana-1630	91	40	.	.	PUNCT
cana-1630	92	1	since	since	SCONJ
cana-1630	92	2	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	92	3	)	)	PUNCT
cana-1630	92	4	is	be	AUX
cana-1630	92	5	normalized	normalize	VERB
cana-1630	92	6	,	,	PUNCT
cana-1630	92	7	assume	assume	VERB
cana-1630	92	8	,	,	PUNCT
cana-1630	92	9	𝜇(𝑥0	𝜇(𝑥0	ADJ
cana-1630	92	10	)	)	PUNCT
cana-1630	92	11	=	=	SYM
cana-1630	92	12	1	1	NUM
cana-1630	92	13	,	,	PUNCT
cana-1630	92	14	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1630	92	15	𝜇′(𝑥	𝜇′(𝑥	PUNCT
cana-1630	92	16	)	)	PUNCT
cana-1630	92	17	=	=	SYM
cana-1630	92	18	𝐹(𝜇(𝑥	𝐹(𝜇(𝑥	NOUN
cana-1630	92	19	)	)	PUNCT
cana-1630	92	20	,	,	PUNCT
cana-1630	92	21	𝜎	𝜎	X
cana-1630	92	22	)	)	PUNCT
cana-1630	92	23	also	also	ADV
cana-1630	92	24	,	,	PUNCT
cana-1630	92	25	𝜇′(𝑥0	𝜇′(𝑥0	ADJ
cana-1630	92	26	)	)	PUNCT
cana-1630	93	1	=	=	SYM
cana-1630	93	2	𝐹(𝜇(𝑥0	𝐹(𝜇(𝑥0	PROPN
cana-1630	93	3	)	)	PUNCT
cana-1630	93	4	,	,	PUNCT
cana-1630	93	5	𝜎	𝜎	X
cana-1630	93	6	)	)	PUNCT
cana-1630	93	7	=	=	SYM
cana-1630	94	1	𝐹(1	𝐹(1	NUM
cana-1630	94	2	,	,	PUNCT
cana-1630	94	3	𝜎	𝜎	NOUN
cana-1630	94	4	)	)	PUNCT
cana-1630	94	5	=	=	SYM
cana-1630	94	6	1	1	NUM
cana-1630	94	7	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	94	8	(	(	PUNCT
cana-1630	94	9	1	1	NUM
cana-1630	94	10	−	−	PROPN
cana-1630	94	11	𝛼	𝛼	PROPN
cana-1630	94	12	𝜎2	𝜎2	NOUN
cana-1630	94	13	)	)	PUNCT
cana-1630	94	14	exp(−	exp(−	PROPN
cana-1630	94	15	1	1	NUM
cana-1630	94	16	2𝜎2	2𝜎2	NUM
cana-1630	94	17	)	)	PUNCT
cana-1630	94	18	this	this	PRON
cana-1630	94	19	ensures	ensure	VERB
cana-1630	94	20	that	that	SCONJ
cana-1630	94	21	𝜇′(𝑥0	𝜇′(𝑥0	ADJ
cana-1630	94	22	)	)	PUNCT
cana-1630	94	23	is	be	AUX
cana-1630	94	24	well	well	ADV
cana-1630	94	25	defined	define	VERB
cana-1630	94	26	and	and	CCONJ
cana-1630	94	27	bounded	bound	VERB
cana-1630	94	28	.	.	PUNCT
cana-1630	95	1	given	give	VERB
cana-1630	95	2	the	the	DET
cana-1630	95	3	linearity	linearity	NOUN
cana-1630	95	4	and	and	CCONJ
cana-1630	95	5	continuity	continuity	NOUN
cana-1630	95	6	of	of	ADP
cana-1630	95	7	the	the	DET
cana-1630	95	8	filter	filter	NOUN
cana-1630	95	9	𝐹(𝑥	𝐹(𝑥	NOUN
cana-1630	95	10	,	,	PUNCT
cana-1630	95	11	𝜎	𝜎	NOUN
cana-1630	95	12	)	)	PUNCT
cana-1630	95	13	,	,	PUNCT
cana-1630	95	14	it	it	PRON
cana-1630	95	15	preserves	preserve	VERB
cana-1630	95	16	the	the	DET
cana-1630	95	17	convex	convex	PROPN
cana-1630	95	18	combinations	combination	NOUN
cana-1630	95	19	such	such	ADJ
cana-1630	95	20	that	that	DET
cana-1630	95	21	𝜇′(𝜆𝑥1	𝜇′(𝜆𝑥1	NOUN
cana-1630	95	22	+	+	CCONJ
cana-1630	95	23	(	(	PUNCT
cana-1630	95	24	1	1	NUM
cana-1630	95	25	−	−	PROPN
cana-1630	95	26	𝜆)𝑥2	𝜆)𝑥2	PROPN
cana-1630	95	27	)	)	PUNCT
cana-1630	95	28	=	=	PUNCT
cana-1630	96	1	𝐹(𝜇(𝜆𝑥1	𝐹(𝜇(𝜆𝑥1	PUNCT
cana-1630	97	1	+	+	CCONJ
cana-1630	97	2	(	(	PUNCT
cana-1630	97	3	1	1	NUM
cana-1630	97	4	−	−	PROPN
cana-1630	97	5	𝜆)𝑥2	𝜆)𝑥2	PROPN
cana-1630	97	6	)	)	PUNCT
cana-1630	97	7	,	,	PUNCT
cana-1630	97	8	𝜎	𝜎	X
cana-1630	97	9	)	)	PUNCT
cana-1630	97	10	≥	≥	PROPN
cana-1630	97	11	min(𝐹(𝜇(𝑥1	min(𝐹(𝜇(𝑥1	PROPN
cana-1630	97	12	)	)	PUNCT
cana-1630	97	13	,	,	PUNCT
cana-1630	97	14	𝜎	𝜎	X
cana-1630	97	15	)	)	PUNCT
cana-1630	97	16	,	,	PUNCT
cana-1630	97	17	𝐹(𝜇(𝑥2	𝐹(𝜇(𝑥2	NOUN
cana-1630	97	18	)	)	PUNCT
cana-1630	97	19	,	,	PUNCT
cana-1630	97	20	𝜎	𝜎	X
cana-1630	97	21	)	)	PUNCT
cana-1630	97	22	)	)	PUNCT
cana-1630	98	1	thus	thus	ADV
cana-1630	98	2	,	,	PUNCT
cana-1630	98	3	convexity	convexity	NOUN
cana-1630	98	4	is	be	AUX
cana-1630	98	5	preserved	preserve	VERB
cana-1630	98	6	.	.	PUNCT
cana-1630	99	1	since	since	SCONJ
cana-1630	99	2	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	99	3	)	)	PUNCT
cana-1630	99	4	is	be	AUX
cana-1630	99	5	bounded	bound	VERB
cana-1630	99	6	within	within	ADP
cana-1630	99	7	[	[	X
cana-1630	99	8	0,1	0,1	NUM
cana-1630	99	9	]	]	PUNCT
cana-1630	99	10	,	,	PUNCT
cana-1630	99	11	𝜇′(𝑥	𝜇′(𝑥	PUNCT
cana-1630	99	12	)	)	PUNCT
cana-1630	99	13	=	=	SYM
cana-1630	99	14	𝐹(𝜇(𝑥	𝐹(𝜇(𝑥	NOUN
cana-1630	99	15	)	)	PUNCT
cana-1630	99	16	,	,	PUNCT
cana-1630	99	17	𝜎	𝜎	X
cana-1630	99	18	)	)	PUNCT
cana-1630	99	19	remains	remain	VERB
cana-1630	99	20	bounded	bound	VERB
cana-1630	99	21	whose	whose	DET
cana-1630	99	22	proof	proof	NOUN
cana-1630	99	23	is	be	AUX
cana-1630	99	24	similar	similar	ADJ
cana-1630	99	25	to	to	PART
cana-1630	99	26	bibo	bibo	VERB
cana-1630	99	27	stability	stability	NOUN
cana-1630	99	28	.	.	PUNCT
cana-1630	100	1	theorem	theorem	VERB
cana-1630	100	2	3	3	NUM
cana-1630	100	3	:	:	PUNCT
cana-1630	100	4	frequency	frequency	NOUN
cana-1630	100	5	response	response	NOUN
cana-1630	100	6	and	and	CCONJ
cana-1630	100	7	fuzzy	fuzzy	ADJ
cana-1630	100	8	membership	membership	NOUN
cana-1630	100	9	function	function	NOUN
cana-1630	100	10	theorem	theorem	VERB
cana-1630	100	11	when	when	SCONJ
cana-1630	100	12	a	a	DET
cana-1630	100	13	fuzzy	fuzzy	ADJ
cana-1630	100	14	membership	membership	NOUN
cana-1630	100	15	function	function	NOUN
cana-1630	100	16	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	100	17	)	)	PUNCT
cana-1630	100	18	is	be	AUX
cana-1630	100	19	processed	process	VERB
cana-1630	100	20	through	through	ADP
cana-1630	100	21	a	a	DET
cana-1630	100	22	filter	filter	NOUN
cana-1630	100	23	𝐹(𝑥	𝐹(𝑥	NOUN
cana-1630	100	24	,	,	PUNCT
cana-1630	100	25	𝜎	𝜎	X
cana-1630	100	26	)	)	PUNCT
cana-1630	100	27	the	the	DET
cana-1630	100	28	output	output	NOUN
cana-1630	100	29	maintains	maintain	VERB
cana-1630	100	30	properties	property	NOUN
cana-1630	100	31	of	of	ADP
cana-1630	100	32	a	a	DET
cana-1630	100	33	fuzzy	fuzzy	ADJ
cana-1630	100	34	membership	membership	NOUN
cana-1630	100	35	function	function	NOUN
cana-1630	100	36	.	.	PUNCT
cana-1630	101	1	proof	proof	NOUN
cana-1630	101	2	:	:	PUNCT
cana-1630	101	3	the	the	DET
cana-1630	101	4	frequency	frequency	NOUN
cana-1630	101	5	response	response	NOUN
cana-1630	101	6	of	of	ADP
cana-1630	101	7	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	101	8	,	,	PUNCT
cana-1630	101	9	𝜎	𝜎	X
cana-1630	101	10	)	)	PUNCT
cana-1630	101	11	can	can	AUX
cana-1630	101	12	be	be	AUX
cana-1630	101	13	analyzed	analyze	VERB
cana-1630	101	14	using	use	VERB
cana-1630	101	15	the	the	DET
cana-1630	101	16	fourier	fourier	NOUN
cana-1630	101	17	transform	transform	NOUN
cana-1630	101	18	(	(	PUNCT
cana-1630	101	19	ft	ft	NOUN
cana-1630	101	20	):	):	PUNCT
cana-1630	101	21	ℱ{𝐹(𝑥	ℱ{𝐹(𝑥	NUM
cana-1630	101	22	,	,	PUNCT
cana-1630	101	23	𝜎	𝜎	NOUN
cana-1630	101	24	)	)	PUNCT
cana-1630	102	1	}	}	PUNCT
cana-1630	102	2	=	=	SYM
cana-1630	102	3	ℱ	ℱ	PROPN
cana-1630	102	4	{	{	PUNCT
cana-1630	102	5	1	1	NUM
cana-1630	102	6	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	102	7	(	(	PUNCT
cana-1630	102	8	1	1	NUM
cana-1630	102	9	−	−	PROPN
cana-1630	102	10	𝛼.	𝛼.	NOUN
cana-1630	102	11	𝑥	𝑥	PROPN
cana-1630	102	12	𝜎2	𝜎2	NOUN
cana-1630	102	13	)	)	PUNCT
cana-1630	102	14	exp	exp	NOUN
cana-1630	102	15	(	(	PUNCT
cana-1630	102	16	−	−	PROPN
cana-1630	102	17	𝑥2	𝑥2	NOUN
cana-1630	102	18	𝜎2	𝜎2	NOUN
cana-1630	102	19	)	)	PUNCT
cana-1630	102	20	}	}	PUNCT
cana-1630	102	21	using	use	VERB
cana-1630	102	22	properties	property	NOUN
cana-1630	102	23	of	of	ADP
cana-1630	102	24	ft	ft	NOUN
cana-1630	102	25	and	and	CCONJ
cana-1630	102	26	linearity	linearity	NOUN
cana-1630	102	27	communications	communication	NOUN
cana-1630	102	28	on	on	ADP
cana-1630	102	29	applied	apply	VERB
cana-1630	102	30	nonlinear	nonlinear	ADJ
cana-1630	102	31	analysis	analysis	NOUN
cana-1630	102	32	issn	issn	NOUN
cana-1630	102	33	:	:	PUNCT
cana-1630	102	34	1074	1074	NUM
cana-1630	102	35	-	-	PUNCT
cana-1630	102	36	133x	133x	NUM
cana-1630	102	37	vol	vol	NOUN
cana-1630	102	38	32	32	NUM
cana-1630	102	39	no	no	NOUN
cana-1630	102	40	.	.	NOUN
cana-1630	102	41	1	1	NUM
cana-1630	102	42	(	(	PUNCT
cana-1630	102	43	2025	2025	NUM
cana-1630	102	44	)	)	PUNCT
cana-1630	102	45	189	189	NUM
cana-1630	102	46	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	102	47	ℱ{𝐹(𝑥	ℱ{𝐹(𝑥	NUM
cana-1630	102	48	,	,	PUNCT
cana-1630	102	49	𝜎	𝜎	NOUN
cana-1630	102	50	)	)	PUNCT
cana-1630	102	51	}	}	PUNCT
cana-1630	102	52	=	=	SYM
cana-1630	102	53	ℱ	ℱ	PROPN
cana-1630	102	54	{	{	PUNCT
cana-1630	102	55	1	1	NUM
cana-1630	102	56	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	102	57	}	}	PUNCT
cana-1630	102	58	∗	∗	NOUN
cana-1630	102	59	ℱ	ℱ	PROPN
cana-1630	102	60	{	{	PUNCT
cana-1630	102	61	(	(	PUNCT
cana-1630	102	62	1	1	NUM
cana-1630	102	63	−	−	PROPN
cana-1630	102	64	𝛼.	𝛼.	NOUN
cana-1630	102	65	𝑥	𝑥	PROPN
cana-1630	102	66	𝜎2	𝜎2	NOUN
cana-1630	102	67	)	)	PUNCT
cana-1630	102	68	exp	exp	NOUN
cana-1630	102	69	(	(	PUNCT
cana-1630	102	70	−	−	PROPN
cana-1630	102	71	𝑥2	𝑥2	NOUN
cana-1630	102	72	𝜎2	𝜎2	NOUN
cana-1630	102	73	)	)	PUNCT
cana-1630	102	74	}	}	PUNCT
cana-1630	102	75	also	also	ADV
cana-1630	102	76	,	,	PUNCT
cana-1630	102	77	the	the	DET
cana-1630	102	78	filtered	filter	VERB
cana-1630	102	79	output	output	NOUN
cana-1630	102	80	in	in	ADP
cana-1630	102	81	the	the	DET
cana-1630	102	82	frequency	frequency	NOUN
cana-1630	102	83	domain	domain	NOUN
cana-1630	102	84	will	will	AUX
cana-1630	102	85	maintain	maintain	VERB
cana-1630	102	86	the	the	DET
cana-1630	102	87	bounded	bounded	ADJ
cana-1630	102	88	and	and	CCONJ
cana-1630	102	89	smooth	smooth	ADJ
cana-1630	102	90	nature	nature	NOUN
cana-1630	102	91	due	due	ADP
cana-1630	102	92	to	to	ADP
cana-1630	102	93	the	the	DET
cana-1630	102	94	gaussian	gaussian	ADJ
cana-1630	102	95	properties	property	NOUN
cana-1630	102	96	and	and	CCONJ
cana-1630	102	97	linear	linear	PROPN
cana-1630	102	98	filtering	filtering	NOUN
cana-1630	102	99	.	.	PUNCT
cana-1630	103	1	hence	hence	ADV
cana-1630	103	2	the	the	DET
cana-1630	103	3	proof	proof	NOUN
cana-1630	103	4	.	.	PUNCT
cana-1630	104	1	3.2	3.2	NUM
cana-1630	104	2	adaptive	adaptive	ADJ
cana-1630	104	3	gaussian	gaussian	ADJ
cana-1630	104	4	2d	2d	NUM
cana-1630	104	5	filter	filter	NOUN
cana-1630	104	6	and	and	CCONJ
cana-1630	104	7	derivative	derivative	NOUN
cana-1630	104	8	of	of	ADP
cana-1630	104	9	2d	2d	NUM
cana-1630	104	10	filter	filter	NOUN
cana-1630	104	11	(	(	PUNCT
cana-1630	104	12	agd-2d	agd-2d	PROPN
cana-1630	104	13	)	)	PUNCT
cana-1630	104	14	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	104	15	,	,	PUNCT
cana-1630	104	16	𝑦	𝑦	NOUN
cana-1630	104	17	,	,	PUNCT
cana-1630	104	18	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	104	19	,	,	PUNCT
cana-1630	104	20	𝜎𝑦	𝜎𝑦	PRON
cana-1630	104	21	,	,	PUNCT
cana-1630	104	22	𝛼	𝛼	PROPN
cana-1630	104	23	,	,	PUNCT
cana-1630	104	24	𝛽	𝛽	NOUN
cana-1630	104	25	)	)	PUNCT
cana-1630	104	26	=	=	SYM
cana-1630	104	27	1	1	NUM
cana-1630	104	28	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	105	1	[	[	PUNCT
cana-1630	105	2	1	1	NUM
cana-1630	105	3	−	−	PROPN
cana-1630	105	4	𝛼.	𝛼.	NOUN
cana-1630	105	5	𝑥	𝑥	PRON
cana-1630	105	6	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	105	7	2	2	NUM
cana-1630	105	8	−	−	NOUN
cana-1630	105	9	𝛽.	𝛽.	NOUN
cana-1630	105	10	𝑦	𝑦	NOUN
cana-1630	105	11	𝜎𝑦	𝜎𝑦	PRON
cana-1630	105	12	2	2	NUM
cana-1630	105	13	]	]	PUNCT
cana-1630	105	14	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	105	15	(	(	PUNCT
cana-1630	105	16	−	−	PROPN
cana-1630	105	17	𝑥2	𝑥2	NOUN
cana-1630	105	18	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	105	19	2	2	NUM
cana-1630	105	20	−	−	NOUN
cana-1630	105	21	𝑦2	𝑦2	NOUN
cana-1630	105	22	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	105	23	2	2	NUM
cana-1630	105	24	)	)	PUNCT
cana-1630	105	25	(	(	PUNCT
cana-1630	105	26	2	2	X
cana-1630	105	27	)	)	PUNCT
cana-1630	105	28	where	where	SCONJ
cana-1630	105	29	,	,	PUNCT
cana-1630	105	30	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	105	31	=	=	SYM
cana-1630	105	32	√	√	PROPN
cana-1630	105	33	1	1	NUM
cana-1630	105	34	𝑛	𝑛	PRON
cana-1630	105	35	∑	∑	PUNCT
cana-1630	105	36	(	(	PUNCT
cana-1630	105	37	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	105	38	,	,	PUNCT
cana-1630	105	39	𝑗	𝑗	NOUN
cana-1630	105	40	)	)	PUNCT
cana-1630	105	41	−	−	NOUN
cana-1630	105	42	𝑚𝑥)2𝑛	𝑚𝑥)2𝑛	NOUN
cana-1630	105	43	𝑗=1	𝑗=1	PROPN
cana-1630	105	44	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1630	105	45	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	PROPN
cana-1630	105	46	𝑖	𝑖	PROPN
cana-1630	105	47	;	;	PUNCT
cana-1630	105	48	𝜎𝑦	𝜎𝑦	PROPN
cana-1630	105	49	=	=	NOUN
cana-1630	105	50	√	√	NUM
cana-1630	105	51	1	1	NUM
cana-1630	105	52	𝑚	𝑚	NOUN
cana-1630	105	53	∑	∑	PUNCT
cana-1630	105	54	(	(	PUNCT
cana-1630	105	55	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	105	56	,	,	PUNCT
cana-1630	105	57	𝑗	𝑗	NOUN
cana-1630	105	58	)	)	PUNCT
cana-1630	105	59	−	−	PROPN
cana-1630	105	60	𝑚𝑦)2𝑚	𝑚𝑦)2𝑚	NOUN
cana-1630	105	61	𝑖=1	𝑖=1	PROPN
cana-1630	106	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1630	106	2	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	NOUN
cana-1630	106	3	𝑗	𝑗	PROPN
cana-1630	106	4	∝	∝	NOUN
cana-1630	106	5	=	=	SYM
cana-1630	106	6	𝐴(𝑖,𝑗	𝐴(𝑖,𝑗	PROPN
cana-1630	106	7	)	)	PUNCT
cana-1630	107	1	max(|𝐴(𝑖,𝑗)|)𝑟𝑜𝑤	max(|𝐴(𝑖,𝑗)|)𝑟𝑜𝑤	NOUN
cana-1630	107	2	𝑓𝑜𝑟	𝑓𝑜𝑟	AUX
cana-1630	107	3	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	PROPN
cana-1630	107	4	𝑖	𝑖	PROPN
cana-1630	107	5	,	,	PUNCT
cana-1630	107	6	𝑗	𝑗	PROPN
cana-1630	107	7	;	;	PUNCT
cana-1630	107	8	𝛽	𝛽	NOUN
cana-1630	107	9	=	=	SYM
cana-1630	107	10	𝐴(𝑖,𝑗	𝐴(𝑖,𝑗	X
cana-1630	107	11	)	)	PUNCT
cana-1630	107	12	max(|𝐴(𝑖,𝑗)|)𝑐𝑜𝑙𝑢𝑚𝑛	max(|𝐴(𝑖,𝑗)|)𝑐𝑜𝑙𝑢𝑚𝑛	VERB
cana-1630	107	13	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1630	107	14	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	PROPN
cana-1630	107	15	𝑖	𝑖	PROPN
cana-1630	107	16	,	,	PUNCT
cana-1630	107	17	𝑗	𝑗	PROPN
cana-1630	107	18	theorem	theorem	NOUN
cana-1630	107	19	4	4	NUM
cana-1630	107	20	:	:	PUNCT
cana-1630	107	21	bibo	bibo	VERB
cana-1630	107	22	stability	stability	NOUN
cana-1630	107	23	the	the	DET
cana-1630	107	24	adaptive	adaptive	ADJ
cana-1630	107	25	gaussian	gaussian	ADJ
cana-1630	107	26	2d	2d	NUM
cana-1630	107	27	filter	filter	NOUN
cana-1630	107	28	given	give	VERB
cana-1630	107	29	by	by	ADP
cana-1630	107	30	the	the	DET
cana-1630	107	31	function	function	NOUN
cana-1630	107	32	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	107	33	,	,	PUNCT
cana-1630	107	34	𝑦	𝑦	NOUN
cana-1630	107	35	,	,	PUNCT
cana-1630	107	36	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	107	37	,	,	PUNCT
cana-1630	107	38	𝜎𝑦	𝜎𝑦	PROPN
cana-1630	107	39	,	,	PUNCT
cana-1630	107	40	𝛼	𝛼	X
cana-1630	107	41	,	,	PUNCT
cana-1630	107	42	𝛽	𝛽	NOUN
cana-1630	107	43	)	)	PUNCT
cana-1630	107	44	is	be	AUX
cana-1630	107	45	bibo	bibo	ADJ
cana-1630	107	46	stable	stable	ADJ
cana-1630	107	47	,	,	PUNCT
cana-1630	107	48	provided	provide	VERB
cana-1630	107	49	that	that	SCONJ
cana-1630	107	50	the	the	DET
cana-1630	107	51	input	input	NOUN
cana-1630	107	52	image	image	NOUN
cana-1630	107	53	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	107	54	,	,	PUNCT
cana-1630	107	55	𝑗	𝑗	NOUN
cana-1630	107	56	)	)	PUNCT
cana-1630	107	57	is	be	AUX
cana-1630	107	58	bounded	bound	VERB
cana-1630	107	59	.	.	PUNCT
cana-1630	108	1	proof	proof	NOUN
cana-1630	108	2	:	:	PUNCT
cana-1630	108	3	let	let	VERB
cana-1630	108	4	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	108	5	,	,	PUNCT
cana-1630	108	6	𝑗	𝑗	AUX
cana-1630	108	7	)	)	PUNCT
cana-1630	108	8	be	be	AUX
cana-1630	108	9	a	a	DET
cana-1630	108	10	bounded	bounded	ADJ
cana-1630	108	11	input	input	NOUN
cana-1630	108	12	image	image	NOUN
cana-1630	108	13	.	.	PUNCT
cana-1630	109	1	then	then	ADV
cana-1630	109	2	there	there	PRON
cana-1630	109	3	exist	exist	VERB
cana-1630	109	4	a	a	DET
cana-1630	109	5	constant	constant	ADJ
cana-1630	109	6	m	m	NOUN
cana-1630	109	7	such	such	ADJ
cana-1630	109	8	that	that	SCONJ
cana-1630	109	9	|𝐴(𝑖	|𝐴(𝑖	PROPN
cana-1630	109	10	,	,	PUNCT
cana-1630	109	11	𝑗)|	𝑗)|	VERB
cana-1630	109	12	≤	≤	NUM
cana-1630	109	13	𝑀	𝑀	PROPN
cana-1630	109	14	∀(𝑖	∀(𝑖	NUM
cana-1630	109	15	,	,	PUNCT
cana-1630	109	16	𝑗	𝑗	NOUN
cana-1630	109	17	)	)	PUNCT
cana-1630	109	18	.	.	PUNCT
cana-1630	110	1	since	since	SCONJ
cana-1630	110	2	𝐴(𝑖	𝐴(𝑖	ADV
cana-1630	110	3	,	,	PUNCT
cana-1630	110	4	𝑗	𝑗	X
cana-1630	110	5	)	)	PUNCT
cana-1630	110	6	is	be	AUX
cana-1630	110	7	bounded	bound	VERB
cana-1630	110	8	,	,	PUNCT
cana-1630	110	9	∝	∝	PROPN
cana-1630	110	10	,	,	PUNCT
cana-1630	110	11	𝛽	𝛽	PROPN
cana-1630	110	12	are	be	AUX
cana-1630	110	13	also	also	ADV
cana-1630	110	14	bounded	bound	VERB
cana-1630	110	15	.	.	PUNCT
cana-1630	111	1	also	also	ADV
cana-1630	111	2	,	,	PUNCT
cana-1630	111	3	|∝|	|∝|	VERB
cana-1630	111	4	≤	≤	ADV
cana-1630	111	5	1	1	NUM
cana-1630	111	6	and	and	CCONJ
cana-1630	111	7	|𝛽|	|𝛽|	ADJ
cana-1630	111	8	≤	≤	ADJ
cana-1630	111	9	1	1	NUM
cana-1630	111	10	because	because	SCONJ
cana-1630	111	11	they	they	PRON
cana-1630	111	12	are	be	AUX
cana-1630	111	13	normalized	normalize	VERB
cana-1630	111	14	by	by	ADP
cana-1630	111	15	the	the	DET
cana-1630	111	16	maximum	maximum	ADJ
cana-1630	111	17	absolute	absolute	ADJ
cana-1630	111	18	value	value	NOUN
cana-1630	111	19	in	in	ADP
cana-1630	111	20	their	their	PRON
cana-1630	111	21	respective	respective	ADJ
cana-1630	111	22	rows	row	NOUN
cana-1630	111	23	and	and	CCONJ
cana-1630	111	24	columns	column	NOUN
cana-1630	111	25	.	.	PUNCT
cana-1630	112	1	the	the	DET
cana-1630	112	2	gaussian	gaussian	ADJ
cana-1630	112	3	filter	filter	NOUN
cana-1630	112	4	component	component	NOUN
cana-1630	112	5	of	of	ADP
cana-1630	112	6	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	112	7	,	,	PUNCT
cana-1630	112	8	𝑦	𝑦	NOUN
cana-1630	112	9	,	,	PUNCT
cana-1630	112	10	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	112	11	,	,	PUNCT
cana-1630	112	12	𝜎𝑦	𝜎𝑦	PROPN
cana-1630	112	13	,	,	PUNCT
cana-1630	112	14	𝛼	𝛼	X
cana-1630	112	15	,	,	PUNCT
cana-1630	112	16	𝛽	𝛽	NOUN
cana-1630	112	17	)	)	PUNCT
cana-1630	112	18	given	give	VERB
cana-1630	112	19	by	by	ADP
cana-1630	112	20	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	112	21	(	(	PUNCT
cana-1630	112	22	−	−	PROPN
cana-1630	112	23	𝑥2	𝑥2	NOUN
cana-1630	112	24	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	112	25	2	2	NUM
cana-1630	112	26	−	−	NOUN
cana-1630	112	27	𝑦2	𝑦2	NOUN
cana-1630	112	28	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	112	29	2	2	NUM
cana-1630	112	30	)	)	PUNCT
cana-1630	112	31	is	be	AUX
cana-1630	112	32	always	always	ADV
cana-1630	112	33	nonnegative	nonnegative	ADJ
cana-1630	112	34	and	and	CCONJ
cana-1630	112	35	integrates	integrate	NOUN
cana-1630	112	36	to	to	ADP
cana-1630	112	37	1	1	NUM
cana-1630	112	38	over	over	ADP
cana-1630	112	39	entire	entire	ADJ
cana-1630	112	40	(	(	PUNCT
cana-1630	112	41	x	x	NOUN
cana-1630	112	42	,	,	PUNCT
cana-1630	112	43	y	y	NOUN
cana-1630	112	44	)	)	PUNCT
cana-1630	112	45	plane	plane	NOUN
cana-1630	112	46	.	.	PUNCT
cana-1630	113	1	the	the	DET
cana-1630	113	2	output	output	NOUN
cana-1630	113	3	of	of	ADP
cana-1630	113	4	the	the	DET
cana-1630	113	5	convolution	convolution	NOUN
cana-1630	113	6	of	of	ADP
cana-1630	113	7	the	the	DET
cana-1630	113	8	input	input	NOUN
cana-1630	113	9	image	image	NOUN
cana-1630	113	10	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	113	11	,	,	PUNCT
cana-1630	113	12	𝑗	𝑗	NOUN
cana-1630	113	13	)	)	PUNCT
cana-1630	113	14	with	with	ADP
cana-1630	113	15	the	the	DET
cana-1630	113	16	gaussian	gaussian	ADJ
cana-1630	113	17	filter	filter	NOUN
cana-1630	113	18	𝐹(𝑥	𝐹(𝑥	PROPN
cana-1630	113	19	,	,	PUNCT
cana-1630	113	20	𝑦	𝑦	NOUN
cana-1630	113	21	,	,	PUNCT
cana-1630	113	22	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	113	23	,	,	PUNCT
cana-1630	113	24	𝜎𝑦	𝜎𝑦	PRON
cana-1630	113	25	,	,	PUNCT
cana-1630	113	26	𝛼	𝛼	PROPN
cana-1630	113	27	,	,	PUNCT
cana-1630	113	28	𝛽	𝛽	NOUN
cana-1630	113	29	)	)	PUNCT
cana-1630	113	30	can	can	AUX
cana-1630	113	31	be	be	AUX
cana-1630	113	32	expressed	express	VERB
cana-1630	113	33	as	as	ADP
cana-1630	113	34	:	:	PUNCT
cana-1630	113	35	𝑔(𝑥	𝑔(𝑥	PROPN
cana-1630	113	36	,	,	PUNCT
cana-1630	113	37	𝑦	𝑦	NOUN
cana-1630	113	38	)	)	PUNCT
cana-1630	113	39	=	=	SYM
cana-1630	114	1	∑∑𝐴(𝑖	∑∑𝐴(𝑖	NOUN
cana-1630	114	2	,	,	PUNCT
cana-1630	114	3	𝑗	𝑗	X
cana-1630	114	4	)	)	PUNCT
cana-1630	114	5	𝑗	𝑗	PROPN
cana-1630	114	6	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	114	7	−	−	PROPN
cana-1630	114	8	𝑢	𝑢	PROPN
cana-1630	114	9	,	,	PUNCT
cana-1630	114	10	𝑦	𝑦	NOUN
cana-1630	114	11	−	−	NOUN
cana-1630	114	12	𝑣	𝑣	NOUN
cana-1630	114	13	,	,	PUNCT
cana-1630	114	14	𝜎𝑥	𝜎𝑥	INTJ
cana-1630	114	15	,	,	PUNCT
cana-1630	114	16	𝜎𝑦	𝜎𝑦	DET
cana-1630	114	17	,	,	PUNCT
cana-1630	114	18	𝛼	𝛼	PROPN
cana-1630	114	19	,	,	PUNCT
cana-1630	114	20	𝛽	𝛽	NOUN
cana-1630	114	21	)	)	PUNCT
cana-1630	114	22	𝑖	𝑖	VERB
cana-1630	114	23	now	now	ADV
cana-1630	114	24	to	to	PART
cana-1630	114	25	prove	prove	VERB
cana-1630	114	26	,	,	PUNCT
cana-1630	114	27	g(x	g(x	PROPN
cana-1630	114	28	,	,	PUNCT
cana-1630	114	29	y	y	NOUN
cana-1630	114	30	)	)	PUNCT
cana-1630	114	31	is	be	AUX
cana-1630	114	32	bounded	bound	VERB
cana-1630	114	33	.	.	PUNCT
cana-1630	115	1	since	since	SCONJ
cana-1630	115	2	,	,	PUNCT
cana-1630	115	3	𝐴(𝑖	𝐴(𝑖	CCONJ
cana-1630	115	4	,	,	PUNCT
cana-1630	115	5	𝑗	𝑗	X
cana-1630	115	6	)	)	PUNCT
cana-1630	115	7	is	be	AUX
cana-1630	115	8	bounded	bound	VERB
cana-1630	115	9	and	and	CCONJ
cana-1630	115	10	the	the	DET
cana-1630	115	11	properties	property	NOUN
cana-1630	115	12	of	of	ADP
cana-1630	115	13	gaussian	gaussian	ADJ
cana-1630	115	14	filter	filter	NOUN
cana-1630	115	15	then	then	ADV
