FACTA UNIVERSITATIS
Series: Mechanical Engineering Vol. 17, N
o
3, 2019, pp. 385 - 396
https://doi.org/10.22190/FUME190327034D
© 2019 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND
Original scientific paper
NEW CLASS OF DIGITAL MALMQUIST-TYPE ORTHOGONAL
FILTERS BASED ON THE GENERALIZED INNER PRODUCT;
APPLICATION TO THE MODELING DPCM SYSTEM
Nikola Danković, Dragan Antić, Saša Nikolić,
Marko Milojković, Staniša Perić
University of Niš, Faculty of Electronic Engineering, Department of Control Systems, Serbia
Abstract. A new class of cascade digital orthogonal filters of the Malmquist type based
on bilinear transformation for mapping poles to zeroes and vice versa is presented in
this paper. In a way, it is a generalization of the majority of the classical orthogonal
filters and some newly designed filters as well. These filters are orthogonal with
respect to the generalized inner product which is actually a generalization of the
classical inner product. Outputs of these filters are obtained by using polynomials
orthogonal with respect to the new inner product. The main quality of these filters is
that they are parametric adaptive. The filter with six sections is practically realized in
the Laboratory for Modeling, Simulation and Control Systems. Performances of the
designed filter are proved on modeling and identification of the system for differential
pulse code modulation. Real response and response from the proposed filter are
compared with regard to the chosen criteria function. Also, a comparative analysis of
the proposed filter with some existing filters is performed.
Key Words: Digital Orthogonal Filters, Malmquist Functions, Müntz Polynomials,
Bilinear Transformation, Inner Product, Differential Pulse Code Modulation
1. INTRODUCTION
The majority of the already existing classes of orthogonal rational functions are obtained
by using one of the two simple transformations. The first one is a linear transformation
s→as+b which was used for obtaining classical orthogonal functions as Legendre, Müntz-
Legendre, Jacobi, and some other classes of orthogonal functions and appropriate filters as
almost orthogonal [1–3], quasi-orthogonal [4, 5], and some generalized classes of
Received March 27, 2019 / Accepted July 25, 2019
Corresponding author: Nikola Danković
University of Niš, Faculty of Electronic Engineering, Department of Control Systems, Aleksandra Medvedeva
14, 18000 Niš, Republic of Serbia
E-mail: nikola.dankovic@elfak.ni.ac.rs
386 N. DANKOVIĆ, D. ANTIĆ, S. NIKOLIĆ, M. MILOJKOVIĆ, S. PERIĆ
orthogonal functions and filters [6]. The second transformation is a reciprocal transformation
of poles to zeroes s→1/s (Malmquist, Laguerre functions) [7, 8].
In [9] and [10] the relation between generalized Malmquist functions and new Müntz
polynomials is given. In [11] this relation is generalized using bilinear transformation for
mapping poles to zeroes and vice versa. A new class of Müntz polynomials, orthogonal
with respect to the inner product obtained by this transformation of poles to zeroes, is
derived. The filters based on the generalized Malmquist orthogonal functions are designed in
[12]. These filters are orthogonal with respect to the new inner product which is derived
from symmetric reciprocal transformation s→b/(cs) which is a generalization of the
reciprocal transformation given in [7, 8]. Two types of these filters are designed in [12],
namely, analogue and digital ones. Further generalization of these orthogonal functions is
based on the above mentioned symmetric bilinear transformation s→(as+b)/(cs-a) [11]. This
transformation includes the above mentioned linear and reciprocal transformations. In [13],
analogue orthogonal filters based on these functions are designed.
Since the first pulse-code modulation transmission of digitally quantized speech, in
World War II, digital signal processing (DSP) began to proliferate to all areas of human life.
