Time-varying two-dimensional state-space structures

D.C. McLernon

Indexing terms: State-space structure, State-transition matrix, 2-Dfilters, Time-varying matrices

Abstract: The paper considers a two-dimensional state-space structure with periodically time- varying matrices. A new equivalent state- transition matrix for such a system is first developed, and this is used to show how the general response formula can be evaluated. Finally, an equivalent time-invariant block struc- ture is presented, with a simplified graphical method for its determination. This form can not only be used as a new means of implementation, but also to easily evaluate the stability of the overall time-varying system.

1 Introduction

1 . I The first recorded demonstration of what could be described as a periodically time-varying system was per- formed by Faraday in 1831 when he produced wave motion in fluids, such as oil and water, by vibrating a plate or membrane in contact with the fluid. This, and other historical perspectives on continuous-time, linear, periodically time-varying (LPTV) systems are described in Reference 1.

Some of the earliest published work in the field of discrete-time periodically time-varying systems, and in particular digital filters, goes back over three decades [2-51. Traditionally, digital filters have been considered as linear time-invariant systems, but the option of allow- ing the filter coefficients to vary periodically gives an extra degree of freedom, which is one of the main attrac- tions.

The first unified analysis of one-dimensional (1-D) LPTV digital filters was given by Meyer and Burrus [SI, and many other papers have appeared dealing with the fundamental problems of the design of such filters [7-91, their stability [lo], time-invariant equivalent realisations [ll, 123, z-transform analysis [13], etc.

The renewed interest in 1-D LPTV filters has come about because the availble DSP technology has made many more applications possible, for example TDM/FDM transmultiplexers, modelling and filtering of cyclostationary random processes, speech scrambling, general periodic filters, etc. [S, 9, 14-16]. In addition, many other applications exist which can also be analysed as LPTV systems, not necessarily because of the explicit

Background to periodically time-varying systems

0 IEE, 1995 Paper 182OG (ElO), first received 1st August and in revised form 20th December 1994 The author is with the Department of Electronic and Electrical Engin- eerin& The University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, United Kingdom

periodic change in a parameter, but because of the incorporation of modulators, multipliers, up-samplers, down-samplers, etc., e.g. polyphase networks, N-path filters, multirate systems, interpolators/decimators, etc.

However, very little has been written about two- dimensional (2-D) LPTV filters. The theory of 2-D multi- rate filters, which mathematically is concerned with LFTV ideas, has recently been published [20]. And for 2-D structures with explicit periodic variations in the filter coefficients, these have been analysed in direct-form configurations [21], have been considered as an altern- ative means of implementing 2-D time-invariant filters [22], and have been used for removing 2-D limit cycle oscillations [23]. Other potential applications for 2-D LPTV filters include image scrambling and processing of images with cyclostationary noise. Computationally eff- cient structures for such operations are presently being investigated*.

In this paper, we will consider a popular state-space structure often used in 2-D implementations, and let the state matrices vary periodically. An equivalent state- transition matrix will be defined, and a general response formula derived. Then a time-invariant block structure will be constructed, which also allows determination of the system's stability.

1.2 2- D time-varying state-space structures Consider the well-known Roesser 2-D state-space model

[6,17-191.

~241 R(m + 1, n) = A,R(m, n) + A,qrn, n) + B,x(m, n) q m . n + 1) = A3R(m, n) + A,S(m, n) + B,x(rn, n)

y(m, n) = C,R(m, n) + C, S(m, n) + Dx(m, n) (1) This can be re-written in the following condensed form

w(m, n) = A"w(m - 1, n) + Ao'w(m, n - 1)

+ B'Ox(m - 1, n) + Bo'x(m, n - 1) (24 y(m, n) = Cw(m, n) + Dx(m, n)

where

* Private communication with Prof. T. Bose, Dept. of Electrical Engin- eering, University of Colorado, Denver, USA

120 I E E Proc.-Circuits Deuices Syst., Vol. 142, No. 2, April 1995

Let the model in eqn. 2a have periodically time-varying matrices with period (M, N) . This can now be written in two formst

