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ORTHOGONAL FILTERS

BIBLIOTHEEK TU Delft P 1674 4313

ORTHOGONAL FILTERS

Proefschrift

ter verkrijging van de graad van doctor in de

technische wetenschappen aan de Technische

Hogeschool te Delft,

op gezag van de rector magnificus,

Prof. ir. B. P. Th. Veltman

voor een commissie aangewezen door het

college van dekanen te verdedigen op dinsdag

13 oktober 1981 te 14.00 uur.

door

Edmond Deprettere

electrotechnisch ingenieur

Dit proefschrift is goedgekeurd door

de promotor prof. dr. ir. P.M. Dewilde

I wish to express my gratitude to all those who have contributed to the final achievement of this thesis. I dedicate it to my parents. A n d to Christine, her encouraging style of living.

5

In this study, we obtain some results concerning the subject of multivariate trans-fer function synthesis and realization for the stationary discrete-time case.

Our realizations are strongly related to continuous-time lossless filters, well-known in classical analog network theory. They are generalizations of certain filters, called wave digital filters, that are discrete-time counterparts of analog ladder filters and lattice filters, and that emerge as a scalar subclass of the class of multivariate filters presented hereafter.

We devote most of our attention to the filter realizations and realization structures, to construction procedures and algorithms, and to certain important properties related to some useful interpretations. The structures are highly modular, with de-lay-free interconnectable modules and orthogonal model realizations. This is attrac-tive from both a (hardware) implementation and a numerical analysis point of view. The construction procedures provide recursive and norm preserving algorithms, ex-tracting modules from primary data in some optimal way.

We heavily lean on the theory of lossless and J-lossless embedding, through which we are led naturally to a scattering frame work. The implication of this approach is twofold. First, the scattering interpretation of the filter construction algorithms suggest a number of potential applications in the domain of inverse filtering, in particular inverse scattering, for recursive and orthogonal system identification, es-timation and modeling. Secondly, some results of scattering theory can be applied to a number of seemingly unrelated problems dealing with interpolation, spectral factorization, least-squares approximation and linear estimation, to which fast and robust recursive solutions can be found that are, not surprisingly in retrospective, akin to our filter construction algorithms.

In chapter I, we introduce the filters studied in this work in a fairly general way. That is, both as rational multivariate, segmented orthogonal models of linear, shift-invariant, passive multiports or multichannel systems with unitary input-output map , as well as numerically well-conditioned cascade realizations of embeddings of contrative transfer functions, (dual) spectral factors and positive real matrices. We expose in some details the relation between contractive scattering matrices and J-expansive chain scattering matrices as well as the relations between the various quantities embedded in such matrices.

By invoking some standard results from classical network theory, among which the notion of degree reduction by factorization is quite fundamental, we show in chap-ter II that any square scatchap-tering matrix, which is of even dimension, strictly contrac-tive outside the unit disc and unitary on the unit circle can be realized in a dead-lock-free and norm preserving cascade of lowest possible complexity. We pay parti-cular attention to the orthogonal state-space realizations and realization structures of the various filter sections involved. Construction procedures and algorithms form one of the subjects of the next chapter.

Chapter III is devoted to exact and approximating minimal embedding, with special emphasis on J-expansive/J-unitary embedding, to exact and approximating factori-zations of embedding matrices, and to exact and partial realifactori-zations of embedded functions. The chapter explores and exploits an important and useful connection

between generalized Darlington synthesis of lossless filters and Schur parametriza-tion of bounded series.

We show that the recursive filter construction algorithms are useful in time domain approximating modeling, interpolation and factorization problems. A number of potential applications is briefly discussed.

More connections between the results presented in this work and related theories and problems, although not explicitly stated and exposed, have been touched upon in some of the examples scattered throughout.