cana-1630	115	16	,	,	PUNCT
cana-1630	115	17	𝑔(𝑥	𝑔(𝑥	PROPN
cana-1630	115	18	,	,	PUNCT
cana-1630	115	19	𝑦	𝑦	NOUN
cana-1630	115	20	)	)	PUNCT
cana-1630	115	21	≤	≤	NOUN
cana-1630	115	22	∑∑|𝐴(𝑖	∑∑|𝐴(𝑖	NOUN
cana-1630	115	23	,	,	PUNCT
cana-1630	115	24	𝑗)|	𝑗)|	NOUN
cana-1630	115	25	𝑗	𝑗	INTJ
cana-1630	116	1	|	|	ADV
cana-1630	116	2	1	1	NUM
cana-1630	116	3	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	116	4	[	[	PUNCT
cana-1630	116	5	1	1	NUM
cana-1630	116	6	−	−	NOUN
cana-1630	116	7	𝛼	𝛼	PART
cana-1630	116	8	𝑥	𝑥	NOUN
cana-1630	116	9	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	116	10	2	2	NUM
cana-1630	116	11	−	−	NOUN
cana-1630	116	12	𝛽	𝛽	NOUN
cana-1630	116	13	𝑦	𝑦	NOUN
cana-1630	116	14	𝜎𝑦	𝜎𝑦	PRON
cana-1630	116	15	2	2	NUM
cana-1630	116	16	]	]	PUNCT
cana-1630	116	17	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	116	18	(	(	PUNCT
cana-1630	116	19	−	−	PROPN
cana-1630	116	20	𝑥2	𝑥2	NOUN
cana-1630	116	21	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	116	22	2	2	NUM
cana-1630	116	23	−	−	NOUN
cana-1630	117	1	𝑦2	𝑦2	NOUN
cana-1630	117	2	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	117	3	2	2	NUM
cana-1630	117	4	)	)	PUNCT
cana-1630	117	5	|	|	ADV
cana-1630	117	6	𝑖	𝑖	X
cana-1630	117	7	given	give	VERB
cana-1630	117	8	|𝐴(𝑖	|𝐴(𝑖	PROPN
cana-1630	117	9	,	,	PUNCT
cana-1630	117	10	𝑗)|	𝑗)|	NOUN
cana-1630	117	11	≤	≤	NUM
cana-1630	117	12	𝑀	𝑀	PROPN
cana-1630	117	13	,	,	PUNCT
cana-1630	117	14	|∝|	|∝|	VERB
cana-1630	117	15	≤	≤	ADV
cana-1630	117	16	1	1	NUM
cana-1630	117	17	and	and	CCONJ
cana-1630	117	18	|𝛽|	|𝛽|	ADJ
cana-1630	117	19	≤	≤	NUM
cana-1630	117	20	1	1	NUM
cana-1630	117	21	then	then	ADV
cana-1630	117	22	,	,	PUNCT
cana-1630	117	23	𝑔(𝑥	𝑔(𝑥	PROPN
cana-1630	117	24	,	,	PUNCT
cana-1630	117	25	𝑦	𝑦	NOUN
cana-1630	117	26	)	)	PUNCT
cana-1630	117	27	≤	≤	NOUN
cana-1630	118	1	∑∑𝑀	∑∑𝑀	PROPN
cana-1630	119	1	𝑗	𝑗	INTJ
cana-1630	119	2	|	|	ADV
cana-1630	119	3	1	1	NUM
cana-1630	119	4	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	119	5	[	[	PUNCT
cana-1630	119	6	1	1	NUM
cana-1630	119	7	−	−	PART
cana-1630	119	8	𝑥	𝑥	PRON
cana-1630	119	9	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	119	10	2	2	NUM
cana-1630	119	11	−	−	NOUN
cana-1630	119	12	𝑦	𝑦	NOUN
cana-1630	119	13	𝜎𝑦	𝜎𝑦	PRON
cana-1630	119	14	2	2	NUM
cana-1630	119	15	]	]	PUNCT
cana-1630	119	16	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	119	17	(	(	PUNCT
cana-1630	119	18	−	−	PROPN
cana-1630	119	19	𝑥2	𝑥2	NOUN
cana-1630	119	20	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	119	21	2	2	NUM
cana-1630	119	22	−	−	NOUN
cana-1630	120	1	𝑦2	𝑦2	NOUN
cana-1630	120	2	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	120	3	2	2	NUM
cana-1630	120	4	)	)	PUNCT
cana-1630	120	5	|	|	NOUN
cana-1630	120	6	𝑖	𝑖	ADP
cana-1630	120	7	communications	communication	NOUN
cana-1630	120	8	on	on	ADP
cana-1630	120	9	applied	apply	VERB
cana-1630	120	10	nonlinear	nonlinear	ADJ
cana-1630	120	11	analysis	analysis	NOUN
cana-1630	120	12	issn	issn	NOUN
cana-1630	120	13	:	:	PUNCT
cana-1630	120	14	1074	1074	NUM
cana-1630	120	15	-	-	PUNCT
cana-1630	120	16	133x	133x	NUM
cana-1630	120	17	vol	vol	NOUN
cana-1630	120	18	32	32	NUM
cana-1630	120	19	no	no	NOUN
cana-1630	120	20	.	.	NOUN
cana-1630	120	21	1	1	NUM
cana-1630	120	22	(	(	PUNCT
cana-1630	120	23	2025	2025	NUM
cana-1630	120	24	)	)	PUNCT
cana-1630	120	25	190	190	NUM
cana-1630	120	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	120	27	since	since	SCONJ
cana-1630	120	28	,	,	PUNCT
cana-1630	120	29	the	the	DET
cana-1630	120	30	gaussian	gaussian	ADJ
cana-1630	120	31	function	function	NOUN
cana-1630	120	32	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	120	33	(	(	PUNCT
cana-1630	120	34	−	−	PROPN
cana-1630	120	35	𝑥2	𝑥2	NOUN
cana-1630	120	36	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	120	37	2	2	NUM
cana-1630	120	38	−	−	NOUN
cana-1630	120	39	𝑦2	𝑦2	NOUN
cana-1630	120	40	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	120	41	2	2	NUM
cana-1630	120	42	)	)	PUNCT
cana-1630	120	43	is	be	AUX
cana-1630	120	44	bounded	bound	VERB
cana-1630	120	45	and	and	CCONJ
cana-1630	120	46	the	the	DET
cana-1630	120	47	terms	term	NOUN
cana-1630	120	48	involving	involve	VERB
cana-1630	120	49	∝	∝	PROPN
cana-1630	120	50	and	and	CCONJ
cana-1630	120	51	𝛽	𝛽	PROPN
cana-1630	120	52	do	do	AUX
cana-1630	120	53	not	not	PART
cana-1630	120	54	cause	cause	VERB
cana-1630	120	55	the	the	DET
cana-1630	120	56	expression	expression	NOUN
cana-1630	120	57	to	to	PART
cana-1630	120	58	diverge	diverge	VERB
cana-1630	120	59	,	,	PUNCT
cana-1630	120	60	the	the	DET
cana-1630	120	61	output	output	NOUN
cana-1630	120	62	g	g	PROPN
cana-1630	120	63	(	(	PUNCT
cana-1630	120	64	x	x	PROPN
cana-1630	120	65	,	,	PUNCT
cana-1630	120	66	y	y	NOUN
cana-1630	120	67	)	)	PUNCT
cana-1630	120	68	remains	remain	VERB
cana-1630	120	69	bounded	bound	VERB
cana-1630	120	70	.	.	PUNCT
cana-1630	121	1	therefore	therefore	ADV
cana-1630	121	2	,	,	PUNCT
cana-1630	121	3	|𝑔(𝑥	|𝑔(𝑥	PROPN
cana-1630	121	4	,	,	PUNCT
cana-1630	121	5	𝑦)|	𝑦)|	PROPN
cana-1630	121	6	≤	≤	NOUN
cana-1630	121	7	𝑘	𝑘	PRON
cana-1630	121	8	for	for	ADP
cana-1630	121	9	some	some	DET
cana-1630	121	10	constant	constant	ADJ
cana-1630	121	11	k	k	NOUN
cana-1630	121	12	,	,	PUNCT
cana-1630	121	13	proving	prove	VERB
cana-1630	121	14	that	that	SCONJ
cana-1630	121	15	adaptive	adaptive	ADJ
cana-1630	121	16	gaussian	gaussian	NOUN
cana-1630	121	17	2d	2d	NUM
cana-1630	121	18	filter	filter	NOUN
cana-1630	121	19	is	be	AUX
cana-1630	121	20	bibo	bibo	ADJ
cana-1630	121	21	stable	stable	ADJ
cana-1630	121	22	.	.	PUNCT
cana-1630	122	1	hence	hence	ADV
cana-1630	122	2	the	the	DET
cana-1630	122	3	proof	proof	NOUN
cana-1630	122	4	.	.	PUNCT
cana-1630	123	1	corollary	corollary	ADJ
cana-1630	123	2	if	if	SCONJ
cana-1630	123	3	a	a	DET
cana-1630	123	4	bounded	bounded	ADJ
cana-1630	123	5	input	input	NOUN
cana-1630	123	6	image	image	NOUN
cana-1630	123	7	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	123	8	,	,	PUNCT
cana-1630	123	9	𝑗	𝑗	NOUN
cana-1630	123	10	)	)	PUNCT
cana-1630	123	11	is	be	AUX
cana-1630	123	12	processed	process	VERB
cana-1630	123	13	by	by	ADP
cana-1630	123	14	the	the	DET
cana-1630	123	15	adaptive	adaptive	ADJ
cana-1630	123	16	gaussian	gaussian	ADJ
cana-1630	123	17	2d	2d	NUM
cana-1630	123	18	filter	filter	NOUN
cana-1630	123	19	defined	define	VERB
cana-1630	123	20	by	by	ADP
cana-1630	123	21	,	,	PUNCT
cana-1630	123	22	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	123	23	,	,	PUNCT
cana-1630	123	24	𝑦	𝑦	NOUN
cana-1630	123	25	,	,	PUNCT
cana-1630	123	26	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	123	27	,	,	PUNCT
cana-1630	123	28	𝜎𝑦	𝜎𝑦	DET
cana-1630	123	29	,	,	PUNCT
cana-1630	123	30	𝛼	𝛼	PROPN
cana-1630	123	31	,	,	PUNCT
cana-1630	123	32	𝛽	𝛽	NOUN
cana-1630	123	33	)	)	PUNCT
cana-1630	123	34	=	=	SYM
cana-1630	123	35	1	1	NUM
cana-1630	123	36	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	124	1	[	[	PUNCT
cana-1630	124	2	1	1	NUM
cana-1630	124	3	−	−	NOUN
cana-1630	124	4	𝛼	𝛼	PART
cana-1630	124	5	𝑥	𝑥	NOUN
cana-1630	124	6	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	124	7	2	2	NUM
cana-1630	124	8	−	−	NOUN
cana-1630	124	9	𝛽	𝛽	NOUN
cana-1630	124	10	𝑦	𝑦	NOUN
cana-1630	124	11	𝜎𝑦	𝜎𝑦	DET
cana-1630	124	12	2	2	NUM
cana-1630	124	13	]	]	PUNCT
cana-1630	124	14	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	124	15	(	(	PUNCT
cana-1630	124	16	−	−	PROPN
cana-1630	124	17	𝑥2	𝑥2	NOUN
cana-1630	124	18	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	124	19	2	2	NUM
cana-1630	124	20	−	−	NOUN
cana-1630	124	21	𝑦2	𝑦2	NOUN
cana-1630	124	22	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	124	23	2	2	NUM
cana-1630	124	24	)	)	PUNCT
cana-1630	124	25	the	the	DET
cana-1630	124	26	output	output	NOUN
cana-1630	124	27	image	image	NOUN
cana-1630	124	28	g	g	PROPN
cana-1630	124	29	(	(	PUNCT
cana-1630	124	30	x	x	PROPN
cana-1630	124	31	,	,	PUNCT
cana-1630	124	32	y	y	NOUN
cana-1630	124	33	)	)	PUNCT
cana-1630	124	34	will	will	AUX
cana-1630	124	35	also	also	ADV
cana-1630	124	36	be	be	AUX
cana-1630	124	37	bounded	bound	VERB
cana-1630	124	38	,	,	PUNCT
cana-1630	124	39	preserving	preserve	VERB
cana-1630	124	40	the	the	DET
cana-1630	124	41	stability	stability	NOUN
cana-1630	124	42	of	of	ADP
cana-1630	124	43	the	the	DET
cana-1630	124	44	filtering	filtering	NOUN
cana-1630	124	45	process	process	NOUN
cana-1630	124	46	.	.	PUNCT
cana-1630	125	1	theorem	theorem	ADJ
cana-1630	125	2	5	5	NUM
cana-1630	125	3	:	:	PUNCT
cana-1630	125	4	frequency	frequency	NOUN
cana-1630	125	5	response	response	NOUN
cana-1630	125	6	the	the	DET
cana-1630	125	7	frequency	frequency	NOUN
cana-1630	125	8	response	response	NOUN
cana-1630	125	9	of	of	ADP
cana-1630	125	10	a	a	DET
cana-1630	125	11	filter	filter	NOUN
cana-1630	125	12	describes	describe	VERB
cana-1630	125	13	how	how	SCONJ
cana-1630	125	14	the	the	DET
cana-1630	125	15	filter	filter	NOUN
cana-1630	125	16	modifies	modify	VERB
cana-1630	125	17	the	the	DET
cana-1630	125	18	amplitude	amplitude	NOUN
cana-1630	125	19	and	and	CCONJ
cana-1630	125	20	phase	phase	NOUN
cana-1630	125	21	of	of	ADP
cana-1630	125	22	input	input	NOUN
cana-1630	125	23	signals	signal	NOUN
cana-1630	125	24	at	at	ADP
cana-1630	125	25	different	different	ADJ
cana-1630	125	26	frequencies	frequency	NOUN
cana-1630	125	27	.	.	PUNCT
cana-1630	126	1	for	for	ADP
cana-1630	126	2	the	the	DET
cana-1630	126	3	adaptive	adaptive	ADJ
cana-1630	126	4	gaussian	gaussian	ADJ
cana-1630	126	5	2d	2d	NUM
cana-1630	126	6	filter	filter	NOUN
cana-1630	126	7	defined	define	VERB
cana-1630	126	8	by	by	ADP
cana-1630	126	9	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	126	10	,	,	PUNCT
cana-1630	126	11	𝑦	𝑦	NOUN
cana-1630	126	12	,	,	PUNCT
cana-1630	126	13	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	126	14	,	,	PUNCT
cana-1630	126	15	𝜎𝑦	𝜎𝑦	DET
cana-1630	126	16	,	,	PUNCT
cana-1630	126	17	𝛼	𝛼	PROPN
cana-1630	126	18	,	,	PUNCT
cana-1630	126	19	𝛽	𝛽	NOUN
cana-1630	126	20	)	)	PUNCT
cana-1630	126	21	=	=	SYM
cana-1630	126	22	1	1	NUM
cana-1630	126	23	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	126	24	[	[	PUNCT
cana-1630	126	25	1	1	NUM
cana-1630	126	26	−	−	NOUN
cana-1630	126	27	𝛼	𝛼	PART
cana-1630	126	28	𝑥	𝑥	NOUN
cana-1630	126	29	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	126	30	2	2	NUM
cana-1630	126	31	−	−	NOUN
cana-1630	126	32	𝛽	𝛽	NOUN
cana-1630	126	33	𝑦	𝑦	NOUN
cana-1630	126	34	𝜎𝑦	𝜎𝑦	PRON
cana-1630	126	35	2	2	NUM
cana-1630	126	36	]	]	PUNCT
cana-1630	126	37	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	126	38	(	(	PUNCT
cana-1630	126	39	−	−	PROPN
cana-1630	126	40	𝑥2	𝑥2	NOUN
cana-1630	126	41	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	126	42	2	2	NUM
cana-1630	126	43	−	−	NOUN
cana-1630	126	44	𝑦2	𝑦2	NOUN
cana-1630	126	45	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	126	46	2	2	NUM
cana-1630	126	47	)	)	PUNCT
cana-1630	126	48	the	the	DET
cana-1630	126	49	frequency	frequency	NOUN
cana-1630	126	50	response	response	NOUN
cana-1630	126	51	can	can	AUX
cana-1630	126	52	be	be	AUX
cana-1630	126	53	found	find	VERB
cana-1630	126	54	by	by	ADP
cana-1630	126	55	taking	take	VERB
cana-1630	126	56	the	the	DET
cana-1630	126	57	fourier	fourier	ADJ
cana-1630	126	58	transform	transform	NOUN
cana-1630	126	59	of	of	ADP
cana-1630	126	60	f	f	PROPN
cana-1630	126	61	(	(	PUNCT
cana-1630	126	62	x	x	PROPN
cana-1630	126	63	,	,	PUNCT
cana-1630	126	64	y	y	PROPN
cana-1630	126	65	)	)	PUNCT
cana-1630	126	66	.	.	PUNCT
cana-1630	127	1	proof	proof	NOUN
cana-1630	127	2	:	:	PUNCT
cana-1630	127	3	the	the	DET
cana-1630	127	4	2d	2d	NUM
cana-1630	127	5	fourier	fourier	NOUN
cana-1630	127	6	transform	transform	NOUN
cana-1630	127	7	of	of	ADP
cana-1630	127	8	f(x	f(x	PROPN
cana-1630	127	9	.y	.y	PROPN
cana-1630	127	10	)	)	PUNCT
cana-1630	127	11	is	be	AUX
cana-1630	127	12	given	give	VERB
cana-1630	127	13	by	by	ADP
cana-1630	127	14	ℱ{𝐹(𝑥	ℱ{𝐹(𝑥	NUM
cana-1630	127	15	,	,	PUNCT
cana-1630	127	16	𝑦	𝑦	NOUN
cana-1630	127	17	)	)	PUNCT
cana-1630	127	18	}	}	PUNCT
cana-1630	128	1	=	=	SYM
cana-1630	128	2	∫	∫	PROPN
cana-1630	128	3	∫	∫	PROPN
cana-1630	128	4	𝐹(𝑥	𝐹(𝑥	PROPN
cana-1630	128	5	,	,	PUNCT
cana-1630	128	6	𝑦	𝑦	X
cana-1630	128	7	)	)	PUNCT
cana-1630	128	8	𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥𝑑𝑦	𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥𝑑𝑦	VERB
cana-1630	128	9	∞	∞	PROPN
cana-1630	128	10	−∞	−∞	ADP
cana-1630	128	11	∞	∞	PROPN
cana-1630	128	12	−∞	−∞	ADP
cana-1630	128	13	where	where	SCONJ
cana-1630	128	14	(	(	PUNCT
cana-1630	128	15	u	u	NOUN
cana-1630	128	16	,	,	PUNCT
cana-1630	128	17	v	v	NOUN
cana-1630	128	18	)	)	PUNCT
cana-1630	128	19	are	be	AUX
cana-1630	128	20	the	the	DET
cana-1630	128	21	frequency	frequency	NOUN
cana-1630	128	22	variables	variable	NOUN
cana-1630	128	23	.	.	PUNCT
cana-1630	129	1	the	the	DET
cana-1630	129	2	ft	ft	PROPN
cana-1630	129	3	of	of	ADP
cana-1630	129	4	the	the	DET
cana-1630	129	5	gaussian	gaussian	ADJ
cana-1630	129	6	component	component	NOUN
cana-1630	129	7	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	129	8	(	(	PUNCT
cana-1630	129	9	−	−	PROPN
cana-1630	129	10	𝑥2	𝑥2	NOUN
cana-1630	129	11	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	129	12	2	2	NUM
cana-1630	129	13	−	−	NOUN
cana-1630	129	14	𝑦2	𝑦2	NOUN
cana-1630	129	15	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	129	16	2	2	NUM
cana-1630	129	17	)	)	PUNCT
cana-1630	129	18	is	be	AUX
cana-1630	129	19	another	another	DET
cana-1630	129	20	gaussian	gaussian	ADJ
cana-1630	129	21	function	function	NOUN
cana-1630	129	22	:	:	PUNCT
cana-1630	129	23	ℱ	ℱ	PROPN
cana-1630	129	24	{	{	PUNCT
cana-1630	129	25	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	129	26	(	(	PUNCT
cana-1630	129	27	−	−	PROPN
cana-1630	129	28	𝑥2	𝑥2	NOUN
cana-1630	129	29	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	129	30	2	2	NUM
cana-1630	129	31	−	−	NOUN
cana-1630	129	32	𝑦2	𝑦2	NOUN
cana-1630	129	33	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	129	34	2	2	NUM
cana-1630	129	35	)	)	PUNCT
cana-1630	129	36	}	}	PUNCT
cana-1630	129	37	=	=	SYM
cana-1630	129	38	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	129	39	exp(−2𝜋2𝜎𝑥	exp(−2𝜋2𝜎𝑥	NOUN
cana-1630	129	40	2𝑢2	2𝑢2	NUM
cana-1630	129	41	−	−	PROPN
cana-1630	129	42	2𝜋2𝜎𝑦	2𝜋2𝜎𝑦	NUM
cana-1630	129	43	2𝑣2	2𝑣2	NOUN
cana-1630	129	44	)	)	PUNCT
cana-1630	129	45	the	the	DET
cana-1630	129	46	ft	ft	PROPN
cana-1630	129	47	of	of	ADP
cana-1630	129	48	linear	linear	PROPN
cana-1630	129	49	terms	term	NOUN
cana-1630	129	50	𝑥	𝑥	X
cana-1630	129	51	.	.	PUNCT
cana-1630	130	1	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	130	2	(	(	PUNCT
cana-1630	130	3	−	−	PROPN
cana-1630	130	4	𝑥2	𝑥2	NOUN
cana-1630	130	5	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	130	6	2	2	NUM
cana-1630	130	7	−	−	NOUN
cana-1630	131	1	𝑦2	𝑦2	NOUN
cana-1630	131	2	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	131	3	2	2	NUM
cana-1630	131	4	)	)	PUNCT
cana-1630	131	5	and	and	CCONJ
cana-1630	131	6	𝑦.	𝑦.	PROPN
cana-1630	131	7	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	131	8	(	(	PUNCT
cana-1630	131	9	−	−	PROPN
cana-1630	131	10	𝑥2	𝑥2	NOUN
cana-1630	131	11	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	131	12	2	2	NUM
cana-1630	131	13	−	−	NOUN
cana-1630	131	14	𝑦2	𝑦2	NOUN
cana-1630	131	15	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	131	16	2	2	NUM
cana-1630	131	17	)	)	PUNCT
cana-1630	131	18	introduce	introduce	VERB
cana-1630	131	19	additional	additional	ADJ
cana-1630	131	20	frequency	frequency	NOUN
cana-1630	131	21	dependent	dependent	ADJ
cana-1630	131	22	terms	term	NOUN
cana-1630	131	23	.	.	PUNCT
cana-1630	132	1	the	the	DET
cana-1630	132	2	overall	overall	ADJ
cana-1630	132	3	frequency	frequency	NOUN
cana-1630	132	4	response	response	NOUN
cana-1630	132	5	h(u	h(u	PROPN
cana-1630	132	6	,	,	PUNCT
cana-1630	132	7	v	v	NOUN
cana-1630	132	8	)	)	PUNCT
cana-1630	132	9	will	will	AUX
cana-1630	132	10	be	be	AUX
cana-1630	132	11	a	a	DET
cana-1630	132	12	combination	combination	NOUN
cana-1630	132	13	of	of	ADP
cana-1630	132	14	these	these	DET
cana-1630	132	15	terms	term	NOUN
cana-1630	132	16	:	:	PUNCT
cana-1630	132	17	𝐻(𝑢	𝐻(𝑢	ADP
cana-1630	132	18	,	,	PUNCT
cana-1630	132	19	𝑣	𝑣	NOUN
cana-1630	132	20	)	)	PUNCT
cana-1630	132	21	=	=	SYM
cana-1630	132	22	(	(	PUNCT
cana-1630	133	1	1	1	NUM
cana-1630	133	2	−	−	NOUN
cana-1630	133	3	𝑗	𝑗	PRON
cana-1630	133	4	2𝜋𝜎𝑥𝑢𝛼	2𝜋𝜎𝑥𝑢𝛼	PROPN
cana-1630	133	5	−	−	PROPN
cana-1630	133	6	𝑗2𝜋𝜎𝑦𝑣𝛽	𝑗2𝜋𝜎𝑦𝑣𝛽	NOUN
cana-1630	133	7	)	)	PUNCT
cana-1630	133	8	exp(−2𝜋2𝜎𝑥	exp(−2𝜋2𝜎𝑥	PROPN
cana-1630	133	9	2𝑢2	2𝑢2	NUM
cana-1630	134	1	−	−	PROPN
cana-1630	134	2	2𝜋2𝜎𝑦	2𝜋2𝜎𝑦	NUM
cana-1630	134	3	2𝑣2	2𝑣2	NOUN
cana-1630	134	4	)	)	PUNCT
cana-1630	134	5	hence	hence	ADV
cana-1630	134	6	the	the	DET
cana-1630	134	7	proof	proof	NOUN
cana-1630	134	8	.	.	PUNCT
cana-1630	135	1	theorem	theorem	VERB
cana-1630	135	2	6	6	NUM
cana-1630	135	3	:	:	PUNCT
cana-1630	135	4	preservation	preservation	NOUN
cana-1630	135	5	of	of	ADP
cana-1630	135	6	fuzzy	fuzzy	ADJ
cana-1630	135	7	membership	membership	NOUN
cana-1630	135	8	distribution	distribution	NOUN
cana-1630	135	9	if	if	SCONJ
cana-1630	135	10	a	a	DET
cana-1630	135	11	fuzzy	fuzzy	ADJ
cana-1630	135	12	membership	membership	NOUN
cana-1630	135	13	function	function	NOUN
cana-1630	135	14	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	135	15	,	,	PUNCT
cana-1630	135	16	𝑦	𝑦	NOUN
cana-1630	135	17	)	)	PUNCT
cana-1630	135	18	is	be	AUX
cana-1630	135	19	processed	process	VERB
cana-1630	135	20	by	by	ADP
cana-1630	135	21	an	an	DET
cana-1630	135	22	adaptive	adaptive	ADJ
cana-1630	135	23	gaussian	gaussian	ADJ
cana-1630	135	24	2d	2d	NUM
cana-1630	135	25	filter	filter	NOUN
cana-1630	135	26	defined	define	VERB
cana-1630	135	27	by	by	ADP
cana-1630	135	28	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	135	29	,	,	PUNCT
cana-1630	135	30	𝑦	𝑦	NOUN
cana-1630	135	31	,	,	PUNCT
cana-1630	135	32	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	135	33	,	,	PUNCT
cana-1630	135	34	𝜎𝑦	𝜎𝑦	PROPN
cana-1630	135	35	,	,	PUNCT
cana-1630	135	36	𝛼	𝛼	X
cana-1630	135	37	,	,	PUNCT
cana-1630	135	38	𝛽	𝛽	NOUN
cana-1630	135	39	)	)	PUNCT
cana-1630	135	40	=	=	SYM
cana-1630	135	41	1	1	NUM
cana-1630	135	42	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	136	1	[	[	PUNCT
cana-1630	136	2	1	1	NUM
cana-1630	136	3	−	−	NOUN
cana-1630	136	4	𝛼	𝛼	PART
cana-1630	136	5	𝑥	𝑥	NOUN
cana-1630	136	6	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	136	7	2	2	NUM
cana-1630	136	8	−	−	NOUN
cana-1630	136	9	𝛽	𝛽	NOUN
cana-1630	136	10	𝑦	𝑦	PRON
cana-1630	136	11	𝜎𝑦	𝜎𝑦	PRON
cana-1630	136	12	2	2	NUM
cana-1630	136	13	]	]	PUNCT
cana-1630	136	14	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	136	15	(	(	PUNCT
cana-1630	136	16	−	−	PROPN
cana-1630	136	17	𝑥2	𝑥2	NOUN
cana-1630	136	18	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	136	19	2	2	NUM
cana-1630	136	20	−	−	NOUN
cana-1630	136	21	𝑦2	𝑦2	NOUN
cana-1630	136	22	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	136	23	2	2	NUM
cana-1630	136	24	)	)	PUNCT
cana-1630	136	25	the	the	DET
cana-1630	136	26	resulting	result	VERB
cana-1630	136	27	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	NOUN
cana-1630	136	28	𝑦	𝑦	NOUN
cana-1630	136	29	)	)	PUNCT
cana-1630	136	30	will	will	AUX
cana-1630	136	31	also	also	ADV
cana-1630	136	32	be	be	AUX
cana-1630	136	33	a	a	DET
cana-1630	136	34	valid	valid	ADJ
cana-1630	136	35	fuzzy	fuzzy	ADJ
cana-1630	136	36	membership	membership	NOUN
cana-1630	136	37	function	function	NOUN
cana-1630	136	38	,	,	PUNCT
cana-1630	136	39	preserving	preserve	VERB
cana-1630	136	40	its	its	PRON
cana-1630	136	41	essential	essential	ADJ
cana-1630	136	42	properties	property	NOUN
cana-1630	136	43	.	.	PUNCT
cana-1630	137	1	communications	communication	NOUN
cana-1630	137	2	on	on	ADP
cana-1630	137	3	applied	apply	VERB
cana-1630	137	4	nonlinear	nonlinear	ADJ
cana-1630	137	5	analysis	analysis	NOUN
cana-1630	137	6	issn	issn	NOUN
cana-1630	137	7	:	:	PUNCT
cana-1630	137	8	1074	1074	NUM
cana-1630	137	9	-	-	PUNCT
cana-1630	137	10	133x	133x	NUM
cana-1630	137	11	vol	vol	NOUN
cana-1630	137	12	32	32	NUM
cana-1630	137	13	no	no	NOUN
cana-1630	137	14	.	.	NOUN
cana-1630	137	15	1	1	NUM
cana-1630	137	16	(	(	PUNCT
cana-1630	137	17	2025	2025	NUM
cana-1630	137	18	)	)	PUNCT
cana-1630	137	19	191	191	NUM
cana-1630	138	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	138	2	proof	proof	NOUN
cana-1630	138	3	:	:	PUNCT
cana-1630	138	4	the	the	DET
cana-1630	138	5	fuzzy	fuzzy	ADJ
cana-1630	138	6	membership	membership	NOUN
cana-1630	138	7	function	function	NOUN
cana-1630	138	8	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	138	9	,	,	PUNCT
cana-1630	138	10	𝑦	𝑦	NOUN
cana-1630	138	11	)	)	PUNCT
cana-1630	138	12	is	be	AUX
cana-1630	138	13	typically	typically	ADV
cana-1630	138	14	bounded	bound	VERB
cana-1630	138	15	between	between	ADP
cana-1630	138	16	0	0	NUM
cana-1630	138	17	and	and	CCONJ
cana-1630	138	18	1	1	NUM
cana-1630	138	19	.	.	X
cana-1630	139	1	for	for	ADP
cana-1630	139	2	any	any	DET
cana-1630	139	3	fuzzy	fuzzy	ADJ
cana-1630	139	4	membership	membership	NOUN
cana-1630	139	5	function	function	NOUN
cana-1630	139	6	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	139	7	,	,	PUNCT
cana-1630	139	8	𝑦)it	𝑦)it	PROPN
cana-1630	139	9	holds	hold	VERB
cana-1630	139	10	that	that	SCONJ
