Relationship between an LPTV system and the equivalent LTI MIMO structure

D.C. McLernon

Abstract: It is understood how to transform a linear periodically time-varying (LPTV) filter/ difference equation into an equivalent multiple-input/multiple-output (MIMO) structure, or transfer matrix, with linear time-invariant (LTI) elements, but no published method exists for the reverse operation. The paper presents a technique to transform from the LTI MIMO structure to the original single-input/single-output LPTV difference equation and discusses its implica- tions. In addition, it is shown how this new result is then used to represent parallel and cascade connections of LPTV systems as single LPTV filters, implement order reduction of an LPTV difference equation, and finally obtain an LPTV difference equation representation that is equivalent to an LPTV state-space structure.

1 Introduction

While the modern origins of research into linear periodi- cally time-varying (LPTV) continuous-time filters/systems go back over 40 years [ l , 21, the first investigation of LPTV digital filters was a short (not widely referenced) paper by Fjallbrant [3] in 1970. While these filters can be either one- or two-dimensional [4-81, the growing interest in one-dimensional 'LPTV filters/systems has been driven by applications in communications where signals with cyclostationary statistics are often observed and processed. Applications include the filtering of cyclostationary processes [9], channel equalisation [lo, 1 I], blind signal separation [12] and use in multiple access [ I 31. Even H,, a framework traditionally used by control engineers to address robustness issues, has recently been employed in conjunction with LPTV filters [14]. For other applications and a more comprehensive review, see [6].

Also in [6], all the interrelationships between the various equivalent structures for an LPTV filter (realised by a difference equation) were derived, and examples were given to show how this could assist the analysis of LPTV cascade, parallel and inverse filtering configurations that could arise in the applications mentioned above. Again in [6], one well-known equivalent structure for an LPTV filter was derived-the multiple-input/multiple-output (MIMO) realisation, H(z), or the transfer matrix. However, what has not been understood before is how to go backwards from H(z) to the equivalent LPTV difference equation (see Fig. 1). This important new result will now be derived and its implication in analysing and simplifying different LPTV configurations will also be presented.

0 IEE, 2003 IEE Proceedings online no. 20030509 DOI: 10.1049/ip-vis:20030509 Paper first received 2 I st June 2002 and in revised form 13th February 2003 The author is with the Institute of Integrated Information Systems, School of Electronic and Electrical Engineering, The University of Leeds, Leeds LS2 9JT, UK

IEE Proc.-Vis. Image Signal Process., Vol. 150, No. 3, June 2003

2 Statement of problem

Consider the following general LPTV difference equation for the direct-form-I structure (see Fig. la):

MI M Z

A n ) = c a;(n>x(n - 4 + c bj(nlY(n -A

U ~ ( H + N i ) = u;(II), bj(n + N,) = bj(n)

(1)

(2) i=O j= I

where the fundamental period is N, with N being the lowest common multiple of { { N i } 2 o , {N,}z Following the approach in [6], if we define

xk(n) = X(nN + k ) - 00 5 II 5 00

yk(n) = y(nN 4- k ) 0 5 k 5 N - 1

aik = ~ ; ( k ) = a;(nN + k ) 0 5 i 5 MI bjk = bj(k) = bj(nN + k ) t 1 5 j 5 M2

(3)

then we may re-write the single LPTV difference equation in (1) as N cross-coupled difference equations, but where each equation now has different time-invariant coefficients. Then, after taking the z-transform and solving, we obtain

Y(z ) = B-'(z)A(z)X(z) = H(z)X(z) (4)

where:

133

(LPTV to MIMO - known)

~ '(O) ' N - l ; " _ 1 1 Y(O) ON-7 Y(N) "'

b,l -

Ox(1) ON-, x(N+I) " ' Oy(1) 0N-l y(N+1) " '

x(n) fi (9) - 0N-l X(N-1) 0N-l "' ON-l y(N-1) ON-7 "'

(MIMO to LPTV - unknown) [ON-l =(N-1) zeros]

f i (zN) is closely related to H(z) in (4). For N=2, for example, if

b ai@) =ai(n+N) bj(n) =b,(n +N)

a

Fig. 1 U LPTV structure b MIMO structure

Given the MIMO model (H(z) or H(p)), how do we return to the LPTV direct form-I structure?

with coefficients age formed from { aU} and {by} , and N,,(z) = i: aj'z-' = a01 + (a21 - aolb20 + alobl,)z- '

i=O

k=l

with each coefficient P k formed from {by} . So we have gone from the LPTV difference equation

in (1) to the MIMO equivalent structure H(z) in (4)-see Fig. Ib. That much is well known. But what is not understood is, if we know the elements of H(z) in (4), how we find the LPTV coefficients { a q } and {bv} in (1).