7 C O N T E N T S I N T R O D U C T I O N ' 8 C H A P T E R I M U L T I P O R T U N I T A R Y A N D J - U N I T A R Y F I L T E R S 1.0. Introduction 13 1.1. Preliminaries: multiports 13

1.2. Scattering matrices and chain scattering matrices 16 1.3. Degree and factorization of rational matrices 19 1.4. Positive hermitian matrices, spectral factoring and

minimal embedding 23 1.5. Inverse m-ports and dual 2m-ports 28

1.6. Lossless (S-)filters and J-lossless (0-)filters 32 C H A P T E R II O R T H O G O N A L C A S C A D E R E A L I Z A T I O N O F R E A L

M U L T I P O R T D I G I T A L F I L T E R S

2.1. Introduction 37 2.2. Standard degree 1 factors 38

2.3. Computable cascades 41 2.4. Real chain scattering matrices of degree 2 47

2.5. Scalar applications and 2-channel example 51 A p p e n d i x 1 Degree reduction conditions 55 A p p e n d i x 2 Standard matched unipolar factors and variants 55

A p p e n d i x 3 Real factors 58 C H A P T E R III R A T I O N A L A P P R O X I M A T I O N O F T I M E D O M A I N D A T A

V I A M I N I M A L E M B E D D I N G

3.0. Introduction 62 3.1. M i n i m a l para-unitary and para-J-unitary embedding of

scattering matrices 62 3.2. Parametrical representations of minimal f-invertible

embeddings 65 3.3. Recursive Schur extractions of Darlington sections 69

3.3.1. Schur lemma 69 3.3.2. Singularities 70 3.3.3. Schur/Darlington recursion 72

3.3.4. Inverse scattering 78 3.3.5. Once more the factorization theorem 81

3.4. Rational approximation 83 3.4.1. Introduction 83 3.4.2. The state space containing the approximation

information 85 3.4.3. Projections and normal equations 86

3.4.4. Approximating embeddings and partial realizations 90

3.4.5. Application notes 92 A p p e n d i x 1 M i n i m a l embedding, remarks and examples 98

A p p e n d i x 2 M i n i m a l embedding, non full-rank cases 102

A p p e n d i x 3 106 Appendix 4 Proof of Schur lemma 106

8

Introduction

In this thesis, we shall study the problem of realizing a stable discrete-time, station-ary transfer function T(z) in a modular and numerically well conditioned filter. A n y discrete-time filter has a 'state' vector x(n) representing the content of the filter memory at time n. x(n) w i l l have finite dimension throughout.

The filter, then, can be represented by a system ( A , B , C , D ) [ 1,2] defining a proces-sing map which assigns to the input vector u(n) from input space U and the state vector x(n) from state space X , the output vector y(n) i n output space Y

y(n) = Du(n) + Cx(n) ( l a )

and which supplies an update for the state vector

x(n + 1) = Bu(n) + A x ( n ) ( l b ) where A , B , C and D are matrices of appropriate dimensions, which are constant

in the stationary case.

Assuming that (1) is asymptotically stable [ 3 ] , the matrix D + C(z — A )_ 1B , where z = r el 9 w i t h |r| > 1, is called (the state space realization of) the stable trans-fer function of the filter. It is minimal i f A has the lowest possible dimension. We shall say that the filter (1) is an orthogonal filter i f the system (processor) matrix

D C

B A

(2)

in the state-variable description (realization) (1) is unitary. It is well k n o w n [4] that unitary transformations have great numerical stability properties. Hence, it may be expected that filters implemented with these transformations will also have very desirable mathematical properties.

N o w the transferfunction of an orthogonal filter, E ( z ) say, is stable (has poles in-side the unit disc D = {z : |z| < 1}) and is unitary on the unit circle

T = {z : |z| = 1} i n the complex z-plane. Obviously, not any stable transfer function T(z) is the transferfunction of an orthogonal filter, and i f we wish, none the less, to realize any stable T(z) in an orthogonal filter, we shall have to follow a more sophisticated strategy, called embedding. That is, we search for a transfer-function 2 ( z ) of an orthogonal filter, such that T(z) is realized by applying appro-priate boundary relations on the inputs and outputs of 2 ( z ) . E.g., T(z) might be one of the block-entries of 2 ( z ) . Such an approach w i l l indeed allow us to realize any stable T(z) i n an orthogonal filter.