cana-1630	139	11	0	0	NUM
cana-1630	139	12	≤	≤	NUM
cana-1630	139	13	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	139	14	,	,	PUNCT
cana-1630	139	15	𝑦	𝑦	NOUN
cana-1630	139	16	)	)	PUNCT
cana-1630	139	17	≤	≤	NUM
cana-1630	139	18	1	1	NUM
cana-1630	139	19	.	.	PUNCT
cana-1630	140	1	the	the	DET
cana-1630	140	2	filtered	filter	VERB
cana-1630	140	3	membership	membership	NOUN
cana-1630	140	4	function	function	NOUN
cana-1630	140	5	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	PROPN
cana-1630	140	6	𝑦	𝑦	NOUN
cana-1630	140	7	)	)	PUNCT
cana-1630	140	8	is	be	AUX
cana-1630	140	9	obtained	obtain	VERB
cana-1630	140	10	by	by	ADP
cana-1630	140	11	convolving	convolve	VERB
cana-1630	140	12	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	140	13	,	,	PUNCT
cana-1630	140	14	𝑦	𝑦	NOUN
cana-1630	140	15	)	)	PUNCT
cana-1630	140	16	with	with	ADP
cana-1630	140	17	the	the	DET
cana-1630	140	18	adaptive	adaptive	ADJ
cana-1630	140	19	gaussian	gaussian	ADJ
cana-1630	140	20	filter	filter	NOUN
cana-1630	140	21	𝐹(𝑥	𝐹(𝑥	PROPN
cana-1630	140	22	,	,	PUNCT
cana-1630	140	23	𝑦	𝑦	NOUN
cana-1630	140	24	)	)	PUNCT
cana-1630	140	25	:	:	PUNCT
cana-1630	141	1	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	NOUN
cana-1630	141	2	𝑦	𝑦	X
cana-1630	141	3	)	)	PUNCT
cana-1630	141	4	=	=	SYM
cana-1630	142	1	∫	∫	PROPN
cana-1630	142	2	∫	∫	PROPN
cana-1630	142	3	𝜇(𝑢	𝜇(𝑢	PROPN
cana-1630	142	4	,	,	PUNCT
cana-1630	142	5	𝑣	𝑣	NOUN
cana-1630	142	6	)	)	PUNCT
cana-1630	142	7	∞	∞	PROPN
cana-1630	143	1	−∞	−∞	X
cana-1630	143	2	𝐹(𝑥	𝐹(𝑥	PUNCT
cana-1630	143	3	−	−	PROPN
cana-1630	143	4	𝑢	𝑢	PROPN
cana-1630	143	5	,	,	PUNCT
cana-1630	143	6	𝑦	𝑦	NOUN
cana-1630	143	7	−	−	NOUN
cana-1630	143	8	𝑣	𝑣	NOUN
cana-1630	143	9	,	,	PUNCT
cana-1630	143	10	𝜎𝑥	𝜎𝑥	INTJ
cana-1630	143	11	,	,	PUNCT
cana-1630	143	12	𝜎𝑦	𝜎𝑦	PRON
cana-1630	143	13	,	,	PUNCT
cana-1630	143	14	𝛼	𝛼	PROPN
cana-1630	143	15	,	,	PUNCT
cana-1630	143	16	𝛽)𝑑𝑢	𝛽)𝑑𝑢	NOUN
cana-1630	143	17	𝑑𝑣	𝑑𝑣	ADP
cana-1630	143	18	∞	∞	PROPN
cana-1630	143	19	−∞	−∞	ADP
cana-1630	143	20	since	since	SCONJ
cana-1630	143	21	𝜇(𝑢	𝜇(𝑢	PROPN
cana-1630	143	22	,	,	PUNCT
cana-1630	143	23	𝑣	𝑣	NOUN
cana-1630	143	24	)	)	PUNCT
cana-1630	143	25	is	be	AUX
cana-1630	143	26	bounded	bound	VERB
cana-1630	143	27	between	between	ADP
cana-1630	143	28	0	0	NUM
cana-1630	143	29	and	and	CCONJ
cana-1630	143	30	1	1	NUM
cana-1630	143	31	and	and	CCONJ
cana-1630	143	32	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	143	33	,	,	PUNCT
cana-1630	143	34	𝑦	𝑦	X
cana-1630	143	35	)	)	PUNCT
cana-1630	143	36	is	be	AUX
cana-1630	143	37	non	non	ADJ
cana-1630	143	38	-	-	ADJ
cana-1630	143	39	negative	negative	ADJ
cana-1630	143	40	and	and	CCONJ
cana-1630	143	41	integrates	integrate	NOUN
cana-1630	143	42	to	to	ADP
cana-1630	143	43	1	1	NUM
cana-1630	143	44	,	,	PUNCT
cana-1630	143	45	then	then	ADV
cana-1630	143	46	,	,	PUNCT
cana-1630	143	47	0	0	NUM
cana-1630	143	48	≤	≤	NUM
cana-1630	143	49	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	PROPN
cana-1630	143	50	𝑦	𝑦	NOUN
cana-1630	143	51	)	)	PUNCT
cana-1630	143	52	≤	≤	NUM
cana-1630	144	1	∫	∫	PROPN
cana-1630	144	2	∫	∫	PROPN
cana-1630	145	1	𝜇(𝑢	𝜇(𝑢	PROPN
cana-1630	145	2	,	,	PUNCT
cana-1630	145	3	𝑣	𝑣	NOUN
cana-1630	145	4	)	)	PUNCT
cana-1630	145	5	∞	∞	PROPN
cana-1630	146	1	−∞	−∞	X
cana-1630	146	2	𝐹(𝑥	𝐹(𝑥	PUNCT
cana-1630	146	3	−	−	PROPN
cana-1630	146	4	𝑢	𝑢	PROPN
cana-1630	146	5	,	,	PUNCT
cana-1630	146	6	𝑦	𝑦	NOUN
cana-1630	146	7	−	−	NOUN
cana-1630	146	8	𝑣	𝑣	NOUN
cana-1630	146	9	,	,	PUNCT
cana-1630	146	10	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	146	11	,	,	PUNCT
cana-1630	146	12	𝜎𝑦	𝜎𝑦	PRON
cana-1630	146	13	,	,	PUNCT
cana-1630	146	14	𝛼	𝛼	PROPN
cana-1630	146	15	,	,	PUNCT
cana-1630	146	16	𝛽)𝑑𝑢	𝛽)𝑑𝑢	NOUN
cana-1630	146	17	𝑑𝑣	𝑑𝑣	ADP
cana-1630	146	18	∞	∞	PROPN
cana-1630	146	19	−∞	−∞	ADP
cana-1630	146	20	≤	≤	NUM
cana-1630	146	21	1	1	NUM
cana-1630	146	22	the	the	DET
cana-1630	146	23	adaptive	adaptive	ADJ
cana-1630	146	24	gaussian	gaussian	ADJ
cana-1630	146	25	filter	filter	NOUN
cana-1630	146	26	preserves	preserve	VERB
cana-1630	146	27	the	the	DET
cana-1630	146	28	smoothness	smoothness	NOUN
cana-1630	146	29	and	and	CCONJ
cana-1630	146	30	continuity	continuity	NOUN
cana-1630	146	31	of	of	ADP
cana-1630	146	32	the	the	DET
cana-1630	146	33	membership	membership	NOUN
cana-1630	146	34	function	function	NOUN
cana-1630	146	35	.	.	PUNCT
cana-1630	147	1	since	since	SCONJ
cana-1630	147	2	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	147	3	,	,	PUNCT
cana-1630	147	4	𝑦	𝑦	X
cana-1630	147	5	)	)	PUNCT
cana-1630	147	6	is	be	AUX
cana-1630	147	7	smooth	smooth	ADJ
cana-1630	147	8	and	and	CCONJ
cana-1630	147	9	the	the	DET
cana-1630	147	10	convolution	convolution	NOUN
cana-1630	147	11	operation	operation	NOUN
cana-1630	147	12	with	with	ADP
cana-1630	147	13	𝐹(𝑥	𝐹(𝑥	NUM
cana-1630	147	14	,	,	PUNCT
cana-1630	147	15	𝑦	𝑦	NOUN
cana-1630	147	16	)	)	PUNCT
cana-1630	147	17	maintain	maintain	VERB
cana-1630	147	18	this	this	DET
cana-1630	147	19	smoothness	smoothness	NOUN
cana-1630	147	20	,	,	PUNCT
cana-1630	147	21	the	the	DET
cana-1630	147	22	filtered	filter	VERB
cana-1630	147	23	membership	membership	NOUN
cana-1630	147	24	function	function	NOUN
cana-1630	147	25	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	PROPN
cana-1630	147	26	𝑦	𝑦	NUM
cana-1630	147	27	)	)	PUNCT
cana-1630	147	28	will	will	AUX
cana-1630	147	29	also	also	ADV
cana-1630	147	30	be	be	AUX
cana-1630	147	31	smooth	smooth	ADJ
cana-1630	147	32	.	.	PUNCT
cana-1630	148	1	the	the	DET
cana-1630	148	2	filtered	filter	VERB
cana-1630	148	3	output	output	NOUN
cana-1630	148	4	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	VERB
cana-1630	148	5	𝑦	𝑦	NOUN
cana-1630	148	6	)	)	PUNCT
cana-1630	148	7	remains	remain	VERB
cana-1630	148	8	a	a	DET
cana-1630	148	9	valid	valid	ADJ
cana-1630	148	10	fuzzy	fuzzy	ADJ
cana-1630	148	11	membership	membership	NOUN
cana-1630	148	12	function	function	NOUN
cana-1630	148	13	,	,	PUNCT
cana-1630	148	14	preserving	preserve	VERB
cana-1630	148	15	the	the	DET
cana-1630	148	16	essential	essential	ADJ
cana-1630	148	17	properties	property	NOUN
cana-1630	148	18	such	such	ADJ
cana-1630	148	19	as	as	ADP
cana-1630	148	20	boundedness	boundedness	NOUN
cana-1630	148	21	,	,	PUNCT
cana-1630	148	22	normalization	normalization	NOUN
cana-1630	148	23	and	and	CCONJ
cana-1630	148	24	smoothness	smoothness	NOUN
cana-1630	148	25	.	.	PUNCT
cana-1630	149	1	hence	hence	ADV
cana-1630	149	2	the	the	DET
cana-1630	149	3	proof	proof	NOUN
cana-1630	149	4	.	.	PUNCT
cana-1630	150	1	corollary	corollary	ADJ
cana-1630	150	2	:	:	PUNCT
cana-1630	150	3	if	if	SCONJ
cana-1630	150	4	a	a	DET
cana-1630	150	5	fuzzy	fuzzy	ADJ
cana-1630	150	6	membership	membership	NOUN
cana-1630	150	7	function	function	NOUN
cana-1630	150	8	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	150	9	,	,	PUNCT
cana-1630	150	10	𝑦	𝑦	NOUN
cana-1630	150	11	)	)	PUNCT
cana-1630	150	12	is	be	AUX
cana-1630	150	13	processed	process	VERB
cana-1630	150	14	by	by	ADP
cana-1630	150	15	the	the	DET
cana-1630	150	16	adaptive	adaptive	ADJ
cana-1630	150	17	gaussian	gaussian	ADJ
cana-1630	150	18	2d	2d	NUM
cana-1630	150	19	filter	filter	NOUN
cana-1630	150	20	,	,	PUNCT
cana-1630	150	21	the	the	DET
cana-1630	150	22	resulting	result	VERB
cana-1630	150	23	function	function	NOUN
cana-1630	150	24	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥.	NOUN
cana-1630	150	25	𝑦	𝑦	NOUN
cana-1630	150	26	)	)	PUNCT
cana-1630	150	27	will	will	AUX
cana-1630	150	28	not	not	PART
cana-1630	150	29	only	only	ADV
cana-1630	150	30	be	be	AUX
cana-1630	150	31	bounded	bound	VERB
cana-1630	150	32	between	between	ADP
cana-1630	150	33	0	0	NUM
cana-1630	150	34	and	and	CCONJ
cana-1630	150	35	1	1	NUM
cana-1630	150	36	but	but	CCONJ
cana-1630	150	37	will	will	AUX
cana-1630	150	38	also	also	ADV
cana-1630	150	39	maintain	maintain	VERB
cana-1630	150	40	the	the	DET
cana-1630	150	41	essential	essential	ADJ
cana-1630	150	42	properties	property	NOUN
cana-1630	150	43	of	of	ADP
cana-1630	150	44	a	a	DET
cana-1630	150	45	fuzzy	fuzzy	ADJ
cana-1630	150	46	membership	membership	NOUN
cana-1630	150	47	function	function	NOUN
cana-1630	150	48	,	,	PUNCT
cana-1630	150	49	ensuring	ensure	VERB
cana-1630	150	50	that	that	SCONJ
cana-1630	150	51	the	the	DET
cana-1630	150	52	concept	concept	NOUN
cana-1630	150	53	represented	represent	VERB
cana-1630	150	54	by	by	ADP
cana-1630	150	55	the	the	DET
cana-1630	150	56	membership	membership	NOUN
cana-1630	150	57	function	function	NOUN
cana-1630	150	58	is	be	AUX
cana-1630	150	59	preserved	preserve	VERB
cana-1630	150	60	in	in	ADP
cana-1630	150	61	the	the	DET
cana-1630	150	62	filtered	filter	VERB
cana-1630	150	63	output	output	NOUN
cana-1630	150	64	.	.	PUNCT
cana-1630	151	1	3.3	3.3	NUM
cana-1630	151	2	adaptive	adaptive	ADJ
cana-1630	151	3	fuzzy	fuzzy	ADJ
cana-1630	151	4	gaussian	gaussian	NOUN
cana-1630	151	5	1d	1d	NUM
cana-1630	151	6	filter	filter	NOUN
cana-1630	151	7	and	and	CCONJ
cana-1630	151	8	derivative	derivative	ADJ
cana-1630	151	9	fuzzy	fuzzy	ADJ
cana-1630	151	10	gaussian	gaussian	NOUN
cana-1630	151	11	1d	1d	NUM
cana-1630	151	12	filter	filter	NOUN
cana-1630	151	13	(	(	PUNCT
cana-1630	151	14	afgd-1d	afgd-1d	NOUN
cana-1630	151	15	)	)	PUNCT
cana-1630	151	16	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	151	17	,	,	PUNCT
cana-1630	151	18	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	151	19	,	,	PUNCT
cana-1630	151	20	𝛼	𝛼	NOUN
cana-1630	151	21	)	)	PUNCT
cana-1630	151	22	=	=	SYM
cana-1630	151	23	1	1	NUM
cana-1630	151	24	√2𝜋𝜎	√2𝜋𝜎	INTJ
cana-1630	151	25	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	151	26	(	(	PUNCT
cana-1630	151	27	−	−	PROPN
cana-1630	151	28	(	(	PUNCT
cana-1630	151	29	𝑥−𝑚)2	𝑥−𝑚)2	NOUN
cana-1630	151	30	2𝜎2	2𝜎2	NUM
cana-1630	151	31	)	)	PUNCT
cana-1630	152	1	[	[	X
cana-1630	152	2	1	1	NUM
cana-1630	152	3	−	−	PROPN
cana-1630	152	4	𝛼.	𝛼.	NOUN
cana-1630	152	5	(	(	PUNCT
cana-1630	152	6	𝑥−𝑚	𝑥−𝑚	NOUN
cana-1630	152	7	𝜎2	𝜎2	NOUN
cana-1630	152	8	)	)	PUNCT
cana-1630	152	9	]	]	PUNCT
cana-1630	152	10	(	(	PUNCT
cana-1630	152	11	3	3	X
cana-1630	152	12	)	)	PUNCT
cana-1630	152	13	where	where	SCONJ
cana-1630	152	14	,	,	PUNCT
cana-1630	152	15	𝑚	𝑚	X
cana-1630	152	16	=	=	SYM
cana-1630	152	17	1	1	NUM
cana-1630	152	18	𝑚𝑛	𝑚𝑛	NOUN
cana-1630	152	19	∑	∑	PROPN
cana-1630	152	20	∑	∑	PROPN
cana-1630	152	21	𝐴	𝐴	PROPN
cana-1630	152	22	(	(	PUNCT
cana-1630	152	23	𝑖	𝑖	NOUN
cana-1630	152	24	,	,	PUNCT
cana-1630	152	25	𝑗)𝑛	𝑗)𝑛	PUNCT
cana-1630	152	26	𝑗=1	𝑗=1	PROPN
cana-1630	152	27	𝑚	𝑚	X
cana-1630	152	28	𝑖=1	𝑖=1	PROPN
cana-1630	152	29	.	.	PUNCT
cana-1630	152	30	theorem	theorem	ADJ
cana-1630	152	31	7	7	NUM
cana-1630	152	32	:	:	PUNCT
cana-1630	152	33	bibo	bibo	VERB
cana-1630	152	34	stability	stability	NOUN
cana-1630	152	35	the	the	DET
cana-1630	152	36	adaptive	adaptive	ADJ
cana-1630	152	37	gaussian	gaussian	ADJ
cana-1630	152	38	filter	filter	NOUN
cana-1630	152	39	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	152	40	,	,	PUNCT
cana-1630	152	41	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	152	42	,	,	PUNCT
cana-1630	152	43	𝛼	𝛼	NOUN
cana-1630	152	44	)	)	PUNCT
cana-1630	152	45	is	be	AUX
cana-1630	152	46	bibo	bibo	ADJ
cana-1630	152	47	stable	stable	ADJ
cana-1630	152	48	.	.	PUNCT
cana-1630	153	1	that	that	PRON
cana-1630	153	2	is	be	AUX
cana-1630	153	3	,	,	PUNCT
cana-1630	153	4	for	for	ADP
cana-1630	153	5	any	any	DET
cana-1630	153	6	bounded	bounded	ADJ
cana-1630	153	7	input	input	NOUN
cana-1630	153	8	image	image	NOUN
cana-1630	153	9	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	153	10	,	,	PUNCT
cana-1630	153	11	𝑗	𝑗	NOUN
cana-1630	153	12	)	)	PUNCT
cana-1630	153	13	the	the	DET
cana-1630	153	14	output	output	NOUN
cana-1630	153	15	image	image	NOUN
cana-1630	153	16	will	will	AUX
cana-1630	153	17	also	also	ADV
cana-1630	153	18	be	be	AUX
cana-1630	153	19	bounded	bound	VERB
cana-1630	153	20	.	.	PUNCT
cana-1630	154	1	proof	proof	NOUN
cana-1630	154	2	:	:	PUNCT
cana-1630	154	3	let	let	VERB
cana-1630	154	4	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	154	5	,	,	PUNCT
cana-1630	154	6	𝑗	𝑗	AUX
cana-1630	154	7	)	)	PUNCT
cana-1630	154	8	be	be	AUX
cana-1630	154	9	a	a	DET
cana-1630	154	10	bounded	bounded	ADJ
cana-1630	154	11	input	input	NOUN
cana-1630	154	12	image	image	NOUN
cana-1630	154	13	,	,	PUNCT
cana-1630	154	14	then	then	ADV
cana-1630	154	15	there	there	PRON
cana-1630	154	16	exist	exist	VERB
cana-1630	154	17	a	a	DET
cana-1630	154	18	constant	constant	ADJ
cana-1630	154	19	m	m	NOUN
cana-1630	154	20	such	such	ADJ
cana-1630	154	21	that	that	SCONJ
cana-1630	154	22	|	|	ADV
cana-1630	154	23	𝐴(𝑖	𝐴(𝑖	ADV
cana-1630	154	24	,	,	PUNCT
cana-1630	154	25	𝑗)|	𝑗)|	NOUN
cana-1630	154	26	≤	≤	NOUN
cana-1630	154	27	𝑀∀(𝑖	𝑀∀(𝑖	PROPN
cana-1630	154	28	,	,	PUNCT
cana-1630	154	29	𝑗	𝑗	NOUN
cana-1630	154	30	)	)	PUNCT
cana-1630	154	31	where	where	SCONJ
cana-1630	154	32	𝜎	𝜎	PROPN
cana-1630	154	33	is	be	AUX
cana-1630	154	34	standard	standard	ADJ
cana-1630	154	35	deviation	deviation	NOUN
cana-1630	154	36	,	,	PUNCT
cana-1630	154	37	m	m	VERB
cana-1630	154	38	is	be	AUX
cana-1630	154	39	mean	mean	NOUN
cana-1630	154	40	of	of	ADP
cana-1630	154	41	the	the	DET
cana-1630	154	42	image	image	NOUN
cana-1630	154	43	which	which	PRON
cana-1630	154	44	is	be	AUX
cana-1630	154	45	bounded	bound	VERB
cana-1630	154	46	because	because	SCONJ
cana-1630	154	47	𝐴(𝑖	𝐴(𝑖	CCONJ
cana-1630	154	48	,	,	PUNCT
cana-1630	154	49	𝑗	𝑗	NOUN
cana-1630	154	50	)	)	PUNCT
cana-1630	154	51	is	be	AUX
cana-1630	154	52	bounded	bound	VERB
cana-1630	154	53	,	,	PUNCT
cana-1630	154	54	𝛼	𝛼	PROPN
cana-1630	154	55	is	be	AUX
cana-1630	154	56	adaptive	adaptive	ADJ
cana-1630	154	57	parameter	parameter	NOUN
cana-1630	154	58	which	which	PRON
cana-1630	154	59	can	can	AUX
cana-1630	154	60	be	be	AUX
cana-1630	154	61	bounded	bound	VERB
cana-1630	154	62	based	base	VERB
cana-1630	154	63	on	on	ADP
cana-1630	154	64	the	the	DET
cana-1630	154	65	range	range	NOUN
cana-1630	154	66	of	of	ADP
cana-1630	154	67	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	154	68	,	,	PUNCT
cana-1630	154	69	𝑗	𝑗	NOUN
cana-1630	154	70	)	)	PUNCT
cana-1630	154	71	.	.	PUNCT
cana-1630	155	1	the	the	DET
cana-1630	155	2	gaussian	gaussian	ADJ
cana-1630	155	3	part	part	NOUN
cana-1630	155	4	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	155	5	(	(	PUNCT
cana-1630	155	6	−	−	PROPN
cana-1630	155	7	(	(	PUNCT
cana-1630	155	8	𝑥−𝑚)2	𝑥−𝑚)2	NOUN
cana-1630	155	9	2𝜎2	2𝜎2	NUM
cana-1630	155	10	)	)	PUNCT
cana-1630	155	11	is	be	AUX
cana-1630	155	12	bounded	bound	VERB
cana-1630	155	13	by	by	ADP
cana-1630	155	14	1	1	NUM
cana-1630	155	15	and	and	CCONJ
cana-1630	155	16	the	the	DET
cana-1630	155	17	term	term	NOUN
cana-1630	155	18	1	1	NUM
cana-1630	155	19	−	−	NOUN
cana-1630	155	20	𝛼.	𝛼.	NOUN
cana-1630	155	21	(	(	PUNCT
cana-1630	155	22	𝑥−𝑚	𝑥−𝑚	NOUN
cana-1630	155	23	𝜎2	𝜎2	NOUN
cana-1630	155	24	)	)	PUNCT
cana-1630	155	25	is	be	AUX
cana-1630	155	26	also	also	ADV
cana-1630	155	27	bounded	bound	VERB
cana-1630	155	28	because	because	SCONJ
cana-1630	155	29	𝛼	𝛼	PROPN
cana-1630	155	30	and	and	CCONJ
cana-1630	155	31	(	(	PUNCT
cana-1630	155	32	𝑥−𝑚	𝑥−𝑚	NOUN
cana-1630	155	33	𝜎2	𝜎2	NOUN
cana-1630	155	34	)	)	PUNCT
cana-1630	155	35	are	be	AUX
cana-1630	155	36	bounded	bound	VERB
cana-1630	155	37	.	.	PUNCT
cana-1630	156	1	the	the	DET
cana-1630	156	2	output	output	NOUN
cana-1630	156	3	of	of	ADP
cana-1630	156	4	the	the	DET
cana-1630	156	5	filter	filter	NOUN
cana-1630	156	6	applied	apply	VERB
cana-1630	156	7	to	to	ADP
cana-1630	156	8	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	156	9	,	,	PUNCT
cana-1630	156	10	𝑗	𝑗	X
cana-1630	156	11	)	)	PUNCT
cana-1630	156	12	can	can	AUX
cana-1630	156	13	be	be	AUX
cana-1630	156	14	expressed	express	VERB
cana-1630	156	15	as	as	ADP
cana-1630	156	16	a	a	DET
cana-1630	156	17	convolution	convolution	NOUN
cana-1630	156	18	:	:	PUNCT
cana-1630	156	19	communications	communication	NOUN
cana-1630	156	20	on	on	ADP
cana-1630	156	21	applied	apply	VERB
cana-1630	156	22	nonlinear	nonlinear	ADJ
cana-1630	156	23	analysis	analysis	NOUN
cana-1630	156	24	issn	issn	NOUN
cana-1630	156	25	:	:	PUNCT
cana-1630	156	26	1074	1074	NUM
cana-1630	156	27	-	-	PUNCT
cana-1630	156	28	133x	133x	NUM
cana-1630	156	29	vol	vol	NOUN
cana-1630	156	30	32	32	NUM
cana-1630	156	31	no	no	NOUN
cana-1630	156	32	.	.	NOUN
cana-1630	156	33	1	1	NUM
cana-1630	156	34	(	(	PUNCT
cana-1630	156	35	2025	2025	NUM
cana-1630	156	36	)	)	PUNCT
cana-1630	156	37	192	192	NUM
cana-1630	156	38	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	156	39	𝑔(𝑥	𝑔(𝑥	NUM
cana-1630	156	40	)	)	PUNCT
cana-1630	157	1	=	=	SYM
cana-1630	158	1	(	(	PUNCT
cana-1630	158	2	𝐴	𝐴	PROPN
cana-1630	158	3	∗	∗	NOUN
cana-1630	158	4	𝐺)(𝑥	𝐺)(𝑥	PROPN
cana-1630	158	5	)	)	PUNCT
cana-1630	159	1	=	=	SYM
cana-1630	159	2	∫	∫	PROPN
cana-1630	159	3	𝐴(𝑡	𝐴(𝑡	PROPN
cana-1630	159	4	)	)	PUNCT
cana-1630	160	1	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	160	2	−	−	NOUN
cana-1630	160	3	𝑡	𝑡	PROPN
cana-1630	160	4	,	,	PUNCT
cana-1630	160	5	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	160	6	,	,	PUNCT
cana-1630	160	7	𝛼)𝑑𝑡	𝛼)𝑑𝑡	PROPN
cana-1630	160	8	∞	∞	PROPN
cana-1630	161	1	−∞	−∞	ADP
cana-1630	161	2	given	give	VERB
cana-1630	161	3	the	the	DET
cana-1630	161	4	boundedness	boundedness	NOUN
cana-1630	161	5	of	of	ADP
cana-1630	161	6	𝐴(𝑡	𝐴(𝑡	PROPN
cana-1630	161	7	)	)	PUNCT
cana-1630	161	8	and	and	CCONJ
cana-1630	161	9	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	161	10	,	,	PUNCT
cana-1630	161	11	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	161	12	,	,	PUNCT
cana-1630	161	13	𝛼	𝛼	NOUN
cana-1630	161	14	)	)	PUNCT
cana-1630	161	15	,	,	PUNCT
cana-1630	161	16	then	then	ADV
cana-1630	161	17	|𝑔(𝑥)|	|𝑔(𝑥)|	VERB
cana-1630	161	18	≤	≤	NUM
cana-1630	161	19	∫	∫	NOUN
cana-1630	161	20	|𝐴(𝑡)||𝐺(𝑥	|𝐴(𝑡)||𝐺(𝑥	NOUN
cana-1630	161	21	−	−	PROPN
cana-1630	161	22	𝑡	𝑡	PROPN
cana-1630	161	23	,	,	PUNCT
cana-1630	161	24	𝜎	𝜎	PROPN
cana-1630	161	25	,	,	PUNCT
cana-1630	161	26	𝑚	𝑚	PROPN
cana-1630	161	27	,	,	PUNCT
cana-1630	161	28	𝛼)|	𝛼)|	PROPN
cana-1630	161	29	𝑑𝑡	𝑑𝑡	ADP
cana-1630	161	30	∞	∞	PROPN
cana-1630	161	31	−∞	−∞	ADP
cana-1630	161	32	≤	≤	ADJ
cana-1630	161	33	𝑀	𝑀	PROPN
cana-1630	161	34	∫	∫	NOUN
cana-1630	161	35	|𝐺(𝑥	|𝐺(𝑥	VERB
cana-1630	161	36	−	−	PROPN
cana-1630	161	37	𝑡	𝑡	PROPN
cana-1630	161	38	,	,	PUNCT
cana-1630	161	39	𝜎,𝑚	𝜎,𝑚	PROPN
cana-1630	161	40	,	,	PUNCT
cana-1630	161	41	𝛼)|	𝛼)|	PROPN
cana-1630	161	42	𝑑𝑡	𝑑𝑡	ADP
cana-1630	161	43	∞	∞	PROPN
cana-1630	161	44	−∞	−∞	PUNCT
cana-1630	161	45	since	since	SCONJ
cana-1630	161	46	,	,	PUNCT
cana-1630	161	47	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	161	48	,	,	PUNCT
cana-1630	161	49	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	161	50	,	,	PUNCT
cana-1630	161	51	𝛼	𝛼	NOUN
cana-1630	161	52	)	)	PUNCT
cana-1630	161	53	is	be	AUX
cana-1630	161	54	a	a	DET
cana-1630	161	55	bounded	bounded	ADJ
cana-1630	161	56	function	function	NOUN
cana-1630	161	57	that	that	PRON
cana-1630	161	58	integrates	integrate	VERB
cana-1630	161	59	to	to	ADP
cana-1630	161	60	a	a	DET
cana-1630	161	61	finite	finite	ADJ
cana-1630	161	62	value	value	NOUN
cana-1630	161	63	,	,	PUNCT
cana-1630	161	64	the	the	DET
cana-1630	161	65	output	output	NOUN
cana-1630	161	66	g(x	g(x	NOUN
cana-1630	161	67	)	)	PUNCT
cana-1630	161	68	is	be	AUX
cana-1630	161	69	also	also	ADV
cana-1630	161	70	bounded	bound	VERB
cana-1630	161	71	.	.	PUNCT
cana-1630	162	1	therefore	therefore	ADV
cana-1630	162	2	,	,	PUNCT
cana-1630	162	3	the	the	DET
cana-1630	162	4	adaptive	adaptive	ADJ
cana-1630	162	5	filter	filter	NOUN
cana-1630	162	6	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	162	7	,	,	PUNCT
cana-1630	162	8	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	162	9	,	,	PUNCT
cana-1630	162	10	𝛼	𝛼	NOUN
cana-1630	162	11	)	)	PUNCT
cana-1630	162	12	is	be	AUX
cana-1630	162	13	bibo	bibo	ADJ
cana-1630	162	14	stable	stable	ADJ
cana-1630	162	15	.	.	PUNCT
cana-1630	163	1	corollary	corollary	ADJ
cana-1630	163	2	:	:	PUNCT
cana-1630	163	3	stability	stability	NOUN
cana-1630	163	4	of	of	ADP
cana-1630	163	5	the	the	DET
cana-1630	163	6	adaptive	adaptive	ADJ
cana-1630	163	7	gaussian	gaussian	ADJ
cana-1630	163	8	filter	filter	NOUN
cana-1630	163	9	if	if	SCONJ
cana-1630	163	10	a	a	DET
cana-1630	163	11	bounded	bounded	ADJ
cana-1630	163	12	input	input	NOUN
cana-1630	163	13	image	image	NOUN
cana-1630	163	14	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	163	15	,	,	PUNCT
cana-1630	163	16	𝑗	𝑗	NOUN
cana-1630	163	17	)	)	PUNCT
cana-1630	163	18	is	be	AUX
cana-1630	163	19	processed	process	VERB
cana-1630	163	20	by	by	ADP
cana-1630	163	21	the	the	DET
cana-1630	163	22	adaptive	adaptive	ADJ
cana-1630	163	23	gaussian	gaussian	ADJ
cana-1630	163	24	filter	filter	NOUN
cana-1630	163	25	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	163	26	,	,	PUNCT
cana-1630	163	27	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	163	28	,	,	PUNCT
cana-1630	163	29	𝛼	𝛼	NOUN
cana-1630	163	30	)	)	PUNCT
cana-1630	163	31	the	the	DET
cana-1630	163	32	output	output	NOUN
cana-1630	163	33	will	will	AUX
cana-1630	163	34	remain	remain	VERB
cana-1630	163	35	bounded	bounded	ADJ
cana-1630	163	36	,	,	PUNCT
cana-1630	163	37	ensuring	ensure	VERB
cana-1630	163	38	the	the	DET
cana-1630	163	39	stability	stability	NOUN
cana-1630	163	40	of	of	ADP