Let us deal first with the obvious approach, and for ease of understanding (without loss of generality) let the LPTV filter in (1) be second-order with period N= 2. Then from (4)

H(z ) = B-l(z)A(z)

- a21 b20~-2 2

i=O D(z) = biz-; = 1 - (b20 + b,, + bl0b,,)z-'

+ b20b21z-2 (7)

So ifwe are given H(z), i.e. {ay; i = O:Ml, ke = 0, O:N, N> and {Pi; i = O M 2 } , then one approach would be to attempt to solve 12 nonlinear equations in 10 unknowns ( {aq} and {bq}), i.e.:

a00 = c@

(a20 - aoob21 + allblo) = El

@20 + 621 + bl0bll) = PI

b2ob21 = P 2

00

(8)

This has been tried by the author using various nonlinear optimisation algorithms. Even for only a second-order LPTV filter with period N = 2 (as in (S)), it is cumbersome and convergence is not guaranteed, but for higher-order filters and larger Ni t becomes totally impractical. So this is the aim of this paper-to find an effective analytic solution to (S), generalised for any order and any periodicity N.

3 Solution to problem

This new result is presented in the Appendix (Section 9.1) for the most general case. However, precisely because of its generality, the underlying principle is somewhat hidden. So the basic reasoning is expounded here in Section 3, with the derivation for the particular case of a second-order LPTV filter in (1) with period N = 2. So let us expand A(z), B(z) and H(z) in (4) and (6) as matrix polynomials in z-':

where HU (z) = N u (z)/D(z), with

2 00 -i -

. .- a20b21z-2

NOO(Z) = c a; z - a00 + (a20 - aoob2, + a,,b,o)z-' i=O

No,(z) = =&ayz-; = (a10 + aolblo)z-'

1 i=O A(z) =

+ (a21b10 - a10b21)z-2 2

i=O N , ~ ( z ) = c ~ ~ O Z - ~ = (all + ~ ~ ~ b ~ , )

134 IEE Pr0c:fi.s. Image Signal Process.. Vol. 150. No. 3. June 2003

1 - b20Z-l -bloz-l 1 B(z) = [ -b l , 1 - b 2 1 z - ' ] = [ -611 f ]

N ( z ) H(z) = B- (z)A(z) = ~

j;z-i 2

i=O

k PIZ-' r=O

(1 1) - N o + Nlz- ' + N 2 ~ - 2 -

2

1=0 c BIZ-'

Then, from (9)-(11) (see also in the Appendix (Section 9.2) for an alternative derivation),

2

I=o (A, + A , z d ) pIz-' = (B, +BIZ- ' )

x ( N o +Nlz - ' +N2z-*) (12)

and by equating powers of z-l we can partition the resulting linear equations in terms of the unknown block matrices [Ao A I ] and [Bo B , ] :

Solving for [Bo B1] gives

B M = [ B , B l ]M

where

After re-arranging, this produces the following eight equa- tions in four unknowns:

where ri is the ith row of M. From the Appendix (Section 9.3), let the solution be

b = Pzbl (18)

where P;f represents the pseudo-inverse of Pb. Now consider (1 5) . The two rows of B in BM= 0 are

clearly two linearly independent left eigenvectors of M corresponding to two zero eigenvalues. This implies that the

IEE Proc.-Vis. Image Signal Process., Vol. 150, No. 3, Juwe 2003

rank o f M must be at least two less than its maximum possible rank. Since M is 4 x 4 we can say that if (15) is true then

rank M 5 2

So (19) is a necessary condition [Note 11 for an arbitrary second-order MIMO H(z) to be realised by a second-order LPTV difference equation in (1) with the feedback coeffi- cients calculated from (18). Conversely, if (19) does not hold, then (1) cannot be used to model H(z). Now using this solution for b from (1 8) in (1 3) and (1 4), we can then solve for [A0

(19)

A I ] (eight equations in six unknowns):

