Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation
Main objective of this paper is to present the general solution of the first order matrix difference system X (n+1)=AX (n)B+C(n)U(n)D(n), and then study the stability and sensitivity analysis of the digital filters via eigenvalue sensitivity and normal form transformations.
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irk-123456789-618072014-05-12T03:01:53Z Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation Murty, K.N. Reddy, K.V. Математические методы и модели Main objective of this paper is to present the general solution of the first order matrix difference system X (n+1)=AX (n)B+C(n)U(n)D(n), and then study the stability and sensitivity analysis of the digital filters via eigenvalue sensitivity and normal form transformations. Предложено общее решение разностной системы матриц первого порядка X (n+1)=AX (n)B+C(n)U(n)D(n). Исследована стабильность и чувствительность цифровых фильтров посредством анализа чувствительности собственного значения и преобразования нормальной формы. Запропоновано загальний розв’язок різницевої системи матриць першого порядку X (n+1)=AX (n)B+C(n)U(n)D(n). Досліджено стабільність і чутливість цифрових фільтрів за допомогою аналізу чутливості власного значення та перетворення нормальної форми. 2012 Article Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation / K.N. Murty, K.V. Reddy // Электронное моделирование. — 2012 — Т. 34, № 1. — С. 3-13. — Бібліогр.: 5 назв. — англ. 0204-3572 http://dspace.nbuv.gov.ua/handle/123456789/61807 en Электронное моделирование Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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Математические методы и модели Математические методы и модели Murty, K.N. Reddy, K.V. Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation Электронное моделирование |
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Main objective of this paper is to present the general solution of the first order matrix difference system X (n+1)=AX (n)B+C(n)U(n)D(n), and then study the stability and sensitivity analysis of the digital filters via eigenvalue sensitivity and normal form transformations. |
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Murty, K.N. Reddy, K.V. |
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Murty, K.N. Reddy, K.V. |
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Murty, K.N. |
| title |
Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation |
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Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation |
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Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation |
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Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation |
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Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation |
| title_sort |
stability and sensitivity analysis of digital filters under finite word length effects via normal form transformation |
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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2012 |
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Математические методы и модели |
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| citation_txt |
Stability and Sensitivity Analysis of Digital Filters under Finite Word Length Effects via Normal Form Transformation / K.N. Murty, K.V. Reddy // Электронное моделирование. — 2012 — Т. 34, № 1. — С. 3-13. — Бібліогр.: 5 назв. — англ. |
| series |
Электронное моделирование |
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AT murtykn stabilityandsensitivityanalysisofdigitalfiltersunderfinitewordlengtheffectsvianormalformtransformation AT reddykv stabilityandsensitivityanalysisofdigitalfiltersunderfinitewordlengtheffectsvianormalformtransformation |
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2025-07-05T12:45:33Z |
| last_indexed |
2025-07-05T12:45:33Z |
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1836811062556491776 |
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K. N. Murty *, K. V. Reddy **
Sreenidhi Institute of Science & Technology
*Department of Humanities and Science (Mathematics),
**Department of Electronics and Communication Engineering,
(Yamnampet, Ghatkesar, Hyderabad - 501 301. A.P. India,
*e-mail: nkanuri@hotmail.com)
Stability and Sensitivity
Analysis of Digital Filters under Finite Word
Length Effects via Normal Form Transformation
Main objective of this paper is to present the general solution of the first order matrix difference
system X n AX n B C n U n D n( ) ( ) ( ) ( ) ( )� � �1 , and then study the stability and sensitivity ana-
lysis of the digital filters via eigenvalue sensitivity and normal form transformations.
Ïðåäëîæåíî îáùåå ðåøåíèå ðàçíîñòíîé ñèñòåìû ìàòðèö ïåðâîãî ïîðÿäêà X n( )� �1
� �AX n B C n U n D n( ) ( ) ( ) ( ). Èññëåäîâàíà ñòàáèëüíîñòü è ÷óâñòâèòåëüíîñòü öèôðîâûõ
ôèëüòðîâ ïîñðåäñòâîì àíàëèçà ÷óâñòâèòåëüíîñòè ñîáñòâåííîãî çíà÷åíèÿ è ïðåîáðàçîâà-
íèÿ íîðìàëüíîé ôîðìû.
