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IIUM Engineering Journal, Vol. 2, No. 2, 2001 A. Hossen
21
A NEW FAST APPROXIMATE HILBERT TRANSFORM WITH DIFFERENT
APPLICATIONS
Abdulnasir Hossen
Information Engineering Department, Sultan Qaboos University, P.O.Box 33 Al-Khod, 123
Sultanate of Oman
abhossen@squ.edu.om
Abstract: A new and fast approximate Hilbert
transform based on subband decomposition is
presented. This new algorithm is called the subband
(SB)-Hilbert transform. The reduction in complexity is
obtained for narrow-band signal applications by
considering only the band of most energy. Different
properties of the SB-Hilbert transform are discussed
with simulation examples. The new algorithm is
compared with the full band Hilbert transform in terms
of complexity and accuracy. The aliasing errors taking
place in the algorithm are found by applying the
Hilbert transform to the inverse FFT (time signal) of
the aliasing errors of the SB-FFT of the input signal.
Different examples are given to find the analytic signal
using SB-Hilbert transform with a varying number of
subbands. Applications of the new algorithm are given
in single-sideband amplitude modulation and in
demodulating frequency-modulated signals in
communication systems.
Key Words: Fast Algorithms, Hilbert Transform,
Analytic Signal Processing.
1. INTRODUCTION
The subband decomposition idea applied to both
FFT[1,2] and DCT[3,4] is used in this paper for computing
the Hilbert transform. The discrete Hilbert transform is
related to the discrete Fourier transform, and both of
them can be computed using the fast Fourier transform
FFT [5]. The Hilbert transform of a signal is equivalent
to a ±90° phase shift in all frequency components of
the signal. Considering a signal x(t) with Fourier
transform X(f), the Hilbert transform of x(t), denoted
by ˆ( )x t , is defined [6]:
1 ( )ˆ( )
x
x t d
t
(1)
From this equation, we note that the Hilbert transform
ˆ( )x t may be interpreted as the convolution of x(t) with
the time function h(t) =1/(π t). So for a discrete
sequence x(n), Eq. (1) can be represented as:
0
ˆ( ) ( ) ( )
i
i
x n h i x n i
(2)
Also we know that the convolution of two functions in
time domain is transformed into the multiplication of
their Fourier transforms in the frequency domain [7].
For the time function 1/(πt), we have a Fourier
transform of -jsgn(f), where sgn(f) is the signum or sign
function defined as
1 0,
sgn( ) 0 0,
1 0.
f
f f
f
Then, the Fourier transform ˆ ( )X f of ˆ( )x t is given by
ˆ ( ) sgn( ) ( )X f j f X f (3)
The Hilbert transform signal ˆ( )x t is obtained by taking
the inverse Fourier transform of ˆ ( )X f . The analytic
signal for a discrete sequence x(n) has a one sided-
Fourier transform since negative frequencies are zero.
It can be determined by calculating the FFT of the
input sequence, replacing those FFT coefficients that
correspond to negative frequencies with zeros, and
calculating the inverse FFT of the result. The
analytical signal of a real sequence is a complex
sequence with a real part, which is the original data,
and an imaginary part that contains the Hilbert
transform [5].
The following procedure is to be followed to compute
the Hilbert transform for a discrete signal[8]:
1. Calculating the FFT X(k) of the input
sequence x(n).
2. Creating a vector h(k) whose elements have
the values:
1 0,
2
( ) 2 1, 2, , 1
2
0 1, 2, , 1
2 2
N
k
N
h k k
N N
k N
3. Calculating the element-wise product of X
and h.
4. Calculating the inverse FFT (IFFT) of the
sequence obtained in step 3.
5. Taking the imaginary part of the result of
the above step.
For narrow-band signals, the subband FFT [1], [2] can be
obtained by decomposing the input sequence into two
IIUM Engineering Journal, Vol. 2, No. 2, 2001 A. Hossen
22
bands corresponding to low- and high-pass sequences.
The band with the greater energy is transformed while
the other band is ignored leading to a fast and
approximate FFT. In this work the subband
decomposition idea is applied to compute the Hilbert
transform.
The organization of the paper is as follows: In the next
section, the idea of the subband FFT[1], [2] is reviewed.
Section 3 introduces the basic idea of the fast
approximate Hilbert transform with its properties.