Classical digital systems are known to possess poor parameters under finite-precision
arithmetic, like frequency response sensitivity to changes in the structural parameters, noise,
inner oscillations, and limit cycles [14]. These effects have led to development of wave
filters [15] and orthogonal filters. Rapid development of the digital orthogonal filters started
in early 1980's [16, 17]. The most common approaches to the orthogonal filter synthesis are
transfer function decomposition [16, 18] and the state space approach [19, 20]. Various
realizations of the digital orthogonal filters developed in the last several decades. For
example, in [18], the development of the pipelined structure of digital orthogonal filters
started. It continued in the 1990's till present days [14]. When the two most common
approaches are compared, state-space realization has advantages in the case of MIMO
filters; it provides a better insight into their structure. However, in this paper the authors use
transfer function realization of the proposed filters that is suitable for our purpose of
modeling real systems. It was noted that IIR digital filter can be applied to modern DSP
applications (e.g. mobile communication), to multichannel prediction, etc. Because of all
this, the main goal of this paper is design and later practical realization of a new class of
digital orthogonal filters.
In this paper a bilinear transformation for mapping poles to zeroes and vice versa in
the case of digital system is performed: z→(az+b)/(cz-a). In this way, the filters, orthogonal
with respect to a more generalized inner product, are obtained both in z-domain and discrete-
time domain. The class of discrete orthogonal polynomials derived from [9, 10] will be used for
determining outputs of the proposed filters. The proof of this orthogonality is given in the
Appendix. In this way, the designed orthogonal digital filter is a generalization of the majority
of classical orthogonal filters (Legendre, Müntz-Legendre, Laguerre, Jacobi, Malmquist) as
well as the most recently designed filter based on the reciprocal transformation [12]. The
generalized Malmquist filters designed in this paper are parametric adaptive.
The proposed digital orthogonal filter is practically realized in the Laboratory for
Modeling, Simulation and Control Systems [21] and it will be applied to modeling of the
well-known digital system for signal transmission, Differential Pulse Code Modulation
(DPCM) system [22–24]. Performances of the new filter are verified by comparing them
with some other classes of digital orthogonal filters.
New Class of Digital Malmquist-Type Orthogonal Filters Based on Generalized Inner Product... 387
The rest of the paper is organized as follows. In Section 2, new digital orthogonal filters
based on bilinear transformation are developed; first theoretically, later by simulation; finally,
they are practically realized. The proposed filter is used to modeling a linear part of the DPCM
system in Section 3. Comparison between the practically realized filter and some other filters is
performed in Section 4. In Section 5, the authors give conclusions and discuss a possibility for
further development in this area. Finally, at the end, the Appendix includes the proof of
orthogonality of these filters in discrete-time domain, and also relations for the inner products
and norms.
2. DIGITAL MALMQUIST-TYPE ORTHOGONAL FILTERS:
MATHEMATICAL BACKGROUND, DESIGN AND PRACTICAL REALIZATION
The generalized Malmquist filters based on bilinear transformation in discrete-time
domain can be derived from corresponding generalized Malmquist filters in continuous-
time domain [13] with the same procedure given in [12]. Namely, operator s (operator of
differentiation) is substituted by operator z (operator of prediction) to design corresponding
digital filters. The transfer function of new digital filters has the following form:
*
*1
10
( ) , , .
n
k k
n k k
k k k
z a bz
W z R
z z c a
(1)
Cascade scheme based on (1) is given in Fig. 1, where K is the number of samples,
and T is a sample period. The authors assume the sample period is one second because it
is not important for further analysis, i.e. K≡KT (discrete time).
Fig. 1 Block diagram of a digital orthogonal filter based on bilinear transformation
A sequence of functions on the outputs of cascades of the proposed digital filter (1)
obtained mathematically corresponds to responses by simulation and from practically realized
orthogonal digital filter. These outputs for the specific case a=0, b=1, c=1 (reciprocal
transformation) are already given (α0=1/2, α1=1/3, α2=1/4, α3=1/5) in [12] illustratively.
These filters are orthogonal in complex z-plane:
* 2
,
1
( ( ) ( )) ( ) ( )
2
n m n m n n m
W z W z W z W z ds N
i
, (2)
where
1
1
10
( )
k
n
k
n
k k
a b
z
c az
W z
z z
, ,n m represents Kronecker symbol, and contour Г
surrounds all the poles of Wn(s).