Orthogonal filters become even more attractive, both from an implementational and an arithmetical point of view i f there is a simple, modular procedure to build them. If the modules are themselves orthogonal, then it will become easy to control the processing of internal quantities locally, so as to ensure stable, that is close to unitary though contractive, computations i n practice. Such a procedure indeed exists and is called cascading. If two orthogonal filters with transfer functions 2t( z ) and 22( z ) are cascaded, then a new orthogonal filter results, now w i t h transfer function S ( z ) = S2( z ) * 2 j ( z ) , where * indicates the so-called Redheffer star product [5.].

9

A n easier representation for the transfer properties of the product is obtained by reordering, if possible, inputs and outputs on the filter structure and converting the stable transfer function E ( z ) into a modified, not necessarily stable, transfer function 0(z), with the property that the Redheffer product 2(z) = £2(z) * ^ i (z) is transformed into a standard matrix product 0(z) = 02(z)0j(z).

Under certain conditions on the dimension of S(z) and the invertibility of its block-entries, we will find [6] that 0(z) is unitary with respect to an indefinite metric described by a signature matrix J = Im + (—Im), where m is half the dimen-sion of 2(z). The construction, then, of cascades is most likely and basically tackled by invoking some factorization techniques [7, and the references therein] supply-ing the cascade modules, preferably in a fast recursive way.

Unitary realizations of z-plane Redheffer cascades are discrete-time counterparts of continuous-time lossless filters [9ab], and are generalizations of so-called wave digital filters [10, and references therein]. Therefore, we shall sometimes call 2(z) a lossless transfer function. Likewise, we shall sometimes call 0(z) a J-lossless trans-fer matrix.

Conventionally, some rational contractive transfer matrix is embedded in a lossless transfer matrix for which a realization is constructed via degree reduction by fac-torization {1.1 ] , whereby, i n order to obtain a true cascade, the facfac-torization must be carried out on a corresponding J-lossless transfer matrix [ 1 2 ] , as described above.

However, as introduced here, 0(z) appears to be only a convenient mathematical description for cascading purposes. More can be said about J-lossless transfer func-tions. Indeed, although 0(z) need not be stable, it may be so and in some impor-tant cases it will be stable and play an essential role as stable transfer function of a filter i n which certain inverses of transfer functions are realized. We shall see that it is possible to construct orthogonal filters on non-rational string data via the con-struction of cascade filters which have a system matrix R defined via (2) which is J-unitary and realizes a stable J-lossless transfer matrix 0(z). Examples where this happens to occur are among others, spectral factorization [ 1 3 ] , innovations factori-zation and stochastic least squares estimation [ 14, and references therein], [ 8 ] , in-verse filtering and inin-verse scattering [ 3 1 ] . Therefore, we shall also study the realiza-tion of stable J-lossless transfer funcrealiza-tions — a generalizarealiza-tion of the theory for lossless functions.

A t this point a difficulty, which has been overcome in this thesis, must be mentioned. When orthogonal filters are used in digital signal processing applications, iterative com-putational schemes are not likely to be permissible. However, when one attempts to cascade orthogonal filters, then computational deadlocks at the filter interfaces are the very situations which are most likely to occur. We show that such deadlocks can always be avoided by bringing into play some degree of freedom. Moreover, in appli-cations where data is processed by means of (growing) orthogonal cascade filters which are concurrently constructed from this data, then deadlock-free cascades not only allow fast processing of the data but also guarantee the uniqueness for the result-ing transfer functions realized i n such filters.

Thus, our main problem can be stated as follows.

Obtain a realization R * (A,B,C,D) for a given filtering problem in such a way that the implementation is a deadlock-free cascade of elementary unitary (J-unitary) modules; preferably, one should be able to compute the elements of the cascade recur-sively from primary data.