BIZ P2Z

1 0 -b20 No NI = [ - b l l 1 0 Ihb::]([ 0 N o ]

where ii is the ith row of

and F i is the ith row of

NI N2

So again from Section 9.3 in the Appendix, let the solution to (21) be

a = P,'al (22) What are the implications of (1 8) and (22)? Firstly, if we have the elements of the MIMO model H(z) that we know originated from an LPTV filter representation in (l), then (1 8) and (22) will give us the exact LPTV coefficients in (1). Secondly, if we are given the elements of an arbitrary MIMO model H(z), then we can immediately check if an exact LPTV representation might exist by perfonning a rank examination as in (19). If H(z) fails this test, then perhaps (1 8) and (22) would still give us coefficients for an LPTV filter in (1) that might in some way 'approximate' H(z)? Thirdly, using (18) and (22) with the recent results from [6], we can now replace cascade and parallel combi- nations of any different LPTV structures with a single

Note 1 : The fact that (19) is theoretically only necessary and not sufficient for H(z) to be realised by the LPTV difference equation in (I) , follows on two counts. Firstly, if a LPTV filter in (1) produces the MIMO structure H(z), this implies that (15) holds-not the other way around. Secondly, if (1 5) holds this implies that rank M 5 2. There may, however, be cases where rank M s 2, but one cannot find suitable coefficients so that (19) holds and also with the appropriate structure for [Bo B , ] in (IO). But in practice, after many simulations, we have observed (without exception) that (19) appears to be also a sufficient condition for the existence of a second-order LPTV difference equation in (1) that will exactly realise H(z). Note that we have carefully said 'sufficient condition for the existence of', because one case has been observed where (19) was true (and we also knew a priori that coefficients for ( I ) did actually exist to realise H(z)), but the algorithm did not retum these correct coefficients-see observation 2 in Section 5.

135

equivalent LPTV difference as in (1). Finally, order reduc- tion and LPTV state-space to LPTV difference equation transformation can now be carried out. Before the deriva- tion of (18) and (22), none of the above could have been achieved. This will now be illustrated with some examples.

4 Examples

Let the period N = 2 in all the following examples.

Example 1: Obtaining the exact LPTV equivalent for a MIMO system, H(z): Consider an arbitrary second-order MIMO structure H(z) as in (6) and (7). The problem is to examine whether this MIMO filtering operation can be more efficiently carried out using an LPTV filter structure as in (1). So let H(z) be

N~,(z) = 1 + 9.2z-I - 1.2zF2, NoI(z) = 6 . 4 ~ ~ ' + ~ . O Z - ~

NI,(z) = 4.5 - 9.32-', N I I (z) = 4 + 1 . 8 ~ - 1 - 6 . 4 ~ ~ ~ (23) D(z) = 1 + 0 . 4 5 ~ ~ ' + 0 . 3 2 ~ ~ ~

From (16) we confirm that rank M = 2 , and so from (19) H(z) may have an exact second-order LPTV filter realisa- tion. So from (1 8) and (22) we obtain the following LPTV coefficients for (1):

But since (1 9) is only necessary and not sufficient for (24) to realise (23), we need to check this result. So from (6) and (7) the LPTV coefficiects in (24) are equivalent to a MIMO system we will call H(z), which if calculated turns out to be identical to H(z) in (23). So this confirms that (24) is indeed the LPTV filter equivalent of H(z) in (23).

This result has other implications. Consider two LPTV systems. From [6] the cascade structure in Fig. 2 has an equivalent MIMO representation, H(z) = H2(z)H, (z) (with H(z) =Hl(z) + H2(z) for a parallel arrangement). Now knowing H(z) we can find a single equivalent LPTV differ- ence equation. The important difference now is that we do not need to apply (1 9)-it will always be true-as H(z) will always have an exact LPTV difference equation realisation via (l), when H(z) comes from a cascade or parallel structure of LPTV filters-see the Appendix (Section 9.4) for a proof.

Example 2: Can we use this method to obtain the LPTV approximation for a MIMO system, H(z)? Repeat example 1, where now we have for H(z)

" ( z ) = 1 + 22-I + 32-2, NO,(z) = 1 + 5z-I + 3z-2

NI,(z) = 6 + 4z-I + 7zP2, NIl (z ) = 5 + 72-I + S Z - ~ (25) D(z) = 1 + 1 . 7 ~ ~ ' + 0 . 9 ~ ~ ~

Fig. 2 Two LPTV filters in cascade-see example I and Section 9.4

136

From (1 6), rank M = 4, and so from (1 9) H(z) will not have an exact second-order LPTV filter realisation, but from (18) and (22) we still obtain the following LPTV coeffi- cients for use in (1):

[a00 a10 a20 h o b2o]