K e y w o r d s: digital filters, difference equations, exponentiation of a matrix, stability and sensi-
tivity analysis.
1. Introduction. Digital filters are widely used in many branches of science and
engineering. More specifically, in communication systems and in control of lin-
ear and nonlinear systems the theory of digital filters is of immense importance
due to discrete nature of the signals. Further, the study of discretization methods
for differential equations has also increased the scope of the theory and applica-
tions of difference equations. In recent years, the investigation of the theory of
difference equations has assumed greater significance as a well-deserved inde-
pendent discipline due to various applications in signal processing and control
systems. The control system theory in fact first gained considerable maturity in
the discipline of engineering and has been successfully applied in a variety of
branches of engineering, particularly receiving great impetus from aerospace en-
gineering [1]. Most of the analog signals cannot be exactly implemented by the
digital schemes, since the undesirable finite world length effect will occur dur-
ing A/D process and / or in the registers of the CPU. Further, in some applica-
tions, the signals are faint and result in a degenerate accuracy of the digitaliza-
tion. For example, in space makers the signals captured from a human body are
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 1 3
ÌÀÒÅÌÀÒÈ×ÅÑÊÈÅ
ÌÅÒÎÄÛ È ÌÎÄÅËÈ
discrete and hence finite word length (FWL) effects are very essential. The finite
word length effects have been an important issue in hardware implementation
and are widely discussed in literature [2, 3].
A novel approach is presented in [2] to analyze and minimize fixed point
arithmetic errors for digital filter implementation based on eigenvalue sensitivi-
ty analysis. Many sensitive methods proposed for finding the optimal state-
space realization are effective to deal with all linear systems under the FWL effects,
especially with the digital filter and digital control systems [4]. In the year 2003 and
2004 in [3] an analytically algebraic method is proposed for solving an optimal
transformation to achieve the minimal sensitivity subject to the FWL effects for dig-
ital control systems. For more information in this area of research we refer to [3].
Our paper is organized as follows. Section 2 presents criteria for constructing
the general solution of the homogeneous system X n AX n B( ) ( )� �1 in terms of
two fundamental matrix solutions of X n AX n( ) ( )� �1 and X n( )�1 � B X n* ( ).
The general solution of the first order matrix system
X n AX n B C n U n D n( ) ( ) ( ) ( ) ( )� � �1 , X X( )0 0� , (1)
state equation
R n EX n FU n( ) ( ) ( )� � , (2)
output equation, where A and B are ( )k k� non-singular constant matrices and
X n R k k( )� � (or C k k� ) are the components defined on N n n nn0 0 0 1 0 2
�
� ��{ , , ,...
..., , ...}n k0� , where k N� � and n N0 � , N being the set of integers. U n( ) is the
control function and is of the order ( )s k� , C (n) being k s� matrix, and the output
signal R is ( )r k� . Standard terminology is that the difference system (1) is said
to be time invariant, if the coefficient matrices A and B do not vary with time.
In section 3, we present a set of sufficient conditions for the stability analy-
sis of the system (1) and then show through examples that the system X n( )� �1
� AX n( ) need not to be stable and the matrix B can be chosen so that the first or-
der difference system X n AX n B( ) ( )� �1 be stable. Section 4 is concerned with
sensitivity analysis of the linear system and eigenvalue sensitivity minimization
and the optimally similar transformation. Section 5 is concerned with eigenvalue
sensitivity and normal form transformation.
2. Preliminaries. In this section, we shall be concerned with the general so-
lution of the first order matrix difference system
X n AX n B C n U n D n( ) ( ) ( ) ( ) ( )� � �1 , X X( )0 0� (3)
in terms of the two fundamental matrix solutions of the system X n AX n( ) ( )� �1
and X n B X n( ) ( )*� �1 and then present a technique to compute An. For a linear
system of the first order difference system, the computation of An is the analo-
K. N. Murty, K. V. Reddy
4 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 1
gous computation of eAt in the first order system of ordinary differential equa-
tions. From (3) we have
X X( )0 0� ,
X AX B C U D( ) ( ) ( ) ( )1 0 0 00� � ,
X AX B C U D( ) ( ) ( ) ( )2 1 1 11� � �
� � �A X B AC U D B C U D2
0
2 0 0 0 1 1 1( ) ( ) ( ) ( ) ( ) ( ),
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
X n A X B A C j U j D j Bn n
j
n
n j n j( ) ( ) ( ) ( )� � � � �
�
�
� � � ��0
0
1
1 11 1 1 .