10

Here is how our treatment of this problem is organized. In chapter I, we briefly re-view the set up, via properly chosen input and output spaces and input-output maps, the scattering frame work i n which the various results will be presented. We investigate in some detail the relation between lossless transfer functions and J-lossless transfer functions, as well as the various functions which can be embedded i n such matrices. Positive real functions, familiar to network theorists [ 15ab, 9a, 16] will play a central role. We summarize a few results from degree theory [17, 18] and give standard ma-trix degree reduction conditions and generally applicable factorization rules [9a, 11,

12]. We give precise definitions of, and some useful relations between what we shall call lossless (E-)filters and J-lossless (Q-)filters. These are deadlock free, minimal uni-tary and J-uniuni-tary realizations of lossless Redheffer cascades (S(z) = Sn( z ) * . . . * S j ( z ) ) and J-lossless asymptotically stable Potapov cascades (@(z) = @n( z ) 0n_ 4( z ) . . . © j ( z ) ) respectively.

In chapter II, we w i l l concentrate on the elernents 2j(z) and @;(z) of least possible complexity (degree 1) which may occur i n such cascades. We obtain their minimal realizations, unitary and J-unitary respectively, and we make these realizations unique to meet the deadlock condition as stated above. We also include some flow-diagram representations of the elements system map. These are not unique and variants can be easily given, although these may reduce the economy in the arithmetic operations. In chapter HI, we will be concerned with less conventional interpretations and appli-cations of the notions of unitary and J-unitary embedding, factorization and cascade realization. We first restate a classical theorem on unitary embedding [ 19, 9 a ] , now in terms of J-unitary embedding, and we pay attention to non full rank cases. We show that the factorization of a minimal, asymptotically stable, J-lossless transfer matrix, in particular the construction of a J-lossless (0-)filter, can be viewed as a sort of (Gram-Schmidt) orthogonalization and triangularization procedure that supplies a recursive solution to certain so-called normal equations [ 2 0 , 2 1 , 3 6 ] . These are generali-zations of the so-called Wiener-Hopf equations, involving the inversion o f a Toeplitz-Gram matrix. This establishes the relation between orthogonal filter theory and con-struction algorithms on the one hand, and least-squares optimization or best (rational) approximation theory [28, chapter I X ; 23] and certain well conditioned matrix in-version and decomposition algorithms [21, 24, 36, 4] on the other hand. One example is i n [ 2 5 ] . A n other example is the realization of A R M A images, under a generalized Kolmogorov isometric mapping, of innovations processes [ 1 4 ] . It has been in this context [26] that the above mentioned relationship was first pointed out explicitly. See also [ 3 7 ] . Other examples will be briefly discussed.

We shall study the problem of constructing J-lossless (0-)filters, hence lossless (2-) filters, directly from string data, called waves, as a way of obtaining embedding reali-zations of approximating solutions to normal equations. In this case, the cascade ele-ments, though 'extracted' from the data i n some optimal way, will not i n general be degree reducing. Such realizations w i l l be called partial realizations.

The key problem i n this context is spectral factorization. The method is essentially co-prime factorization in the sense of [27: section 6, and reference [23] therein], and the result is actually dual spectral factorization. See also [ 2 8 ] . We shall find, however, that many interesting features of the method go back to a paper of Schur [29]. See also [ 3 0 ] .

Since lossless (S-)filters can be viewed as numerical analogues of physical conservative transmission and scattering systems, the filter construction algorithms presented in this chapter can also be related to problems to which some results of scattering

theo-11

ry do apply. The prototype problem, then, is the so-called inverse scattering problem [31, 32, 33] which will be exposured in some detail. See also [ 3 4 ] .

However, only preliminary results w i l l be given for the general case dealing with sys-tems exhibiting real frequency zeros of transmission and/or non full rankness. Needless to say that the theory presented is by no means complete. The emphasis is on realization and construction algorithms, and besides on some interpretations and properties related to problems and applications i n least-squares approximating time-domain factorization. A l t h o u g h we show that the approximations produced with the recursive algorithms are unique and show up decreasing approximating errors with increasing approximating order, we shall not give a rigorous treatment of convergence conditions and properties. This problem is dealed w i t h i n [35ab] in relation to the linear least squares estimation problem.