[ % I all a21 61' b211

= [0.8466 4.1407 2.1174 -0.2190 -0.24281

= [4.2983 6.1635 5.2848 -0.7440 -0.37441

(26) From (6) and (7), the LPTV coefficients in (26) are equivalent to a MIMO H(z), which if calculated is (as expected) not identical to H(z) in (25):

kOi,,(z) = 0.8446 + 1.084722-1 + O.7927zf2

&'~I(z) = 3 . 1 9 9 5 ~ ~ ' + 0 . 3 9 3 0 ~ - ~

fiIo(z) = 5.5336 - O.O788z-' (27)

GII(z) = 4.2983 + 3.24782-' + 1 . 2 8 3 2 ~ - ~

b(z) = 1 + 0 . 4 5 4 3 ~ ~ ' + 0 . 0 9 0 9 ~ ~ ~

So can we now say that (26) in some way 'optimally approximates' (25) via (27)? Tempting as this may appear, no such claim should be made, and while for certain H(z) and particular inputs (usually narrowband) we have observed that the LPTV filter implementation may be, to varying degrees, 'close' (in the time-domain) to the MIMO realisation, little confidence can be placed in such an approach. We are currently investigating modifi- cations to the algorithm to see if such a method can be developed.

Example 3: Order reduction for an LPTVfilter: Consider a second-order LPTV filter as in (1) with the following coefficients:

[aoo ain a20 bio b20]

[a01 all a21 bll b211

= [ 1 3.3260 1.7680 0.3580 0.39781

= [ 3 4.5488 1.0976 0.3512 0.43901 (28)

Can we replace this second-order LPTV filter with a lower- order filter? From (6) and (7) we get the second-order MIMO equivalent H(z), where:

Checking terms of

N00(z) = 1 + 2 . 9 5 7 4 ~ ~ ' - 0 . 7 7 6 2 ~ ~ ~

NO, (z) = 4 . 4 ~ ~ ' - 1 . 0 6 7 2 2 ~ ~

N,,(Z) = 4.9 - 1.1885zY' (29)

NI 1 (z) = 3 + 1.0723~-I - 0 . 4 3 6 6 ~ ~ ~

D(z) = 1 - 0 . 9 6 2 6 ~ ~ ' + 0 . 1 7 4 6 ~ ~ ~

the zeros of the numerator and denominator H(z) in (29) shows a common factor of

lA - O.2425zp', which we can cancel out to give a new H(z) with:

fioO(z) = 1 + 3.2z-1, f iOI(Z) = 4.42-I

fiIO(Z) = 4.9, GI (z) = 3 + 1 .sz-l (30) b(z ) = 1 - 0 . 7 2 ~ ~ '

IEE Proc -VIS Imuge Signal Process, Vol 150, No 3, June 2003

via (7)

via (18) and (22)

!B LPTV filter in (1) 4 MIMO structure H(z) in (6)

I arbitrary 1st-order coefficients

I 2nd-order 1

I 2nd-order I F

Fig. 3 See discussion in Section 5

First- and second-order LPTVjil ter equivalence

So, solving the first-order [Note 21 ?quivalent of (17) and (2 1) (i.e. configured for a first-order H(z) and not a second- order MIMO structure to avoid the problem described in Fig. 3 in Section 5) , then rank M = 1, and we get the first- order (see (1)) LPTV filter equivalent of (28):

Remembering that rank M = 1 satisfies only the necessary condition for a first-order LPTV realisation to exist, then from (6)Aand (7) we can confirm that (31) does indeed realise If(.).

Example 4: From an LPTV state-space structure to the equivalent LPTV difference equation: Consider the follow- ing SISO LPTV state-space structure:

where:

A(n) = A(n + 2), C(n) = C(n + 2) and D(n) = D(n + 2)

B(n) = B(n + 2)

A(0) =

C(O) = [-3 41, D(O) = 2

-0.7 -0.2 (33)

[ -:.5 -0.9 A ( l ) =

C(1) = [ 1 11, D(1) = -4

Note 2: For example, (15) and (16) for a first-order LPTV filter now become:

[ -b , , 1 0 1 0 0 -blo 0 ]M=O

equivalence due to cancellation of

different common factors (1 +az-') each time between numerators and denominator

in H(z)

It is well known [ 151 how to obtain the transfer matrix H(z) from (32):

Noo(z) = 2 - 6 . 5 6 ~ ~ ' - 2.5zY2

N~,(Z) = -10~-' - 1 6 . 7 2 ~ ~ '

Nl0(z) = 2 + 0.222-', Nl , (z ) = -4 - 5 . 8 8 ~ ~ ' - 4 . 7 6 ~ ~ ~

D(z) = 1 + 1.022-' + 0 . 3 5 ~ ~ ~ (34)

So from (1 8), (22) and (34) we get the coefficients for the equivalent LPTV difference equation in (1):

[a00 a10 a20 bI0 b201

[a01 all a21 bll b21 I = [ 2 -174.4 57.5 -41 8.051

= [ -4 1.5565 0.5913 0.2217 0.04351 (35)

5 Discussion of results

From the previous theoretical results and the four examples in the preceding two Sections, it is important to make a number of observations and comments.