cana-1630	163	41	the	the	DET
cana-1630	163	42	filtering	filtering	NOUN
cana-1630	163	43	process	process	NOUN
cana-1630	163	44	.	.	PUNCT
cana-1630	164	1	proof	proof	NOUN
cana-1630	164	2	:	:	PUNCT
cana-1630	164	3	same	same	ADJ
cana-1630	164	4	as	as	SCONJ
cana-1630	164	5	bibo	bibo	PROPN
cana-1630	164	6	stability	stability	NOUN
cana-1630	164	7	.	.	PUNCT
cana-1630	165	1	theorem	theorem	VERB
cana-1630	165	2	8	8	NUM
cana-1630	165	3	:	:	PUNCT
cana-1630	165	4	preservation	preservation	NOUN
cana-1630	165	5	of	of	ADP
cana-1630	165	6	fuzzy	fuzzy	ADJ
cana-1630	165	7	membership	membership	NOUN
cana-1630	165	8	distribution	distribution	NOUN
cana-1630	165	9	if	if	SCONJ
cana-1630	165	10	a	a	DET
cana-1630	165	11	fuzzy	fuzzy	ADJ
cana-1630	165	12	membership	membership	NOUN
cana-1630	165	13	function	function	NOUN
cana-1630	165	14	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	165	15	)	)	PUNCT
cana-1630	165	16	is	be	AUX
cana-1630	165	17	processed	process	VERB
cana-1630	165	18	by	by	ADP
cana-1630	165	19	the	the	DET
cana-1630	165	20	adaptive	adaptive	ADJ
cana-1630	165	21	gaussian	gaussian	ADJ
cana-1630	165	22	filter	filter	NOUN
cana-1630	165	23	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	165	24	,	,	PUNCT
cana-1630	165	25	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	165	26	,	,	PUNCT
cana-1630	165	27	𝛼	𝛼	NOUN
cana-1630	165	28	)	)	PUNCT
cana-1630	165	29	the	the	DET
cana-1630	165	30	resulting	result	VERB
cana-1630	165	31	function	function	NOUN
cana-1630	165	32	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	165	33	)	)	PUNCT
cana-1630	165	34	is	be	AUX
cana-1630	165	35	also	also	ADV
cana-1630	165	36	a	a	DET
cana-1630	165	37	valid	valid	ADJ
cana-1630	165	38	fuzzy	fuzzy	ADJ
cana-1630	165	39	membership	membership	NOUN
cana-1630	165	40	function	function	NOUN
cana-1630	165	41	,	,	PUNCT
cana-1630	165	42	preserving	preserve	VERB
cana-1630	165	43	its	its	PRON
cana-1630	165	44	essential	essential	ADJ
cana-1630	165	45	properties	property	NOUN
cana-1630	165	46	.	.	PUNCT
cana-1630	166	1	proof	proof	NOUN
cana-1630	166	2	:	:	PUNCT
cana-1630	166	3	the	the	DET
cana-1630	166	4	fuzzy	fuzzy	ADJ
cana-1630	166	5	membership	membership	NOUN
cana-1630	166	6	function	function	NOUN
cana-1630	166	7	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	166	8	)	)	PUNCT
cana-1630	166	9	is	be	AUX
cana-1630	166	10	typically	typically	ADV
cana-1630	166	11	bounded	bound	VERB
cana-1630	166	12	between	between	ADP
cana-1630	166	13	0	0	NUM
cana-1630	166	14	and	and	CCONJ
cana-1630	166	15	1	1	NUM
cana-1630	166	16	.	.	X
cana-1630	167	1	for	for	ADP
cana-1630	167	2	any	any	DET
cana-1630	167	3	fuzzy	fuzzy	ADJ
cana-1630	167	4	membership	membership	NOUN
cana-1630	167	5	function	function	NOUN
cana-1630	167	6	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	167	7	)	)	PUNCT
cana-1630	167	8	,	,	PUNCT
cana-1630	167	9	it	it	PRON
cana-1630	167	10	holds	hold	VERB
cana-1630	167	11	that	that	SCONJ
cana-1630	167	12	0	0	NUM
cana-1630	167	13	≤	≤	NUM
cana-1630	167	14	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	167	15	)	)	PUNCT
cana-1630	167	16	≤	≤	NUM
cana-1630	167	17	1	1	NUM
cana-1630	167	18	.	.	PUNCT
cana-1630	168	1	the	the	DET
cana-1630	168	2	filtered	filter	VERB
cana-1630	168	3	membership	membership	NOUN
cana-1630	168	4	function	function	NOUN
cana-1630	168	5	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	168	6	)	)	PUNCT
cana-1630	168	7	is	be	AUX
cana-1630	168	8	obtained	obtain	VERB
cana-1630	168	9	by	by	ADP
cana-1630	168	10	convolving	convolve	VERB
cana-1630	168	11	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	168	12	)	)	PUNCT
cana-1630	168	13	with	with	ADP
cana-1630	168	14	the	the	DET
cana-1630	168	15	adaptive	adaptive	ADJ
cana-1630	168	16	gaussian	gaussian	ADJ
cana-1630	168	17	filter	filter	NOUN
cana-1630	168	18	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	168	19	,	,	PUNCT
cana-1630	168	20	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	168	21	,	,	PUNCT
cana-1630	168	22	𝛼	𝛼	NOUN
cana-1630	168	23	):	):	PUNCT
cana-1630	168	24	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	NOUN
cana-1630	168	25	)	)	PUNCT
cana-1630	169	1	=	=	SYM
cana-1630	169	2	∫	∫	PROPN
cana-1630	169	3	𝜇(𝑡)𝐺(𝑥	𝜇(𝑡)𝐺(𝑥	PROPN
cana-1630	169	4	−	−	PROPN
cana-1630	169	5	𝑡	𝑡	PROPN
cana-1630	169	6	,	,	PUNCT
cana-1630	169	7	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	169	8	,	,	PUNCT
cana-1630	169	9	𝛼)𝑑𝑡	𝛼)𝑑𝑡	PROPN
cana-1630	169	10	∞	∞	PROPN
cana-1630	169	11	−∞	−∞	PUNCT
cana-1630	169	12	since	since	SCONJ
cana-1630	169	13	,	,	PUNCT
cana-1630	169	14	𝜇(𝑡	𝜇(𝑡	PROPN
cana-1630	169	15	)	)	PUNCT
cana-1630	169	16	is	be	AUX
cana-1630	169	17	bounded	bound	VERB
cana-1630	169	18	between	between	ADP
cana-1630	169	19	0	0	NUM
cana-1630	169	20	and	and	CCONJ
cana-1630	169	21	1	1	NUM
cana-1630	169	22	and	and	CCONJ
cana-1630	169	23	g(x	g(x	NOUN
cana-1630	169	24	)	)	PUNCT
cana-1630	169	25	is	be	AUX
cana-1630	169	26	non	non	ADJ
cana-1630	169	27	-	-	ADJ
cana-1630	169	28	negative	negative	ADJ
cana-1630	169	29	and	and	CCONJ
cana-1630	169	30	integrates	integrate	NOUN
cana-1630	169	31	to	to	ADP
cana-1630	169	32	a	a	DET
cana-1630	169	33	finite	finite	ADJ
cana-1630	169	34	value	value	NOUN
cana-1630	169	35	then	then	ADV
cana-1630	169	36	,	,	PUNCT
cana-1630	169	37	0	0	NUM
cana-1630	169	38	≤	≤	NUM
cana-1630	169	39	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	169	40	)	)	PUNCT
cana-1630	169	41	≤	≤	NUM
cana-1630	169	42	∫	∫	NOUN
cana-1630	170	1	𝜇(𝑡)𝐺(𝑥	𝜇(𝑡)𝐺(𝑥	PROPN
cana-1630	170	2	−	−	PROPN
cana-1630	170	3	𝑡	𝑡	PROPN
cana-1630	170	4	,	,	PUNCT
cana-1630	170	5	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	170	6	,	,	PUNCT
cana-1630	170	7	𝛼)𝑑𝑡	𝛼)𝑑𝑡	PROPN
cana-1630	170	8	∞	∞	PROPN
cana-1630	170	9	−∞	−∞	ADP
cana-1630	170	10	≤	≤	NUM
cana-1630	170	11	1	1	NUM
cana-1630	170	12	the	the	DET
cana-1630	170	13	adaptive	adaptive	ADJ
cana-1630	170	14	gaussian	gaussian	ADJ
cana-1630	170	15	filter	filter	NOUN
cana-1630	170	16	preserves	preserve	VERB
cana-1630	170	17	the	the	DET
cana-1630	170	18	smoothness	smoothness	NOUN
cana-1630	170	19	and	and	CCONJ
cana-1630	170	20	continuity	continuity	NOUN
cana-1630	170	21	of	of	ADP
cana-1630	170	22	the	the	DET
cana-1630	170	23	membership	membership	NOUN
cana-1630	170	24	function	function	NOUN
cana-1630	170	25	.	.	PUNCT
cana-1630	171	1	since	since	SCONJ
cana-1630	171	2	,	,	PUNCT
cana-1630	171	3	g(x	g(x	NOUN
cana-1630	171	4	)	)	PUNCT
cana-1630	171	5	is	be	AUX
cana-1630	171	6	smooth	smooth	ADJ
cana-1630	171	7	and	and	CCONJ
cana-1630	171	8	the	the	DET
cana-1630	171	9	convolution	convolution	NOUN
cana-1630	171	10	operation	operation	NOUN
cana-1630	171	11	with	with	ADP
cana-1630	171	12	g(x	g(x	NOUN
cana-1630	171	13	)	)	PUNCT
cana-1630	171	14	maintains	maintain	VERB
cana-1630	171	15	the	the	DET
cana-1630	171	16	smoothness	smoothness	NOUN
cana-1630	171	17	,	,	PUNCT
cana-1630	171	18	the	the	DET
cana-1630	171	19	filtered	filter	VERB
cana-1630	171	20	membership	membership	NOUN
cana-1630	171	21	function	function	NOUN
cana-1630	171	22	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	171	23	)	)	PUNCT
cana-1630	171	24	will	will	AUX
cana-1630	171	25	also	also	ADV
cana-1630	171	26	be	be	AUX
cana-1630	171	27	smooth	smooth	ADJ
cana-1630	171	28	.	.	PUNCT
cana-1630	172	1	thus	thus	ADV
cana-1630	172	2	,	,	PUNCT
cana-1630	172	3	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	172	4	)	)	PUNCT
cana-1630	172	5	remains	remain	VERB
cana-1630	172	6	a	a	DET
cana-1630	172	7	valid	valid	ADJ
cana-1630	172	8	fuzzy	fuzzy	ADJ
cana-1630	172	9	membership	membership	NOUN
cana-1630	172	10	function	function	NOUN
cana-1630	172	11	,	,	PUNCT
cana-1630	172	12	preserving	preserve	VERB
cana-1630	172	13	the	the	DET
cana-1630	172	14	essential	essential	ADJ
cana-1630	172	15	properties	property	NOUN
cana-1630	172	16	such	such	ADJ
cana-1630	172	17	as	as	ADP
cana-1630	172	18	boundedness	boundedness	NOUN
cana-1630	172	19	,	,	PUNCT
cana-1630	172	20	normalization	normalization	NOUN
cana-1630	172	21	and	and	CCONJ
cana-1630	172	22	smoothness	smoothness	NOUN
cana-1630	172	23	.	.	PUNCT
cana-1630	173	1	corollary	corollary	ADJ
cana-1630	173	2	:	:	PUNCT
cana-1630	173	3	preservation	preservation	NOUN
cana-1630	173	4	of	of	ADP
cana-1630	173	5	fuzzy	fuzzy	ADJ
cana-1630	173	6	membership	membership	NOUN
cana-1630	173	7	properties	property	NOUN
cana-1630	173	8	if	if	SCONJ
cana-1630	173	9	a	a	DET
cana-1630	173	10	fuzzy	fuzzy	ADJ
cana-1630	173	11	membership	membership	NOUN
cana-1630	173	12	function	function	NOUN
cana-1630	173	13	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	173	14	)	)	PUNCT
cana-1630	173	15	is	be	AUX
cana-1630	173	16	processed	process	VERB
cana-1630	173	17	by	by	ADP
cana-1630	173	18	the	the	DET
cana-1630	173	19	adaptive	adaptive	ADJ
cana-1630	173	20	gaussian	gaussian	ADJ
cana-1630	173	21	filter	filter	NOUN
cana-1630	173	22	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	173	23	,	,	PUNCT
cana-1630	173	24	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	173	25	,	,	PUNCT
cana-1630	173	26	𝛼	𝛼	NOUN
cana-1630	173	27	)	)	PUNCT
cana-1630	173	28	the	the	DET
cana-1630	173	29	resulting	result	VERB
cana-1630	173	30	function	function	NOUN
cana-1630	173	31	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	173	32	)	)	PUNCT
cana-1630	173	33	will	will	AUX
cana-1630	173	34	not	not	PART
cana-1630	173	35	only	only	ADV
cana-1630	173	36	be	be	AUX
cana-1630	173	37	bounded	bound	VERB
cana-1630	173	38	between	between	ADP
cana-1630	173	39	0	0	NUM
cana-1630	173	40	and	and	CCONJ
cana-1630	173	41	1	1	NUM
cana-1630	173	42	but	but	CCONJ
cana-1630	173	43	will	will	AUX
cana-1630	173	44	also	also	ADV
cana-1630	173	45	maintain	maintain	VERB
cana-1630	173	46	the	the	DET
cana-1630	173	47	essential	essential	ADJ
cana-1630	173	48	properties	property	NOUN
cana-1630	173	49	of	of	ADP
cana-1630	173	50	a	a	DET
cana-1630	173	51	fuzzy	fuzzy	ADJ
cana-1630	173	52	membership	membership	NOUN
cana-1630	173	53	function	function	NOUN
cana-1630	173	54	.	.	PUNCT
cana-1630	174	1	proof	proof	NOUN
cana-1630	174	2	:	:	PUNCT
cana-1630	174	3	follows	follow	VERB
cana-1630	174	4	directly	directly	ADV
cana-1630	174	5	from	from	ADP
cana-1630	174	6	the	the	DET
cana-1630	174	7	preservation	preservation	NOUN
cana-1630	174	8	of	of	ADP
cana-1630	174	9	fuzzy	fuzzy	ADJ
cana-1630	174	10	membership	membership	NOUN
cana-1630	174	11	distribution	distribution	NOUN
cana-1630	174	12	theorem	theorem	NOUN
cana-1630	174	13	.	.	PUNCT
cana-1630	175	1	communications	communication	NOUN
cana-1630	175	2	on	on	ADP
cana-1630	175	3	applied	apply	VERB
cana-1630	175	4	nonlinear	nonlinear	ADJ
cana-1630	175	5	analysis	analysis	NOUN
cana-1630	175	6	issn	issn	NOUN
cana-1630	175	7	:	:	PUNCT
cana-1630	175	8	1074	1074	NUM
cana-1630	175	9	-	-	PUNCT
cana-1630	175	10	133x	133x	NUM
cana-1630	175	11	vol	vol	NOUN
cana-1630	175	12	32	32	NUM
cana-1630	175	13	no	no	NOUN
cana-1630	175	14	.	.	NOUN
cana-1630	175	15	1	1	NUM
cana-1630	175	16	(	(	PUNCT
cana-1630	175	17	2025	2025	NUM
cana-1630	175	18	)	)	PUNCT
cana-1630	175	19	193	193	NUM
cana-1630	175	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	175	21	theorem	theorem	VERB
cana-1630	175	22	9	9	NUM
cana-1630	175	23	:	:	PUNCT
cana-1630	175	24	frequency	frequency	NOUN
cana-1630	175	25	response	response	NOUN
cana-1630	175	26	the	the	DET
cana-1630	175	27	frequency	frequency	NOUN
cana-1630	175	28	response	response	NOUN
cana-1630	175	29	of	of	ADP
cana-1630	175	30	the	the	DET
cana-1630	175	31	adaptive	adaptive	ADJ
cana-1630	175	32	gaussian	gaussian	ADJ
cana-1630	175	33	filter	filter	NOUN
cana-1630	175	34	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	175	35	,	,	PUNCT
cana-1630	175	36	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	175	37	,	,	PUNCT
cana-1630	175	38	𝛼)can	𝛼)can	AUX
cana-1630	175	39	be	be	AUX
cana-1630	175	40	derived	derive	VERB
cana-1630	175	41	from	from	ADP
cana-1630	175	42	its	its	PRON
cana-1630	175	43	fourier	fourier	NOUN
cana-1630	175	44	transform	transform	NOUN
cana-1630	175	45	.	.	PUNCT
cana-1630	176	1	proof	proof	NOUN
cana-1630	176	2	:	:	PUNCT
cana-1630	176	3	the	the	DET
cana-1630	176	4	fourier	fourier	NOUN
cana-1630	176	5	transform	transform	NOUN
cana-1630	176	6	of	of	ADP
cana-1630	176	7	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	176	8	,	,	PUNCT
cana-1630	176	9	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	176	10	,	,	PUNCT
cana-1630	176	11	𝛼	𝛼	NOUN
cana-1630	176	12	)	)	PUNCT
cana-1630	176	13	is	be	AUX
cana-1630	176	14	given	give	VERB
cana-1630	176	15	:	:	PUNCT
cana-1630	176	16	ℱ{𝐺(𝑥	ℱ{𝐺(𝑥	NUM
cana-1630	176	17	,	,	PUNCT
cana-1630	176	18	𝜎,𝑚	𝜎,𝑚	PROPN
cana-1630	176	19	,	,	PUNCT
cana-1630	176	20	𝛼	𝛼	NOUN
cana-1630	176	21	)	)	PUNCT
cana-1630	176	22	}	}	PUNCT
cana-1630	177	1	=	=	SYM
cana-1630	177	2	∫	∫	PROPN
cana-1630	177	3	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	177	4	,	,	PUNCT
cana-1630	177	5	𝜎,𝑚	𝜎,𝑚	NOUN
cana-1630	177	6	,	,	PUNCT
cana-1630	177	7	𝛼)𝑒−𝑗2𝜋𝑢𝑥	𝛼)𝑒−𝑗2𝜋𝑢𝑥	ADV
cana-1630	177	8	𝑑𝑥	𝑑𝑥	VERB
cana-1630	177	9	∞	∞	PROPN
cana-1630	177	10	−∞	−∞	ADP
cana-1630	177	11	the	the	DET
cana-1630	177	12	fourier	fourier	NOUN
cana-1630	177	13	transform	transform	NOUN
cana-1630	177	14	of	of	ADP
cana-1630	177	15	the	the	DET
cana-1630	177	16	gaussian	gaussian	ADJ
cana-1630	177	17	component	component	NOUN
cana-1630	177	18	1	1	NUM
cana-1630	177	19	√2𝜋𝜎	√2𝜋𝜎	ADJ
cana-1630	177	20	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	177	21	(	(	PUNCT
cana-1630	177	22	−	−	PROPN
cana-1630	177	23	(	(	PUNCT
cana-1630	177	24	𝑥−𝑚)2	𝑥−𝑚)2	NOUN
cana-1630	177	25	2𝜎2	2𝜎2	NUM
cana-1630	177	26	)	)	PUNCT
cana-1630	177	27	is	be	AUX
cana-1630	177	28	:	:	PUNCT
cana-1630	177	29	ℱ	ℱ	PROPN
cana-1630	177	30	{	{	PUNCT
cana-1630	177	31	1	1	NUM
cana-1630	177	32	√2𝜋𝜎	√2𝜋𝜎	INTJ
cana-1630	177	33	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	177	34	(	(	PUNCT
cana-1630	177	35	−	−	PROPN
cana-1630	177	36	(	(	PUNCT
cana-1630	177	37	𝑥−𝑚)2	𝑥−𝑚)2	NOUN
cana-1630	177	38	2𝜎2	2𝜎2	NUM
cana-1630	177	39	)	)	PUNCT
cana-1630	177	40	}	}	PUNCT
cana-1630	177	41	=	=	SYM
cana-1630	178	1	exp(−2𝜋2𝜎2𝑢2)𝑒−𝑗2𝜋𝑚𝑢	exp(−2𝜋2𝜎2𝑢2)𝑒−𝑗2𝜋𝑚𝑢	PROPN
cana-1630	178	2	the	the	DET
cana-1630	178	3	linear	linear	ADJ
cana-1630	178	4	term	term	NOUN
cana-1630	178	5	[	[	X
cana-1630	178	6	1	1	NUM
cana-1630	178	7	−	−	PROPN
cana-1630	178	8	𝛼.	𝛼.	NOUN
cana-1630	178	9	(	(	PUNCT
cana-1630	178	10	𝑥−𝑚	𝑥−𝑚	NOUN
cana-1630	178	11	𝜎2	𝜎2	NOUN
cana-1630	178	12	)	)	PUNCT
cana-1630	178	13	]	]	PUNCT
cana-1630	178	14	introduces	introduce	VERB
cana-1630	178	15	additional	additional	ADJ
cana-1630	178	16	frequency	frequency	NOUN
cana-1630	178	17	components	component	NOUN
cana-1630	178	18	.	.	PUNCT
cana-1630	179	1	the	the	DET
cana-1630	179	2	overall	overall	ADJ
cana-1630	179	3	frequency	frequency	ADJ
cana-1630	179	4	response	response	NOUN
cana-1630	179	5	h(u	h(u	PROPN
cana-1630	179	6	)	)	PUNCT
cana-1630	179	7	is	be	AUX
cana-1630	179	8	a	a	DET
cana-1630	179	9	combination	combination	NOUN
cana-1630	179	10	of	of	ADP
cana-1630	179	11	these	these	DET
cana-1630	179	12	terms	term	NOUN
cana-1630	179	13	:	:	PUNCT
cana-1630	179	14	𝐻(𝑢	𝐻(𝑢	ADP
cana-1630	179	15	)	)	PUNCT
cana-1630	179	16	=	=	SYM
cana-1630	179	17	(	(	PUNCT
cana-1630	179	18	1	1	NUM
cana-1630	179	19	−	−	PROPN
cana-1630	179	20	𝑗2𝜋𝛼𝜎𝑢	𝑗2𝜋𝛼𝜎𝑢	NOUN
cana-1630	179	21	)	)	PUNCT
cana-1630	180	1	exp(−2𝜋2𝜎2𝑢2)𝑒−𝑗2𝜋𝑚𝑢	exp(−2𝜋2𝜎2𝑢2)𝑒−𝑗2𝜋𝑚𝑢	PROPN
cana-1630	180	2	hence	hence	ADV
cana-1630	180	3	the	the	DET
cana-1630	180	4	proof	proof	NOUN
cana-1630	180	5	.	.	PUNCT
cana-1630	181	1	corollary	corollary	ADJ
cana-1630	181	2	:	:	PUNCT
cana-1630	181	3	the	the	DET
cana-1630	181	4	frequency	frequency	NOUN
cana-1630	181	5	response	response	NOUN
cana-1630	181	6	of	of	ADP
cana-1630	181	7	the	the	DET
cana-1630	181	8	adaptive	adaptive	ADJ
cana-1630	181	9	gaussian	gaussian	ADJ
cana-1630	181	10	filter	filter	NOUN
cana-1630	181	11	shows	show	VERB
cana-1630	181	12	that	that	SCONJ
cana-1630	181	13	it	it	PRON
cana-1630	181	14	acts	act	VERB
cana-1630	181	15	as	as	ADP
cana-1630	181	16	a	a	DET
cana-1630	181	17	low	low	ADJ
cana-1630	181	18	-	-	PUNCT
cana-1630	181	19	pass	pass	NOUN
cana-1630	181	20	filter	filter	NOUN
cana-1630	181	21	,	,	PUNCT
cana-1630	181	22	attenuating	attenuate	VERB
cana-1630	181	23	high	high	ADJ
cana-1630	181	24	-	-	PUNCT
cana-1630	181	25	frequency	frequency	NOUN
cana-1630	181	26	components	component	NOUN
cana-1630	181	27	and	and	CCONJ
cana-1630	181	28	preserving	preserve	VERB
cana-1630	181	29	low	low	ADJ
cana-1630	181	30	-	-	PUNCT
cana-1630	181	31	frequency	frequency	NOUN
cana-1630	181	32	components	component	NOUN
cana-1630	181	33	of	of	ADP
cana-1630	181	34	the	the	DET
cana-1630	181	35	input	input	NOUN
cana-1630	181	36	signal	signal	NOUN
cana-1630	181	37	.	.	PUNCT
cana-1630	182	1	proof	proof	NOUN
cana-1630	182	2	:	:	PUNCT
cana-1630	182	3	follows	follow	VERB
cana-1630	182	4	directly	directly	ADV
cana-1630	182	5	from	from	ADP
cana-1630	182	6	the	the	DET
cana-1630	182	7	frequency	frequency	NOUN
cana-1630	182	8	response	response	NOUN
cana-1630	182	9	derivation	derivation	NOUN
cana-1630	182	10	,	,	PUNCT
cana-1630	182	11	where	where	SCONJ
cana-1630	182	12	the	the	DET
cana-1630	182	13	term	term	NOUN
cana-1630	182	14	exp(−2𝜋2𝜎2𝑢2	exp(−2𝜋2𝜎2𝑢2	PROPN
cana-1630	182	15	)	)	PUNCT
cana-1630	182	16	indicates	indicate	VERB
cana-1630	182	17	attenuation	attenuation	NOUN
cana-1630	182	18	of	of	ADP
cana-1630	182	19	higher	high	ADJ
cana-1630	182	20	frequencies	frequency	NOUN
cana-1630	182	21	.	.	PUNCT
cana-1630	183	1	3.4	3.4	NUM
cana-1630	183	2	adaptive	adaptive	ADJ
cana-1630	183	3	fuzzy	fuzzy	ADJ
cana-1630	183	4	gaussian	gaussian	ADJ
cana-1630	183	5	2d	2d	NUM
cana-1630	183	6	filter	filter	NOUN
cana-1630	183	7	and	and	CCONJ
cana-1630	183	8	derivative	derivative	NOUN
cana-1630	183	9	of	of	ADP
cana-1630	183	10	fuzzy	fuzzy	ADJ
cana-1630	183	11	gaussian	gaussian	ADJ
cana-1630	183	12	2d	2d	NOUN
cana-1630	183	13	(	(	PUNCT
cana-1630	183	14	afgd-2d	afgd-2d	NOUN
cana-1630	183	15	)	)	PUNCT
cana-1630	183	16	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	183	17	,	,	PUNCT
cana-1630	183	18	𝑦	𝑦	NOUN
cana-1630	183	19	,	,	PUNCT
cana-1630	183	20	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	183	21	,	,	PUNCT
cana-1630	183	22	𝜎𝑦	𝜎𝑦	DET
cana-1630	183	23	,	,	PUNCT
cana-1630	183	24	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	183	25	,	,	PUNCT
cana-1630	183	26	𝑚𝑦	𝑚𝑦	ADP
cana-1630	183	27	,	,	PUNCT
cana-1630	183	28	𝛼	𝛼	PROPN
cana-1630	183	29	,	,	PUNCT
cana-1630	183	30	𝛽	𝛽	NOUN
cana-1630	183	31	)	)	PUNCT
cana-1630	183	32	=	=	SYM
cana-1630	183	33	1	1	NUM
cana-1630	183	34	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	183	35	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	184	1	[	[	X
cana-1630	184	2	−	−	X
cana-1630	184	3	(	(	PUNCT
cana-1630	184	4	𝑥−𝑚𝑥)2	𝑥−𝑚𝑥)2	NUM
cana-1630	184	5	2𝜎𝑥	2𝜎𝑥	ADJ
cana-1630	184	6	2	2	NUM
cana-1630	184	7	−	−	PROPN
cana-1630	184	8	(	(	PUNCT
cana-1630	184	9	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	184	10	)	)	PUNCT
cana-1630	184	11	2	2	NUM
cana-1630	184	12	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	184	13	2	2	NUM
cana-1630	184	14	]	]	PUNCT
cana-1630	185	1	[	[	X
cana-1630	185	2	1	1	NUM
cana-1630	185	3	−	−	PROPN
cana-1630	185	4	𝛼.	𝛼.	NOUN
cana-1630	185	5	(	(	PUNCT
cana-1630	185	6	𝑥−𝑚𝑥	𝑥−𝑚𝑥	NOUN
cana-1630	185	7	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	185	8	2	2	NUM
cana-1630	185	9	)	)	PUNCT
cana-1630	185	10	−	−	PROPN
cana-1630	185	11	𝛽.	𝛽.	NOUN
cana-1630	185	12	(	(	PUNCT
cana-1630	185	13	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	185	14	𝜎𝑦	𝜎𝑦	DET
cana-1630	185	15	2	2	NUM
cana-1630	185	16	)	)	PUNCT
cana-1630	185	17	]	]	PUNCT
cana-1630	185	18	(	(	PUNCT
cana-1630	185	19	4	4	X
cana-1630	185	20	)	)	PUNCT
cana-1630	185	21	for	for	ADP
cana-1630	185	22	2d	2d	NUM
cana-1630	185	23	the	the	DET
cana-1630	185	24	constants	constant	NOUN
cana-1630	185	25	are	be	AUX
cana-1630	185	26	calculated	calculate	VERB
cana-1630	185	27	using	use	VERB
cana-1630	185	28	formulas	formula	NOUN
cana-1630	185	29	for	for	ADP
cana-1630	185	30	a	a	DET
cana-1630	185	31	given	give	VERB
cana-1630	185	32	matrix	matrix	NOUN
cana-1630	185	33	a	a	PRON
cana-1630	185	34	of	of	ADP
cana-1630	185	35	size	size	NOUN
cana-1630	185	36	m	m	PROPN
cana-1630	185	37	x	x	SYM
cana-1630	185	38	n	n	CCONJ
cana-1630	185	39	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	185	40	=	=	SYM
cana-1630	185	41	1	1	NUM
cana-1630	185	42	𝑛	𝑛	PRON
cana-1630	185	43	∑	∑	PUNCT
cana-1630	185	44	𝐴(𝑖	𝐴(𝑖	ADV
cana-1630	185	45	,	,	PUNCT
cana-1630	185	46	𝑗)𝑛	𝑗)𝑛	PUNCT
cana-1630	186	1	𝑗=1	𝑗=1	PROPN
cana-1630	187	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1630	187	2	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	NOUN
cana-1630	187	3	𝑖	𝑖	X
cana-1630	187	4	;	;	PUNCT
cana-1630	187	5	𝑚𝑦	𝑚𝑦	ADP
cana-1630	187	6	=	=	SYM
cana-1630	187	7	1	1	NUM
cana-1630	187	8	𝑚	𝑚	NOUN
cana-1630	187	9	∑	∑	PUNCT
cana-1630	187	10	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	187	11	,	,	PUNCT
cana-1630	187	12	𝑗)𝑚	𝑗)𝑚	X
cana-1630	187	13	𝑖=1	𝑖=1	PROPN
cana-1630	188	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1630	188	2	𝑒𝑎𝑐ℎ	𝑒𝑎𝑐ℎ	PROPN
cana-1630	188	3	𝑗	𝑗	PROPN
cana-1630	188	4	theorem	theorem	ADJ
cana-1630	188	5	10	10	NUM