Furthermore, X n( ) can be computed with the following
X n n n X n n( ) ( , ) ( , )*� � 0 0 0
� � � � � �
�
�
�
j
n
n j C j U j D j n j
0
1
0 01 1 1 1 1 ( , ) ( ) ( ) ( ) ( , )*
and, if X n( ) is a particular solution of (3), then any solution of (3) is given by
X n n n X n n X n( ) ( , ) ( , ) ( )*� � 0 0 0
, where ( , )n n A
n n
0
0� �
and
* ( , )n n0 �
� �
B
n n0 . It is a well-known fact that the exponentiation of the matrix A defined by
e I At
At At
n
At
n
� � � � � �( )
( )
!
...
( )
!
...
2
2
is a solution of � �y Ay (y being an n-vector) and any solution of � �y Ay, y (0) = y0
is given by eAt y0 . The next theorem is useful for the stability analysis of digital
filters under the FWL effects via normal-form transformation.
Theorem 1. Let A be a constant non-singular ( )k k� matrix with characteristic
polynomial � � � � � � � � � � ��I A C C Ck k
k
k� � � � � � � � ��1
1
1 0 1 2... ( )( ) ...( ).
Then
A x n I x n A x n A x n An
k
k� � � � � �
1 2 3
2 1( ) ( ) ( ) ... ( ) ,
where x1, x2, ..., xk are the k-linearly independent solutions of the kth order scalar
differential equation X n k C X n k C X n C X nk( ) ( ) ... ( ) ( )� � � � � � � � ��1 1 01 1 0
satisfying the initial conditions
x1 0 1( ) � , x2 0 0( ) � , ..., xk ( )0 0� ,
x2 1 0( ) � , x2 1 1( ) � , ..., xk ( )1 0� ,
(4)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x n1 1 0( )� � , x n2 1 0( )� � , ..., x nk ( )� �1 1.
Stability and Sensitivity Analysis of Digital Filters under Finite Word
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 1 5
P r o o f. Let A be a constant non-singular ( )k k� matrix and let G ( )� �
� � � � � � ��� � � � ��I A C C Ck k
k
1
1
1 0 0... be its characteristic polynomial. De-
fine
( ) ( ) ( ) ( ) ... ( )n x n I x n A x n A x n Ak
k� � � � � �
1 2 3
2 1, where x1, x2, ..., xk are
the k-linearly independent solutions of the kth order scalar difference equation
X n k C X n k C X n C X nk( ) ( ) ... ( ) ( )� � � � � � � � ��1 1 01 1 0 satisfying the initial
conditions (4). Then
( ) ( ) ... ( ) ( )n k C n k C n C nk� � � � � � � � ��1 1 01 1
� � � � � � � � � ��[ ( ) ( ) ... ( ) ( )]x n k C x n k C x n C x nk1 1 1 1 1 0 11 1
� � � � � � � � � ��[ ( ) ( ) ... ( ) ( )]x n k C x n k C x n C x n Ak2 1 2 1 2 0 21 1
+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +
� � � � � � � � ��
�[ ( ) ( ) ... ( ) ( )]x n k C x n k C x n C x n Ak k k k k
k
1 1 0
11 1 �
� � � � ��0 0 00 01. . ... .I A Ak .
Thus,
( ) ( ) ... ( ) ( )n k C n k C n C nk� � � � � � � � ��1 1 01 1 0 for all n N� 0
� and
( ) ( ) ( ) ( ) ... ( )0 0 0 0 01 2 3
2 1� � � � � ��x I x A x A x A Ik
k ,
( ) ( ) ( ) ( ) ... ( )1 1 1 1 11 2 3
2 1� � � � � ��x I x A x A x A Ak
k ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
( ) ( ) ( ) ( ) ... ( )k x k I x k A x k A x k Ak
k� � � � � � � � � � �1 1 1 1 11 2 3
2 1 � �Ak 1.