Observation 1: Referring to Fig. 3, consider a first-order LPTV filter as in (l), and via (7) generate the first-order equivalent MIMO H(z). If we now use H(z) in (1 8) and (22), where these equations are configured for a second-order LPTV filter in (l), then we get a set of second-order LPTV coefficients for (l), and not the original first-order filter coefficients. However, once again via (7), this second-crder LPTV filter can be shown to produce a second-order H(z), which will be identical (through pole/zero cancellation) to the original H(z). This is intuitively what we would expect to happen. Finally, the whole process can be repeated ad infinitum (see Fig. 3) to generate a set of different second- order LPTV filters which are all equivalent to the original first-order LPTV filter.

As an example, consider the following coefficients for a first-order LPTV filter:

[a00 a10 a20 61, bzo] = [ 1 2 0 0.8 01 [a,, a,, a21 bl, b21] = [ 3 4 0 0.9 01

(36)

IEE Proc.-vis. Image Signal Process., Vol. 150, No. 3, June 2003 137

via (7)

via (18) and (22) LPTV filter in (1) ’ MIMO structure H(z) in (6) <

I arbitrary 2nd-order coeffi arbitrary 2nd-order coefficients 2 (nearly always) rank Ms2

rank Ms2 no equivalence

Fig. 4 How algorithm in Section 3 can occasionally fail See discussion in Section 5. Numbers represent the order of the sequence of events ’

From Fig. 3, this will produce the following first order H(z):

No0(z) = 1 + 3.2~-’ , Nol (z) = 4 . 4 ~ ~ ’

N~,(z) = 4.9,NiI(z) zz 3 + 1.8z-I (37) D(z) = 1 - 0.72~-I

which in turn gives the following coefficients for a second- order LPTV filter:

. . A

[2oo a10 a20 g i n 6 2 0 1

b o 1 4 1 221 4, 6 2 1 1

= [ 1 3.326 1.768 0.3580 0.39781

= [ 3 4.5488 1.0976 0.3512 0.43901 (38)

These are equivalent to a second-order MIMO system H(z):

fio0(z) = 1 + 2.9574~-I - 0.7762~-~

&no,(z) = 4 . 4 ~ ~ ’ - 1 . 0 6 7 2 ~ ~ ~

filO(z) = 4.9 - 1.1885z-’ (39) GI (z) = 3 + 1 .O723zp1 - 0 . 4 3 6 6 ~ ~ ~

fi(z) = 1 - 0.9626~-I + 0.1746~-’

Finally, it is clear that k(z) = (1 - 0.2425z-’)H(z), as predicted via Fig. 3. The conclusion to be drawn from all this is that when we use (18) and (22) we need to consider in advance (and adjust the equations of the algorithm accordingly) the order of the equivalent LPTV filter that we wish to derive. Otherwise we will end up with a LPTV filter that will model the MIMO structure H(z), but whose order is too large. So, in this particular case, we should not have used (18) and (20) (based on a second-order LPTV filter), but their first-order equivalents [Note 21 and this would then have returned the original LPTV coefficients of (36), but as first-order vectors.

Observation 2: Referring to Fig. 4, let us suppose that we have a second-order LPTV filter in (1) and we generate an equivalent second-order MIMO H(z) via (7)-see path 1. If we subsequently use (18) and (22) to return to a second- order LPTV filter realisation (via path 2 in Fig. 4), we get (as expected) the original second-order coefficients of (1). However, through simulations, rare exceptions have been observed where we do not return to the original coefficients for (1) (see path 3), and this new LPTV filter also does not have the same ‘impulse responses’ as the original LPTV

138

filter, so there is no ‘equivalence’. One such rare exception is for the coefficients below (albeit unstable, but this is irrelevant):

[aoo a,, a2, b10 b2,1 = [ 1 2 3 4 51 [a,, ail a21 611 b21] = [ 6 7 8 9 101

(40)