388 N. DANKOVIĆ, D. ANTIĆ, S. NIKOLIĆ, M. MILOJKOVIĆ, S. PERIĆ
The proof of orthogonality is similar as in the case of continuous-time systems [11, 13].
If in transfer function Wn(z) bilinear transformation is performed, the authors obtain Wn
*
(z)
whose poles are equal to zeroes of Wn(z) and zeroes of Wn
*
(z) are equal to poles of Wn(z).
Thereby, all poles of Wn
*
(z) are outside contour Г, and zeroes of Wn
*
(z) are inside contour Г.
If m≠n due to symmetry of the bilinear transformation, all the poles of the integrand
(2) that lie inside contour Г are annulled with appropriate zeros of Wm
*
(s), so the contour
integral (2) is equal to zero. In the case of m=n, there exists one first-order pole inside
contour Г. After applying the Cauchy theorem, the following expression is obtained:
(Wn(s), Wm(s))=Nn
2
≠0. Finally, all the expressions stated above imply (2).
Practical realization of the generalized Malmquist digital orthogonal filter is shown in
Fig. 2.
Fig. 2 Practical realization of the generalized digital orthogonal filter based on bilinear
transformation – printed circuit board
Outputs from the filter in z-domain Φl(z)=U(z)Wl(z), l=0, 1, 2, ..., n are orthogonal [12,
13, 25]:
2
,
( ( ) ( ))
n m n n m
z z N . (3)
Outputs from this digital filter in time domain are obtained using inverse z-transformation:
1 11
( ) { ( )} ( )
2
K
l l l
K Ζ z z z dz
i
, (4)
where contour Γ surrounds all the poles of Wl(z). This contour can be obtained by
mapping s- to z-domain: Γ={c|z|
2
-2aRez-b=0}[25].
A new inner product for the filter outputs is:
2* * * *
, ,
1
( ( ), ( )) ( ( ), ( )) ( ) ( )
n m n m n m n m n n m
K
J K K K K K K N
. (5)
New Class of Digital Malmquist-Type Orthogonal Filters Based on Generalized Inner Product... 389
The relation for Jn,m and Nn
2
is given in the Appendix. Otherwise, this inner product is
generated in the practically realized filter thanks to its structure.
These filters are adjustable. It means that their parameters can be changed: numerical values
of poles and parameters of bilinear transformation. For example, for c=0 well-known classical
orthogonal filters based on linear transformation are obtained, and for a=0 orthogonal filters
based on the reciprocal transformation (Malmquist type). Finally, in the case of a≠0, b≠0, c≠0
the most generalized orthogonal digital filter can be obtained.
This digital cascade orthogonal filter will be practically applied to modeling a prediction
filter in a well-known digital system in telecommunications, the DPCM system [22]. Because of
the cascade structure of the realized filter, i.e. a possibility to append more sections, the authors
suppose that the filter will be suitable for modeling DPCM systems which can theoretically
have an arbitrary high order predictor. A cascade structure is already verified in the case of
appropriate class of analogue generalized Malmquist filters [4].
3. APPLICATION TO MODELING DPCM SYSTEM
A new digital cascade orthogonal filter based on bilinear transformation will be
applied to modeling and signal identification of a linear part of the DPCM system [22].
The DPCM is a well-known and commonly used technique for signal transmission in
telecommunications. This system has a wide usage in different areas, starting from speech
and image coding to the latest medical research [23]. An estimate, i.e. a prediction of the
present value of the input signal is based on the knowledge of its earlier values [24]. That
is why one of the most important parts of every DPCM and ADPCM (Adaptive
Differential Pulse Code Modulation) is a predictor (a linear part of the system).
For modeling of the prediction filter the authors use an adjustable model based on the
proposed generalized digital filter (Fig. 1). A block diagram of the digital orthogonal adjustable
model based on bilinear transformation is shown in Fig. 3. In this case the authors use a
filter with six sections and real poles αk
*
=(aαk+b)/(cαk-a), k=0, 1, ..., n.