Clearly
( ) ( ) ( ) ( ) ... ( )n k x n I x n A x n A x n Ak
k� � � � � � �
1 2 3
2 1 satisfies the ini-
tial value problem
( ) ( ) ... ( ) ( )n k C n k C n C nk� � � � � � � � ��1 1 01 1 0 and
( )0 � I,
( )1 � A,
( )2 2� A , ...,
( )k Ak� � �1 1.
Thus, it follows from the uniqueness of initial value problems that
( )n is
the unique solution of (3) and A x n I x n A x n A x n An
k
k� � � � � �
1 2 3
2 1( ) ( ) ( ) ... ( ) .
To illustrate the importance of the above theorem, we consider the following
example.
Consider the matrix A given by
A � �
�
�
�
�
�
�
�
�
�
�
�
0 1 1
2 3 1
3 1 4
.
The characteristic polynomialG ( )� is given by G I A( ) ( ) ( )� � � � � �� � � � � �2 0.
Therefore, the eigenvalues of the matrix are given by 2, 2, 3. Clearly An �
K. N. Murty, K. V. Reddy
6 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 1
� � �x n I x n A x n A1 2 3
2( ) ( ) ( ) . Let X n C C n Cn n n( ) ( ) ( ) ( )� � �1 2 32 2 3 . To deter-
mine the constants C1, C2 and C3 we have
x1 0 1( ) � ; x2 0 0( ) � ; x3 0 1( ) � ,
x1 1 0( ) � ; x2 1 1( ) � ; x3 1 0( ) � ,
x1 2 0( ) � ; x2 2 0( ) � ; x3 2 1( ) � .
Now, the three linearly independent solutions X1(n), X2(n) and X3(n) are
given by
x n n n n
1 3 2 3 2 4 3( ) ( ) ( ) ( )� � � � ,
x n nn n n
2 4 2 25 2 4 3( ) ( ) . ( ) ( )� � � ,
x n nn n n
3 2 05 2 3( ) ( ) . ( ) ( )� � � � .
Therefore,
A x n I x n A x n An � � � �1 2 3
2( ) ( ) ( )
�
� � � �
� �
� � �( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) (
2 3 2 2 2 3
2 3 2
1 1 1n n n n n n
n n
n n
n ) ( )( ) ( ) ( )
( ) ( ) ( ) ( )
n n n n
n n n n
n
n n
� �
� �
� � �
� �
1 1
1 1
2 2 2 3
2 2 3 2 2 � � �
�
�
�
�
�
�
�
�
�
�1 2 2 3( ) ( )n n
.
Now consider the homogeneous system X n AX n B( ) ( )� �1 , X I( )0 � , where A
is the matrix given above and
B �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
4
0 0
0
1
4
0
0 0
1
4
,
we have
B n
n
n
n
�
�
�
�
�
�
�
�
�
�
�
�
�
�
2 0 0
0 2 0
0 0 2
2
2
2
,
and hence X (n) = An � (0) Bn is a solution of the system X n AX n B( ) ( )� �1 .
3. Stability analysis. Stability is a very important concept in signal / image
processing. A small perturbation in the initial data effects a substantial deviation
in the image processing of the signal, then such a system is not acceptable even
approximately. In general, stability in a system means that small changes in the
Stability and Sensitivity Analysis of Digital Filters under Finite Word
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 1 7
system inputs or in initial conditions or in the system parameters do not effect
a substantial change in the system behaviour. Most of the working systems are
designed in such a way that they are perfectly stable [5] (stability means ability
of the system to come back to the original state after a small perturbation). We
need to investigate in this section conditions under which the homogeneous sys-
tem is stable and asymptotically stable homogeneous system.
One of the most important requirements in the performance of control sys-
tems is stability. This is true for continuous-data systems as well as digital cont-
rol systems.
Example 1. Consider the difference system
X n X n AX n( ) ( ) ( )� �
� �
�
��
�
��
�1
0 1
1 2
.
The eigenvalues of the matrix A are given by� �1 2 1� � � . The two linearly inde-
pendent solutions are given by
X n n n
1 1 1( ) ( )( )� � � ,
X n n n
2 1( ) ( )� � � .
Therefore a solution matrix � ( )n is given by
� ( ) ( ) ( )
( )
( )
( )n X n I X n A A
n n
n n
n n� � � �
� �
�
�
��
�
��
�1 2
1
1
1 .