Note also that Pa in (22) is now unusually not full rank. Also, note that here rank M = 2 (see (19)), but consistent with [Note 11, rank M = 2 correctly implies the existence of second-order LPTV coefficients for (1) that will exactly realise H(z). However, in this case the algorithm does not actually return these coefficients of (40), but gives (via path 3 in Fig. 4) the following coefficients which do not realise H(z) , because some of the feedforward values are incorrect:

[aoO a10 a20 b10 b2,1 = [-1 2 1 4 51 [a01 all Q21 b,, b,,] = [-I 7 1 9 101

(41)

Strangely, just by changing the coefficient 10 in (40) to 10.000 1, the algorithm performs correctly, re-generating the original LPTV coefficients in (40) via path 2 in Fig. 4, with Pa in (22) now full rank.

These cases are rare-in fact this was the only one observed after numerous simulations were performed. At present we do not have a theoretical explanation, and the search for one is currently the purpose of ongoing research.

Observation 3: Finally, in the case of example 2 we considered a stable MIMO H(z), and derived the coeffi- cients for an ‘approximate’ LPTV equivalent filter in (1). It is impossible to predict in advance of applying the algorithm whether this ‘approximate’ LPTV filter will be stable or unstable, although in simulations it is usually stable. While there does not appear any theoretical a priori method to ensure stability, this is not a problem, as example 2 was only included to show the limitations of trying to ‘approximate’ a MIMO system using the proposed algorithm. Thus this approach is not recom- mended and so stability in example 2 is not an issue, and we will not attempt a theoretical stability analysis.

6 Conclusions

In a recent paper [6] a summary of many LPTV filter/system applications in communications was

IEE Proc.-Vis. Image Signal Process., Vol. 150, No. 3, June 2003

given, and all the interrelationships between the various equivalent structures for an LPTV filter were derived. While other researchers [ 15, 161 have tackled the problem of obtaining the equivalent minimal LPTV state-space representation when given the elements of the transfer function matrix H(z), one question has remained unsolved-how to return from the MIMO structure H(z) in (6) to the equivalent LPTV difference equation in (1). A solution has now been derived for any order of LPTV filter with any period N, and a number of examples have been presented to show how this result may be applied in practice.

In addition, this result will now allow both the simpli- fication and the analysis of interconnections of various LTI and LPTV structures (in communication systems) within a consistent mathematical framework, where issues of complexity, sensitivity, etc. may now be dealt with. The LTI structures may be represented as LPTV systems with period N, but where the coefficients do not alter. Even the LPTV structures can each also have different periods (say N I to NQ), and analysis can now be carried out ov,er the whole interconnection using period N=N, where N is the lowest common multiple of {Ni}?= I .

Finally, future work will deal with the issues raised in Section 5, as well as attempting to extend the results in this paper for use in the analysis and representations of the important emerging area of MIMO communication systems [ 171.

7 Acknowledgments

The author would like to thank Dr D. Wilson for his help with the development of this paper.

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OROZCO-LUGO, A.G., and MCLERNON, D.C.: ‘A blind signal separation method for SDMA based on periodically time-varying modulation’. Proceedings of IEE Nat. Conf. on Antennas and propaga- tion, University of York, UK, 3 1 March-1 April 1999, IEE Conf: Publ. 461, pp. 182-186 ROVIRAS, D., LACAZE, B., and THOMAS, N.: ‘Effects of discrete LPTV on stationary signals’. Proc. IEEE Int. Conf. on Acoustics, speech and signal processing, Orlando, USA, 13-17 May 2002, pp. 1217-1220 XIE, L.S., WANG, C., and ZHANG, S.: ‘H, deconvolution of periodic channels’, Signal Process., 2000, 80, pp. 2365-2378 LIN, C.-A., and KING, C-W.: ‘Minimal periodic realisations of transfer matrices’, IEEE Trans. Autom. Control, 1993, 38, pp. 462466 COLANERI, P., and LONGHI, S.: ‘The realisation problem for linear periodic systems’, Automatica, 1995, 31, pp. 775-779 Proceedings of IEE Seminar on MIMO: communications systems from concept to implementation, IEE, Savoy Place, London, 12 December 2001, Publication no. 2001/175

Appendix

9.1 This is the generalisation of the solution presented in Section 3 now for an LPTV filter of any order and any period N. To simplify nomenclature, let P = M I = M2 in (l), and then expand the matrices in (4) as matrix poly- nomials where we can show that [Note 31