cana-1630	188	6	:	:	PUNCT
cana-1630	188	7	bibo	bibo	VERB
cana-1630	188	8	stability	stability	NOUN
cana-1630	188	9	the	the	DET
cana-1630	188	10	adaptive	adaptive	ADJ
cana-1630	188	11	2d	2d	NUM
cana-1630	188	12	gaussian	gaussian	ADJ
cana-1630	188	13	filter	filter	NOUN
cana-1630	188	14	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	188	15	,	,	PUNCT
cana-1630	188	16	𝑦	𝑦	NOUN
cana-1630	188	17	,	,	PUNCT
cana-1630	188	18	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	188	19	,	,	PUNCT
cana-1630	188	20	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	188	21	,	,	PUNCT
cana-1630	188	22	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	188	23	,	,	PUNCT
cana-1630	188	24	𝑚𝑦	𝑚𝑦	ADP
cana-1630	188	25	,	,	PUNCT
cana-1630	188	26	𝛼	𝛼	PROPN
cana-1630	188	27	,	,	PUNCT
cana-1630	188	28	𝛽	𝛽	NOUN
cana-1630	188	29	)	)	PUNCT
cana-1630	188	30	is	be	AUX
cana-1630	188	31	bounded	bound	VERB
cana-1630	188	32	input	input	NOUN
cana-1630	188	33	bounded	bound	VERB
cana-1630	188	34	output	output	NOUN
cana-1630	188	35	(	(	PUNCT
cana-1630	188	36	bibo	bibo	ADJ
cana-1630	188	37	)	)	PUNCT
cana-1630	188	38	stable	stable	ADJ
cana-1630	188	39	.	.	PUNCT
cana-1630	189	1	proof	proof	NOUN
cana-1630	189	2	:	:	PUNCT
cana-1630	189	3	let	let	VERB
cana-1630	189	4	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	189	5	,	,	PUNCT
cana-1630	189	6	𝑗	𝑗	AUX
cana-1630	189	7	)	)	PUNCT
cana-1630	189	8	be	be	AUX
cana-1630	189	9	a	a	DET
cana-1630	189	10	bounded	bounded	ADJ
cana-1630	189	11	input	input	NOUN
cana-1630	189	12	matrix	matrix	NOUN
cana-1630	189	13	.	.	PUNCT
cana-1630	190	1	then	then	ADV
cana-1630	190	2	there	there	PRON
cana-1630	190	3	exist	exist	VERB
cana-1630	190	4	a	a	DET
cana-1630	190	5	constant	constant	ADJ
cana-1630	190	6	m	m	NOUN
cana-1630	190	7	such	such	ADJ
cana-1630	190	8	that	that	SCONJ
cana-1630	190	9	|𝐴(𝑖	|𝐴(𝑖	PROPN
cana-1630	190	10	,	,	PUNCT
cana-1630	190	11	𝑗)|	𝑗)|	VERB
cana-1630	190	12	≤	≤	NUM
cana-1630	190	13	𝑀	𝑀	PROPN
cana-1630	190	14	∀(𝑖	∀(𝑖	NUM
cana-1630	190	15	,	,	PUNCT
cana-1630	190	16	𝑗	𝑗	NOUN
cana-1630	190	17	)	)	PUNCT
cana-1630	190	18	.	.	PUNCT
cana-1630	191	1	also	also	ADV
cana-1630	191	2	,	,	PUNCT
cana-1630	191	3	𝜎𝑥and	𝜎𝑥and	VERB
cana-1630	191	4	𝜎𝑦	𝜎𝑦	PRON
cana-1630	191	5	are	be	AUX
cana-1630	191	6	standard	standard	ADJ
cana-1630	191	7	deviations	deviation	NOUN
cana-1630	191	8	,	,	PUNCT
cana-1630	191	9	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	191	10	and	and	CCONJ
cana-1630	191	11	𝑚𝑦	𝑚𝑦	NOUN
cana-1630	191	12	are	be	AUX
cana-1630	191	13	the	the	DET
cana-1630	191	14	means	mean	NOUN
cana-1630	191	15	of	of	ADP
cana-1630	191	16	the	the	DET
cana-1630	191	17	image	image	NOUN
cana-1630	191	18	along	along	ADP
cana-1630	191	19	rows	row	NOUN
cana-1630	191	20	and	and	CCONJ
cana-1630	191	21	columns	column	NOUN
cana-1630	191	22	respectively	respectively	ADV
cana-1630	191	23	which	which	PRON
cana-1630	191	24	are	be	AUX
cana-1630	191	25	bounded	bound	VERB
cana-1630	191	26	because	because	SCONJ
cana-1630	191	27	𝐴(𝑖	𝐴(𝑖	CCONJ
cana-1630	191	28	,	,	PUNCT
cana-1630	191	29	𝑗	𝑗	NOUN
cana-1630	191	30	)	)	PUNCT
cana-1630	191	31	is	be	AUX
cana-1630	191	32	bounded	bound	VERB
cana-1630	191	33	,	,	PUNCT
cana-1630	191	34	𝛼	𝛼	X
cana-1630	191	35	and	and	CCONJ
cana-1630	191	36	𝛽	𝛽	NOUN
cana-1630	191	37	are	be	AUX
cana-1630	191	38	adaptive	adaptive	ADJ
cana-1630	191	39	parameters	parameter	NOUN
cana-1630	191	40	which	which	PRON
cana-1630	191	41	can	can	AUX
cana-1630	191	42	be	be	AUX
cana-1630	191	43	bounded	bound	VERB
cana-1630	191	44	based	base	VERB
cana-1630	191	45	on	on	ADP
cana-1630	191	46	the	the	DET
cana-1630	191	47	range	range	NOUN
cana-1630	191	48	of	of	ADP
cana-1630	191	49	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	191	50	,	,	PUNCT
cana-1630	191	51	𝑗	𝑗	NOUN
cana-1630	191	52	)	)	PUNCT
cana-1630	191	53	the	the	DET
cana-1630	191	54	gaussian	gaussian	ADJ
cana-1630	191	55	part	part	NOUN
cana-1630	191	56	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-1630	192	1	[	[	X
cana-1630	192	2	−	−	X
cana-1630	192	3	(	(	PUNCT
cana-1630	192	4	𝑥−𝑚𝑥)2	𝑥−𝑚𝑥)2	AUX
cana-1630	192	5	2𝜎𝑥	2𝜎𝑥	ADJ
cana-1630	192	6	2	2	NUM
cana-1630	192	7	−	−	NOUN
cana-1630	192	8	communications	communication	NOUN
cana-1630	192	9	on	on	ADP
cana-1630	192	10	applied	apply	VERB
cana-1630	192	11	nonlinear	nonlinear	ADJ
cana-1630	192	12	analysis	analysis	NOUN
cana-1630	192	13	issn	issn	NOUN
cana-1630	192	14	:	:	PUNCT
cana-1630	192	15	1074	1074	NUM
cana-1630	192	16	-	-	PUNCT
cana-1630	192	17	133x	133x	NUM
cana-1630	192	18	vol	vol	NOUN
cana-1630	192	19	32	32	NUM
cana-1630	192	20	no	no	NOUN
cana-1630	192	21	.	.	NOUN
cana-1630	192	22	1	1	NUM
cana-1630	192	23	(	(	PUNCT
cana-1630	192	24	2025	2025	NUM
cana-1630	192	25	)	)	PUNCT
cana-1630	192	26	194	194	NUM
cana-1630	192	27	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	192	28	(	(	PUNCT
cana-1630	192	29	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	192	30	)	)	PUNCT
cana-1630	192	31	2	2	NUM
cana-1630	192	32	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	192	33	2	2	NUM
cana-1630	192	34	]	]	PUNCT
cana-1630	192	35	is	be	AUX
cana-1630	192	36	bounded	bound	VERB
cana-1630	192	37	by	by	ADP
cana-1630	192	38	1	1	NUM
cana-1630	192	39	,	,	PUNCT
cana-1630	192	40	and	and	CCONJ
cana-1630	192	41	the	the	DET
cana-1630	192	42	term	term	NOUN
cana-1630	192	43	[	[	X
cana-1630	192	44	1	1	NUM
cana-1630	192	45	−	−	PROPN
cana-1630	192	46	𝛼.	𝛼.	NOUN
cana-1630	192	47	(	(	PUNCT
cana-1630	192	48	𝑥−𝑚𝑥	𝑥−𝑚𝑥	NOUN
cana-1630	192	49	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	192	50	2	2	NUM
cana-1630	192	51	)	)	PUNCT
cana-1630	192	52	−	−	PROPN
cana-1630	192	53	𝛽.	𝛽.	NOUN
cana-1630	192	54	(	(	PUNCT
cana-1630	192	55	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	192	56	𝜎𝑦	𝜎𝑦	DET
cana-1630	192	57	2	2	NUM
cana-1630	192	58	)	)	PUNCT
cana-1630	192	59	]	]	PUNCT
cana-1630	192	60	is	be	AUX
cana-1630	192	61	also	also	ADV
cana-1630	192	62	bounded	bound	VERB
cana-1630	192	63	because	because	SCONJ
cana-1630	192	64	𝛼	𝛼	X
cana-1630	192	65	,	,	PUNCT
cana-1630	192	66	𝛽	𝛽	NOUN
cana-1630	192	67	,	,	PUNCT
cana-1630	192	68	(	(	PUNCT
cana-1630	192	69	𝑥−𝑚𝑥	𝑥−𝑚𝑥	NOUN
cana-1630	192	70	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	192	71	2	2	NUM
cana-1630	192	72	)	)	PUNCT
cana-1630	192	73	and	and	CCONJ
cana-1630	192	74	(	(	PUNCT
cana-1630	192	75	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	192	76	𝜎𝑦	𝜎𝑦	DET
cana-1630	192	77	2	2	NUM
cana-1630	192	78	)	)	PUNCT
cana-1630	192	79	are	be	AUX
cana-1630	192	80	bounded	bound	VERB
cana-1630	192	81	.	.	PUNCT
cana-1630	193	1	the	the	DET
cana-1630	193	2	output	output	NOUN
cana-1630	193	3	of	of	ADP
cana-1630	193	4	the	the	DET
cana-1630	193	5	filter	filter	NOUN
cana-1630	193	6	applied	apply	VERB
cana-1630	193	7	to	to	ADP
cana-1630	193	8	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	193	9	,	,	PUNCT
cana-1630	193	10	𝑗	𝑗	X
cana-1630	193	11	)	)	PUNCT
cana-1630	193	12	can	can	AUX
cana-1630	193	13	be	be	AUX
cana-1630	193	14	expressed	express	VERB
cana-1630	193	15	as	as	ADP
cana-1630	193	16	a	a	DET
cana-1630	193	17	convolution	convolution	NOUN
cana-1630	193	18	,	,	PUNCT
cana-1630	193	19	𝑔(𝑥	𝑔(𝑥	PROPN
cana-1630	193	20	,	,	PUNCT
cana-1630	193	21	𝑦	𝑦	NOUN
cana-1630	193	22	)	)	PUNCT
cana-1630	193	23	=	=	SYM
cana-1630	194	1	(	(	PUNCT
cana-1630	194	2	𝐴	𝐴	PROPN
cana-1630	194	3	∗	∗	VERB
cana-1630	194	4	𝐺)(𝑥	𝐺)(𝑥	PROPN
cana-1630	194	5	,	,	PUNCT
cana-1630	194	6	𝑦	𝑦	NOUN
cana-1630	194	7	)	)	PUNCT
cana-1630	194	8	=	=	SYM
cana-1630	195	1	∫	∫	PROPN
cana-1630	195	2	∫	∫	PROPN
cana-1630	195	3	𝐴(𝑢	𝐴(𝑢	PROPN
cana-1630	195	4	,	,	PUNCT
cana-1630	195	5	𝑣)𝐺(𝑥	𝑣)𝐺(𝑥	NOUN
cana-1630	195	6	−	−	NOUN
cana-1630	195	7	𝑢	𝑢	PROPN
cana-1630	195	8	,	,	PUNCT
cana-1630	195	9	𝑦	𝑦	NOUN
cana-1630	195	10	−	−	NOUN
cana-1630	196	1	𝑣	𝑣	NOUN
cana-1630	196	2	,	,	PUNCT
cana-1630	196	3	𝜎𝑥	𝜎𝑥	INTJ
cana-1630	196	4	,	,	PUNCT
cana-1630	196	5	𝜎𝑦	𝜎𝑦	PROPN
cana-1630	196	6	,	,	PUNCT
cana-1630	196	7	𝑚𝑥,𝑚𝑦	𝑚𝑥,𝑚𝑦	X
cana-1630	196	8	,	,	PUNCT
cana-1630	196	9	𝛼	𝛼	X
cana-1630	196	10	,	,	PUNCT
cana-1630	196	11	𝛽)𝑑𝑢	𝛽)𝑑𝑢	NOUN
cana-1630	196	12	𝑑𝑣	𝑑𝑣	ADP
cana-1630	196	13	∞	∞	PROPN
cana-1630	196	14	−∞	−∞	ADP
cana-1630	196	15	∞	∞	PROPN
cana-1630	196	16	−∞	−∞	ADP
cana-1630	196	17	given	give	VERB
cana-1630	196	18	the	the	DET
cana-1630	196	19	boundedness	boundedness	NOUN
cana-1630	196	20	of	of	ADP
cana-1630	196	21	a	a	DET
cana-1630	196	22	(	(	PUNCT
cana-1630	196	23	u	u	NOUN
cana-1630	196	24	,	,	PUNCT
cana-1630	196	25	v	v	NOUN
cana-1630	196	26	)	)	PUNCT
cana-1630	196	27	and	and	CCONJ
cana-1630	196	28	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	196	29	,	,	PUNCT
cana-1630	196	30	𝑦	𝑦	NOUN
cana-1630	196	31	,	,	PUNCT
cana-1630	196	32	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	196	33	,	,	PUNCT
cana-1630	196	34	𝜎𝑦	𝜎𝑦	PROPN
cana-1630	196	35	,	,	PUNCT
cana-1630	196	36	𝑚𝑥,𝑚𝑦	𝑚𝑥,𝑚𝑦	X
cana-1630	196	37	,	,	PUNCT
cana-1630	196	38	𝛼	𝛼	X
cana-1630	196	39	,	,	PUNCT
cana-1630	196	40	𝛽	𝛽	NOUN
cana-1630	196	41	)	)	PUNCT
cana-1630	196	42	then	then	ADV
cana-1630	196	43	,	,	PUNCT
cana-1630	196	44	|𝑔(𝑥	|𝑔(𝑥	PROPN
cana-1630	196	45	,	,	PUNCT
cana-1630	196	46	𝑦)|	𝑦)|	PROPN
cana-1630	196	47	≤	≤	PROPN
cana-1630	196	48	∫	∫	PROPN
cana-1630	196	49	∫	∫	PROPN
cana-1630	196	50	|𝐴(𝑢	|𝐴(𝑢	PROPN
cana-1630	196	51	,	,	PUNCT
cana-1630	196	52	𝑣)||𝐺(𝑥	𝑣)||𝐺(𝑥	NOUN
cana-1630	196	53	−	−	PROPN
cana-1630	196	54	𝑢	𝑢	PROPN
cana-1630	196	55	,	,	PUNCT
cana-1630	196	56	𝑦	𝑦	NOUN
cana-1630	196	57	−	−	NOUN
cana-1630	196	58	𝑣	𝑣	NOUN
cana-1630	196	59	,	,	PUNCT
cana-1630	196	60	𝜎𝑥	𝜎𝑥	INTJ
cana-1630	196	61	,	,	PUNCT
cana-1630	196	62	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	196	63	,	,	PUNCT
cana-1630	196	64	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	196	65	,	,	PUNCT
cana-1630	196	66	𝑚𝑦	𝑚𝑦	ADP
cana-1630	196	67	,	,	PUNCT
cana-1630	196	68	𝛼	𝛼	VERB
cana-1630	196	69	,	,	PUNCT
cana-1630	196	70	𝛽)|𝑑𝑢	𝛽)|𝑑𝑢	NOUN
cana-1630	196	71	𝑑𝑣	𝑑𝑣	NUM
cana-1630	196	72	∞	∞	PROPN
cana-1630	196	73	−∞	−∞	ADP
cana-1630	196	74	∞	∞	PROPN
cana-1630	196	75	−∞	−∞	ADP
cana-1630	196	76	≤	≤	ADJ
cana-1630	196	77	𝑀	𝑀	PROPN
cana-1630	196	78	∫	∫	PROPN
cana-1630	196	79	∫	∫	PROPN
cana-1630	196	80	|𝐺(𝑥	|𝐺(𝑥	VERB
cana-1630	196	81	−	−	PROPN
cana-1630	196	82	𝑢	𝑢	NOUN
cana-1630	196	83	,	,	PUNCT
cana-1630	196	84	𝑦	𝑦	NOUN
cana-1630	196	85	−	−	NOUN
cana-1630	196	86	𝑣	𝑣	NOUN
cana-1630	196	87	,	,	PUNCT
cana-1630	196	88	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	196	89	,	,	PUNCT
cana-1630	196	90	𝜎𝑦	𝜎𝑦	DET
cana-1630	196	91	,	,	PUNCT
cana-1630	196	92	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	196	93	,	,	PUNCT
cana-1630	196	94	𝑚𝑦	𝑚𝑦	ADP
cana-1630	196	95	,	,	PUNCT
cana-1630	196	96	𝛼	𝛼	VERB
cana-1630	196	97	,	,	PUNCT
cana-1630	196	98	𝛽)|𝑑𝑢	𝛽)|𝑑𝑢	NOUN
cana-1630	196	99	𝑑𝑣	𝑑𝑣	NUM
cana-1630	196	100	∞	∞	PROPN
cana-1630	196	101	−∞	−∞	ADP
cana-1630	196	102	∞	∞	PROPN
cana-1630	196	103	−∞	−∞	PUNCT
cana-1630	196	104	since	since	SCONJ
cana-1630	196	105	,	,	PUNCT
cana-1630	196	106	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	196	107	,	,	PUNCT
cana-1630	196	108	𝑦	𝑦	NOUN
cana-1630	196	109	,	,	PUNCT
cana-1630	196	110	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	196	111	,	,	PUNCT
cana-1630	196	112	𝜎𝑦	𝜎𝑦	DET
cana-1630	196	113	,	,	PUNCT
cana-1630	196	114	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	196	115	,	,	PUNCT
cana-1630	196	116	𝑚𝑦	𝑚𝑦	ADP
cana-1630	196	117	,	,	PUNCT
cana-1630	196	118	𝛼	𝛼	PROPN
cana-1630	196	119	,	,	PUNCT
cana-1630	196	120	𝛽	𝛽	NOUN
cana-1630	196	121	)	)	PUNCT
cana-1630	196	122	is	be	AUX
cana-1630	196	123	a	a	DET
cana-1630	196	124	bounded	bounded	ADJ
cana-1630	196	125	function	function	NOUN
cana-1630	196	126	that	that	PRON
cana-1630	196	127	integrates	integrate	VERB
cana-1630	196	128	to	to	ADP
cana-1630	196	129	a	a	DET
cana-1630	196	130	finite	finite	ADJ
cana-1630	196	131	value	value	NOUN
cana-1630	196	132	,	,	PUNCT
cana-1630	196	133	the	the	DET
cana-1630	196	134	output	output	NOUN
cana-1630	196	135	g(x	g(x	PROPN
cana-1630	196	136	,	,	PUNCT
cana-1630	196	137	y	y	NOUN
cana-1630	196	138	)	)	PUNCT
cana-1630	196	139	is	be	AUX
cana-1630	196	140	also	also	ADV
cana-1630	196	141	bounded	bound	VERB
cana-1630	196	142	.	.	PUNCT
cana-1630	197	1	therefore	therefore	ADV
cana-1630	197	2	,	,	PUNCT
cana-1630	197	3	the	the	DET
cana-1630	197	4	adaptive	adaptive	ADJ
cana-1630	197	5	2d	2d	NUM
cana-1630	197	6	gaussian	gaussian	ADJ
cana-1630	197	7	filter	filter	NOUN
cana-1630	197	8	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	197	9	,	,	PUNCT
cana-1630	197	10	𝑦	𝑦	NOUN
cana-1630	197	11	,	,	PUNCT
cana-1630	197	12	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	197	13	,	,	PUNCT
cana-1630	197	14	𝜎𝑦	𝜎𝑦	DET
cana-1630	197	15	,	,	PUNCT
cana-1630	197	16	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	197	17	,	,	PUNCT
cana-1630	197	18	𝑚𝑦	𝑚𝑦	ADP
cana-1630	197	19	,	,	PUNCT
cana-1630	197	20	𝛼	𝛼	PROPN
cana-1630	197	21	,	,	PUNCT
cana-1630	197	22	𝛽	𝛽	NOUN
cana-1630	197	23	)	)	PUNCT
cana-1630	197	24	is	be	AUX
cana-1630	197	25	bibo	bibo	ADJ
cana-1630	197	26	stable	stable	ADJ
cana-1630	197	27	.	.	PUNCT
cana-1630	198	1	corollary	corollary	ADJ
cana-1630	198	2	:	:	PUNCT
cana-1630	198	3	if	if	SCONJ
cana-1630	198	4	a	a	DET
cana-1630	198	5	bounded	bounded	ADJ
cana-1630	198	6	input	input	NOUN
cana-1630	198	7	matrix	matrix	NOUN
cana-1630	198	8	𝐴(𝑖	𝐴(𝑖	ADP
cana-1630	198	9	,	,	PUNCT
cana-1630	198	10	𝑗	𝑗	NOUN
cana-1630	198	11	)	)	PUNCT
cana-1630	198	12	is	be	AUX
cana-1630	198	13	processed	process	VERB
cana-1630	198	14	by	by	ADP
cana-1630	198	15	the	the	DET
cana-1630	198	16	adaptive	adaptive	ADJ
cana-1630	198	17	2d	2d	NUM
cana-1630	198	18	gaussian	gaussian	ADJ
cana-1630	198	19	filter	filter	NOUN
cana-1630	198	20	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	198	21	,	,	PUNCT
cana-1630	198	22	𝑦	𝑦	NOUN
cana-1630	198	23	,	,	PUNCT
cana-1630	198	24	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	198	25	,	,	PUNCT
cana-1630	198	26	𝜎𝑦	𝜎𝑦	DET
cana-1630	198	27	,	,	PUNCT
cana-1630	198	28	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	198	29	,	,	PUNCT
cana-1630	198	30	𝑚𝑦	𝑚𝑦	ADP
cana-1630	198	31	,	,	PUNCT
cana-1630	198	32	𝛼	𝛼	PROPN
cana-1630	198	33	,	,	PUNCT
cana-1630	198	34	𝛽	𝛽	NOUN
cana-1630	198	35	)	)	PUNCT
cana-1630	198	36	the	the	DET
cana-1630	198	37	output	output	NOUN
cana-1630	198	38	will	will	AUX
cana-1630	198	39	remain	remain	VERB
cana-1630	198	40	bounded	bounded	ADJ
cana-1630	198	41	ensuring	ensure	VERB
cana-1630	198	42	the	the	DET
cana-1630	198	43	stability	stability	NOUN
cana-1630	198	44	of	of	ADP
cana-1630	198	45	the	the	DET
cana-1630	198	46	filtering	filtering	NOUN
cana-1630	198	47	process	process	NOUN
cana-1630	198	48	.	.	PUNCT
cana-1630	199	1	proof	proof	NOUN
cana-1630	199	2	:	:	PUNCT
cana-1630	199	3	follows	follow	VERB
cana-1630	199	4	directly	directly	ADV
cana-1630	199	5	from	from	ADP
cana-1630	199	6	bibo	bibo	ADJ
cana-1630	199	7	stability	stability	NOUN
cana-1630	199	8	proof	proof	NOUN
cana-1630	199	9	.	.	PUNCT
cana-1630	200	1	theorem	theorem	ADJ
cana-1630	200	2	11	11	NUM
cana-1630	200	3	:	:	PUNCT
cana-1630	200	4	preservation	preservation	NOUN
cana-1630	200	5	of	of	ADP
cana-1630	200	6	fuzzy	fuzzy	ADJ
cana-1630	200	7	membership	membership	NOUN
cana-1630	200	8	distribution	distribution	NOUN
cana-1630	200	9	if	if	SCONJ
cana-1630	200	10	the	the	DET
cana-1630	200	11	fuzzy	fuzzy	ADJ
cana-1630	200	12	membership	membership	NOUN
cana-1630	200	13	function	function	NOUN
cana-1630	200	14	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	200	15	,	,	PUNCT
cana-1630	200	16	𝑦	𝑦	NOUN
cana-1630	200	17	)	)	PUNCT
cana-1630	200	18	is	be	AUX
cana-1630	200	19	processed	process	VERB
cana-1630	200	20	by	by	ADP
cana-1630	200	21	the	the	DET
cana-1630	200	22	adaptive	adaptive	ADJ
cana-1630	200	23	2d	2d	NUM
cana-1630	200	24	gaussian	gaussian	ADJ
cana-1630	200	25	filter	filter	NOUN
cana-1630	200	26	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	200	27	,	,	PUNCT
cana-1630	200	28	𝑦	𝑦	NOUN
cana-1630	200	29	,	,	PUNCT
cana-1630	200	30	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	200	31	,	,	PUNCT
cana-1630	200	32	𝜎𝑦	𝜎𝑦	DET
cana-1630	200	33	,	,	PUNCT
cana-1630	200	34	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	200	35	,	,	PUNCT
cana-1630	200	36	𝑚𝑦	𝑚𝑦	ADP
cana-1630	200	37	,	,	PUNCT
cana-1630	200	38	𝛼	𝛼	PROPN
cana-1630	200	39	,	,	PUNCT
cana-1630	200	40	𝛽	𝛽	NOUN
cana-1630	200	41	)	)	PUNCT
cana-1630	200	42	the	the	DET
cana-1630	200	43	resulting	result	VERB
cana-1630	200	44	function	function	NOUN
cana-1630	200	45	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	200	46	,	,	PUNCT
cana-1630	200	47	𝑦	𝑦	NOUN
cana-1630	200	48	)	)	PUNCT
cana-1630	200	49	is	be	AUX
cana-1630	200	50	also	also	ADV
cana-1630	200	51	a	a	DET
cana-1630	200	52	valid	valid	ADJ
cana-1630	200	53	fuzzy	fuzzy	ADJ
cana-1630	200	54	membership	membership	NOUN
cana-1630	200	55	function	function	NOUN
cana-1630	200	56	,	,	PUNCT
cana-1630	200	57	preserving	preserve	VERB
cana-1630	200	58	its	its	PRON
cana-1630	200	59	essential	essential	ADJ
cana-1630	200	60	properties	property	NOUN
cana-1630	200	61	.	.	PUNCT
cana-1630	201	1	proof	proof	NOUN
cana-1630	201	2	:	:	PUNCT
cana-1630	201	3	the	the	DET
cana-1630	201	4	fuzzy	fuzzy	ADJ
cana-1630	201	5	membership	membership	NOUN
cana-1630	201	6	function	function	NOUN
cana-1630	201	7	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	201	8	,	,	PUNCT
cana-1630	201	9	𝑦	𝑦	NOUN
cana-1630	201	10	)	)	PUNCT
cana-1630	201	11	is	be	AUX
cana-1630	201	12	typically	typically	ADV
cana-1630	201	13	bounded	bound	VERB
cana-1630	201	14	between	between	ADP
cana-1630	201	15	0	0	NUM
cana-1630	201	16	and	and	CCONJ
cana-1630	201	17	1	1	NUM
cana-1630	201	18	.	.	X
cana-1630	202	1	for	for	ADP
cana-1630	202	2	any	any	DET
cana-1630	202	3	fuzzy	fuzzy	ADJ
cana-1630	202	4	membership	membership	NOUN
cana-1630	202	5	function	function	NOUN
cana-1630	202	6	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	202	7	,	,	PUNCT
cana-1630	202	8	𝑦)it	𝑦)it	PROPN
cana-1630	202	9	holds	hold	VERB
cana-1630	202	10	that	that	SCONJ
cana-1630	202	11	0	0	NUM
cana-1630	202	12	≤	≤	NUM
cana-1630	202	13	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	202	14	,	,	PUNCT
cana-1630	202	15	𝑦	𝑦	NOUN
cana-1630	202	16	)	)	PUNCT
cana-1630	202	17	≤	≤	NUM
cana-1630	202	18	1	1	NUM
cana-1630	202	19	.	.	PUNCT
cana-1630	203	1	the	the	DET
cana-1630	203	2	filtered	filter	VERB
cana-1630	203	3	membership	membership	NOUN
cana-1630	203	4	function	function	PROPN
cana-1630	203	5	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	203	6	,	,	PUNCT
cana-1630	203	7	𝑦	𝑦	NOUN
cana-1630	203	8	)	)	PUNCT
cana-1630	203	9	is	be	AUX
cana-1630	203	10	obtained	obtain	VERB
cana-1630	203	11	by	by	ADP
cana-1630	203	12	convolving	convolve	VERB
cana-1630	203	13	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	203	14	,	,	PUNCT
cana-1630	203	15	𝑦	𝑦	NOUN
cana-1630	203	16	)	)	PUNCT
cana-1630	203	17	with	with	ADP
cana-1630	203	18	the	the	DET
cana-1630	203	19	adaptive	adaptive	ADJ
cana-1630	203	20	gaussian	gaussian	ADJ
cana-1630	203	21	filter	filter	NOUN
cana-1630	203	22	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	203	23	,	,	PUNCT
cana-1630	203	24	𝑦	𝑦	NOUN
cana-1630	203	25	,	,	PUNCT
cana-1630	203	26	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	203	27	,	,	PUNCT
cana-1630	203	28	𝜎𝑦	𝜎𝑦	DET
cana-1630	203	29	,	,	PUNCT
cana-1630	203	30	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	203	31	,	,	PUNCT
cana-1630	203	32	𝑚𝑦	𝑚𝑦	ADP
cana-1630	203	33	,	,	PUNCT
cana-1630	203	34	𝛼	𝛼	PROPN
cana-1630	203	35	,	,	PUNCT
cana-1630	203	36	𝛽	𝛽	NOUN
cana-1630	203	37	)	)	PUNCT
cana-1630	203	38	:	:	PUNCT
cana-1630	204	1	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	NUM
cana-1630	204	2	,	,	PUNCT
cana-1630	204	3	𝑦	𝑦	NOUN
cana-1630	204	4	)	)	PUNCT
cana-1630	204	5	=	=	SYM
cana-1630	205	1	∫	∫	PROPN
cana-1630	205	2	∫	∫	PROPN
cana-1630	205	3	𝜇(𝑢	𝜇(𝑢	PROPN
cana-1630	205	4	,	,	PUNCT
cana-1630	205	5	𝑣)𝐺(𝑥	𝑣)𝐺(𝑥	NOUN
cana-1630	205	6	−	−	NOUN
cana-1630	205	7	𝑢	𝑢	PROPN
cana-1630	205	8	,	,	PUNCT
cana-1630	205	9	𝑦	𝑦	NOUN
cana-1630	205	10	−	−	NOUN
cana-1630	205	11	𝑣	𝑣	NOUN
cana-1630	205	12	,	,	PUNCT
cana-1630	205	13	𝜎𝑥	𝜎𝑥	INTJ
cana-1630	205	14	,	,	PUNCT
cana-1630	205	15	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	205	16	,	,	PUNCT
cana-1630	205	17	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	205	18	,	,	PUNCT
cana-1630	205	19	𝑚𝑦	𝑚𝑦	ADP
cana-1630	205	20	,	,	PUNCT
cana-1630	205	21	𝛼	𝛼	X
cana-1630	205	22	,	,	PUNCT
cana-1630	205	23	𝛽)𝑑𝑢	𝛽)𝑑𝑢	NOUN
cana-1630	205	24	𝑑𝑣	𝑑𝑣	ADP
cana-1630	205	25	∞	∞	PROPN
cana-1630	205	26	−∞	−∞	ADP
cana-1630	205	27	∞	∞	PROPN
cana-1630	205	28	−∞	−∞	ADP
cana-1630	205	29	since	since	SCONJ
cana-1630	205	30	,	,	PUNCT
cana-1630	205	31	𝜇(𝑢	𝜇(𝑢	PROPN
cana-1630	205	32	,	,	PUNCT
cana-1630	205	33	𝑣	𝑣	NOUN
cana-1630	205	34	)	)	PUNCT
cana-1630	205	35	is	be	AUX
cana-1630	205	36	bounded	bound	VERB
cana-1630	205	37	between	between	ADP
cana-1630	205	38	0	0	NUM
cana-1630	205	39	and	and	CCONJ
cana-1630	205	40	1	1	NUM
cana-1630	205	41	and	and	CCONJ
cana-1630	205	42	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	205	43	,	,	PUNCT
cana-1630	205	44	𝑦)is	𝑦)is	ADJ
cana-1630	205	45	non	non	ADJ
cana-1630	205	46	-	-	ADJ
cana-1630	205	47	negative	negative	ADJ