Example 2. Consider the difference system
X n AX n X n( ) ( ) ( )� � �
�
�
��
�
��
1
0 1
2 3
.
The eigenvalues of the matrix A are �1 1� and � 2 2� , and hence the system is
unstable in fact
An
n n n n
n n n n
�
� � �
� � �
�
�
�
�
�
2 1 2 1 2
2 1 2 2 1 2 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
� �
� � �
� � �
�
�
�
�
�
�
2 2 1 2
2 2 2 1 2 2
( ) ( ) ( )
( ) ( ) ( )
n n n
n n n
.
As n� �, the eigenvalues of An do not lie inside the unit circle and hence the
system is unstable. On the other hand consider
X n AX n B X n( ) ( ) ( )� � �
�
�
��
�
��
�
�
�
�
�
�
�
�
�
�
�
�
1
0 1
2 3
1
2
0
0
1
2
2
2
K. N. Murty, K. V. Reddy
8 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 1
with X (0) = I, where I is the unit matrix. Then X (n) = An � (0) Bn implies
X n
n n n n
n n n n
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
�
� � �
� � �
�
�
�
2 1 2 1 2
2 1 2 2 1 2 2
�
�
�
�
�
�
�
�
�
�
�
�
�
2 0
0 2
2
2
n
n
.
One can easily see that lim ( )
n
X n
��
�0, and hence the system is stable and in fact
asymptotically stable.
Theorem 2. Let
( )n and
( )n be fundamental matrix solutions of T n( )� �1
� AT n( ) and T n B T n( ) ( )*� �1 satisfying
( ) ( )n n I0 0� � . Then the system
T n AT n B( ) ( )� �1 is stable if, and only if there exists an M > 0 such that
( )n M� and
( )n M� for all n n� 0. In addition if
( )n � 0as n� � or
( )n � 0as n� �, then the systemT n AT n B( ) ( )� �1 is asymptotically stable.
P r o o f. First, we note that
( )n An� and
( )n B n� . Suppose there exists an
M > 0 such that
( )n M� and
( )n M� for all n n� 0. Then any solution of
T n AT n B( ) ( )� �1 is of the formT n n T n( ) ( ) ( ) ( )�
0 . Let � > 0, then T ( )0 �
�
�
M 2
implies T n( ) � � for all n n� 0. Hence the system is stable. Conversely, sup-
pose the system is stable. Therefore, for a given � > 0, there exists a � > 0 such that
T ( )0 � � implies T n( ) � � for all n n� 0. This implies
�( ) ( ) ( )n T n0 � for all
n n� 0. A simple argument shows that
� � �
�
( ) ( ) ( )n T n m
m
ij
il
0 2
2
� � �
�
�
!
"
#
#�
.
This is true from all i, j = 1, 2, ..., n. Hence T n
n
M( ) � �
2 �
�
. Hence, the proof of
the theorem is complete.
Further, if
( )n � 0 as n� �, and
( )n M� or
( )n M� and
( )n � 0, then T n( ) � 0as n� �. Thus in both cases the system is asymp-
totically stable.
4. Sensitivity analysis. In the section, we present sensitivity analysis of the
signal processing of the first order matrix difference system.
X n AX n B C n U n( ) ( ) ( ) ( )� � �1 . (5)
The main reason for discussing sensitivity analysis is two-fold.
i) The mathematical models of several process are effected by measurement
errors and for these reasons it is important to study the set of all solutions of ad-
missible perturbations of the parameters.
ii) If a backward numerically stable algorithm is implemented to solve the
general first order matrix difference system, then the computed solution will be
exact solution of a slightly perturbed system. In fact, the sensitivity analysis
gives a perturbation bound for the solution as a function of the perturbation in the
Stability and Sensitivity Analysis of Digital Filters under Finite Word
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 1 9
signal processing data and also gives the estimate for the actual error in the
computed solution.
Suppose the coefficient matrices in (5) are subject to small perturbation, say
A A A A� � �� � ; B B B B� � �� � ; C C C C� � �� � ;U U U U� � �� � .