Generalisation of solution in Section 3

-[-PI -[-PI B(z) = i=O 2 Biz-’, A(z) = i=O 1 Aiz-’

P NizP

H(z) = L 5 piz-‘ i=O

So the problem is, given {N‘} and { P I } for H(z) in (42), how do we find the matrices { A , } and { B , ) that contain the LPTV coefficients { a , } and {b,} in (1). So from (4) and (42) we can write

-[-PI

i=O i=O 1 Equating powers of z-’ gives:

(44)

This represents K = 1 + P - [-PI equations in (1 - [-PI) unknowns. Note that if K is even (as in Section 3 ) we can split (44) into two complementary (disjoint) sets of K/2 equations each. In the case where K is odd, (44) can be split into two sets of (K + 1)/2 equations, with each set having

Note 3: Let X=mN+n, O s n s N - 1, where X , m, n and N are all integers, with Nand n non-negative. Then we can define [XI = m (i.e. based on modulo-N arithmetic). So, for example, if X = P = 2 and N = 2 ( X = m N + n j -2 =(-1)2 + 0 j [-2]= -I) , then -[-PI = -[-2] = - ( - - I ) = I , whichfrom(42)impliesBo,BI (andAo,Al)asin(9)and(10).

139 IEE Plot.-Vis. huge Signul Process., 61. 150, No. 3 . .June 2003

one equation in common. Without loss of generality, assume that K is even, and then we obtain

P I =

P 2 =

p3 =

=+ BM = 0 for M = [P2P;'P3 - P4] API = B P 2

A P 3 = BP4

-POI P I 1 P 2 1 P 3 I . . . P K I 2 - 1 1 - 0 P O z P I z P 2 I " ' PKl2-2'

0 0 P o l &I . . . P K / 2 - 3 I

- 0 0 . " P(K/2+[-P]- I )z -

- N o NI N 2 N 3 . . . N K / 2 - 1

N K / 2 - 2 0 No NI N 2 ' . .

0 0 No NI . . . N K / 2 - 3

-

. . . . .

- 0 0 ' . ' N ( K / 2 + [ - P ] - l ) -

- P K / 2 I PK/2+ I 1 P K / 2 + 2 1

P K / 2 - 1 1 P K / 2 1 B K / 2 + I J

PKl2-2' P K / 2 - 1 1 PK/2'

- P(K/2+[-P])' fi(K/2+[-P]+I)' P(K/2+[-P]+2)'

(45)

bK/2+3' ' " - PK/2+2J . . . 0

PK/2+1' . . . 0

P(K/2+[-Pl+3)J ' . . P P I -

with:

NK/2+3 ' ' ' 0 - N K / 2 + 2 . . . 0 NK/2+l ' ' ' 0

NK/2+[-P]+3 ' ' ' N P -

(46)

where K = 1 + P - [-PI, A and B are N x N( 1 - [-PI) and Pi is N(l - [-PI) x (NK/2) .

Now since B in (45) is a matrix of not only the time- varying feedback coefficients { bv} in (l), but also contains some constant elements (i.e. ones and zeros-see (15)), then we can re-structure (45) to get a set of over- determined linear equations, as in Section 3:

P b b = b, j b = Pbfb, (47) 140

where the NP x 1 column vector b = [b10b20...bP0 Pb is an (N2K/2) x NP

matrix, bl is an N2K/2 colwnn vector and 6 is the pseudo- inverse of p b . Both p b and b are formed from the appropriate rows of M = [P2P11P3-P4]. Now using the elements of b in (47) to reconstruct B, then from (3 1) we get A(Pl + P3) = B(P2 + P,), which can be re-structured to give another set of over-determined linear equations and an appropriate solution

P,a = a, + a = P,+a, (48) where the N(P+ 1) x 1 column vector a = [ a o o a l o ~ ~ ~ a p O a O l a l 1 "'ap1 ...aO,N- la 1 ,N- 1 . . . u ~ N - 1 l T , P, is an (N2K/2) x N(P+ 1) matrix formed from the rows of (PI +P3), and al is an N2K/2 x 1 column vector formed from B(P2 + P4). Finally, following similar arguments as in Section 3, then from (45) the generalisation of (19) gives

(49) which is a necessary condition for (1) to exactly realise an arbitrary MIMO system H(z). This generalisation of (19) follows from the fact that the N rows of B in (45), which are the left eigenvectors ofMcorresponding to zero eigenvalues, are linearly independent. The proof of the linear indepen- dence of the eigenvectors will not be formally given here. It follows directly from the imposed structure of B and is obvious from the examples in Table 1, given the relative positions of the ones and the zeros in the rows of B.