cana-1630	205	48	and	and	CCONJ
cana-1630	205	49	integrates	integrate	NOUN
cana-1630	205	50	to	to	ADP
cana-1630	205	51	a	a	DET
cana-1630	205	52	finite	finite	ADJ
cana-1630	205	53	value	value	NOUN
cana-1630	205	54	,	,	PUNCT
cana-1630	205	55	then	then	ADV
cana-1630	205	56	,	,	PUNCT
cana-1630	205	57	communications	communication	NOUN
cana-1630	205	58	on	on	ADP
cana-1630	205	59	applied	apply	VERB
cana-1630	205	60	nonlinear	nonlinear	ADJ
cana-1630	205	61	analysis	analysis	NOUN
cana-1630	205	62	issn	issn	NOUN
cana-1630	205	63	:	:	PUNCT
cana-1630	205	64	1074	1074	NUM
cana-1630	205	65	-	-	PUNCT
cana-1630	205	66	133x	133x	NUM
cana-1630	205	67	vol	vol	NOUN
cana-1630	205	68	32	32	NUM
cana-1630	205	69	no	no	NOUN
cana-1630	205	70	.	.	NOUN
cana-1630	205	71	1	1	NUM
cana-1630	205	72	(	(	PUNCT
cana-1630	205	73	2025	2025	NUM
cana-1630	205	74	)	)	PUNCT
cana-1630	205	75	195	195	NUM
cana-1630	205	76	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	205	77	0	0	PUNCT
cana-1630	205	78	≤	≤	NUM
cana-1630	205	79	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	205	80	,	,	PUNCT
cana-1630	205	81	𝑦	𝑦	NOUN
cana-1630	205	82	)	)	PUNCT
cana-1630	205	83	≤	≤	NUM
cana-1630	205	84	∫	∫	PROPN
cana-1630	205	85	∫	∫	PROPN
cana-1630	205	86	𝜇(𝑢	𝜇(𝑢	PROPN
cana-1630	205	87	,	,	PUNCT
cana-1630	205	88	𝑣)𝐺(𝑥	𝑣)𝐺(𝑥	NOUN
cana-1630	205	89	−	−	NOUN
cana-1630	205	90	𝑢	𝑢	PROPN
cana-1630	205	91	,	,	PUNCT
cana-1630	205	92	𝑦	𝑦	NOUN
cana-1630	205	93	−	−	NOUN
cana-1630	205	94	𝑣	𝑣	NOUN
cana-1630	205	95	,	,	PUNCT
cana-1630	205	96	𝜎𝑥	𝜎𝑥	INTJ
cana-1630	205	97	,	,	PUNCT
cana-1630	205	98	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	205	99	,	,	PUNCT
cana-1630	205	100	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	205	101	,	,	PUNCT
cana-1630	205	102	𝑚𝑦	𝑚𝑦	ADP
cana-1630	205	103	,	,	PUNCT
cana-1630	205	104	𝛼	𝛼	X
cana-1630	205	105	,	,	PUNCT
cana-1630	205	106	𝛽)𝑑𝑢	𝛽)𝑑𝑢	NOUN
cana-1630	205	107	𝑑𝑣	𝑑𝑣	ADP
cana-1630	205	108	∞	∞	PROPN
cana-1630	205	109	−∞	−∞	ADP
cana-1630	205	110	≤	≤	NUM
cana-1630	205	111	1	1	NUM
cana-1630	205	112	∞	∞	NUM
cana-1630	205	113	−∞	−∞	ADP
cana-1630	205	114	the	the	DET
cana-1630	205	115	adaptive	adaptive	ADJ
cana-1630	205	116	gaussian	gaussian	ADJ
cana-1630	205	117	filter	filter	NOUN
cana-1630	205	118	preserves	preserve	VERB
cana-1630	205	119	the	the	DET
cana-1630	205	120	smoothness	smoothness	NOUN
cana-1630	205	121	and	and	CCONJ
cana-1630	205	122	continuity	continuity	NOUN
cana-1630	205	123	of	of	ADP
cana-1630	205	124	the	the	DET
cana-1630	205	125	membership	membership	NOUN
cana-1630	205	126	function	function	NOUN
cana-1630	205	127	.	.	PUNCT
cana-1630	206	1	since	since	SCONJ
cana-1630	206	2	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	206	3	,	,	PUNCT
cana-1630	206	4	𝑦	𝑦	NOUN
cana-1630	206	5	)	)	PUNCT
cana-1630	206	6	is	be	AUX
cana-1630	206	7	smooth	smooth	ADJ
cana-1630	206	8	and	and	CCONJ
cana-1630	206	9	the	the	DET
cana-1630	206	10	convolution	convolution	NOUN
cana-1630	206	11	operation	operation	NOUN
cana-1630	206	12	with	with	ADP
cana-1630	206	13	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	206	14	,	,	PUNCT
cana-1630	206	15	𝑦	𝑦	NOUN
cana-1630	206	16	)	)	PUNCT
cana-1630	206	17	maintains	maintain	VERB
cana-1630	206	18	this	this	DET
cana-1630	206	19	smoothness	smoothness	NOUN
cana-1630	206	20	,	,	PUNCT
cana-1630	206	21	the	the	DET
cana-1630	206	22	filtered	filter	VERB
cana-1630	206	23	membership	membership	NOUN
cana-1630	206	24	function	function	PROPN
cana-1630	206	25	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	206	26	,	,	PUNCT
cana-1630	206	27	𝑦	𝑦	NOUN
cana-1630	206	28	)	)	PUNCT
cana-1630	206	29	will	will	AUX
cana-1630	206	30	also	also	ADV
cana-1630	206	31	be	be	AUX
cana-1630	206	32	smooth	smooth	ADJ
cana-1630	206	33	.	.	PUNCT
cana-1630	207	1	thus	thus	ADV
cana-1630	207	2	,	,	PUNCT
cana-1630	207	3	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	207	4	,	,	PUNCT
cana-1630	207	5	𝑦	𝑦	NOUN
cana-1630	207	6	)	)	PUNCT
cana-1630	207	7	remains	remain	VERB
cana-1630	207	8	a	a	DET
cana-1630	207	9	valid	valid	ADJ
cana-1630	207	10	fuzzy	fuzzy	ADJ
cana-1630	207	11	membership	membership	NOUN
cana-1630	207	12	function	function	NOUN
cana-1630	207	13	,	,	PUNCT
cana-1630	207	14	preserving	preserve	VERB
cana-1630	207	15	the	the	DET
cana-1630	207	16	essential	essential	ADJ
cana-1630	207	17	properties	property	NOUN
cana-1630	207	18	such	such	ADJ
cana-1630	207	19	as	as	ADP
cana-1630	207	20	boundedness	boundedness	NOUN
cana-1630	207	21	,	,	PUNCT
cana-1630	207	22	normalization	normalization	NOUN
cana-1630	207	23	and	and	CCONJ
cana-1630	207	24	smoothness	smoothness	NOUN
cana-1630	207	25	.	.	PUNCT
cana-1630	208	1	corollary	corollary	ADJ
cana-1630	208	2	:	:	PUNCT
cana-1630	208	3	if	if	SCONJ
cana-1630	208	4	a	a	DET
cana-1630	208	5	fuzzy	fuzzy	ADJ
cana-1630	208	6	membership	membership	NOUN
cana-1630	208	7	function	function	NOUN
cana-1630	208	8	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1630	208	9	,	,	PUNCT
cana-1630	208	10	𝑦	𝑦	NOUN
cana-1630	208	11	)	)	PUNCT
cana-1630	208	12	is	be	AUX
cana-1630	208	13	processed	process	VERB
cana-1630	208	14	by	by	ADP
cana-1630	208	15	the	the	DET
cana-1630	208	16	adaptive	adaptive	ADJ
cana-1630	208	17	2d	2d	NUM
cana-1630	208	18	gaussian	gaussian	ADJ
cana-1630	208	19	filter	filter	NOUN
cana-1630	208	20	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	208	21	,	,	PUNCT
cana-1630	208	22	𝑦	𝑦	NOUN
cana-1630	208	23	,	,	PUNCT
cana-1630	208	24	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	208	25	,	,	PUNCT
cana-1630	208	26	𝜎𝑦	𝜎𝑦	DET
cana-1630	208	27	,	,	PUNCT
cana-1630	208	28	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	208	29	,	,	PUNCT
cana-1630	208	30	𝑚𝑦	𝑚𝑦	ADP
cana-1630	208	31	,	,	PUNCT
cana-1630	208	32	𝛼	𝛼	PROPN
cana-1630	208	33	,	,	PUNCT
cana-1630	208	34	𝛽	𝛽	NOUN
cana-1630	208	35	)	)	PUNCT
cana-1630	208	36	the	the	DET
cana-1630	208	37	resulting	result	VERB
cana-1630	208	38	function	function	NOUN
cana-1630	208	39	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥	PROPN
cana-1630	208	40	,	,	PUNCT
cana-1630	208	41	𝑦	𝑦	NOUN
cana-1630	208	42	)	)	PUNCT
cana-1630	208	43	will	will	AUX
cana-1630	208	44	not	not	PART
cana-1630	208	45	only	only	ADV
cana-1630	208	46	be	be	AUX
cana-1630	208	47	bounded	bound	VERB
cana-1630	208	48	between	between	ADP
cana-1630	208	49	0	0	NUM
cana-1630	208	50	and	and	CCONJ
cana-1630	208	51	1	1	NUM
cana-1630	208	52	but	but	CCONJ
cana-1630	208	53	will	will	AUX
cana-1630	208	54	also	also	ADV
cana-1630	208	55	maintain	maintain	VERB
cana-1630	208	56	the	the	DET
cana-1630	208	57	essential	essential	ADJ
cana-1630	208	58	properties	property	NOUN
cana-1630	208	59	of	of	ADP
cana-1630	208	60	a	a	DET
cana-1630	208	61	fuzzy	fuzzy	ADJ
cana-1630	208	62	membership	membership	NOUN
cana-1630	208	63	function	function	NOUN
cana-1630	208	64	.	.	PUNCT
cana-1630	209	1	proof	proof	NOUN
cana-1630	209	2	:	:	PUNCT
cana-1630	209	3	follows	follow	VERB
cana-1630	209	4	directly	directly	ADV
cana-1630	209	5	from	from	ADP
cana-1630	209	6	the	the	DET
cana-1630	209	7	preservation	preservation	NOUN
cana-1630	209	8	of	of	ADP
cana-1630	209	9	fuzzy	fuzzy	ADJ
cana-1630	209	10	membership	membership	NOUN
cana-1630	209	11	distribution	distribution	NOUN
cana-1630	209	12	theorem	theorem	NOUN
cana-1630	209	13	.	.	PUNCT
cana-1630	210	1	theorem	theorem	ADJ
cana-1630	210	2	12	12	NUM
cana-1630	210	3	:	:	PUNCT
cana-1630	210	4	frequency	frequency	NOUN
cana-1630	210	5	response	response	NOUN
cana-1630	210	6	the	the	DET
cana-1630	210	7	frequency	frequency	NOUN
cana-1630	210	8	response	response	NOUN
cana-1630	210	9	of	of	ADP
cana-1630	210	10	the	the	DET
cana-1630	210	11	adaptive	adaptive	ADJ
cana-1630	210	12	2d	2d	NUM
cana-1630	210	13	gaussian	gaussian	ADJ
cana-1630	210	14	filter	filter	NOUN
cana-1630	210	15	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	210	16	,	,	PUNCT
cana-1630	210	17	𝑦	𝑦	NOUN
cana-1630	210	18	,	,	PUNCT
cana-1630	210	19	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	210	20	,	,	PUNCT
cana-1630	210	21	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	210	22	,	,	PUNCT
cana-1630	210	23	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	210	24	,	,	PUNCT
cana-1630	210	25	𝑚𝑦	𝑚𝑦	ADP
cana-1630	210	26	,	,	PUNCT
cana-1630	210	27	𝛼	𝛼	PROPN
cana-1630	210	28	,	,	PUNCT
cana-1630	210	29	𝛽	𝛽	NOUN
cana-1630	210	30	)	)	PUNCT
cana-1630	210	31	can	can	AUX
cana-1630	210	32	be	be	AUX
cana-1630	210	33	derived	derive	VERB
cana-1630	210	34	from	from	ADP
cana-1630	210	35	its	its	PRON
cana-1630	210	36	fourier	fourier	NOUN
cana-1630	210	37	transform	transform	NOUN
cana-1630	210	38	.	.	PUNCT
cana-1630	211	1	proof	proof	NOUN
cana-1630	211	2	:	:	PUNCT
cana-1630	211	3	the	the	DET
cana-1630	211	4	fourier	fourier	NOUN
cana-1630	211	5	transform	transform	NOUN
cana-1630	211	6	of	of	ADP
cana-1630	211	7	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	211	8	,	,	PUNCT
cana-1630	211	9	𝑦	𝑦	NOUN
cana-1630	211	10	,	,	PUNCT
cana-1630	211	11	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	211	12	,	,	PUNCT
cana-1630	211	13	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	211	14	,	,	PUNCT
cana-1630	211	15	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	211	16	,	,	PUNCT
cana-1630	211	17	𝑚𝑦	𝑚𝑦	ADP
cana-1630	211	18	,	,	PUNCT
cana-1630	211	19	𝛼	𝛼	PROPN
cana-1630	211	20	,	,	PUNCT
cana-1630	211	21	𝛽	𝛽	NOUN
cana-1630	211	22	)	)	PUNCT
cana-1630	211	23	is	be	AUX
cana-1630	211	24	:	:	PUNCT
cana-1630	211	25	ℱ{𝐺(𝑥	ℱ{𝐺(𝑥	NUM
cana-1630	211	26	,	,	PUNCT
cana-1630	211	27	𝑦	𝑦	NOUN
cana-1630	211	28	,	,	PUNCT
cana-1630	211	29	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	211	30	,	,	PUNCT
cana-1630	211	31	𝜎𝑦	𝜎𝑦	DET
cana-1630	211	32	,	,	PUNCT
cana-1630	211	33	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	211	34	,	,	PUNCT
cana-1630	211	35	𝑚𝑦	𝑚𝑦	ADP
cana-1630	211	36	,	,	PUNCT
cana-1630	211	37	𝛼	𝛼	PROPN
cana-1630	211	38	,	,	PUNCT
cana-1630	211	39	𝛽	𝛽	NOUN
cana-1630	211	40	)	)	PUNCT
cana-1630	211	41	}	}	PUNCT
cana-1630	212	1	=	=	SYM
cana-1630	212	2	∫	∫	PROPN
cana-1630	212	3	∫	∫	PROPN
cana-1630	212	4	𝐺(𝑥	𝐺(𝑥	NOUN
cana-1630	212	5	,	,	PUNCT
cana-1630	212	6	𝑦	𝑦	NOUN
cana-1630	212	7	,	,	PUNCT
cana-1630	212	8	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	212	9	,	,	PUNCT
cana-1630	212	10	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	212	11	,	,	PUNCT
cana-1630	212	12	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	212	13	,	,	PUNCT
cana-1630	212	14	𝑚𝑦	𝑚𝑦	ADP
cana-1630	212	15	,	,	PUNCT
cana-1630	212	16	𝛼	𝛼	X
cana-1630	212	17	,	,	PUNCT
cana-1630	212	18	𝛽)𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥	𝛽)𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥	ADJ
cana-1630	212	19	𝑑𝑦	𝑑𝑦	ADP
cana-1630	212	20	∞	∞	PROPN
cana-1630	212	21	−∞	−∞	X
cana-1630	212	22	∞	∞	PROPN
cana-1630	212	23	−∞	−∞	ADP
cana-1630	212	24	where	where	SCONJ
cana-1630	212	25	(	(	PUNCT
cana-1630	212	26	u	u	NOUN
cana-1630	212	27	,	,	PUNCT
cana-1630	212	28	v	v	NOUN
cana-1630	212	29	)	)	PUNCT
cana-1630	212	30	are	be	AUX
cana-1630	212	31	the	the	DET
cana-1630	212	32	frequency	frequency	NOUN
cana-1630	212	33	variables	variable	NOUN
cana-1630	212	34	.	.	PUNCT
cana-1630	213	1	the	the	DET
cana-1630	213	2	fourier	fourier	NOUN
cana-1630	213	3	transform	transform	NOUN
cana-1630	213	4	of	of	ADP
cana-1630	213	5	the	the	DET
cana-1630	213	6	gaussian	gaussian	ADJ
cana-1630	213	7	component	component	NOUN
cana-1630	213	8	1	1	NUM
cana-1630	213	9	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	213	10	𝑒𝑥𝑝	𝑒𝑥𝑝	NOUN
cana-1630	214	1	[	[	X
cana-1630	214	2	−	−	X
cana-1630	214	3	(	(	PUNCT
cana-1630	214	4	𝑥−𝑚𝑥)2	𝑥−𝑚𝑥)2	NUM
cana-1630	214	5	2𝜎𝑥	2𝜎𝑥	ADJ
cana-1630	214	6	2	2	NUM
cana-1630	214	7	−	−	PROPN
cana-1630	214	8	(	(	PUNCT
cana-1630	214	9	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	214	10	)	)	PUNCT
cana-1630	214	11	2	2	NUM
cana-1630	214	12	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	214	13	2	2	NUM
cana-1630	214	14	]	]	PUNCT
cana-1630	214	15	is	be	AUX
cana-1630	214	16	ℱ	ℱ	PROPN
cana-1630	214	17	{	{	PUNCT
cana-1630	214	18	1	1	NUM
cana-1630	214	19	2𝜋𝜎𝑥𝜎𝑦	2𝜋𝜎𝑥𝜎𝑦	NUM
cana-1630	214	20	𝑒𝑥𝑝	𝑒𝑥𝑝	NOUN
cana-1630	215	1	[	[	X
cana-1630	215	2	−	−	X
cana-1630	215	3	(	(	PUNCT
cana-1630	215	4	𝑥	𝑥	PRON
cana-1630	215	5	−	−	NUM
cana-1630	215	6	𝑚𝑥	𝑚𝑥	NOUN
cana-1630	215	7	)	)	PUNCT
cana-1630	215	8	2	2	NUM
cana-1630	215	9	2𝜎𝑥	2𝜎𝑥	NOUN
cana-1630	215	10	2	2	NUM
cana-1630	215	11	−	−	NOUN
cana-1630	215	12	(	(	PUNCT
cana-1630	215	13	𝑦	𝑦	NOUN
cana-1630	215	14	−	−	NOUN
cana-1630	215	15	𝑚𝑦	𝑚𝑦	ADP
cana-1630	215	16	)	)	PUNCT
cana-1630	215	17	2	2	NUM
cana-1630	215	18	2𝜎𝑦	2𝜎𝑦	NOUN
cana-1630	215	19	2	2	NUM
cana-1630	215	20	]	]	PUNCT
cana-1630	215	21	}	}	PUNCT
cana-1630	215	22	=	=	SYM
cana-1630	215	23	exp	exp	NOUN
cana-1630	215	24	[	[	X
cana-1630	215	25	−2𝜋2(𝜎𝑥	−2𝜋2(𝜎𝑥	PROPN
cana-1630	215	26	2𝑢2	2𝑢2	NUM
cana-1630	215	27	+	+	CCONJ
cana-1630	215	28	𝜎𝑦	𝜎𝑦	DET
cana-1630	215	29	2𝑣2)]𝑒−𝑗2𝜋(𝑚𝑥𝑢+𝑚𝑣𝑣	2𝑣2)]𝑒−𝑗2𝜋(𝑚𝑥𝑢+𝑚𝑣𝑣	NUM
cana-1630	215	30	)	)	PUNCT
cana-1630	215	31	the	the	DET
cana-1630	215	32	linear	linear	ADJ
cana-1630	215	33	terms	term	NOUN
cana-1630	215	34	[	[	X
cana-1630	215	35	1	1	NUM
cana-1630	215	36	−	−	PROPN
cana-1630	215	37	𝛼.	𝛼.	NOUN
cana-1630	215	38	(	(	PUNCT
cana-1630	215	39	𝑥−𝑚𝑥	𝑥−𝑚𝑥	NOUN
cana-1630	215	40	𝜎𝑥	𝜎𝑥	NOUN
cana-1630	215	41	2	2	NUM
cana-1630	215	42	)	)	PUNCT
cana-1630	215	43	−	−	PROPN
cana-1630	215	44	𝛽.	𝛽.	NOUN
cana-1630	215	45	(	(	PUNCT
cana-1630	215	46	𝑦−𝑚𝑦	𝑦−𝑚𝑦	PROPN
cana-1630	215	47	𝜎𝑦	𝜎𝑦	DET
cana-1630	215	48	2	2	NUM
cana-1630	215	49	)	)	PUNCT
cana-1630	215	50	]	]	PUNCT
cana-1630	215	51	introduce	introduce	VERB
cana-1630	215	52	additional	additional	ADJ
cana-1630	215	53	frequency	frequency	NOUN
cana-1630	215	54	components	component	NOUN
cana-1630	215	55	.	.	PUNCT
cana-1630	216	1	the	the	DET
cana-1630	216	2	overall	overall	ADJ
cana-1630	216	3	frequency	frequency	NOUN
cana-1630	216	4	response	response	NOUN
cana-1630	216	5	h	h	NOUN
cana-1630	216	6	(	(	PUNCT
cana-1630	216	7	u	u	NOUN
cana-1630	216	8	,	,	PUNCT
cana-1630	216	9	v	v	NOUN
cana-1630	216	10	)	)	PUNCT
cana-1630	216	11	is	be	AUX
cana-1630	216	12	a	a	DET
cana-1630	216	13	combination	combination	NOUN
cana-1630	216	14	of	of	ADP
cana-1630	216	15	these	these	DET
cana-1630	216	16	terms	term	NOUN
cana-1630	216	17	:	:	PUNCT
cana-1630	216	18	𝐻(𝑢	𝐻(𝑢	ADP
cana-1630	216	19	,	,	PUNCT
cana-1630	216	20	𝑣	𝑣	NOUN
cana-1630	216	21	)	)	PUNCT
cana-1630	216	22	=	=	SYM
cana-1630	216	23	(	(	PUNCT
cana-1630	216	24	1	1	NUM
cana-1630	216	25	−	−	PROPN
cana-1630	216	26	𝑗2𝜋𝛼𝜎𝑥𝑢	𝑗2𝜋𝛼𝜎𝑥𝑢	PROPN
cana-1630	216	27	−	−	NOUN
cana-1630	216	28	𝑗2𝜋𝛽𝜎𝑦𝑣)exp	𝑗2𝜋𝛽𝜎𝑦𝑣)exp	PROPN
cana-1630	217	1	[	[	X
cana-1630	217	2	−2𝜋2(𝜎𝑥	−2𝜋2(𝜎𝑥	PROPN
cana-1630	217	3	2𝑢2	2𝑢2	NUM
cana-1630	217	4	+	+	CCONJ
cana-1630	217	5	𝜎𝑦	𝜎𝑦	DET
cana-1630	217	6	2𝑣2)]𝑒−𝑗2𝜋(𝑚𝑥𝑢+𝑚𝑦𝑣	2𝑣2)]𝑒−𝑗2𝜋(𝑚𝑥𝑢+𝑚𝑦𝑣	NUM
cana-1630	217	7	)	)	PUNCT
cana-1630	217	8	hence	hence	ADV
cana-1630	217	9	the	the	DET
cana-1630	217	10	proof	proof	NOUN
cana-1630	217	11	.	.	PUNCT
cana-1630	218	1	corollary	corollary	ADJ
cana-1630	218	2	:	:	PUNCT
cana-1630	218	3	frequency	frequency	NOUN
cana-1630	218	4	response	response	NOUN
cana-1630	218	5	properties	property	NOUN
cana-1630	218	6	the	the	DET
cana-1630	218	7	frequency	frequency	NOUN
cana-1630	218	8	response	response	NOUN
cana-1630	218	9	of	of	ADP
cana-1630	218	10	the	the	DET
cana-1630	218	11	adaptive	adaptive	ADJ
cana-1630	218	12	2d	2d	NOUN
cana-1630	218	13	gaussian	gaussian	ADJ
cana-1630	218	14	filter	filter	NOUN
cana-1630	218	15	shows	show	VERB
cana-1630	218	16	that	that	SCONJ
cana-1630	218	17	it	it	PRON
cana-1630	218	18	acts	act	VERB
cana-1630	218	19	as	as	ADP
cana-1630	218	20	a	a	DET
cana-1630	218	21	low	low	ADJ
cana-1630	218	22	-	-	PUNCT
cana-1630	218	23	pass	pass	NOUN
cana-1630	218	24	filter	filter	NOUN
cana-1630	218	25	,	,	PUNCT
cana-1630	218	26	attenuating	attenuate	VERB
cana-1630	218	27	high	high	ADJ
cana-1630	218	28	-	-	PUNCT
cana-1630	218	29	frequency	frequency	NOUN
cana-1630	218	30	components	component	NOUN
cana-1630	218	31	and	and	CCONJ
cana-1630	218	32	preserving	preserve	VERB
cana-1630	218	33	low	low	ADJ
cana-1630	218	34	-	-	PUNCT
cana-1630	218	35	frequency	frequency	NOUN
cana-1630	218	36	components	component	NOUN
cana-1630	218	37	of	of	ADP
cana-1630	218	38	the	the	DET
cana-1630	218	39	input	input	NOUN
cana-1630	218	40	signal	signal	NOUN
cana-1630	218	41	.	.	PUNCT
cana-1630	219	1	proof	proof	NOUN
cana-1630	219	2	:	:	PUNCT
cana-1630	219	3	follows	follow	VERB
cana-1630	219	4	directly	directly	ADV
cana-1630	219	5	from	from	ADP
cana-1630	219	6	the	the	DET
cana-1630	219	7	frequency	frequency	NOUN
cana-1630	219	8	response	response	NOUN
cana-1630	219	9	deviation	deviation	NOUN
cana-1630	219	10	,	,	PUNCT
cana-1630	219	11	where	where	SCONJ
cana-1630	219	12	the	the	DET
cana-1630	219	13	term	term	NOUN
cana-1630	219	14	exp[−2𝜋2(𝜎𝑥	exp[−2𝜋2(𝜎𝑥	PROPN
cana-1630	219	15	2𝑢2	2𝑢2	NUM
cana-1630	219	16	+	+	CCONJ
cana-1630	219	17	𝜎𝑦	𝜎𝑦	DET
cana-1630	219	18	2𝑣2	2𝑣2	NUM
cana-1630	219	19	)	)	PUNCT
cana-1630	219	20	]	]	PUNCT
cana-1630	219	21	indicates	indicate	VERB
cana-1630	219	22	attenuation	attenuation	NOUN
cana-1630	219	23	of	of	ADP
cana-1630	219	24	higher	high	ADJ
cana-1630	219	25	frequencies	frequency	NOUN
cana-1630	219	26	.	.	PUNCT
cana-1630	220	1	communications	communication	NOUN
cana-1630	220	2	on	on	ADP
cana-1630	220	3	applied	apply	VERB
cana-1630	220	4	nonlinear	nonlinear	ADJ
cana-1630	220	5	analysis	analysis	NOUN
cana-1630	220	6	issn	issn	NOUN
cana-1630	220	7	:	:	PUNCT
cana-1630	220	8	1074	1074	NUM
cana-1630	220	9	-	-	PUNCT
cana-1630	220	10	133x	133x	NUM
cana-1630	220	11	vol	vol	NOUN
cana-1630	220	12	32	32	NUM
cana-1630	220	13	no	no	NOUN
cana-1630	220	14	.	.	NOUN
cana-1630	220	15	1	1	NUM
cana-1630	220	16	(	(	PUNCT
cana-1630	220	17	2025	2025	NUM
cana-1630	220	18	)	)	PUNCT
cana-1630	220	19	196	196	NUM
cana-1630	220	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	220	21	3	3	X
cana-1630	220	22	.	.	PUNCT
cana-1630	220	23	numerical	numerical	PROPN
cana-1630	220	24	example	example	NOUN
cana-1630	220	25	let	let	VERB
cana-1630	220	26	us	we	PRON
cana-1630	220	27	consider	consider	VERB
cana-1630	220	28	the	the	DET
cana-1630	220	29	5x5	5x5	NUM
cana-1630	220	30	matrix	matrix	NOUN
cana-1630	220	31	sample	sample	NOUN
cana-1630	220	32	from	from	ADP
cana-1630	220	33	a	a	DET
cana-1630	220	34	image	image	NOUN
cana-1630	220	35	pixel	pixel	NOUN
cana-1630	220	36	matrix	matrix	NOUN
cana-1630	220	37	.	.	PUNCT
cana-1630	221	1	three	three	NUM
cana-1630	221	2	different	different	ADJ
cana-1630	221	3	sample	sample	NOUN
cana-1630	221	4	pixel	pixel	NOUN
cana-1630	221	5	matrix	matrix	NOUN
cana-1630	221	6	is	be	AUX
cana-1630	221	7	considered	consider	VERB
cana-1630	221	8	which	which	PRON
cana-1630	221	9	is	be	AUX
cana-1630	221	10	given	give	VERB
cana-1630	221	11	below	below	ADV
cana-1630	221	12	:	:	PUNCT
cana-1630	221	13	𝐴	𝐴	PROPN
cana-1630	221	14	=	=	PUNCT
cana-1630	222	1	[	[	PUNCT
cana-1630	222	2	1	1	NUM
cana-1630	222	3	2	2	NUM
cana-1630	222	4	3	3	NUM
cana-1630	222	5	4	4	NUM
cana-1630	222	6	5	5	NUM
cana-1630	222	7	5	5	NUM
cana-1630	222	8	4	4	NUM
cana-1630	222	9	3	3	NUM
cana-1630	222	10	2	2	NUM
cana-1630	222	11	1	1	NUM
cana-1630	222	12	1	1	NUM
cana-1630	222	13	4	4	NUM
cana-1630	222	14	2	2	NUM
cana-1630	222	15	3	3	NUM
cana-1630	222	16	2	2	NUM
cana-1630	222	17	5	5	NUM
cana-1630	222	18	5	5	NUM
cana-1630	222	19	1	1	NUM
cana-1630	222	20	4	4	NUM
cana-1630	222	21	2	2	NUM
cana-1630	222	22	5	5	NUM
cana-1630	222	23	3	3	NUM
cana-1630	222	24	4	4	NUM
cana-1630	222	25	3	3	NUM
cana-1630	222	26	1	1	NUM
cana-1630	222	27	]	]	PUNCT
cana-1630	222	28	;	;	PUNCT
cana-1630	222	29	𝐵	𝐵	NOUN
cana-1630	222	30	=	=	PUNCT
cana-1630	222	31	[	[	PUNCT
cana-1630	222	32	255	255	NUM
cana-1630	222	33	255	255	NUM
cana-1630	222	34	249	249	NUM
cana-1630	222	35	255	255	NUM
cana-1630	222	36	252	252	NUM
cana-1630	222	37	251	251	NUM
cana-1630	222	38	253	253	NUM
cana-1630	222	39	255	255	NUM
cana-1630	222	40	248	248	NUM
cana-1630	222	41	255	255	NUM
cana-1630	222	42	247	247	NUM
cana-1630	222	43	255	255	NUM
cana-1630	222	44	251	251	NUM
cana-1630	222	45	255	255	NUM
cana-1630	222	46	255	255	NUM
cana-1630	222	47	252	252	NUM
cana-1630	222	48	254	254	NUM
cana-1630	222	49	255	255	NUM
cana-1630	222	50	255	255	NUM
cana-1630	222	51	255	255	NUM
cana-1630	222	52	252	252	NUM
cana-1630	222	53	255	255	NUM
cana-1630	222	54	247	247	NUM
cana-1630	222	55	255	255	NUM
cana-1630	222	56	251	251	NUM
cana-1630	222	57	]	]	PUNCT
cana-1630	222	58	;	;	PUNCT
cana-1630	222	59	𝐶	𝐶	PROPN
cana-1630	222	60	=	=	PUNCT
cana-1630	222	61	[	[	PUNCT
cana-1630	222	62	11	11	NUM
cana-1630	222	63	5	5	NUM
cana-1630	222	64	1	1	NUM
cana-1630	222	65	2	2	NUM
cana-1630	222	66	1	1	NUM
cana-1630	222	67	23	23	NUM
cana-1630	222	68	17	17	NUM
cana-1630	222	69	7	7	NUM
cana-1630	222	70	2	2	NUM
cana-1630	222	71	3	3	NUM
cana-1630	222	72	34	34	NUM
cana-1630	222	73	33	33	NUM
cana-1630	222	74	25	25	NUM
cana-1630	222	75	28	28	NUM
cana-1630	222	76	24	24	NUM
cana-1630	222	77	12	12	NUM
cana-1630	222	78	15	15	NUM
cana-1630	222	79	15	15	NUM
cana-1630	222	80	9	9	NUM
cana-1630	222	81	4	4	NUM
cana-1630	222	82	5	5	NUM
cana-1630	222	83	3	3	NUM
cana-1630	222	84	6	6	NUM
cana-1630	222	85	7	7	NUM
cana-1630	222	86	6	6	NUM
cana-1630	222	87	]	]	PUNCT