Now, the corresponding perturbed system is given by
( ( ) ( )( ( )( )X X n A A X X n B B� � � � � � �� $ � � $ �1
� � � � �( ( )( )( )( )( )C C n U U n D D n� $ � �
� � � � � � �AX n B C n U n A X AX n A X n B B( ) ( ) ( ) [ ( ) ( )]( )� � � � �
� � �C n U n D n C n U n( ) ) ( ) ( ) )� % � � %
� � � � � � �X n A A X n B B C n U n C n U n( ) ( ) ( )( ) ( ) ( ) ( ) ( )1 � � � � � .
Thus
� � � � � �X n A A X n B B C n U n C n U n( ) ( ) ( )( ) ( ) ( ) ( ) ( )� � � � � � �1
� � � � � �� � � � � � �C n U n A A X n B B C n U n C n U n( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ).
Since the perturbations are small, we neglect the term� �C n U n( ) ( )
� � � � � �X n A A X n B B C n C n U n F n U( ) ( ) ( )( ) ( ( ) ( ) ( )) ( , (� � � � � � �1 n)) �
� � �� � �( ( ) ( ) ( , ( ))A X n B C U n F n U n� � ,
�
�
� %X n n n X n n n j C j U j
j
n
( ) ( , ) ( , ) ( , ) ( )� � �
�
�
��0 0 0
0
1
0 1&' ) �
� � �
�
�
�
' ( , ) ( , ( ))n j F j U j
j
0
0
1 .
If � �A � 1, � �B � and F satisfies Lipschitz condition, then we have
� �
�
X n L X n n n X n n( ) ( )( ) ( , ) ( , )� � �0 0
'
� � � �
�
�
�
�
� �
j
n
j
n j C j U j n j F j U j
0
1
0 0
0
1 1
� %
( , ) ( ) ) ( , ) ( , ( )* ).
Let (
�Sup
n
n n n n{ ( , ) ( , )}*
0 0 and )
�Sup cond cond (
n
n n n n{ ( ( , )) ( , ))}*
0 0 .
Then
L X n C C j U j F j U j( )( ) ( ) ) ( , ( ))� ( ) � %� � � .
Now, for any �X n( ) and �Y n( ), we have
L X n L Y n k X n Y n( )( ) ( )( ) ( ) ( )� � � �� � � .
K. N. Murty, K. V. Reddy
10 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 1
Thus L is a contraction map and hence by the Banach fixed point theorem
the operator L has a unique fixed point and this fixed point is the solution of the
perturbed control system.
5. Eigenvalue sensitivity and normal form transformation. In [3] the
question is considered of eigenvalue sensitivity and normal form transformation
to the difference system
x n Ax n Bu n( ) ( ) ( )� � �1 ,
y n Cx n Du n( ) ( ) ( )� � ,
where x n( ), u n( ) and y n( ) are the state, input and output vectors, respectively.
The above system is known as control of linear multivariate system.
In this section, we consider the more general systems (1) and (2) and analyze
the sensitivity and normal form transformation by using eigenvalues and eigen-
vectors of the system (1). The main purpose in this section is to reduce the FWL
effects via the sensitivity minimization approach, since the eigenvalues of an
ideal digital filter may be perturbed under the FWL effects resulting in poor per-
formances or even instability for actual system implementation. Further, sensi-
tivity minimization becomes one of the effective schemes against the FWL
effects of a state space realization.
It may be noted that if � and � are the eigenvalues of the matrix A and the
perturbed matrix A+ and if( and ( are the corresponding eigenvalues of the ma-
trix B and B+, then (assuming that the perturbations are small) the system
X n AX n B( ) ( )� �1 is said to be stable, if � � �� � and ( ( �� � for all suffi-
ciently small � > 0.
Theorem 3. Suppose the system matrices A and B in (5) are diagonalizable
and let � k and (k be the kth eigenvalues of A and B, respectively. Then
� �
�
� (
�
� (
� (
k k k k k k k
H
k
H
k kA B
u y v x
* �
Re ( )
,
where xk and yk are the kth non-zero right and left eigenvectors of A and uk, vk are
the non-zero right and left eigenvectors of B and vk
H , xk
H are the Hermetian
transpose vectors of vk and xk, respectively.
In all practical purposes, all eigenvalues of a digital filter will be perturbed
by the FWL effect and hence the sensitivity measure is considered as follows:
+ �
k
n
k
A�
�
1 2
, �
,
, + �-
k
n
k
B�
�
1 2
, (
,
and + � y x
2 2
, +- � u v
2 2
where x and y are the sets of right and left
eigenvalues of A, and U, and V are the sets of right and left eigenvectors of B.