Section 4 compares the complexity of the subband
Hilbert transform with that of the full-band Hilbert
transform. The aliasing errors in the subband Hilbert
transform are discussed in section 5. Section 6
contains applications of the fast approximate Hilbert
transform in analytic-signal generation, in SSB-
modulation, and in FM demodulation. In section 7,
concluding remarks are given.
2. SUBBAND-FFT
In Fig. 1, a(n) and b(n) are the low-pass and high-pass
filtered versions of x(n), with g(n) and h(n) denoting
their factor-2 down-sampled versions, respectively:
)].12()2([
2
1)(
)]12()2([
2
1)(
nxnxnh
nxnxng
(4)
Fig. 1: Two-band decomposition of the subband DFT.
The full-band size-N DFT X(k) can be obtained by
combining the FFTs of g(n) and h(n)[1], [2]:
).k(
h
F)
k
N
W1()k(gF)
k
N
W1()k(X
(5)
Equation (5) is approximated for calculating only the
low-pass band:
( ) (1 ) ( ) , (0, 1, , / 4 1).N
kX k W F k k Ng» + Î -K (6)
The decomposition process in Fig. 1 can be repeated m
times to get M = 2m subbands, out of which only one
band is to be computed depending on information
known or derived adaptively [9] about the input signal
power distribution. Two types of approximation errors
(linear distortion and aliasing) are resulted from the
decomposition and approximation processes [2].
Substituting Fg(k) in Eq. (6) by the DFT of g(n), and
taking g(n) from Eq.(4), we get:
/ 2
/ 2 1
1ˆ ( ) (1 ) ( (2 ) (2 1)) .2
0
k kn
N N
N
X k W x n x n W
n
-
= + + +
=
å
(7)
Equation (7) can be also written as:
2 2
2 2
1 1
0 0
1ˆ ( ) (2 ) (2 1)2
N N
N N
kn k kn
N
n n
X k x n W W x n W
- -
= =
ì üï ïï ï= + +í ýï ïï ïî þ
å å
2 2
2 2
1 1
0 0
1 (2 ) (2 1)2
N N
N N
k kn kn
N
n n
W x n W x n W
- -
= =
ì üï ïï ï+ + +í ýï ïï ïî þ
å å (8)
The two types of approximation errors in )(ˆ kX can be
taken into account in the following equation [2]:
0 1
ˆ ( ) ( ) ( / 2)X k A X k A X k N= + + (9)
where A0 and A1 are two coefficients relating to linear
distortions in the exact transform X(k) and aliasing
error due to X(k+N/2), respectively. Noticing the first
half of Eq. (8) shows that it is the exact transform X(k)
multiplied by a factor of 1/2. The value of X(k+N/2)
can be found from X(k) by replacing each k in X(k)
with k+N/2. Now substituting for X(k) and X(k+N/2)
into Eq. (9) yields:
2 2
2 2
1 1
0
0 0
ˆ ( ) (2 ) (2 1)
N N
N N
kn k kn
N
n n
X k A x n W W x n W
- -
= =
ì üï ïï ïï ï= + +í ýï ïï ïï ïî þ
å å
2 2
2 2
1 1
0 0
(2 ) (2 1) .1
N N
N N
kn k kn
N
n n
A x n W W x n W
- -
= =
ì üï ïï ïï ï+ - +í ýï ïï ïï ïî þ
å å (10)
By comparing Eqs. 8 and 10 we obtain the following:
0 1
1 (1 ),2
k
NA A W+ = + (11)
0 1
1 (1 )2
k
NA A W
-- = + . (12)
Solving Eqs. 11 and 12 yields:
0
1 (1 cos(2 / )) ,2A k Np= + (13)
and
1
2sin( ) .2
j kA N
p= - (14)
Thus the one-stage approximation in Eq. (6) can be
expressed as
X1 2 2ˆ ( ) [1 cos( )] ( ) sin( ) ( )2 2 2
jk k NX k X k kN N
p p= + - +
(15)
where ˆ ( )X k is the approximate, and X(k) is the true
signal spectrum.