Fig. 3 Block diagram of an adjustable model with the proposed orthogonal digital filter
based on bilinear transformation
It can be noticed from Fig. 3 that the orthogonal model output is:
0
( ) ( )
n
M k k
k
y K b K
, (6)
where K is the number of samples, n=5 in our case.
390 N. DANKOVIĆ, D. ANTIĆ, S. NIKOLIĆ, M. MILOJKOVIĆ, S. PERIĆ
The desired model of the linear part of DPCM system (exactly, the prediction filter in the
encoder) is obtained by adjusting the following parameters: αk (k=0,1,…,5), summation
coefficients bk (k=0,1,…,5), and parameters of bilinear transformation a, b, and c. In the case
of modeling a particular unknown system, the parameters of the model should be adjusted in
such a way that the model (Fig. 3) corresponds as closely as possible to the unknown system.
The process of modeling is performed in the well-known manner by introducing the same
input to the system itself and to its adjustable model based on the new cascade orthogonal
digital filter [12]. This input signal is shown in Fig. 4.
Fig. 4 The input of DPCM linear part and the adjustable model
The next step is measuring the outputs from system ys(t) and filter yM(t) and calculating the
mean squared error (criteria function):
2
0
1
( ( ) ( ))
N
S M
K
J y K y K
N
. (7)
Optimal values of unknown parameters which lead to minimization of mean squared
error can be obtained by using genetic algorithm with J is fitness function.
The specific genetic algorithm used in experiments has the following parameters: initial
population of 1000 individuals, a number of generations of 300, a stochastic uniform selection,
a reproduction with 12 elite individuals, and Gaussian mutation with shrinking. The used
structure of chromosome was with eight parameters coded by real numbers: a, b, c, b0, b1, b2, b3,
b4, and b5 (Eq. 1). Poles a0, a1, a2, a3, a4, a5 are also adjustable, and in these experiments their
values are fixed. A series of experiments can be performed with other values of poles until the
obtained model fill requirements in advance. The main goal of the experiment was to obtain the
best model of the unknown system in regard to the criteria function, i.e. mean squared error.
The original signal (output from the DPCM linear part) and the signal from the
adjustable model based on the orthogonal digital filter of the generalized Malmquist type are
given in Fig. 5.
New Class of Digital Malmquist-Type Orthogonal Filters Based on Generalized Inner Product... 391
Fig. 5 Outputs from the DPCM linear part and the adjustable model
The authors performed experiments with six sections. In the case of the digital
orthogonal filter based on reciprocal transformation the filter with six sections is verified
as better related to the criteria function than one with four or five sections [12].
Obtained optimal values for parameters of adjustable orthogonal model are presented
in Table 1. Mean squared error is: Jmin=5.7216∙10
-3
.
Table 1 Values for parameters of the adjustable orthogonal model
Parameters Numerical values
b0 0.56713
b1 0.59111
b2 0.32604
b3 0.61674
b4 -0.21408
b5 0.25053
α0 0.91286
α1 0.87552
α2 0.77119
α3 0.82101
α4 -0.15647
α5 0.79444
A 0.28327
B 0.09449
C 0.73071
In Fig. 5 one representative sample is zoomed for illustrative purposes because of very
small differences between sample values. Measured outputs in discrete-time periods (sample
periods) of the prediction filter and the adjustable model based on the new digital orthogonal
filter are given in Table 2.
392 N. DANKOVIĆ, D. ANTIĆ, S. NIKOLIĆ, M. MILOJKOVIĆ, S. PERIĆ
Table 2 Obtained outputs from the DPCM prediction filter and the adjustable model based
on the digital orthogonal filter
K
Output from the DPCM
prediction filter
Output from the adjustable
orthogonal model
0 0.82500 0.82500
1 2.09125 2.09215
2 3.68806 3.68806
3 5.71962 5.70912
4 7.18302 7.18302
5 9.53761 9.51500
6 10.89502 10.86011
7 12.05639 12.03209
8 14.99215 14.97022
9 16.44441 16.39001
10 17.93314 17.91510
11 19.39597 19.36826
12 19.98442 19.97384
13 20.04930 20.03347
14 19.59736 19.57832
15 20.33942 20.31451
From Fig. 5 and Table 2 a high level of matching can be noticed between signals from
the DPCM linear part and the proposed orthogonal digital filter.