A l t h o u g h we have restricted ourselves to the stationary case, some useful extensions of the results presented in this thesis can be given to cover time-varying problems and adaptive filtering applications. See e.g. [ 3 6 ] .

References

1. Rosenbrock, H., State-space and multivariable theory. Wiley, New York (1970). 2. Kailath, T., Linear systems Prentice Hall, Inc., Englewood Cliffs, N.J., (1980). 3. Popov, VM.,Hyperstability of control systems. Springer, Berlin (1973).

4. Householder, A.S., The theory of matrices in numerical analysis. Dover Publications, Inc., New York (1964).

5. Redheffer, R., 'On the relation of transmission-line theory to scattering and transfer', J. Math. Phys., Vol. X L I , 1-41 (1962).

6. Potapov, V., 'The multiplicative structure of J-contractive matrix functions', Amer. Math. Soc. Transl. Ser. 2, vol. 15, 131-244 (1960).

7. Van Dooren, P.M., and P. Dewilde, 'Minimal cascade factorization of real and complex rational transfer matrices', IEEE Trans. Circuits Syst., Vol. CAS-28(5), 390-400 (1981). 8. Darlington, S., 'Linear least-squares smoothing and prediction, with applications', The

Bell Syst. Tech. J., 37(5), 1221-1294 (1958).

9a. Belewitch, V., Classical Network Theory. San Francisco, C.A. Holden Day (1968). 9b. Belewitch, V„ 'Summary of the history of circuit theory', Proc. IRE, 50(5), 848-855

(1962).

10. Nouta, R., 'Studies in wave digital filter theory and design', Ph.D. Dissertation, Dept. of Electrical Engineering, Delft Univ. of Technology, Delft, The Netherlands.

11. Dewilde P., V. Belewitch and R.W. Newcouw, 'On the problem of degree reduction of a scattering matrix by factorization', J.of the Franklin Institute, 291(5), 387-401 (1971). 12. Dewilde, P., 'Cascade scattering matrix synthesis', Stanford Univ., Stanford Calif., Tech.

Rep. 6560-21 (1970).

13. Helson, H., Lectures on invariant subspaces. Academic Press (1964).

14. Kailath, T., ' A view of three decades of linear filtering theory' in Linear least-squares es-timation (ed. Thomas Kailath). Dowden, Hutchinson & Ross, Inc. (1977) - reprinted from IEEE Trans. Inf. Theory IT- 20(2), 146-181(1974).

15a. Brune, O., 'Synthesis of a finite two terminal network whose driving-point impedance is a prescribed function of frequency'. J. Math, and Phys., 10(3), 191-236 (1931). 15b. Guillemin, E.A., Synthesis of passive networks. John Wiley, Inc., New York (1957). 16. Anderson, B.D.O., and S. Vongpanitlerd, Network analysis and synthesis, a modern

sys-tems theory approach. Prentice Hall, Englewood, Cliffs, N.J. (1973).

17. Youla, D.C., and P. Tissi, 'An explicit formula for the degree of a rational matrix' Poly-technic Inst. Brooklyn, Brooklyn, N.Y., Rep. PIMBRI-1272-65 (1965).

12

18. Kaiman, R.E., 'Irreducible realizations and the degree of a matrix of rational functions', SIAM. J. Appl. Math., 13, 520-544 (1965).

19. Oono, Y., and K. Yasuura, 'Synthesis of finite passive 2n-terminal networks with prescribed scattering matrices', Mem. Fac. Eng. Kyushu Univ., 14(2), 125-177 (1954).

20. Makhoul, J., 'Linear prediction: a tutoral review' Proc. IEEE, 63(4), 561-580 (1975). 21. Kailath, T., A. Vieira and M. Morf, 'Inverse of Toeplitz operators, Innovations and

ortho-gonal polynomials', SIAM Review, 20(1), 106-119 (1978).