For M = 2m bands of decomposition, there will be M-1
aliasing terms, uniformly distributed on the frequency
axis. For the case of retaining only the low-low band
+
++
x(n) ½ + a(n) g(n) Fg(k)
+
+
+
- b(n) h(n) Fh(k)
+
2 FFT
2 FFT
z-1
IIUM Engineering Journal, Vol. 2, No. 2, 2001 A. Hossen
23
out of M=4 subbands in the approximation, Eq. (9) can
be written [2]:
10
ˆ ( ) ( ) ( )4
NX k A X k A X k= + + (16)
2 3 3( ) ( ),2 4
N NA X k A X k+ + + +
And the four aliasing coefficients are found to be:
0
1
2
3
1 2 4( ) [1 cos( )][1 cos( ) ],4
1 4 2 2( ) sin( )[1 cos( ) sin( )] ,8
2 4( ) sin ( )[1 cos( )] ,4
(1 ) 4 2 2( ) sin( ) [1 cos( ) sin( )] .8
k kA k N N
j k k kA k N N N
j k kA k N N
j k k kA k N N N
p p
p p p
p p
p p p
= + +
-= + -
-= +
- +
= + +
(17)
3. SUBBAND HILBERT TRANSFORM
The subband Hilbert transform is obtained using the
following steps:
1. Calculating the SB-FFT X of the input
sequence, this results in a sequence of
length 2
NL = for half-band case,
4
NL = for quarter-band case, or
generally NL M= for M sub-bands.
2. Creating a vector h with length L having
the values:
1 0, 2
( ) 2 1, 2, , 12
0 1, 2, , 12 2
Lk
Lh k k
L Lk L
ìï =ïïïïïï= = -íïïïï = + + -ïïîï
K
K
3. Calculating the element-wise product of
X and h
4. Calculating the IFFT of length L for the
sequence obtained in step 3.
5. Taking the imaginary part of the result of
step 4.
All properties known for the full-band Hilbert
transform are applicable also to the subband Hilbert
transform if the signal to be transformed is a narrow-
band signal. Some of these properties are investigated
in this section with simulation examples. In all these
examples the input signal is plotted with solid-line and
the output signal is plotted with dotted-line.
1. The Hilbert transform of a constant is zero. Fig. 2
shows that both full-band and half-band Hilbert
transforms of a constant is zero.
2. The Hilbert transform of the cosine function
cos(2πfcnT) is equal to sin(2πfcnT). Where T is the
sampling interval. Similarly, the function
sin(2πfcnT) has a Hilbert transform equal to -
cos(2πfcnT). Both the full-band and the SB-Hilbert
transformation produce this results, as shown in
Fig. 3
0 5 10 15 20 25 30 35 40 45 50
-2
-1
0
1
2
Full-band Hilbert Transform
0 5 10 15 20 25 30 35 40 45 50
-2
-1
0
1
2
Half-band Hilbert Transform
Fig. 2: Hilbert of a constant.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
-1.5
-1
-0.5
0
0.5
1
1.5
SB-Hilbert Transform of a cosine function
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
-1.5
-1
-0.5
0
0.5
1
1.5
SB-Hilbert Transform of a sine function
Fig. 3: Hilbert of a cosine function.
3. The energy content of a zero-mean signal is equal
to the energy content of the Hilbert transform. The
property is tested by finding the energy of a
sinusoidal signal (used in property 2) and the
energy of its full-band Hilbert transform and
subband Hilbert transform with M = 2 and M =4
and M = 8. It is found that the energy is almost the
same in all cases depending on the approximation
quality of the subband realization.
4. The signal x(t) and its Hilbert transform are
orthogonal. For the same example used to prove
the previous two properties, it is proved that, if
ˆ( )x t is used as a full-band Hilbert transform, then:
ˆ( ) ( ) 0x t x t dt
¥
- ¥
=ò (18)
and the result is almost zero if subband Hilbert
transforms of M = 2 or M = 4 or M = 8 are used.
5. The Hilbert transformation of a Hilbert transform
produces a phase shift of to the original signal.
In Fig. 4, a sinusoidal signal of 50 Hz is applied to
two stages of Hilbert transform. In each stage
either a full-band or a half-band Hilbert transform
is used.
IIUM Engineering Journal, Vol. 2, No. 2, 2001 A. Hossen
24
0 0.02 0.04 0.06
-1
-0.5
0
0.5
1
Both full-band transforms
0 0.02 0.04 0.06
-1
-0.5
0
0.5
1
Half-band and full-band transforms
0 0.02 0.04 0.06
-1
-0.5
0
0.5
1
Full-band and half-band transforms
0 0.02 0.04 0.06
-1
-0.5
0
0.5
1
Both half-band transforms
Fig. 4: Hilbert of Hilbert transform.