Stability and Sensitivity Analysis of Digital Filters under Finite Word
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 1 11
For convenience, we define the supreme of + +as and supremem + +- -as . To
tackle, the FWL effects of a digital filter and to find the corresponding optimal
state-space realization can be specifically formulated as a minimization problem
as + +min min=
T
and + +-1
1
min min�
T
, where T is the similarity transformation of A,
and T1 is the similarity transformation of B. If some optimality is reached by
similarity transformations, then
A T AT( ) ( ) ( )opt opt opt� �1 , D T D( ) ( )opt opt� �1 ,
B T BT( ) ( ) ( )opt opt opt� �
1
1
1 , E T E( ) ( )opt opt� �1 ,
C T C( ) ( )opt opt� �1 , F T F( ) ( )opt opt� �1
and the reciprocal right and left eigenvectors can also be expressed as
X T X( ) ( )opt opt� �1 and Y T y( ) ( )opt opt� 1
1,
U T U( ) ( )opt opt� �1 andV T v( ) ( )opt opt� 1 .
Theorem 4. Assume X has norms with full rank, U has norms with full
rank. Then the following hold
y x x yH
2 2 2
� , v u U VH
2 2 2
� .
Theorem 5. Let the optimal similarity transformation for minimizing the +
be expressed as T(opt) and for +- be expressed as T-( )opt . Then
T X X H V
( ) ( )opt � � 2 ., T U U H y
1
2
( ) ( )opt � � / ,
where . and / are real orthogonal vectors. It may be noted that XX IH� � , and
UU IH� � .
Theorem 6. If the system matrices A and B of the filter system described by
(5) are transformed to A(opt) and B(opt), respectively, by using T(opt) and T1(opt), the
optimal realizations of A(opt) and B(opt), respectively, are the normal form realiza-
tions, that is
A A( ) ( )( )opt opt2
�0 , B B( ) ( )( )opt opt2
�0 ,
where 0 denotes the spectral radius (the absolute value of all eigenvalues).
P r o o f.
A A AH
( ) ( ) ( )max ( ),opt opt opt2
� � ,
B B BH
( ) ( ) ( )max ( ),opt opt opt2
� ( ,
K. N. Murty, K. V. Reddy
12 ISSN 0204–3572. Electronic Modeling. 2012. V. 34. ¹ 1
therefore
A A
k
k( ) ( )max ( )opt opt2
� �� 0 ,
B B
k
k( ) ( )max ( )opt opt2
� �( 0 .
Çàïðîïîíîâàíî çàãàëüíèé ðîçâ’ÿçîê ð³çíèöåâî¿ ñèñòåìè ìàòðèöü ïåðøîãî ïîðÿäêó X n( )� �1
� �AX n B C n U n D n( ) ( ) ( ) ( ). Äîñë³äæåíî ñòàá³ëüí³ñòü ³ ÷óòëèâ³ñòü öèôðîâèõ ô³ëüòð³â çà
äîïîìîãîþ àíàë³çó ÷óòëèâîñò³ âëàñíîãî çíà÷åííÿ òà ïåðåòâîðåííÿ íîðìàëüíî¿ ôîðìè.
1. Chen C. T. Linear System Theory and Design. Engleweed Cliff. — 3rd edition. — NJ, USA :
Prentice Hall, 1999.
2. Hsien-Ju-Ko. Stability analysis of digital filters under finite-word length effects via normal
forms transformation//Asian J. Health and Information Sciences. — 2006. — 1. — P. 112—
121.
3. Ko H. J., Ko W. S. Sensitivity minimization for control implementation fixed point ap-
proach//Proc. 2004 American Control Conference (ACC 2004). — Boston, MA, USA, 2004.
4. Gopal M. Modern control systems. — New Age International (p) Ltd, Publishers (formerly
Wiley Eastern Ltd), 1995.
5. Chen B. S., Kuo C. T. Stability analysis of digital filters under finite word length effects//
IEEE Proc. — 1989. — 136, N 4. — P.167—172.
Submitted 03.10.11
Stability and Sensitivity Analysis of Digital Filters under Finite Word
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2012. Ò. 34. ¹ 1 13
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/SUO <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>
/SVE <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>
/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|