(3) 1 w(m, 4 = A:,$M(n)N w(m - 1, n)

+ A::)M(">N w(m, n - 1)

+ B?:)M<n)Nx(m. n - 1)

+ D(m>M<n>Nx(m, n)

+ B ; e ) M < n ) N x ( m - '9 n,

Am, n) = C < m > M ( n > N w ( ~ n)

or alternatively, letting m -+ mM + k and n -+ nN + I , we may write

(4) I w ( m M + k , n N + I )

= AL,%(mM + k - 1,nN + r )

+ AE,'w(mM + k, nN + 1 - 1)

+ B:fx(mM + k - 1, nN + r)

+ BE/x(mM + k, nN + 1 - 1)

y ( m M + k , n N + r )

= ck,w(mM + k ,nN + r )

+ D,jX(mM + k, nN + I) fork = 0, 1, 2, ..., M - 1; 1 = 0, 1, 2, ..., N - 1

2 State-transition matrix and general response formula

2.1 Definition of state-transition matrix, A ;" It is well known that if we define A for eqn. 2 as

then the state-transition matrix is the two-tuple power of A , denoted by A"", and is given as [24]

AlOAm-1. I + AolAm*"-' +Z6(m, n) m, n 2 0 otherwise

(6)

A,, A:: + A;/ (7)

A" 42 {o

Consider eqn. 4, and define

Let us now give an expression for the state-transition matrix of a LPTV 2-D system, as the two-tuple power of A,, , which we will define as

A:/A"- 1 , II ( k - l ) M l + AE/AF(I'T?)~

A,"; + Z6(m, n) m , n > O otherwise

(0,O) < (k, 0 < ( M - 1, N - 1) (8)

i o

where 6(m, n) in eqns. 6 and 8 is the 2-D unit sample sequence [25].

2.2 Derivation of the general response formula Lemma I : Consider a single non-zero state-vector w(i, j ) in eqn. 3. The contribution of this state-vector w(i, j ) to any other state vector w(m, n) can now be written as

(9) m)M<n>Nw(i,j) W, 4 2 ( i d w(m, ,,) = A:-i.n-j

t Let X = mN + n, 0 6 n 6 N - 1, and where X, N , m and n are all non-negative integers. Then we can define: (X>, = n and [ X I N = rn for modulo-N arithmetic.

IEE Proc.-Circuits Devices Syst., Vol. 142, No. 2, April 1995

where the two-tuple power of A,, is defined in eqn. 8.

Proof of Lemma I: Clearly eqn. 9 is true for (m, n) = (i, j). Thus assume that

,,qm, n) = , q - i * n - j m ) M ( n ) N 44.8 W , j ) G (m, 4 < @, 4) (10)

From eqn. 3 we may write

a. 4) = A : ; ) M ( q ) N a - 1 9 4) + A ? ; > M < 4 > N a , 4 - 1)

+ A<P>M(q>N [ A ? A , ' C ~ { ~ w ( i , j ) l

= A;;),,,,,CA?~~;;L~,;', w(i,j)l 01

(from eqn. 10)

(1 1) Thus eqn. 10 is also true for (m. n) = (p, 4). This is an effective inductive proof of eqn. 9 because an enumer- ation can be found (e.g. the diagonal enumeration (0, 0),

(p, 4) in eqn. 10 is reached, but not before all (m, n) < @, 4). Q.E.D.