6. The Hilbert transform is a linear process. An input
data sequence x(n) of length 1024 points is used
with three impulses as follows:
( ) ( 25) 1.5 ( 50) 2 ( 75)x n n n nd d d= - + - + -
(19)
Knowing that the Hilbert transform of (t) is /a tp .
Both full-band and SB-Hilbert transforms results in
Fig. 5 prove the linearity theorem.
4. COMPLEXITY
The advantage of using subband decomposition
transforms is the reduction of the complexity in com-
parison to full-band[1-4]. Reductions in complexity are
obtained from all the steps of section 3. by using
sequences of length L instead of N. The subband
Hilbert transform is compared in computational
complexity with the full-band Hilbert transform. Table
1 shows the percentage reduction in complexity
(measured as execution time) of the Hilbert transform,
if SB-Hilbert transforms with different M are to be
used instead of full-band transform.
0 50 100 150 200 250
-1
-0.5
0
0.5
1
Full-band Hilbert Transform
0 50 100 150 200 250
-1
-0.5
0
0.5
1
Half-band Hilbert Transform
Fig. 5: Linearity Property of Hilbert Transform.
Table 1: Complexity reduction results.
5. ACCURACY
For N = 16 and M = 2 and if only the first band is
calculated, Table 2 shows the values of the computed
fast approximate (with aliasing errors) half-band
Hilbert transform ˆ( )x n , assuming that the ratio of the
frequency-transform points causing aliasing to the true
components in the calculated band is fixed to .
The values in Table 2 can be computed by calculating
the aliasing errors of the SB-FFT and then finding the
Hilbert transform of their time signal. The following
numerical example describes this interesting result for
N = 8 and M = 2:
1. For a case of no aliasing error, a frequency signal
of (1,1,0,0,0,0,0,1) is assumed. The half-band
Hilbert transform of the first two-points (the low-
band case) is found to be 0 and 0.5 respectively.
Table 2: Approximate half-band Hilbert transform
with aliasing errors.
)0(x̂ )1(x̂ )2(x̂ )3(x̂
0 0 0.5 0 -0.5
0.1 -0.0457 0.525 -0.0043 -0.475
0.2 -0.0914 0.55 -0.0086 -0.45
0.3 -0.1371 0.575 -0.0129 -0.425
2. If an aliasing component α of 0.1 is assumed
outside the low-frequency band, the corresponding
frequency signal will be:
(1,1,0.1,0.1,0.1,0.1,0.1,1). The half-band Hilbert
transform of the first two-points is found to be -
0.0457 and 0.525 respectively.
3. Comparing the results of 1 & 2, the aliasing errors
in the Hilbert transform of these two points are -
0.0457 and 0.025 respectively.
4. The result in the last step can be found by finding
first the aliasing errors in the subband-FFT from
Eq. (15). These errors are found to be (0, -0.0414j,
-0.1j, -0.2414j) for (k = 0,1,2,3) respectively.
5. The Hilbert transform of the corresponding time
sequence of this error signal is found to be: -
0.0457 and 0.025 for the first two points. The
values of the aliasing errors found from steps 3
and 5 are the same.
The following numerical example describes the
analysis of the aliasing errors in quarter-band SB-
Hilbert transform for N = 16 and M = 4:
M Reduction in Complexity
Number of Input Points
64 128 256 512 1024
2 23.01% 25.91% 26.43% 28.93% 31.48%
4 48.11% 50.45% 52.81% 54.82% 56.55%
8 63.58% 65.61% 67.42% 69.14% 70.73%
16 75.09% 76.38% 77.68% 79.01% 80.22%
IIUM Engineering Journal, Vol. 2, No. 2, 2001 A. Hossen
25
1. For a case of no aliasing error, a frequency signal
of (1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1) is assumed. The
quarter-band Hilbert transform of the first two-
points (the low-low-band case) is found to be 0
and 0.125 respectively.
2. If an aliasing component of 0.1 is assumed outside
the low-low-frequency band, and the frequency
signal is represented as (1,1,0.1,0.1,
0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,1), the
quarter-band Hilbert transform of the first two-
points is found to be -0.0165 and 0.1328
respectively.