22. Gautmacher, F.R., The theory of matrices. Chelsea Publishing Co., New York (1960). 23. Bultheel, A., and P. Dewilde, 'Orthogonal functions related to the Nevanlinna-Pick

problem' in Proc. MTNS conference, Delft, July 1979, (ed. P. Dewilde), Western Periodicals Comp., North Hollywood, 207-211 (1979).

24. Morf. M., 'Fast Algorithms for multivariate systems' Ph.D. dissertation, Dept. of Electrical Engineering, Stanford Univ., Stanford, Calif. (1974).

25. Rhodes, J.D., P.C. Marston and D.C. Youla, 'Explicit solution for the synthesis of two-variable transmission-line networks', IEEE Trans. Circuit theory, CT-20(5), (1973). 26. Dewilde, P., A. Vieira and T. Kailath, ' A generalized Szegö-Levinson realization algorithm

for optimal linear prediction based on a network synthesis approach', IEEE Trans. Circ. and Systems, C A S - 2 5 , 663-675 (1978).

27. Dewilde, P., Input-output description of roomy systems' SIAM J. Control and optimiza-tion, 14(3), 712-736 (1976).

28. Anderson, B.D.O. and T. Kailath, 'Passive networks synthesis via dual spectral factorization' IEEE Trans. Che. and Systems, C A S - 2 6 , 866-873 (1979).

29. Schur, J., 'Uber Potenzreihen, die in Innern des Einheitskreises beschrankt sind', J. für die Reine Angewandte Mathematik, 147, 205-232 (1917).

30. Delsarte, Ph., Y . Genin and Y . Kamp, 'Schur Parametrization of positive definite block-Toeplitz systems' . MBLE Research Laboratories, Report R360, Brussels (1977).

31. Claerbout, J.F., Fundamentals of geophysical data processing. McGraw-Hill, New York (1976).

32. Krein, M.G., 'Solution of the inverse Sturm-Lionville problem' Dokl. Akad. Nauk SSSR, 73, 21-24 (1959).

33. Gopinath, B., and M. Sondhi, 'Inversion of the Telegraph equation and the synthesis of non-uniform lines', Proc. IEEE, 59(3), 383-394 (1971).

34. Dewilde, P., and I. Widyn, 'Signal estimation using inverse scattering techniques', Proc. European Conf. on Circ. Th. and design, The Hague, August 25-28th (1981). 35a. Dewilde, P., and H.Dym, 'Schur recursions, error furmulas and convergence of rational

estimators for stationary stochastic sequences', IEEE Trans. Inf. Theory (to appear). 35b. Dewilde, P., and H. Dym, 'Lossless chain scattering matrices and optimum linear prediction:

the vector case', Int.J. Circ.Th. and App. 9, 135-175 (1981).

36. Deprettere, E., and S.C. Lie, 'Adaptive Schur-Darlington algorithms for lattice-structured matrix inversion/decomposition and stochastic modelling', Delft Univ. Technology, Techn. Report, Delft, The Netherlands (1980).

37. Deprettere E., and P. Dewilde, 'Generalized orthogonal filters for stochastic prediction and modelling', in Digital Signal Processing(eds. V . Cappellini and A.G. Constantinides), Academic Press, 35-45 (1980).

13

I. MULTIPORT UNITARY AND J-UNITARY FILTERS

This chapter is devoted to definitions and properties o f a class o f discrete-time filters. The class we shall consider consists of filter realizations o f input-output maps (I/O maps), o n sequences, which satisfy specific mathematical constraints. In fact, we shall mostly be concerned with filters whose internal behaviour is described by an orthogonal (or unitary) map between internal quantities. However, we shall also consider J-orthogonal (or J-unitary) maps, where J is a signature matrix to be specified later. This w i l l allow us to cover the applications we have in mind. Within the above class o f filters, those which are cascade-realizations w i l l play a central role. Such filters can be viewed as numerical analogues o f physical lossless transmission systems. Aside from the unitarity and J-unitarity constraint, we shall impose, unless otherwise stated, standard constraints such as finiteness, linearity, and shift-invariance (time-invariance).