Lemma 2: The response of the state vector w(m, n) to x(i, j ) alone, and zero initial conditions, can be written as

(12)

- A'-'. 4 - j - < p ) M < q > N w(i, 31 (from eqn. 8)

(1, 01, (0, 11, (2, O), (1, 11, (0, 2), (3, 01, . . .) such that every

w(m, n) = B:$L'(;{~ x(i, j ) (m, n) 2 (i, j )

where

A r l , n B < k - m + 1 ) M ( I - " > N

+ A ; n-lB?i-m>M(I-n+ 1 ) N

{o +Z6(m,n) m , n & O B F A

otherwise

(0,O) < (k, 0 < ( M - 1, N - 1) (13)

Proof of Lemma 2: From eqn. 3, x(i, j ) will contribute an input to two state-vectors, i.e. w(i + 1, j ) and w(i, j + l), which in turn will contribute towards w(m, n) via eqn. 9. Thus from eqn. 3

(14) I w(i + 1,31= Bt ,O , , )M<j )Nx( iJ w(kj + 1) = B : & j + l ) N x ( i J

Then from eqns. 9 and 13

w(m, n) = (A'&;i$>nN-'B:IP+ I > M ( j ) N

+ Am-i.n-j-1 ( n ) M < n > N B?!>i:H<j+ l ) N ) x ( i J

= B'&;$;jN x(i, j ) (15) Q.E.D.

The general response formula for eqn. 3, with zero initial conditions, follows from eqn. 12

w(m, = B%M<n)N * * x(m,

= f t x(i,j)B:-i."-j m ) M ( n ) N i=o j = o

+ D < m > M < n ) N X ( m r n, (16) 3 2-D equivalent time-invariant block structures

3.1 Equivalent time -invariant block equations Can eqns. 3 and 4 be transformed into a time-invariant structure? Consider again eqn. 4 where we have 2MN sets of equations. These may be manipulated into the fol-

121

lowing general block structure

q m , n) = Poli t (m - I, n) + 2O16(m, n - 1)

+ d'Oi(m - 1, n) + Bolf(m, n - I) + dooi(m, n) (17)

(18) j+n, n) = &(m, n) + Dqm, n) n

4(m-I ,n)

As an example, consider the case of M = N = 2 in eqn. 5. From manipulation of these eight equations into the form of eqns. 17 and 18 we obtain the following

+(m,n) I I

I I

Q(m,n-1)

Fig. 1 Pseudo signalflow graph representation of the relationship A'Oflrn - I , n) -k X O ' f l q n - I ) in eqn. I 7 for 6" and 6" as given in e q n 20

where

q m , n)

w(mM, nN) w(mM + 1, nN) w(mM + 2, nN)

w(mM + M - 1, n N ) w(mM, nN + 1)

w(mM + 1, nN + 1) w(mM + 2, nN + 1)

w(mM + M - 1, nN + 1) w(mM, nN + 2)

w(mM + 1, nN + 2) w(mM + 2, nN + 2)

w(mM + M - 1, nN + N - 1). (19)

with similar ordering of the vectors used for i(m, n) and Hm, 4. 122 IEE Proc.-Circuits Devices Syst., Vol. 142, No. 2, April I995

3.2 Generation of the time-invariant block equations via a graphical representation

Manipulating eqn. 4 into the block form of eqns. 17 and 18 is very cumbersome for M , N 3 3. We shall now present a simpler method more amenable to computer implementation.

It is possible to graphically represent the block eqns. 17 and 18. From eqn. 20 this has been done in Fig. 1 for A'Oqm - 1, n) and R'C(m, n - 1) in eqn. 17. A similar graphical representation can also be made for the other parts of eqns. 17 and 18.

What is interesting, though, is that from Fig. 1 one can actually work backwards. From the structure of the pseudo signalflow graph it is now possible to write down directly the associated transmittances simply from inspec- tion, and then use them to construct the block matrices of eqns. 17 and 18. This removes the cumbersome and nonautomated task of manually restructuring 2 M N sets of equations in eqn. 4 into the form of eqns. 17 and 18, in order to obtain expressions for the required block matrices.

So by generalising Fig. 1 for any value of M and N , all the block matrices in eqns. 17 and 18 may be directly written down in terms of the matrices in eqn. 4 as follows.

(a) Draw a graphical representation similar to Fig. 1 where the M N elements of the block vectors C(m, n), q m - 1, n) and y m , n - 1) are clearly displayed, and the elements of q m , n), etc., are ordered according to eqn. 19.