3. Comparing the results of 1 & 2, the aliasing errors
in the Hilbert transform of these two points are -
0.0165 and 0.0078 respectively.
4. The result in the last step can be found by finding
first the aliasing errors in the-FFT from Eq. (17).
These errors are found to be:
(0, -0.0153j, -0.354j, -0.077j) for (k=0,1,2,3)
respectively.
5. The Hilbert transform of the corresponding time
sequence of this error signal is found to be: -
0.0165 and 0.0078 for the first two points, and this
result is the same as that found in step 3 of this
example.
Interpolating the resulting length-N/2 half-band SB-
Hilbert transform by a factor of 2, leads to a length-N
approximated Hilbert transform )(ˆ nx . Table 3 shows
the values of the full-band Hilbert transform with
N=16 and half-band (M=2) subband Hilbert transform
with and without interpolating the results by a factor of
2. The input signal has a frequency transform of:
(1,1,1,1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,1,1,1).
Results show how well the subband Hilbert transform
values approximate those of the full-band Hilbert
transform.
Table 3: Interpolating the approximated Hilbert
transform results.
Index: n Full-band Half-band Half-band Inter.
0 0 -0.0223 -0.0223
1 0.2894 0.2289
2 0.2716 0.3122 0.3122
3 0.1591 0.1717
4 0 -0.0011 -0.0011
5 -0.0103 -0.0127
6 0.0466 0.0539 0.0539
7 0.0699 0.0541
8 0 -0.0006 -0.0006
9 -0.0699 -0.0518
10 -0.0466 -0.0497 -0.0497
11 0.0103 0.0109
12 0 -0.0011 -0.0011
13 -0.1591 -0.1602
14 -0.2716 -0.2914 -0.2914
15 -0.2894 -0.2289
6. APPLICATIONS
6.1 Analytic Signal
There are many applications in communication where
it is necessary to perform a 90° phase shift to the fre-
quency components comprising a signal. One common
application is the generation of the complex-valued
analytic signal ( )x n% . Letting )(ˆ nx be the Hilbert trans-
form of x(n) with all frequency components shifted by
90°, the analytic signal is given by:
)(ˆ)()(~ nxjnxnx (20)
So if x(n)=cos(0nT), )(ˆ nx will be sin(0nT) and
)(~ nx will be cos(0nT) + j sin(0nT), which
corresponds to only the positive-frequency component
of x(n). In Fig. 6, both x(n) (solid line) and )(ˆ nx
(dotted line) are shown for full-band Hilbert transform
and subband Hilbert transform with three different
values of M. In all cases the fundamental frequency is
30 Hz, and the sampling frequency is 1024 Hz.
0 0.05 0.1 0.15
-1
-0.5
0
0.5
1
1.5
Fullband Hilbert Transform
0 0.05 0.1 0.15
-1
-0.5
0
0.5
1
1.5
Subband (M=2) Hilbert Transform
0 0.05 0.1 0.15
-1
-0.5
0
0.5
1
1.5
Subband (M=4) Hilbert Transform
0 0.05 0.1 0.15
-1
-0.5
0
0.5
1
1.5
Subband (M=8) Hilbert Transform
Fig. 6: Analytic signal generation.
6.2 Single Sideband Modulation
A SSB-modulated signal is obtained by multiplying the
Hilbert transform of the information signal x(n) by a
sinusoid of frequency fc and adding (or subtracting) the
result to (or from) the product of the signal x(n) and a
90° phase shifted sinusoid of frequency fc
[11] for upper
and lower SSB signals generation, respectively. Fig. 7
shows the generation of a SSB-signal with fundamental
frequency of 15 Hz and with transmitted carrier
frequency 250 Hz with a sampling frequency of 1024
Hz. Both full-band and SB-Hilbert transforms with
different numbers of subbands are used. The frequency
spectrum of the four generated SSB-signals of Fig. 7
are shown in Fig. 8.
IIUM Engineering Journal, Vol. 2, No. 2, 2001 A. Hossen
26
0 0.1 0.2 0.3 0.4
-1
-0.5
0
0.5
1
Fullband Hilbert Transform
0 0.1 0.2 0.3 0.4
-1
-0.5
0
0.5
1
Half-band Hilbert Transform
0 0.1 0.2 0.3 0.4
-1
-0.5
0
0.5
1
Quarter-band Hilbert Transform
0 0.1 0.2 0.3 0.4
-1
-0.5
0
0.5
1
Subband Hilbert Transform M=8
Fig. 7: Single-sideband modulation examples.