cana-1630	222	88	;	;	PUNCT
cana-1630	222	89	𝐷	𝐷	PROPN
cana-1630	222	90	=	=	SYM
cana-1630	222	91	[	[	PUNCT
cana-1630	222	92	102	102	NUM
cana-1630	222	93	168	168	NUM
cana-1630	222	94	199	199	NUM
cana-1630	222	95	209	209	NUM
cana-1630	222	96	195	195	NUM
cana-1630	222	97	158	158	NUM
cana-1630	222	98	195	195	NUM
cana-1630	222	99	202	202	NUM
cana-1630	222	100	190	190	NUM
cana-1630	222	101	172	172	NUM
cana-1630	222	102	197	197	NUM
cana-1630	222	103	208	208	NUM
cana-1630	222	104	190	190	NUM
cana-1630	222	105	209	209	NUM
cana-1630	222	106	206	206	NUM
cana-1630	222	107	181	181	NUM
cana-1630	222	108	197	197	NUM
cana-1630	222	109	189	189	NUM
cana-1630	222	110	172	172	NUM
cana-1630	222	111	174	174	NUM
cana-1630	222	112	166	166	NUM
cana-1630	222	113	159	159	NUM
cana-1630	222	114	158	158	NUM
cana-1630	222	115	157	157	NUM
cana-1630	222	116	154	154	NUM
cana-1630	222	117	]	]	PUNCT
cana-1630	222	118	;	;	PUNCT
cana-1630	222	119	the	the	DET
cana-1630	222	120	function	function	NOUN
cana-1630	222	121	graph	graph	NOUN
cana-1630	222	122	is	be	AUX
cana-1630	222	123	shown	show	VERB
cana-1630	222	124	in	in	ADP
cana-1630	222	125	following	follow	VERB
cana-1630	222	126	fig.1	fig.1	PROPN
cana-1630	222	127	.	.	PUNCT
cana-1630	223	1	from	from	ADP
cana-1630	223	2	the	the	DET
cana-1630	223	3	graph	graph	NOUN
cana-1630	223	4	its	its	PRON
cana-1630	223	5	concluded	conclude	VERB
cana-1630	223	6	that	that	SCONJ
cana-1630	223	7	as	as	SCONJ
cana-1630	223	8	the	the	DET
cana-1630	223	9	𝜎	𝜎	PROPN
cana-1630	223	10	value	value	NOUN
cana-1630	223	11	increases	increase	VERB
cana-1630	223	12	the	the	DET
cana-1630	223	13	function	function	NOUN
cana-1630	223	14	graph	graph	NOUN
cana-1630	223	15	changes	change	NOUN
cana-1630	223	16	that	that	PRON
cana-1630	223	17	it	it	PRON
cana-1630	223	18	smoothens	smoothen	VERB
cana-1630	223	19	the	the	DET
cana-1630	223	20	intensity	intensity	NOUN
cana-1630	223	21	value	value	NOUN
cana-1630	223	22	accordingly	accordingly	ADV
cana-1630	223	23	.	.	PUNCT
cana-1630	224	1	when	when	SCONJ
cana-1630	224	2	the	the	DET
cana-1630	224	3	standard	standard	ADJ
cana-1630	224	4	deviation	deviation	NOUN
cana-1630	224	5	value	value	NOUN
cana-1630	224	6	is	be	AUX
cana-1630	224	7	less	less	ADJ
cana-1630	224	8	the	the	DET
cana-1630	224	9	image	image	NOUN
cana-1630	224	10	intensity	intensity	NOUN
cana-1630	224	11	value	value	NOUN
cana-1630	224	12	is	be	AUX
cana-1630	224	13	enhanced	enhance	VERB
cana-1630	224	14	.	.	PUNCT
cana-1630	225	1	using	use	VERB
cana-1630	225	2	matlab	matlab	PROPN
cana-1630	225	3	2017b	2017b	NUM
cana-1630	225	4	,	,	PUNCT
cana-1630	225	5	the	the	DET
cana-1630	225	6	results	result	NOUN
cana-1630	225	7	were	be	AUX
cana-1630	225	8	obtained	obtain	VERB
cana-1630	225	9	and	and	CCONJ
cana-1630	225	10	the	the	DET
cana-1630	225	11	standard	standard	ADJ
cana-1630	225	12	deviation	deviation	NOUN
cana-1630	225	13	of	of	ADP
cana-1630	225	14	the	the	DET
cana-1630	225	15	matrices	matrix	NOUN
cana-1630	225	16	a	a	DET
cana-1630	225	17	,	,	PUNCT
cana-1630	225	18	b	b	NOUN
cana-1630	225	19	,	,	PUNCT
cana-1630	225	20	c	c	NOUN
cana-1630	225	21	,	,	PUNCT
cana-1630	225	22	d	d	X
cana-1630	225	23	are	be	AUX
cana-1630	225	24	𝜎𝐴	𝜎𝐴	NOUN
cana-1630	225	25	=	=	SYM
cana-1630	225	26	1.4142	1.4142	NUM
cana-1630	225	27	,	,	PUNCT
cana-1630	225	28	𝜎𝐵	𝜎𝐵	NOUN
cana-1630	225	29	=	=	PUNCT
cana-1630	225	30	2.6880	2.6880	NUM
cana-1630	225	31	,	,	PUNCT
cana-1630	226	1	𝜎𝐶	𝜎𝐶	PROPN
cana-1630	226	2	=	=	SYM
cana-1630	226	3	2.6880	2.6880	NUM
cana-1630	226	4	,	,	PUNCT
cana-1630	226	5	𝜎𝐷	𝜎𝐷	NOUN
cana-1630	226	6	=	=	SYM
cana-1630	226	7	10.1111	10.1111	NUM
cana-1630	226	8	respectively	respectively	ADV
cana-1630	226	9	.	.	PUNCT
cana-1630	227	1	pseudo	pseudo	NOUN
cana-1630	227	2	code	code	NOUN
cana-1630	227	3	:	:	PUNCT
cana-1630	227	4	1	1	X
cana-1630	227	5	.	.	X
cana-1630	227	6	initialize	initialize	VERB
cana-1630	227	7	the	the	DET
cana-1630	227	8	pixel	pixel	NOUN
cana-1630	227	9	matrix	matrix	NOUN
cana-1630	227	10	2	2	NUM
cana-1630	227	11	.	.	PUNCT
cana-1630	227	12	calculate	calculate	VERB
cana-1630	227	13	the	the	DET
cana-1630	227	14	parameters	parameter	NOUN
cana-1630	227	15	of	of	ADP
cana-1630	227	16	matrices	matrix	NOUN
cana-1630	227	17	:	:	PUNCT
cana-1630	227	18	compute	compute	NOUN
cana-1630	227	19	mean	mean	NOUN
cana-1630	227	20	,	,	PUNCT
cana-1630	227	21	standard	standard	ADJ
cana-1630	227	22	deviation	deviation	NOUN
cana-1630	227	23	,	,	PUNCT
cana-1630	227	24	normalized	normalize	VERB
cana-1630	227	25	matrix	matrix	NOUN
cana-1630	227	26	etc	etc	X
cana-1630	227	27	.	.	X
cana-1630	228	1	3	3	X
cana-1630	228	2	.	.	X
cana-1630	228	3	apply	apply	VERB
cana-1630	228	4	the	the	DET
cana-1630	228	5	calculated	calculated	ADJ
cana-1630	228	6	parameters	parameter	NOUN
cana-1630	228	7	to	to	ADP
cana-1630	228	8	the	the	DET
cana-1630	228	9	matrices	matrix	NOUN
cana-1630	228	10	4	4	NUM
cana-1630	228	11	.	.	PUNCT
cana-1630	228	12	define	define	VERB
cana-1630	228	13	1d	1d	NUM
cana-1630	228	14	functions	function	NOUN
cana-1630	228	15	:	:	PUNCT
cana-1630	228	16	𝐹(𝜎	𝐹(𝜎	NUM
cana-1630	228	17	,	,	PUNCT
cana-1630	228	18	𝛼	𝛼	X
cana-1630	228	19	,	,	PUNCT
cana-1630	228	20	𝑥	𝑥	NOUN
cana-1630	228	21	)	)	PUNCT
cana-1630	228	22	,	,	PUNCT
cana-1630	228	23	g(𝜎,𝑚	g(𝜎,𝑚	X
cana-1630	228	24	,	,	PUNCT
cana-1630	228	25	𝛼	𝛼	X
cana-1630	228	26	,	,	PUNCT
cana-1630	228	27	𝑥	𝑥	NOUN
cana-1630	228	28	)	)	PUNCT
cana-1630	228	29	,	,	PUNCT
cana-1630	228	30	5	5	X
cana-1630	228	31	.	.	X
cana-1630	228	32	define	define	VERB
cana-1630	228	33	range	range	NOUN
cana-1630	228	34	of	of	ADP
cana-1630	228	35	x	x	PROPN
cana-1630	228	36	6	6	X
cana-1630	228	37	.	.	PUNCT
cana-1630	228	38	compute	compute	NOUN
cana-1630	228	39	values	value	NOUN
cana-1630	228	40	for	for	ADP
cana-1630	228	41	f	f	PROPN
cana-1630	228	42	,	,	PUNCT
cana-1630	228	43	g	g	PROPN
cana-1630	228	44	and	and	CCONJ
cana-1630	228	45	plot	plot	VERB
cana-1630	228	46	the	the	DET
cana-1630	228	47	results	result	NOUN
cana-1630	228	48	.	.	PUNCT
cana-1630	229	1	7	7	X
cana-1630	229	2	.	.	X
cana-1630	229	3	define	define	VERB
cana-1630	229	4	2d	2d	NUM
cana-1630	229	5	functions	function	NOUN
cana-1630	229	6	:	:	PUNCT
cana-1630	229	7	𝐹(𝜎𝑥	𝐹(𝜎𝑥	PROPN
cana-1630	229	8	,	,	PUNCT
cana-1630	229	9	𝜎𝑦	𝜎𝑦	PRON
cana-1630	229	10	,	,	PUNCT
cana-1630	229	11	𝛼	𝛼	PROPN
cana-1630	229	12	,	,	PUNCT
cana-1630	229	13	𝛽	𝛽	NOUN
cana-1630	229	14	,	,	PUNCT
cana-1630	229	15	𝑥	𝑥	PROPN
cana-1630	229	16	,	,	PUNCT
cana-1630	229	17	𝑦	𝑦	NOUN
cana-1630	229	18	)	)	PUNCT
cana-1630	229	19	,	,	PUNCT
cana-1630	229	20	g(𝜎𝑥	g(𝜎𝑥	PROPN
cana-1630	229	21	,	,	PUNCT
cana-1630	229	22	𝜎𝑦	𝜎𝑦	NOUN
cana-1630	229	23	,	,	PUNCT
cana-1630	229	24	𝑚𝑦	𝑚𝑦	ADP
cana-1630	229	25	,	,	PUNCT
cana-1630	229	26	𝑚𝑦𝛼	𝑚𝑦𝛼	PROPN
cana-1630	229	27	,	,	PUNCT
cana-1630	229	28	𝛽	𝛽	PROPN
cana-1630	229	29	,	,	PUNCT
cana-1630	229	30	𝑥	𝑥	PROPN
cana-1630	229	31	,	,	PUNCT
cana-1630	229	32	𝑦	𝑦	NOUN
cana-1630	229	33	)	)	PUNCT
cana-1630	229	34	8	8	NUM
cana-1630	229	35	.	.	PUNCT
cana-1630	230	1	compute	compute	NOUN
cana-1630	230	2	values	value	NOUN
cana-1630	230	3	for	for	ADP
cana-1630	230	4	f	f	PROPN
cana-1630	230	5	and	and	CCONJ
cana-1630	230	6	g	g	PROPN
cana-1630	230	7	of	of	ADP
cana-1630	230	8	2d	2d	PROPN
cana-1630	230	9	functions	function	NOUN
cana-1630	230	10	.	.	PUNCT
cana-1630	231	1	9	9	X
cana-1630	231	2	.	.	X
cana-1630	231	3	plot	plot	VERB
cana-1630	231	4	the	the	DET
cana-1630	231	5	results	result	NOUN
cana-1630	231	6	communications	communication	NOUN
cana-1630	231	7	on	on	ADP
cana-1630	231	8	applied	apply	VERB
cana-1630	231	9	nonlinear	nonlinear	ADJ
cana-1630	231	10	analysis	analysis	NOUN
cana-1630	231	11	issn	issn	NOUN
cana-1630	231	12	:	:	PUNCT
cana-1630	231	13	1074	1074	NUM
cana-1630	231	14	-	-	PUNCT
cana-1630	231	15	133x	133x	NUM
cana-1630	231	16	vol	vol	NOUN
cana-1630	231	17	32	32	NUM
cana-1630	231	18	no	no	NOUN
cana-1630	231	19	.	.	NOUN
cana-1630	231	20	1	1	NUM
cana-1630	231	21	(	(	PUNCT
cana-1630	231	22	2025	2025	NUM
cana-1630	231	23	)	)	PUNCT
cana-1630	231	24	197	197	NUM
cana-1630	231	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	231	26	(	(	PUNCT
cana-1630	231	27	a	a	NOUN
cana-1630	231	28	)	)	PUNCT
cana-1630	231	29	agd	agd	PROPN
cana-1630	231	30	-1d	-1d	PROPN
cana-1630	231	31	(	(	PUNCT
cana-1630	231	32	b	b	NOUN
cana-1630	231	33	)	)	PUNCT
cana-1630	231	34	afgd	afgd	ADJ
cana-1630	231	35	-1d	-1d	PROPN
cana-1630	231	36	fig.1	fig.1	PROPN
cana-1630	231	37	.	.	PUNCT
cana-1630	232	1	functional	functional	ADJ
cana-1630	232	2	graph	graph	NOUN
cana-1630	232	3	of	of	ADP
cana-1630	232	4	1d	1d	NUM
cana-1630	232	5	filters	filter	NOUN
cana-1630	232	6	where	where	SCONJ
cana-1630	232	7	blue	blue	ADJ
cana-1630	232	8	–	–	PUNCT
cana-1630	232	9	value	value	NOUN
cana-1630	232	10	of	of	ADP
cana-1630	232	11	a	a	DET
cana-1630	232	12	,	,	PUNCT
cana-1630	232	13	greenvalue	greenvalue	NOUN
cana-1630	232	14	of	of	ADP
cana-1630	232	15	b	b	PROPN
cana-1630	232	16	,	,	PUNCT
cana-1630	232	17	black	black	ADJ
cana-1630	232	18	–	–	PUNCT
cana-1630	232	19	value	value	NOUN
cana-1630	232	20	of	of	ADP
cana-1630	232	21	c	c	NOUN
cana-1630	232	22	,	,	PUNCT
cana-1630	232	23	redvalue	redvalue	NOUN
cana-1630	232	24	of	of	ADP
cana-1630	232	25	d	d	PROPN
cana-1630	232	26	(	(	PUNCT
cana-1630	232	27	a	a	PRON
cana-1630	232	28	)	)	PUNCT
cana-1630	232	29	agd-2d	agd-2d	NOUN
cana-1630	232	30	(	(	PUNCT
cana-1630	232	31	b)afgd-2d	b)afgd-2d	NOUN
cana-1630	232	32	fig.2	fig.2	PROPN
cana-1630	232	33	.	.	PUNCT
cana-1630	232	34	functional	functional	ADJ
cana-1630	232	35	graph	graph	NOUN
cana-1630	232	36	of	of	ADP
cana-1630	232	37	2d	2d	NOUN
cana-1630	232	38	filters	filter	NOUN
cana-1630	232	39	where	where	SCONJ
cana-1630	232	40	blue	blue	ADJ
cana-1630	232	41	–	–	PUNCT
cana-1630	232	42	value	value	NOUN
cana-1630	232	43	of	of	ADP
cana-1630	232	44	a	a	DET
cana-1630	232	45	,	,	PUNCT
cana-1630	232	46	greenvalue	greenvalue	NOUN
cana-1630	232	47	of	of	ADP
cana-1630	232	48	b	b	PROPN
cana-1630	232	49	,	,	PUNCT
cana-1630	232	50	black	black	ADJ
cana-1630	232	51	–	–	PUNCT
cana-1630	232	52	value	value	NOUN
cana-1630	232	53	of	of	ADP
cana-1630	232	54	c	c	NOUN
cana-1630	232	55	,	,	PUNCT
cana-1630	232	56	redvalue	redvalue	NOUN
cana-1630	232	57	of	of	ADP
cana-1630	232	58	d	d	PROPN
cana-1630	232	59	4	4	X
cana-1630	232	60	.	.	PUNCT
cana-1630	232	61	conclusion	conclusion	VERB
cana-1630	232	62	the	the	DET
cana-1630	232	63	integration	integration	NOUN
cana-1630	232	64	of	of	ADP
cana-1630	232	65	adaptive	adaptive	ADJ
cana-1630	232	66	gaussian	gaussian	ADJ
cana-1630	232	67	derivative	derivative	ADJ
cana-1630	232	68	filtering	filtering	NOUN
cana-1630	232	69	into	into	ADP
cana-1630	232	70	fuzzy	fuzzy	ADJ
cana-1630	232	71	logic	logic	NOUN
cana-1630	232	72	systems	system	NOUN
cana-1630	232	73	demonstrates	demonstrate	VERB
cana-1630	232	74	a	a	DET
cana-1630	232	75	significant	significant	ADJ
cana-1630	232	76	advancement	advancement	NOUN
cana-1630	232	77	in	in	ADP
cana-1630	232	78	maintaining	maintain	VERB
cana-1630	232	79	the	the	DET
cana-1630	232	80	stability	stability	NOUN
cana-1630	232	81	and	and	CCONJ
cana-1630	232	82	accuracy	accuracy	NOUN
cana-1630	232	83	of	of	ADP
cana-1630	232	84	fuzzy	fuzzy	ADJ
cana-1630	232	85	membership	membership	NOUN
cana-1630	232	86	functions	function	NOUN
cana-1630	232	87	.	.	PUNCT
cana-1630	233	1	this	this	DET
cana-1630	233	2	approach	approach	NOUN
cana-1630	233	3	leverages	leverage	VERB
cana-1630	233	4	the	the	DET
cana-1630	233	5	precision	precision	NOUN
cana-1630	233	6	of	of	ADP
cana-1630	233	7	gaussian	gaussian	ADJ
cana-1630	233	8	derivatives	derivative	NOUN
cana-1630	233	9	to	to	PART
cana-1630	233	10	effectively	effectively	ADV
cana-1630	233	11	capture	capture	VERB
cana-1630	233	12	subtle	subtle	ADJ
cana-1630	233	13	variations	variation	NOUN
cana-1630	233	14	in	in	ADP
cana-1630	233	15	data	datum	NOUN
cana-1630	233	16	,	,	PUNCT
cana-1630	233	17	while	while	SCONJ
cana-1630	233	18	adaptive	adaptive	ADJ
cana-1630	233	19	filtering	filtering	NOUN
cana-1630	233	20	techniques	technique	NOUN
cana-1630	233	21	ensure	ensure	VERB
cana-1630	233	22	that	that	SCONJ
cana-1630	233	23	the	the	DET
cana-1630	233	24	filtering	filter	VERB
cana-1630	233	25	process	process	NOUN
cana-1630	233	26	dynamically	dynamically	ADV
cana-1630	233	27	adjusts	adjust	VERB
cana-1630	233	28	to	to	ADP
cana-1630	233	29	the	the	DET
cana-1630	233	30	specific	specific	ADJ
cana-1630	233	31	characteristics	characteristic	NOUN
cana-1630	233	32	of	of	ADP
cana-1630	233	33	the	the	DET
cana-1630	233	34	input	input	NOUN
cana-1630	233	35	data	datum	NOUN
cana-1630	233	36	.	.	PUNCT
cana-1630	234	1	this	this	DET
cana-1630	234	2	combination	combination	NOUN
cana-1630	234	3	results	result	VERB
cana-1630	234	4	in	in	ADP
cana-1630	234	5	a	a	DET
cana-1630	234	6	robust	robust	ADJ
cana-1630	234	7	filtering	filtering	NOUN
cana-1630	234	8	mechanism	mechanism	NOUN
cana-1630	234	9	capable	capable	ADJ
cana-1630	234	10	of	of	ADP
cana-1630	234	11	preserving	preserve	VERB
cana-1630	234	12	the	the	DET
cana-1630	234	13	integrity	integrity	NOUN
cana-1630	234	14	of	of	ADP
cana-1630	234	15	membership	membership	NOUN
cana-1630	234	16	functions	function	NOUN
cana-1630	234	17	even	even	ADV
cana-1630	234	18	in	in	ADP
cana-1630	234	19	the	the	DET
cana-1630	234	20	presence	presence	NOUN
cana-1630	234	21	of	of	ADP
cana-1630	234	22	noise	noise	NOUN
cana-1630	234	23	and	and	CCONJ
cana-1630	234	24	complex	complex	ADJ
cana-1630	234	25	data	datum	NOUN
cana-1630	234	26	patterns	pattern	NOUN
cana-1630	234	27	.	.	PUNCT
cana-1630	235	1	the	the	DET
cana-1630	235	2	theorem	theorem	NOUN
cana-1630	235	3	and	and	CCONJ
cana-1630	235	4	corollaries	corollary	NOUN
cana-1630	235	5	stated	state	VERB
cana-1630	235	6	above	above	ADP
cana-1630	235	7	establish	establish	VERB
cana-1630	235	8	that	that	SCONJ
cana-1630	235	9	the	the	DET
cana-1630	235	10	adaptive	adaptive	ADJ
cana-1630	235	11	1d	1d	NUM
cana-1630	235	12	and	and	CCONJ
cana-1630	235	13	2d	2d	NOUN
cana-1630	235	14	gaussian	gaussian	ADJ
cana-1630	235	15	filters	filter	NOUN
cana-1630	235	16	is	be	AUX
cana-1630	235	17	bibo	bibo	ADJ
cana-1630	235	18	stable	stable	ADJ
cana-1630	235	19	,	,	PUNCT
cana-1630	235	20	preserves	preserve	VERB
cana-1630	235	21	the	the	DET
cana-1630	235	22	essential	essential	ADJ
cana-1630	235	23	properties	property	NOUN
cana-1630	235	24	of	of	ADP
cana-1630	235	25	fuzzy	fuzzy	ADJ
cana-1630	235	26	membership	membership	NOUN
cana-1630	235	27	function	function	NOUN
cana-1630	235	28	and	and	CCONJ
cana-1630	235	29	acts	act	VERB
cana-1630	235	30	as	as	ADP
cana-1630	235	31	a	a	DET
cana-1630	235	32	low	low	ADJ
cana-1630	235	33	pass	pass	NOUN
cana-1630	235	34	filter	filter	NOUN
cana-1630	235	35	in	in	ADP
cana-1630	235	36	the	the	DET
cana-1630	235	37	frequency	frequency	NOUN
cana-1630	235	38	domain	domain	NOUN
cana-1630	235	39	.	.	PUNCT
cana-1630	236	1	the	the	DET
cana-1630	236	2	mathematical	mathematical	ADJ
cana-1630	236	3	proof	proof	NOUN
cana-1630	236	4	demonstrates	demonstrate	VERB
cana-1630	236	5	the	the	DET
cana-1630	236	6	robustness	robustness	NOUN
cana-1630	236	7	and	and	CCONJ
cana-1630	236	8	reliability	reliability	NOUN
cana-1630	236	9	of	of	ADP
cana-1630	236	10	the	the	DET
cana-1630	236	11	filter	filter	NOUN
cana-1630	236	12	in	in	ADP
cana-1630	236	13	various	various	ADJ
cana-1630	236	14	applications	application	NOUN
cana-1630	236	15	,	,	PUNCT
cana-1630	236	16	ensuring	ensure	VERB
cana-1630	236	17	the	the	DET
cana-1630	236	18	integrity	integrity	NOUN
cana-1630	236	19	and	and	CCONJ
cana-1630	236	20	stability	stability	NOUN
cana-1630	236	21	of	of	ADP
cana-1630	236	22	the	the	DET
cana-1630	236	23	processed	process	VERB
cana-1630	236	24	outputs	output	NOUN
cana-1630	236	25	.	.	PUNCT
cana-1630	237	1	the	the	DET
cana-1630	237	2	research	research	NOUN
cana-1630	237	3	underscores	underscore	VERB
cana-1630	237	4	the	the	DET
cana-1630	237	5	critical	critical	ADJ
cana-1630	237	6	importance	importance	NOUN
cana-1630	237	7	of	of	ADP
cana-1630	237	8	selecting	select	VERB
cana-1630	237	9	appropriate	appropriate	ADJ
cana-1630	237	10	derivative	derivative	ADJ
cana-1630	237	11	orders	order	NOUN
cana-1630	237	12	and	and	CCONJ
cana-1630	237	13	designing	design	VERB
cana-1630	237	14	adaptive	adaptive	ADJ
cana-1630	237	15	mechanisms	mechanism	NOUN
cana-1630	237	16	that	that	PRON
cana-1630	237	17	are	be	AUX
cana-1630	237	18	both	both	PRON
cana-1630	237	19	robust	robust	ADJ
cana-1630	237	20	and	and	CCONJ
cana-1630	237	21	responsive	responsive	ADJ
cana-1630	237	22	.	.	PUNCT
cana-1630	238	1	the	the	DET
cana-1630	238	2	careful	careful	ADJ
cana-1630	238	3	balance	balance	NOUN
cana-1630	238	4	between	between	ADP
cana-1630	238	5	these	these	DET
cana-1630	238	6	elements	element	NOUN
cana-1630	238	7	is	be	AUX
cana-1630	238	8	crucial	crucial	ADJ
cana-1630	238	9	to	to	ADP
cana-1630	238	10	optimizing	optimize	VERB
cana-1630	238	11	the	the	DET
cana-1630	238	12	performance	performance	NOUN
cana-1630	238	13	of	of	ADP
cana-1630	238	14	the	the	DET
cana-1630	238	15	filtering	filtering	NOUN
cana-1630	238	16	process	process	NOUN
cana-1630	238	17	.	.	PUNCT
cana-1630	239	1	the	the	DET
cana-1630	239	2	findings	finding	NOUN
cana-1630	239	3	indicate	indicate	VERB
cana-1630	239	4	that	that	SCONJ
cana-1630	239	5	communications	communication	NOUN
cana-1630	239	6	on	on	ADP
cana-1630	239	7	applied	apply	VERB
cana-1630	239	8	nonlinear	nonlinear	ADJ
cana-1630	239	9	analysis	analysis	NOUN
cana-1630	239	10	issn	issn	NOUN
cana-1630	239	11	:	:	PUNCT
cana-1630	239	12	1074	1074	NUM
cana-1630	239	13	-	-	PUNCT
cana-1630	239	14	133x	133x	NUM
cana-1630	239	15	vol	vol	NOUN
cana-1630	239	16	32	32	NUM
cana-1630	239	17	no	no	NOUN
cana-1630	239	18	.	.	NOUN
cana-1630	239	19	1	1	NUM
cana-1630	239	20	(	(	PUNCT
cana-1630	239	21	2025	2025	NUM
cana-1630	239	22	)	)	PUNCT
cana-1630	239	23	198	198	NUM
cana-1630	239	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-1630	239	25	adaptive	adaptive	ADJ
cana-1630	239	26	gaussian	gaussian	ADJ
cana-1630	239	27	derivative	derivative	ADJ
cana-1630	239	28	filtering	filtering	NOUN
cana-1630	239	29	can	can	AUX
cana-1630	239	30	greatly	greatly	ADV
cana-1630	239	31	enhance	enhance	VERB
cana-1630	239	32	the	the	DET
cana-1630	239	33	reliability	reliability	NOUN
cana-1630	239	34	and	and	CCONJ
cana-1630	239	35	performance	performance	NOUN
cana-1630	239	36	of	of	ADP
cana-1630	239	37	fuzzy	fuzzy	ADJ
cana-1630	239	38	logic	logic	NOUN
cana-1630	239	39	systems	system	NOUN
cana-1630	239	40	,	,	PUNCT
cana-1630	239	41	making	make	VERB
cana-1630	239	42	them	they	PRON
cana-1630	239	43	more	more	ADV
cana-1630	239	44	adept	adept	ADJ
cana-1630	239	45	at	at	ADP
cana-1630	239	46	handling	handle	VERB
cana-1630	239	47	real	real	ADJ
cana-1630	239	48	-	-	PUNCT
cana-1630	239	49	world	world	NOUN
cana-1630	239	50	data	datum	NOUN
cana-1630	239	51	that	that	PRON
cana-1630	239	52	is	be	AUX
cana-1630	239	53	often	often	ADV
cana-1630	239	54	noisy	noisy	ADJ
cana-1630	239	55	and	and	CCONJ
cana-1630	239	56	irregular	irregular	ADJ
cana-1630	239	57	.	.	PUNCT
cana-1630	240	1	these	these	DET
cana-1630	240	2	advancements	advancement	NOUN
cana-1630	240	3	hold	hold	VERB
cana-1630	240	4	considerable	considerable	ADJ
cana-1630	240	5	promise	promise	NOUN
cana-1630	240	6	for	for	ADP
cana-1630	240	7	a	a	DET
cana-1630	240	8	wide	wide	ADJ
cana-1630	240	9	range	range	NOUN
cana-1630	240	10	of	of	ADP
cana-1630	240	11	applications	application	NOUN
cana-1630	240	12	,	,	PUNCT
cana-1630	240	13	from	from	ADP
cana-1630	240	14	image	image	NOUN
cana-1630	240	15	processing	processing	NOUN
cana-1630	240	16	to	to	ADP
cana-1630	240	17	pattern	pattern	NOUN
cana-1630	240	18	recognition	recognition	NOUN
cana-1630	240	19	and	and	CCONJ
cana-1630	240	20	beyond	beyond	ADP
cana-1630	240	21	.	.	PUNCT
cana-1630	241	1	by	by	ADP
cana-1630	241	2	ensuring	ensure	VERB
cana-1630	241	3	the	the	DET
cana-1630	241	4	stability	stability	NOUN
cana-1630	241	5	and	and	CCONJ
cana-1630	241	6	accuracy	accuracy	NOUN
cana-1630	241	7	of	of	ADP
cana-1630	241	8	fuzzy	fuzzy	ADJ
cana-1630	241	9	membership	membership	NOUN
cana-1630	241	10	functions	function	NOUN
cana-1630	241	11	,	,	PUNCT
cana-1630	241	12	this	this	DET
cana-1630	241	13	approach	approach	NOUN
cana-1630	241	14	can	can	AUX
cana-1630	241	15	contribute	contribute	VERB
cana-1630	241	16	to	to	ADP
cana-1630	241	17	more	more	ADV
cana-1630	241	18	reliable	reliable	ADJ
cana-1630	241	19	and	and	CCONJ
cana-1630	241	20	precise	precise	ADJ
cana-1630	241	21	computational	computational	ADJ
cana-1630	241	22	models	model	NOUN
cana-1630	241	23	,	,	PUNCT
cana-1630	241	24	ultimately	ultimately	ADV
cana-1630	241	25	enhancing	enhance	VERB
cana-1630	241	26	the	the	DET
cana-1630	241	27	effectiveness	effectiveness	NOUN
cana-1630	241	28	of	of	ADP
cana-1630	241	29	systems	system	NOUN
cana-1630	241	30	that	that	PRON
cana-1630	241	31	rely	rely	VERB
cana-1630	241	32	on	on	ADP
cana-1630	241	33	fuzzy	fuzzy	ADJ
cana-1630	241	34	logic	logic	NOUN
cana-1630	241	35	for	for	ADP
cana-1630	241	36	decision	decision	NOUN
cana-1630	241	37	-	-	PUNCT
cana-1630	241	38	making	making	NOUN
cana-1630	241	39	and	and	CCONJ
cana-1630	241	40	analysis	analysis	NOUN
cana-1630	241	41	.	.	PUNCT
cana-1630	242	1	the	the	DET