(b) Draw the signal flow lines from every point in qtn, n) to those points in the nearest columns and rows of i(m - 1, n) and i(m, n - 1) with the proviso that every line leaving any point w(mM + k, nN + I) in q m , n) must lie in the south-west quadrant viewed from w(mM + k, nN + I); see Fig. 1 for example.

(c) Now we must add the appropriate transmittances to the signal pow lines. Consider, for example, the signal flow line joining w(mM - 1, nN + p) in i ( m - 1, n) with w(mM + k, nN + I) in q m , n) (I 2 p), as illustrated in Fig.

. . . .

. w(mM-l,nN+p) e

. e . . w(mM-M,nN) w(inM- I , n N ) 0 . . . 0

2. The relationship between the transmittance and the actual position of the two points in their respective block vectors is now clear, and derives from the product of the transmittances associated with the other two sides of the triangle in Fig. 2. Further inspection also identifies where

Similar procedures can be repeated to determine the elements of all the other block matrices in eqns. 17 and 18. Expressed mathematically, the whole procedure is as given in Table 1. Not only can this easily be implemented

A:~I- -PAA'O op W. 111 appear in Z 0 in eqn. 17.

Table 1 : Method for deriving the equivalent time-invariant block matrices in eqns. 17 and 18 from the time-varying matrices in Ban. 4*

Let A"(rn, n) be the (m, n)th element of the block matrixd". and use a similar notation for the other block matriws. Then

A"(i,jM) =A;:A'' B''(i. j M ) =A::;{::::}

for j = 1 , 2 . . . .. N ; i = 1 + - 1)M. 2 + (i - 1)M. .. ., NM

Ao'( i . j+(N-1)M)=A:;Ao' bo' (i, j + (N - 1 ) M ) =A:; By-,),

f o r i = l , 2 ,..., N M ; j = ( N - l ) M , l + ( N - l ) M ,.... l + [ i - 1 1 ,

or else

d ' o ( m , n ) = ~ A o ' ( m , n ) = 0 & ' O ( m , n ) = ~ Bn'(m,n)=O

wherep ( i - l ) , , r 4 [ i - l],and 4 r + 1 - j ,

wherer= ( i - l ) , . s = [ i - l ] , , p = r - 0- l ) M , u = S - D - 11 ,

for ( 1 . 1 ) < ( i , j ) Q (MN. MN).

* The modulo-M arithmetic notation in this table was previously defined for eqn. 3.

by computer, but any high level program (of which there are many) that handles nonscalar functions with complex variables, can immediately determine the stability of a LPTV 2-D state-space structure simply by examining the determinant D(z, w ) = I Z - z- 'al0 - w - ' I o ' I ~ 2 5 1 .

0 0 . . 0 w (mM,nN+N-1) w (mM+M-l,nN+N- 1

I m Fig. 2 which becomes the appropriate entry in A'' in eqn. 17

IEE Proc.-Circuits Deuices Syst., Vol. 142, No. 2, April 1995

Schematic showing how the contribution of w(mM - I , nN + p) in q m - I , n) to w(mM + k, nN + 0 in qm, n), is determined by

123

4 Conclusions

Linear periodically time-varying (LPTV) structures find applications in filter design, multirate systems, polyphase structures, interpolators/decimators, etc., but while the study of discrete LPTV 1-D systems has continued inter- mittently for over thirty years, virtually nothing has been written about LPTV 2-D structures.

This paper has shown that certain aspects of LPTV state-space 2-D configurations, such as the determination of the state transition matrix or the derivation of a time- invariant equivalent structure, can be generalised and computed for any degree of the periodicity (M, N). One consequence is a simple computer method to determine the stability of even the most complex LPTV 2-D struc- ture.

This analysis of, and the derivation of the parameters relating to, LPTV 2-D state-space systems will provide the basis for future work into a more generalised frame- work, and provide a groundwork for consideration of other important related issues including invertibility, observability and controllability, and polyphase represen- tation.

5 References

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