Finally, the model of the prediction filter in the DPCM encoder is formed as:
*5
*1
1
0 0
( ) , 0,
k
i
M k
k i i
z
W z b
z
(8)
where αk
*
=(aαk+b)/(cαk-a), and appropriate values of parameters are given in Table 1.
The proposed filter model (8) using numerical values in Table 1 can be written in the
following form:
5 4 3 2
6 5 4 3 2
0.527 0.661 0.893 1.131 1.432 1
0.619
0.077 0.125 0.592 0.958 1.732 1.214 1
M
z z z z z
W z
z z z z z z
. (9)
The relation (9) for the transfer function of the system is more suitable for control system
theory analysis, while the relation (8) is more suitable for system modeling.
4. COMPARATIVE ANALYSIS BETWEEN THE PROPOSED FILTER
AND SOME OTHER CLASSES OF DIGITAL ORTHOGONAL FILTERS
In order to verify the quality of the model based on the new filter with six sections, a
comparison with the models based on the some already existing digital orthogonal filters
(generalized Legendre and generalized Malmquist filters) is performed. The criteria function is
mean squared error again and the number of sections is six. The transfer functions of all the
filters used in experiments are shown in Table 3 (αi-1 is assumed to have constant value equal to
zero).
New Class of Digital Malmquist-Type Orthogonal Filters Based on Generalized Inner Product... 393
Table 3 Transfer functions of digital orthogonal filters used in experiments
Orthogonal filter type Transfer function
Legendre (Müntz-Legendre)
5
1
0 0
( )
( )
( )
k
i
k
k i i
z
W z b
z
Generalized Malmquist
5
1
0 0
( )
k
i
k
k i i
b
z
W z b
z
Filter based on bilinear transformation
1
5
1
0 0
( )
i
k
i
k
k i i
a b
z
c a
W z b
z
The outputs of the models based on these filters are calculated as
5
0
( ) ( )
M l l
l
y t b t
.
Table 4 Values for parameters of the adjustable orthogonal models based on new and
existing types of digital orthogonal filters
Criteria function value
and orthogonal model
i 0 1 2 3 4 5
J=16.370∙10
-3
orthogonal model with Legendre filter
(λ=0.67)
αi
0.1677 0.3162 0.4777 0.6164 0.7011 0.8168
bi
0.7821 1.6234 0.9102 0.8648 0.5519 0.3271
J=13.022∙10
-3
orthogonal model with generalized
Malmquist filter
(b=0.92)
αi
0.2681 0.1155 -0.1733 0.3124 0.2178 0.1665
bi
3.6410 -0.9131 2.1616 1.9872 0.8764 0.3558
J=5.721∙10
-3
orthogonal model with a new filter
based on bilinear transformation
(a=0.2834, b=0.094, c=0.731)
αi
0.9129 0.8755 0.7712 0.8210 -0.1565 0.7944
bi
0.5671 0.5911 0.3260 0.6167 -0.2141 0.2505
The structure of chromosome that is used in experiments is with 6 standard parameters
coded by real numbers: α0, α1 ,..., α5, and one additional; for Legendre filter λ and for
generalized Malmquist filter b. In Table 4 the obtained parameters for proposed and
existing filters are given.
From Table 4 it can be seen that the mean squared error is much bigger for the
generalized Malmquist [12] and the Legendre digital orthogonal filters than for the filter
presented in this paper. The excellent matching between the system output and the output
of the adjustable model based on the proposed filter has shown the quality and need for
new filters described in this paper.
Of course, the model of the linear part of DPCM system can be derived by using other
methods, but in this paper the authors used new orthogonal filter to verify its performances.