Digital filter theory parallels i n many respects the well established analog filter theory. Moreover, there is some parallelism between the class of filters we have i n mind and the class o f classical lossless filters [ 1 ] , but there are also major differ-ences. Because many notions we w i l l have to use are well k n o w n i n the classical analog theory, we shall introduce and prove them i n an abbreviated way, so long

as their properties are clear or obvious. Where important differences occur, we shall give complete proofs. Examples of notions common to both digital and analog filter domains are the notion of degree of a general rational matrix, factorization and degree reduction.

1.1. Preliminaries

We shall use the prime ('), the bar (—) and the tilde (~) to indicate the transpose, the complex conjugate and the Hermetian conjugate respectively. N o special type-face will be used to distinguish matrices and vestors from scalars. Lower case symbols will mostly denote scalars and vectors. They will also denote vector sequences o n the integer index set I = {. . . —2,-1,0,1,2, . . . }, that is, collections of vectors x(i) with i running through I. We write

x = { x ( i ) } .e I (1)

and x ( . ) = [ xx( . ) . . . x ( , ) ] ' e ( tm, with <t the field of complex numbers. O n ( £m m —

we shall use the Euclidean norm IMIE:||x(i)||F = 2 xk( i ) xR( i ) . x ( . ) being an

k = l

ordered m-tuple, it w i l l be called an m-vector.'

Vector sequences represent discrete-time signals carrying vectors of data which e.g. can be the result o f measurements.

x will be said to be of bounded support i f there exists an index iQ E I such that for all j = 1, . . . , m Xj(i) = 0 for i < iQ.

Definition l a .

A n m-port is an object which to any m-vector sequence x of bounded support, called the 'input', assigns a unique m-vector sequence y , called the 'output'. The map x * y defined by an m-port w i l l be called the I/O-map.

Definition l b .

A n (n + mj-port is an object which to any ordered pair of n-vector and m-vector sequences of bounded support ( xi ;x2) , called the 'input pair', assigns a unique

14 CHAPTER I

ordered pair of n-vector and m-vector sequences (y ,y ), called the 'output pair'. The I/O-map of an (n + m)-port will also be represented by .3f: x + y , whereby x and y are sequences o f ( n + m)-vectors resulting from an obvious concatenation of the sequences i n the input pair and to output pair respectively.

We represent m-ports and (n + m)-ports as i n fig. l a and fig. l b respectively.

Remark

A n m-port will be said to be defectuous when input and output have different dimensions, w i t h m the larger of the t w o vector dimensions involved.

We shall restrict ourselves to multi-ports which are linear, shift-invariant, causal, finite and i n some sense bounded. Before translating these constraints i n terms of properties of the I/O-maps, we introduce further notations and definitions as follows.

I = {0,1, . . . }: index support of the future I = { . . . , 1,0}: index support o f the past

Let I . be the index set which is either I or I+ or I_ . We define ?m( I . ) to be the set of all m-vector sequences o n I . for which the (/2)-norm

||x|| = ( 2 | | x ( i ) | |2) '/ 2 (2)

2 i e i is finite.

Upper case symbols will denote constant matrices or sequences of constant matrices o n I . . In the latter case, we w i l l write

H = { hH( i ) }i e I_ (3)

and hH( i ) = [h.k(i)] j=1 ^ , some n and some m , w i t h e (£. k=l,...,m

Let ||x||, v - { S | | x ( n ) | |2}1 / 2 w i t h v + vn > 0. T o any m-port,*": x * y one (~v0'v> n= - v0 fc 0

can assign a quantity

E(-vn,v) = | | x | |2

0 "( — VQ,V) " ( - V " >

(4)

called the energy adsorbed by the m-port i n the interval [—vj>].

A n m-port is called passive i f for all v > -°° , E(-°°,v) ~> 0. Clearly, a passive m-port has an I/O-map Jf which can be extended to 1^(1) and which is

contractive, i.e., | | ^ | l2 < l | x | |2.

(b) (c) m > n

Fig. 1. Representation of (a) an m-port, (b) an (n + m)-port, and (c) a defectuous m-port. Single arrows indicate data flow.