cana-1630	242	2	potential	potential	NOUN
cana-1630	242	3	for	for	ADP
cana-1630	242	4	improved	improved	ADJ
cana-1630	242	5	robustness	robustness	NOUN
cana-1630	242	6	in	in	ADP
cana-1630	242	7	fuzzy	fuzzy	ADJ
cana-1630	242	8	logic	logic	NOUN
cana-1630	242	9	systems	system	NOUN
cana-1630	242	10	marks	mark	VERB
cana-1630	242	11	a	a	DET
cana-1630	242	12	significant	significant	ADJ
cana-1630	242	13	step	step	NOUN
cana-1630	242	14	forward	forward	ADV
cana-1630	242	15	in	in	ADP
cana-1630	242	16	the	the	DET
cana-1630	242	17	field	field	NOUN
cana-1630	242	18	,	,	PUNCT
cana-1630	242	19	paving	pave	VERB
cana-1630	242	20	the	the	DET
cana-1630	242	21	way	way	NOUN
cana-1630	242	22	for	for	ADP
cana-1630	242	23	future	future	ADJ
cana-1630	242	24	research	research	NOUN
cana-1630	242	25	and	and	CCONJ
cana-1630	242	26	development	development	NOUN
cana-1630	242	27	.	.	PUNCT
cana-1630	243	1	references	reference	NOUN
cana-1630	243	2	[	[	X
cana-1630	243	3	1	1	X
cana-1630	243	4	]	]	PUNCT
cana-1630	243	5	akula	akula	PROPN
cana-1630	243	6	suneetha	suneetha	PROPN
cana-1630	243	7	and	and	CCONJ
cana-1630	243	8	e.	e.	PROPN
cana-1630	243	9	srinivasa	srinivasa	PROPN
cana-1630	243	10	reddy	reddy	PROPN
cana-1630	243	11	:	:	PUNCT
cana-1630	243	12	robust	robust	ADJ
cana-1630	243	13	gaussian	gaussian	ADJ
cana-1630	243	14	noise	noise	NOUN
cana-1630	243	15	detection	detection	NOUN
cana-1630	243	16	and	and	CCONJ
cana-1630	243	17	removal	removal	NOUN
cana-1630	243	18	in	in	ADP
cana-1630	243	19	color	color	NOUN
cana-1630	243	20	images	image	NOUN
cana-1630	243	21	using	use	VERB
cana-1630	243	22	modified	modify	VERB
cana-1630	243	23	fuzzy	fuzzy	ADJ
cana-1630	243	24	set	set	NOUN
cana-1630	243	25	filter	filter	NOUN
cana-1630	243	26	,	,	PUNCT
cana-1630	243	27	journal	journal	NOUN
cana-1630	243	28	of	of	ADP
cana-1630	243	29	intelligence	intelligence	NOUN
cana-1630	243	30	system	system	NOUN
cana-1630	243	31	,	,	PUNCT
cana-1630	243	32	vol.30	vol.30	NOUN
cana-1630	243	33	,	,	PUNCT
cana-1630	243	34	pp	pp	ADJ
cana-1630	243	35	.	.	PUNCT
cana-1630	244	1	240	240	NUM
cana-1630	244	2	-	-	SYM
cana-1630	244	3	257	257	NUM
cana-1630	244	4	(	(	PUNCT
cana-1630	244	5	2021	2021	NUM
cana-1630	244	6	)	)	PUNCT
cana-1630	245	1	[	[	X
cana-1630	245	2	2	2	X
cana-1630	245	3	]	]	PUNCT
cana-1630	245	4	alex	alex	PROPN
cana-1630	245	5	a	a	X
cana-1630	245	6	,	,	PUNCT
cana-1630	245	7	gorodetsky	gorodetsky	NOUN
cana-1630	245	8	,	,	PUNCT
cana-1630	245	9	sertac	sertac	PROPN
cana-1630	245	10	karaman	karaman	PROPN
cana-1630	245	11	,	,	PUNCT
cana-1630	245	12	youssef	youssef	PROPN
cana-1630	245	13	m.	m.	PROPN
cana-1630	245	14	marzouk	marzouk	ADJ
cana-1630	245	15	,	,	PUNCT
cana-1630	245	16	low	low	ADJ
cana-1630	245	17	-	-	PUNCT
cana-1630	245	18	rank	rank	NOUN
cana-1630	245	19	tensor	tensor	NOUN
cana-1630	245	20	integration	integration	NOUN
cana-1630	245	21	for	for	ADP
cana-1630	245	22	gaussian	gaussian	ADJ
cana-1630	245	23	filtering	filtering	NOUN
cana-1630	245	24	of	of	ADP
cana-1630	245	25	continuous	continuous	ADJ
cana-1630	245	26	time	time	NOUN
cana-1630	245	27	nonlinear	nonlinear	PROPN
cana-1630	245	28	systems	systems	PROPN
cana-1630	245	29	,	,	PUNCT
cana-1630	245	30	institute	institute	NOUN
cana-1630	245	31	of	of	ADP
cana-1630	245	32	electrical	electrical	ADJ
cana-1630	245	33	and	and	CCONJ
cana-1630	245	34	electronics	electronic	NOUN
cana-1630	245	35	engineers	engineer	NOUN
cana-1630	245	36	(	(	PUNCT
cana-1630	245	37	ieee	ieee	NOUN
cana-1630	245	38	)	)	PUNCT
cana-1630	245	39	,	,	PUNCT
cana-1630	245	40	(	(	PUNCT
cana-1630	245	41	2017	2017	NUM
cana-1630	245	42	)	)	PUNCT
cana-1630	245	43	.	.	PUNCT
cana-1630	246	1	[	[	X
cana-1630	246	2	3	3	X
cana-1630	246	3	]	]	X
cana-1630	246	4	alex	alex	PROPN
cana-1630	246	5	keilmann	keilmann	PROPN
cana-1630	246	6	,	,	PUNCT
cana-1630	246	7	michael	michael	PROPN
cana-1630	246	8	godehardt	godehardt	PROPN
cana-1630	246	9	,	,	PUNCT
cana-1630	246	10	ali	ali	PROPN
cana-1630	246	11	moghiseh	moghiseh	PROPN
cana-1630	246	12	,	,	PUNCT
cana-1630	246	13	claudia	claudia	PROPN
cana-1630	246	14	redenbach	redenbach	NOUN
cana-1630	246	15	,	,	PUNCT
cana-1630	246	16	katja	katja	PROPN
cana-1630	246	17	schladitz	schladitz	NOUN
cana-1630	246	18	:	:	PUNCT
cana-1630	246	19	improved	improve	VERB
cana-1630	246	20	anisotropic	anisotropic	NOUN
cana-1630	246	21	gaussian	gaussian	ADJ
cana-1630	246	22	filters	filter	NOUN
cana-1630	246	23	,	,	PUNCT
cana-1630	246	24	computer	computer	NOUN
cana-1630	246	25	vision	vision	NOUN
cana-1630	246	26	and	and	CCONJ
cana-1630	246	27	pattern	pattern	NOUN
cana-1630	246	28	recognition	recognition	NOUN
cana-1630	246	29	,	,	PUNCT
cana-1630	246	30	(	(	PUNCT
cana-1630	246	31	2023	2023	NUM
cana-1630	246	32	)	)	PUNCT
cana-1630	247	1	[	[	X
cana-1630	247	2	4	4	NUM
cana-1630	247	3	]	]	X
cana-1630	247	4	animikh	animikh	PROPN
cana-1630	247	5	biswas	biswas	PROPN
cana-1630	247	6	,	,	PUNCT
cana-1630	247	7	michal	michal	PROPN
cana-1630	247	8	branicki	branicki	PROPN
cana-1630	247	9	,	,	PUNCT
cana-1630	247	10	a	a	DET
cana-1630	247	11	unified	unified	ADJ
cana-1630	247	12	framework	framework	NOUN
cana-1630	247	13	for	for	ADP
cana-1630	247	14	the	the	DET
cana-1630	247	15	analysis	analysis	NOUN
cana-1630	247	16	of	of	ADP
cana-1630	247	17	accuracy	accuracy	NOUN
cana-1630	247	18	and	and	CCONJ
cana-1630	247	19	stability	stability	NOUN
cana-1630	247	20	of	of	ADP
cana-1630	247	21	a	a	DET
cana-1630	247	22	class	class	NOUN
cana-1630	247	23	of	of	ADP
cana-1630	247	24	approximate	approximate	ADJ
cana-1630	247	25	gaussian	gaussian	ADJ
cana-1630	247	26	filters	filter	NOUN
cana-1630	247	27	for	for	ADP
cana-1630	247	28	the	the	DET
cana-1630	247	29	navier	navier	NOUN
cana-1630	247	30	-	-	PUNCT
cana-1630	247	31	stokes	stoke	NOUN
cana-1630	247	32	equations	equation	NOUN
cana-1630	247	33	,	,	PUNCT
cana-1630	247	34	feb	feb	X
cana-1630	247	35	(	(	PUNCT
cana-1630	247	36	2024	2024	NUM
cana-1630	247	37	)	)	PUNCT
cana-1630	247	38	.	.	PUNCT
cana-1630	248	1	[	[	X
cana-1630	248	2	5	5	X
cana-1630	248	3	]	]	PUNCT
cana-1630	248	4	chigansky	chigansky	NOUN
cana-1630	248	5	p	p	NOUN
cana-1630	248	6	,	,	PUNCT
cana-1630	248	7	liptser	liptser	NOUN
cana-1630	248	8	r	r	NOUN
cana-1630	248	9	,	,	PUNCT
cana-1630	248	10	van	van	PROPN
cana-1630	248	11	handel	handel	PROPN
cana-1630	248	12	r	r	PROPN
cana-1630	248	13	,	,	PUNCT
cana-1630	248	14	intrinsic	intrinsic	ADJ
cana-1630	248	15	methods	method	NOUN
cana-1630	248	16	in	in	ADP
cana-1630	248	17	filter	filter	NOUN
cana-1630	248	18	stability	stability	NOUN
cana-1630	248	19	,	,	PUNCT
cana-1630	248	20	the	the	DET
cana-1630	248	21	oxford	oxford	PROPN
cana-1630	248	22	handbook	handbook	NOUN
cana-1630	248	23	of	of	ADP
cana-1630	248	24	non	non	ADJ
cana-1630	248	25	-	-	ADJ
cana-1630	248	26	linear	linear	ADJ
cana-1630	248	27	filtering	filtering	NOUN
cana-1630	248	28	,	,	PUNCT
cana-1630	248	29	pp	pp	ADJ
cana-1630	248	30	.	.	PUNCT
cana-1630	249	1	319	319	NUM
cana-1630	249	2	-	-	SYM
cana-1630	249	3	351	351	NUM
cana-1630	249	4	,	,	PUNCT
cana-1630	249	5	(	(	PUNCT
cana-1630	249	6	2011	2011	NUM
cana-1630	249	7	)	)	PUNCT
cana-1630	249	8	.	.	PUNCT
cana-1630	250	1	[	[	X
cana-1630	250	2	6	6	NUM
cana-1630	250	3	]	]	X
cana-1630	250	4	christoph	christoph	PROPN
cana-1630	250	5	d.	d.	PROPN
cana-1630	250	6	mathys	mathys	PROPN
cana-1630	250	7	,	,	PUNCT
cana-1630	250	8	ekaterina	ekaterina	PROPN
cana-1630	250	9	i.lomakina	i.lomakina	PROPN
cana-1630	250	10	,	,	PUNCT
cana-1630	250	11	jean	jean	PROPN
cana-1630	250	12	daunizeau	daunizeau	PROPN
cana-1630	250	13	,	,	PUNCT
cana-1630	250	14	sandra	sandra	PROPN
cana-1630	250	15	iglesias	iglesias	PROPN
cana-1630	250	16	,	,	PUNCT
cana-1630	250	17	kay	kay	PROPN
cana-1630	250	18	h.	h.	PROPN
cana-1630	250	19	broderson	broderson	PROPN
cana-1630	250	20	,	,	PUNCT
cana-1630	250	21	karl	karl	PROPN
cana-1630	250	22	j.	j.	PROPN
cana-1630	250	23	friston	friston	PROPN
cana-1630	250	24	and	and	CCONJ
cana-1630	250	25	klaas	klaas	PROPN
cana-1630	250	26	e.	e.	PROPN
cana-1630	250	27	stephan	stephan	PROPN
cana-1630	250	28	,	,	PUNCT
cana-1630	250	29	uncertainty	uncertainty	NOUN
cana-1630	250	30	in	in	ADP
cana-1630	250	31	perception	perception	NOUN
cana-1630	250	32	and	and	CCONJ
cana-1630	250	33	the	the	DET
cana-1630	250	34	hierarchical	hierarchical	ADJ
cana-1630	250	35	gaussian	gaussian	ADJ
cana-1630	250	36	filter	filter	NOUN
cana-1630	250	37	,	,	PUNCT
cana-1630	250	38	frontiers	frontier	NOUN
cana-1630	250	39	in	in	ADP
cana-1630	250	40	human	human	ADJ
cana-1630	250	41	neuroscience	neuroscience	NOUN
cana-1630	250	42	,	,	PUNCT
cana-1630	250	43	vol	vol	NOUN
cana-1630	250	44	.	.	PROPN
cana-1630	250	45	8	8	NUM
cana-1630	250	46	,	,	PUNCT
cana-1630	250	47	nov	nov	PROPN
cana-1630	250	48	.	.	PROPN
cana-1630	250	49	(	(	PUNCT
cana-1630	250	50	2014	2014	NUM
cana-1630	250	51	)	)	PUNCT
cana-1630	250	52	.	.	PUNCT
cana-1630	251	1	doi:10.3389	doi:10.3389	PROPN
cana-1630	251	2	/	/	SYM
cana-1630	251	3	fnhum.2014.00825	fnhum.2014.00825	PROPN
cana-1630	252	1	[	[	X
cana-1630	252	2	7	7	NUM
cana-1630	252	3	]	]	X
cana-1630	252	4	harishvijey	harishvijey	NOUN
cana-1630	252	5	a	a	X
cana-1630	252	6	,	,	PUNCT
cana-1630	252	7	benadict	benadict	PROPN
cana-1630	252	8	raja	raja	PROPN
cana-1630	252	9	j	j	PROPN
cana-1630	252	10	:	:	PUNCT
cana-1630	252	11	automated	automate	VERB
cana-1630	252	12	technique	technique	NOUN
cana-1630	252	13	for	for	ADP
cana-1630	252	14	eeg	eeg	NOUN
cana-1630	252	15	signal	signal	NOUN
cana-1630	252	16	processing	processing	NOUN
cana-1630	252	17	to	to	PART
cana-1630	252	18	detect	detect	VERB
cana-1630	252	19	seizure	seizure	NOUN
cana-1630	252	20	with	with	ADP
cana-1630	252	21	optimized	optimize	VERB
cana-1630	252	22	variable	variable	ADJ
cana-1630	252	23	gaussian	gaussian	ADJ
cana-1630	252	24	filter	filter	NOUN
cana-1630	252	25	and	and	CCONJ
cana-1630	252	26	fuzzy	fuzzy	ADJ
cana-1630	252	27	rbfelm	rbfelm	NOUN
cana-1630	252	28	classifier	classifier	NOUN
cana-1630	252	29	,	,	PUNCT
cana-1630	252	30	biomedical	biomedical	ADJ
cana-1630	252	31	signal	signal	NOUN
cana-1630	252	32	processing	processing	NOUN
cana-1630	252	33	and	and	CCONJ
cana-1630	252	34	control	control	NOUN
cana-1630	252	35	,	,	PUNCT
cana-1630	252	36	vol	vol	NOUN
cana-1630	252	37	.	.	PROPN
cana-1630	252	38	74	74	NUM
cana-1630	252	39	,	,	PUNCT
cana-1630	252	40	(	(	PUNCT
cana-1630	252	41	2022	2022	NUM
cana-1630	252	42	)	)	PUNCT
cana-1630	252	43	.	.	PUNCT
cana-1630	253	1	10.1016	10.1016	NUM
cana-1630	253	2	/	/	SYM
cana-1630	253	3	j.bspc.2021.103450	j.bspc.2021.103450	NOUN
cana-1630	254	1	[	[	X
cana-1630	254	2	8	8	NUM
cana-1630	254	3	]	]	X
cana-1630	254	4	jin	jin	NOUN
cana-1630	254	5	won	win	VERB
cana-1630	254	6	kim	kim	PROPN
cana-1630	254	7	,	,	PUNCT
cana-1630	254	8	anant	anant	PROPN
cana-1630	254	9	a.	a.	PROPN
cana-1630	254	10	joshi	joshi	PROPN
cana-1630	254	11	,	,	PUNCT
cana-1630	254	12	prashant	prashant	PROPN
cana-1630	254	13	g.	g.	PROPN
cana-1630	254	14	mehta	mehta	PROPN
cana-1630	254	15	,	,	PUNCT
cana-1630	254	16	backward	backward	ADJ
cana-1630	254	17	map	map	NOUN
cana-1630	254	18	for	for	ADP
cana-1630	254	19	filter	filter	NOUN
cana-1630	254	20	stability	stability	NOUN
cana-1630	254	21	analysis	analysis	NOUN
cana-1630	254	22	,	,	PUNCT
cana-1630	254	23	optimization	optimization	NOUN
cana-1630	254	24	and	and	CCONJ
cana-1630	254	25	control,(2024	control,(2024	NUM
cana-1630	254	26	)	)	PUNCT
cana-1630	255	1	[	[	X
cana-1630	255	2	9	9	NUM
cana-1630	255	3	]	]	X
cana-1630	255	4	junfeng	junfeng	PROPN
cana-1630	255	5	zhang	zhang	PROPN
cana-1630	255	6	,	,	PUNCT
cana-1630	255	7	xiao	xiao	PROPN
cana-1630	255	8	he	he	PRON
cana-1630	255	9	,	,	PUNCT
cana-1630	255	10	donghua	donghua	PROPN
cana-1630	255	11	zhou	zhou	PROPN
cana-1630	255	12	:	:	PUNCT
cana-1630	255	13	generalised	generalise	VERB
cana-1630	255	14	proportional	proportional	ADJ
cana-1630	255	15	–	–	PUNCT
cana-1630	255	16	integral	integral	ADJ
cana-1630	255	17	–	–	PUNCT
cana-1630	255	18	derivative	derivative	ADJ
cana-1630	255	19	filter	filter	NOUN
cana-1630	255	20	,	,	PUNCT
cana-1630	255	21	iet	iet	PROPN
cana-1630	255	22	control	control	PROPN
cana-1630	255	23	theory	theory	NOUN
cana-1630	255	24	and	and	CCONJ
cana-1630	255	25	applications	application	NOUN
cana-1630	255	26	,	,	PUNCT
cana-1630	255	27	vol	vol	NOUN
cana-1630	255	28	.	.	PROPN
cana-1630	255	29	10	10	NUM
cana-1630	255	30	,	,	PUNCT
cana-1630	255	31	issue	issue	NOUN
cana-1630	255	32	17	17	NUM
cana-1630	255	33	,	,	PUNCT
cana-1630	255	34	pp	pp	ADJ
cana-1630	255	35	.	.	PUNCT
cana-1630	255	36	2339	2339	NUM
cana-1630	255	37	-	-	SYM
cana-1630	255	38	2347	2347	NUM
cana-1630	255	39	,	,	PUNCT
cana-1630	255	40	(	(	PUNCT
cana-1630	255	41	2016	2016	NUM
cana-1630	255	42	)	)	PUNCT
cana-1630	255	43	.	.	PUNCT
cana-1630	256	1	[	[	X
cana-1630	256	2	10	10	NUM
cana-1630	256	3	]	]	X
cana-1630	256	4	koichiro	koichiro	PROPN
cana-1630	256	5	watanabe	watanabe	PROPN
cana-1630	256	6	,	,	PUNCT
cana-1630	256	7	yoshihiro	yoshihiro	PROPN
cana-1630	256	8	maeda	maeda	PROPN
cana-1630	256	9	,	,	PUNCT
cana-1630	256	10	norishige	norishige	NOUN
cana-1630	256	11	fukushima	fukushima	NOUN
cana-1630	256	12	,	,	PUNCT
cana-1630	256	13	stability	stability	NOUN
cana-1630	256	14	of	of	ADP
cana-1630	256	15	recursive	recursive	ADJ
cana-1630	256	16	gaussian	gaussian	ADJ
cana-1630	256	17	filtering	filtering	NOUN
cana-1630	256	18	for	for	ADP
cana-1630	256	19	piecewise	piecewise	NOUN
cana-1630	256	20	linear	linear	ADJ
cana-1630	256	21	bilateral	bilateral	ADJ
cana-1630	256	22	filtering	filtering	NOUN
cana-1630	256	23	,	,	PUNCT
cana-1630	256	24	international	international	ADJ
cana-1630	256	25	conference	conference	NOUN
cana-1630	256	26	on	on	ADP
cana-1630	256	27	frontiers	frontier	NOUN
cana-1630	256	28	of	of	ADP
cana-1630	256	29	computer	computer	NOUN
cana-1630	256	30	vision	vision	NOUN
cana-1630	256	31	,	,	PUNCT
cana-1630	256	32	(	(	PUNCT
cana-1630	256	33	2018	2018	NUM
cana-1630	256	34	)	)	PUNCT
cana-1630	256	35	.	.	PUNCT
cana-1630	257	1	[	[	X
cana-1630	257	2	11	11	NUM
cana-1630	257	3	]	]	PUNCT
cana-1630	257	4	manuel	manuel	NOUN
cana-1630	257	5	wuthrich	wuthrich	PROPN
cana-1630	257	6	,	,	PUNCT
cana-1630	257	7	sebastian	sebastian	PROPN
cana-1630	257	8	trimpe	trimpe	PROPN
cana-1630	257	9	,	,	PUNCT
cana-1630	257	10	crisina	crisina	PROPN
cana-1630	257	11	garcia	garcia	PROPN
cana-1630	257	12	cifuentes	cifuentes	PROPN
cana-1630	257	13	,	,	PUNCT
cana-1630	257	14	daniel	daniel	PROPN
cana-1630	257	15	kappler	kappler	PROPN
cana-1630	257	16	,	,	PUNCT
cana-1630	257	17	stefan	stefan	PROPN
cana-1630	257	18	schaal	schaal	PROPN
cana-1630	257	19	,	,	PUNCT
cana-1630	257	20	a	a	DET
cana-1630	257	21	new	new	ADJ
cana-1630	257	22	perspective	perspective	NOUN
cana-1630	257	23	and	and	CCONJ
cana-1630	257	24	extension	extension	NOUN
cana-1630	257	25	of	of	ADP
cana-1630	257	26	gaussian	gaussian	ADJ
cana-1630	257	27	filter	filter	NOUN
cana-1630	257	28	,	,	PUNCT
cana-1630	257	29	the	the	DET
cana-1630	257	30	international	international	ADJ
cana-1630	257	31	journal	journal	NOUN
cana-1630	257	32	of	of	ADP
cana-1630	257	33	robotics	robotic	NOUN
cana-1630	257	34	research	research	NOUN
cana-1630	257	35	,	,	PUNCT
cana-1630	257	36	vol.35	vol.35	NOUN
cana-1630	257	37	,	,	PUNCT
cana-1630	257	38	issue	issue	NOUN
cana-1630	257	39	14	14	NUM
cana-1630	257	40	,	,	PUNCT
cana-1630	257	41	pp	pp	ADJ
cana-1630	257	42	.	.	PUNCT
cana-1630	258	1	1731	1731	NUM
cana-1630	258	2	-	-	SYM
cana-1630	258	3	1749	1749	NUM
cana-1630	258	4	,	,	PUNCT
cana-1630	258	5	(	(	PUNCT
cana-1630	258	6	2016	2016	NUM
cana-1630	258	7	)	)	PUNCT
cana-1630	258	8	.	.	PUNCT
cana-1630	259	1	10.1177/0278364916684019	10.1177/0278364916684019	NUM
cana-1630	260	1	[	[	X
cana-1630	260	2	12	12	NUM
cana-1630	260	3	]	]	PUNCT
cana-1630	260	4	qichun	qichun	PROPN
cana-1630	260	5	zhang	zhang	PROPN
cana-1630	260	6	,	,	PUNCT
cana-1630	260	7	yuyang	yuyang	PROPN
cana-1630	260	8	zhou	zhou	PROPN
cana-1630	260	9	,	,	PUNCT
cana-1630	260	10	recent	recent	ADJ
cana-1630	260	11	advances	advance	NOUN
cana-1630	260	12	in	in	ADP
cana-1630	260	13	non	non	ADJ
cana-1630	260	14	-	-	ADJ
cana-1630	260	15	gaussian	gaussian	ADJ
cana-1630	260	16	stochastic	stochastic	NOUN
cana-1630	260	17	systems	system	NOUN
cana-1630	260	18	control	control	NOUN
cana-1630	260	19	theory	theory	NOUN
cana-1630	260	20	and	and	CCONJ
cana-1630	260	21	its	its	PRON
cana-1630	260	22	applications	application	NOUN
cana-1630	260	23	,	,	PUNCT
cana-1630	260	24	international	international	ADJ
cana-1630	260	25	journal	journal	NOUN
cana-1630	260	26	of	of	ADP
cana-1630	260	27	network	network	NOUN
cana-1630	260	28	dynamics	dynamic	NOUN
cana-1630	260	29	and	and	CCONJ
cana-1630	260	30	intelligence	intelligence	NOUN
cana-1630	260	31	,	,	PUNCT
cana-1630	260	32	vol	vol	NOUN
cana-1630	260	33	.	.	NOUN
cana-1630	260	34	1(1	1(1	NUM
cana-1630	260	35	)	)	PUNCT
cana-1630	260	36	,	,	PUNCT
cana-1630	260	37	pp	pp	ADV
cana-1630	260	38	111	111	NUM
cana-1630	260	39	-	-	SYM
cana-1630	260	40	119	119	NUM
cana-1630	260	41	,	,	PUNCT
cana-1630	260	42	(	(	PUNCT
cana-1630	260	43	2022	2022	NUM
cana-1630	260	44	)	)	PUNCT
cana-1630	260	45	.	.	PUNCT
cana-1630	261	1	[	[	X
cana-1630	261	2	13	13	NUM
cana-1630	261	3	]	]	X
cana-1630	261	4	toni	toni	PROPN
cana-1630	261	5	karvonen	karvonen	PROPN
cana-1630	261	6	,	,	PUNCT
cana-1630	261	7	silvere	silvere	NOUN
cana-1630	261	8	bonnabel	bonnabel	NOUN
cana-1630	261	9	,	,	PUNCT
cana-1630	261	10	eric	eric	PROPN
cana-1630	261	11	moulines	moulines	PROPN
cana-1630	261	12	,	,	PUNCT
cana-1630	261	13	simo	simo	PROPN
cana-1630	261	14	sarkka	sarkka	VERB
cana-1630	261	15	,	,	PUNCT
cana-1630	261	16	on	on	ADP
cana-1630	261	17	stability	stability	NOUN
cana-1630	261	18	of	of	ADP
cana-1630	261	19	a	a	DET
cana-1630	261	20	class	class	NOUN
cana-1630	261	21	od	od	PROPN
cana-1630	261	22	filters	filter	NOUN
cana-1630	261	23	non	non	ADJ
cana-1630	261	24	-	-	ADJ
cana-1630	261	25	linear	linear	ADJ
cana-1630	261	26	stochastic	stochastic	NOUN
cana-1630	261	27	systems	system	NOUN
cana-1630	261	28	,	,	PUNCT
cana-1630	261	29	siam	siam	ADJ
cana-1630	261	30	journal	journal	NOUN
cana-1630	261	31	of	of	ADP
cana-1630	261	32	control	control	NOUN
cana-1630	261	33	and	and	CCONJ
cana-1630	261	34	optimization	optimization	NOUN
cana-1630	261	35	,	,	PUNCT
cana-1630	261	36	(	(	PUNCT
cana-1630	261	37	2020	2020	NUM
cana-1630	261	38	)	)	PUNCT
cana-1630	261	39	.	.	PUNCT
cana-1630	262	1	[	[	X
cana-1630	262	2	14	14	NUM
cana-1630	262	3	]	]	X
cana-1630	262	4	wernerson	wernerson	PROPN
cana-1630	262	5	d.	d.	PROPN
cana-1630	262	6	parreira	parreira	PROPN
cana-1630	262	7	,	,	PUNCT
cana-1630	262	8	jose	jose	PROPN
cana-1630	262	9	carlos	carlos	PROPN
cana-1630	262	10	m.	m.	PROPN
cana-1630	262	11	bermudez	bermudez	PROPN
cana-1630	262	12	,	,	PUNCT
cana-1630	262	13	cedric	cedric	PROPN
cana-1630	262	14	richard	richard	PROPN
cana-1630	262	15	,	,	PUNCT
cana-1630	262	16	jean	jean	PROPN
cana-1630	262	17	–	–	PUNCT
cana-1630	262	18	yves	yves	PROPN
cana-1630	262	19	tourneret	tourneret	PROPN
cana-1630	262	20	,	,	PUNCT
cana-1630	262	21	stability	stability	NOUN
cana-1630	262	22	behavior	behavior	NOUN
cana-1630	262	23	analysis	analysis	NOUN
cana-1630	262	24	of	of	ADP
cana-1630	262	25	the	the	DET
cana-1630	262	26	gaussian	gaussian	ADJ
cana-1630	262	27	kernel	kernel	NOUN
cana-1630	262	28	-	-	PUNCT
cana-1630	262	29	least	least	ADJ
cana-1630	262	30	-	-	PUNCT
cana-1630	262	31	mean	mean	ADJ
cana-1630	262	32	-	-	PUNCT
cana-1630	262	33	square	square	NOUN
cana-1630	262	34	algorithm	algorithm	NOUN
cana-1630	262	35	,	,	PUNCT
cana-1630	262	36	ieee	ieee	NOUN
cana-1630	262	37	transactions	transaction	NOUN
cana-1630	262	38	on	on	ADP
cana-1630	262	39	signal	signal	ADJ
cana-1630	262	40	processing	processing	NOUN
cana-1630	262	41	,	,	PUNCT
cana-1630	262	42	vol	vol	NOUN
cana-1630	262	43	.	.	PROPN
cana-1630	262	44	60	60	NUM
cana-1630	262	45	,	,	PUNCT
cana-1630	262	46	pp	pp	ADP
cana-1630	262	47	22082222	22082222	NUM
cana-1630	262	48	,	,	PUNCT
cana-1630	262	49	(	(	PUNCT
cana-1630	262	50	2012	2012	NUM
cana-1630	262	51	)	)	PUNCT
cana-1630	262	52	.	.	PUNCT
cana-1630	263	1	[	[	X
cana-1630	263	2	15	15	NUM
cana-1630	263	3	]	]	X
cana-1630	263	4	zhao	zhao	X
cana-1630	263	5	,	,	PUNCT
cana-1630	263	6	zheng	zheng	PROPN
cana-1630	263	7	,	,	PUNCT
cana-1630	263	8	sarkka	sarkka	VERB
cana-1630	263	9	simo	simo	PROPN
cana-1630	263	10	,	,	PUNCT
cana-1630	263	11	non	non	ADJ
cana-1630	263	12	-	-	ADJ
cana-1630	263	13	linear	linear	ADJ
cana-1630	263	14	gaussian	gaussian	NOUN
cana-1630	263	15	smoothing	smoothing	NOUN
cana-1630	263	16	with	with	ADP
cana-1630	263	17	taylor	taylor	PROPN
cana-1630	263	18	moment	moment	PROPN
cana-1630	263	19	expansion	expansion	NOUN
cana-1630	263	20	,	,	PUNCT
cana-1630	263	21	ieee	ieee	NOUN
cana-1630	263	22	signal	signal	NOUN
cana-1630	263	23	processing	processing	NOUN
cana-1630	263	24	letters	letter	NOUN
cana-1630	263	25	,	,	PUNCT
cana-1630	263	26	vol	vol	NOUN
cana-1630	263	27	.	.	PROPN
cana-1630	263	28	29	29	NUM
cana-1630	263	29	,	,	PUNCT
cana-1630	263	30	pp	pp	ADV
cana-1630	263	31	80	80	NUM
cana-1630	263	32	-	-	SYM
cana-1630	263	33	84	84	NUM
cana-1630	263	34	,	,	PUNCT
cana-1630	263	35	https://doi.org/10.1109/lsp.2021.3125831	https://doi.org/10.1109/lsp.2021.3125831	NOUN