394 N. DANKOVIĆ, D. ANTIĆ, S. NIKOLIĆ, M. MILOJKOVIĆ, S. PERIĆ
5. CONCLUSION AND FUTURE WORK
In this paper the authors gave mathematical background, simulation and practical
realization of new digital orthogonal filters based on bilinear transformation of poles to zeroes
and vice versa. It is an extension of generalizations of traditional orthogonal filters starting with
filters based on reciprocal transformation. All good performances of already existing filters are
included in this class of filters.
The great quality of this filter is parametric adaptivity, i.e., possibility of adjusting values of
poles and parameters of bilinear transformation. Also, it is demonstrated that by setting specific
values for parameters of bilinear transformation, most of the classes of realized filters can be
obtained. In the case when c=0, these filters degenerate into classical orthogonal filters
(Legendre, Laguerre, Jacobi), and when is a=0, they degenerate into classical Malmquist and
generalized Malmquist filters.
In the future work, the authors could try to derive appropriate classes of orthogonal filters
with complex poles which in some practical cases could be even better. Further generalization
could also be in the usage of more general symmetric transformations than the bilinear one. In
this way, the study of generalized class of orthogonal analogue and digital filters will be
concluded.
APPENDIX
Polynomials obtained by using bilinear transformation of poles to zeroes are orthogonal
on the contour:
2
2 Re 0c z a z b . (A1)
Contour (A1) is a circle with radius
2
a b
R
c c
and center in , 0
a
c
.
Using transformation z=Rz
*
+a/c, i.e., z
*
=(za/c)/R, the circle is mapped into the unit
circle with the center at the origin.
The model of these polynomials is shown in Fig. 1.
Outputs φm(z) for the unit input are:
0
10 0
1
( ) ,..., ( ) , 1, 2,...,
i
m
i
m
i i
a b
z
c az
z z m n
z z z
. (A2)
By development of Eq. (A2) in partial fraction it is obtained:
,
0
( )
m
m j
m
j j
A
z
z
, (A3)
where :
,
lim( ) ( )
i
m j j m
z
A z z
,
1
0
,
0
( )
m
i
i i
m j m
j i
i
i j
a b
z
c a
A
.
New Class of Digital Malmquist-Type Orthogonal Filters Based on Generalized Inner Product... 395
By using inverse z-transformation of φm(z) outputs in time-domain are obtained:
( 1)
,
0
( )
m
K
m m j j
j
K A
. (A4)
By applying linear transformation onto linear systems, orthogonality is held.
When linear transformation z=Rz
*
+a/c is used, the region of orthogonality of filters
based on bilinear transformation is mapped into the unit circle with the center at the origin,
i.e. these polynomials are mapped into classical discrete-time orthogonal polynomials where
orthogonality in the classical sense is valid:
1
* * *
( ) ( ) ( ) ( )
n m n m
z z dz z z dz
, (A5)
where:
, ,* *
* *0 0
( ) , ( )
n m
n j m j
n m
j j
j j
A A
z z
a a
Rz Rz
c c
.
By using inverse z-transformation it is obtained:
* ( 1) * ( 1)
, ,
0 0
( ) , ( )
n m
K K
n n j j m m j j
j j
K A K A
. (A6)
The norm for outputs of filters obtained by bilinear transformation is:
2 * * * *
1
( ( ), ( )) ( ( ), ( )) ( ) ( )
n n n n n n n
K
N K K K K K K
. (A7)
Therefore, a new inner product for outputs of the filter is obtained:
2* * * *
, ,
1
( ( ), ( )) ( ( ), ( )) ( ) ( )
n m n m n m n m n n m
K
J K K K K K K N
. (A8)
Let the authors note that relations for the new inner product and norm Nn
2
are used in
mathematical analysis of these filters. In the case when the filters are practically realized
(see Case Study), these relations are calculated thanks to their structure.
Acknowledgements: This paper was realized as a part of the projects III 43007, III 44006 and TR
35005 financed by the Ministry of Education, Science and Technological Development of the
Republic of Serbia.
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