rank M 5 {minimum dimension of M } - N

9.2 An alternative derivation of (12) or (43) It is instructive to note that (12) can also be obtained by a block realisation of an LPTV filter. Consider the particular case of second-order, N = 2 in (1)-but this is easily generalised for (43). Using (1)-(3) we can write

1 0 0 0 0 0

- b 2 0 -blo 1 0 0 0 -41 1

0 -b21 -hi1 1 0 0 . . . 0 0 -b20 -blo 1 0

0 0

=+ [i ;: :o ...I[ i;] = [:, .:: lo ...I[;] 0 0

+BOYn =A&, +A,Xn-, -BIYnpl (50)

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Table 1: Illustration of the linear independence of the rows of B in BM=O in (45) (i.e. left eigenvectors of M corresponding to zero eigenvalues) for different orders of LPTV filter and different periods, N

First-order in (1) Second-order in (1) Third-order i n (1)

N = 2 : B = [ - b 1 , : : -:OI 1 0 - 4 0 -blo 0 - 4 0 1

'=[-bl1 1 -q, -b21 0 0

where Ao, A I , Bo and B1 are defined in (9) and (lo), U, = bo(n) yl(n>lT and Xn = [xo(n) xl(n)lT. Using the defi- nitions in (5), take the z-transform of (50); after rearran- ging this gives

Y (z) = H(z)X(z)

H(z) = (Bo +B,z-')-'(A, +A,z- ' ) - N , + N,z-' + N2z-2

(51) - 2

i=O Biz-'

from which we finally obtain (12).

9.3 Why use PL and P t in (47) and (48)? Consider Pbb = bl in (47), with the following discussion, where appropriate, also applying to (48). This represents an over-determined set of linear equations, and from simulations we have observed three conditions that can and do occur.

Condition I : Consistent set of equations with unique solution. P b is h l l rank and rank[Pb] =rank[Phlbl]. Choose unique solution. This corresponds to examples 1, 3 and 4 in Section 4.

Condition 2: Consistent set of equations with infinite number of solutions. Pb is not full rank and rank[Ph] = rank[Pblbl]. Choose minimum-norm solution. This corresponds to a rare case observed in simulations but not included within section 4.

Condition 3: Inconsistent set of equations with unique least- squares solutions. P b is full rank and rank[Pb] < rank[Pblbl]. Choose unique least-squares solution. This corresponds to example 2 in Section 4.

To accommodate the above three scenarios, we use the pseudo-inverse and write b = Pgbl . Finally, from simula- tions it appears that Pa is rarely rank deficient-see observation 2 in Section 5.

9.4 All cascade and parallel structures of LPTV filters can be realised by a single LPTV filter This claim was made in example 1 of Section 4 to explain why (1 9) (or (49) in general) will always be true for cascade or parallel structures. So consider the cascade structure of two first-order LPTVs in Fig. 2, where each filter has coefficients with period N . For simplicity, use different nomenclature for the coefficients, and so we can write:

~ ( 8 ) = u(H)x(.) + b(n)x(n - 1) + C ( T Z ) Z ( ~ - 1) (52) y(n) = d(n)z(n) + e(n)z(n - I ) +f(n)y(n - 1) (53)

(55)

z(n - 1) =

e(n - l)[a(n - l)x(n - 1) + b(n - l)x(n - 2)]

+c(n - l)b(n - 1) - f ( n - l)y(n - 2)1 (56) e(n - 1) + c(n - l)d(n - 1)

Finally, substitute (55) and (56) into (53), which after re-arranging gives

where

I x [b(n)d(n) + a(n - l)e(n - l)]

-c(n - llf(n - l)[c(n)d(n) + e(n)l c(n - l )d (n - 1) + e(n - 1) b,(n) =

So Fig. 2 is equivalent to the second-order LPTV differ- ence equation in (57) and (58). While the above is a proof for two first-order filters in cascade, due to lack of space a formal proof for higher-order filters will not be given, but it is clear how this approach may be easily generalised. Finally, a parallel structure can also be analysed in a broadly similar manner.

So from [6] and Fig. 2, H(z) = H2(z)Hl(z), and thus we do not need to apply the rank test in (49) to H(z) for cascade (or parallel structures), because we now know that a single LPTV filter can always realise H(z), and so (49) is always true.

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