0 0.05 0.1 0.15 0.2
0
20
40
60
80
100
120
Fullband Hilbert Transform
0 0.05 0.1 0.15 0.2
0
20
40
60
80
100
120
Half-band Hilbert Transform
0 0.05 0.1 0.15 0.2
0
20
40
60
80
100
120
Quarter-band Hilbert Transform
0 0.05 0.1 0.15 0.2
0
20
40
60
80
100
120
Subband Hilbert Transform M=8
Fig. 8: Frequency spectrum of SSB examples.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-0.5
0
0.5
Fullband Hilbert Transform
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-0.5
0
0.5
Halfband Hilbert Transform
Fig. 9: Demodulated FM signal examples.
6.3 FM Demodulation
An FM signal may be demodulated by modulating the
Hilbert transform of the waveform by a complex
exponential of frequency -fc, and obtaining the
instantaneous frequency of the result[8]. In Fig. 9 a
demodulated sinusoidal signal with a frequency 50 Hz
is shown, the carrier frequency being 200 Hz, while the
sampling frequency is 1000 Hz. Both full-band Hilbert
transform and half-band Hilbert transform are used.
7. CONCLUSION
A new application to the subband-decomposition idea
in computing the Hilbert transformation is presented.
The result is a fast and approximate Hilbert transform.
A reduction in complexity of about 20% to 30% in
half-band and about 45% to 55% in quarter--band
Hilbert transforms is obtained with respect to the full-
band Hilbert transform of a discrete sequence of length
N=16 to N=1024. Properties of the approximate
Hilbert transform are examined with different
examples. A very interesting relation between the
aliasing errors of the SB-FFT used for computing the
approximate Hilbert transform and the aliasing errors
of the computed Hilbert transform is found. Different
simulation examples of approximate Hilbert transforms
with different numbers of subbands are given to
describe this relation. Application examples in
communication systems are also included.
ACKNOWLEDGEMENT
The author would like to thank Prof. Ulrich Heute, the
head of the department of network and system theory at
the university of Kiel-Germany, for his valuable
comments.
REFERENCES
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Hossen. "Subbnad DFT-Part I: Definition,
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1995.
[2] A. Hossen, U. Heute, O. Shentov, and S.
Mitra, "Subband DFT-Part II: Accuracy,
Complexity, and Applications", Signal
Processing, Vol. 41, no.3, pp.279--294. Feb.
1995.
[3] S. Jung, S. Mitra, and D. Mukherjee,
"Subband DCT: Definition, Analysis, and
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Systems for Video Technology, Vol.6, no.3,
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[4] A. Hossen and U Heute, "General Adaptive
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Prague, Czech Republic, June 1997.
[5] S. Hann," Hilbert Transforms in Signal
Processing", Artech House, 1996.
[6] S. Haykin, "Communication Systems", John
Wiley and Sons, 1994.
[7] A. V. Oppenheim and R. W. Schafer,
"Discrete-Time Signal Processing", Prentice
Hall, 1989.
[8] "Matlab Signal Processing Toolbox", The
MathWorks, 1994.
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[9] A. Hossen and U. Heute, "Fully Adaptive
Evaluation of Subband-DFT", Proc. of
ISCAS'93, Chicago, pp.655-658, May 1993.
[10] L. Jackson, "Digital Filters and Signal
processing", Kluwer Academic Publishers,
1993.
[11] J. Proakis and M. Salehi, "Communication
Systems Engineering", Prentice Hall, 1994.
BIOGRAPHY
Abdulnasir Hossen received his Dr.-Ing. Degree in
Digital Signal Processing from the Ruhr University of
Bochum, Germany in 1994. In 1997, he completed his
Post Doc. Research at the Technical Faculty of Kiel-
Germany in the same field. Then he joined the Applied
Science University in Jordan as an Assistant Professor
for two years. In 1999, he joined the Information
Engineering Department, at Sultan Qaboos University
in Oman as an Assistant Professor. Dr. Hossen’s
primary research interests are in digital signal
processing, fast algorithms, and their applications in
speech, image, radar and biomedical signal processing.
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