CHAPTER I 15

The I/O-map 3f defined b y a linear and shift-invariant m-port can be represented by an m-dimensional matrix valued sequence o n I, H = { hH( i ) } .g I, which acts o n inputs b y convolution (3f = H * ) :

y(i) = S hHG ) x ( i - j ) (5)

j e i

The convolution sequence H is called the impulse response of the m-port. If the m-port is, moreover causal, then hH( i ) = 0 for all i < 0, hence H = {n H( i ) }i e I • We shall reserve the symbols , / a n d S = { ss( i ) }i G I for linear shift-invariant multi-ports which are causal and contractive, hence passive.

In this case we have

(1) S*, with S = { ss( i ) }i e I + (6a)

||S * a||

(2) \\A\= sup. . . . . 2 < 1 (6b) a = /2( i ) l | a | l2

m v '

A most suitable way to deal w i t h bounded linear shift-invariant I/O-maps is to use Zee-transform representations. Standard notations, definitions and properties are the following.

If z e <t, let ID = {z: | z | < 1}, IE = {z: |z| > 1} and T = {z: |z| = 1}. D = ID u

T

The Zee-transform o f an m-vector sequence x on I is designated X ( z ) and is formally defined by X ( z ) = ,rx= 2 x ( i ) z "i, z e <t (7a) i<EI 1 dz x(i) = § X ( z ) z1 - , j = V 3 ! (7b) 27TJ Z

where the countour encloses the origin and is contained i n the region o f conver-gence o f X ( z ) .

L ^ T T ) (also designated L ^ ( ~-)) is the space of complex m-vector functions o n T

for which the norm | | F ( ej 9) | l = {— I F ( ej e) F ( ej e) d ( 9 }1 / 2 is finite. It is well-known 2 27T _ „

( [ 2 ] , Riesz-Fischer) that the Fourier map : ?^(I) * L^fTT) is a Hilbert space iso-morfism o f I 2( I ) onto L 2(T). Thus for x e / 2( I ) there exists an X ( ej e) e L 2(T)

m ' mv m v ' m

such that x(n) = i - ƒ X ( ej 0) e+ j n ed 6 ) and X ( ej e) = l i m % x ( n ) e -j"e. The

con-27T _n N - . « . - N

vergence o f the Fourier summation is i n the L2—sense. Parseval's theorem [2] asserts that ||x|f2 = | | X ( e J0) | |2.

is the (Hardy-Hilbert) space [ 3 ] , o f complex m-vector functions analytic o n IE

for which the norm ||F(z)|L = l i m [— ƒ"11F(rej<9)11?,dd]1/2 is finite. H 2 is a closed

2 r l l 27T _ „ *• m

and shift-invariant (z ' H 2 C H2 ) subspace of L2 ( T ) which consists of those X(e-"e)

m m r m

for which

*) An m-port is shift-invariant if the I/O-map defined by the m-port commutes with the shift operator.

16 CHAPTER I

x( n ) = — f 'rX ( ej e) ej n ed e = 0 for all n < 0. Hence X ( z ) e H 2 has radial limit almost

27T _T, M

everywhere on TT with inverse Fourier transform x e /m( I+) - X ( z ) is the Poisson integral of X ( ej e) :

X(z) = i - / [Re^ ^ ] X ( eje) d c ) , z e IE (7c)* 2 7 r - i r z - ej e

Replacing I+ by I _ and IE b y ID, one obtains the subspace of L^(T). N o w let H = { hH( i ) }j e J be the impulse response of an m-port with bounded linear shift-invariant I/O-map J K f r o m I tf) into l^W- Taking Fourier transforms ( [ 4 ] , Bochner Chandrasekharan theorem), convolution becomes matrix multipli-cation, that is H * * [H(e^e)] •, and the m-port has an equivalent representation [ H ( ej e) ] - : L^(T) * L^(T) w i t h | U f l | = ess^sup | | H ( ej e) | lF.

Here, | | H ( ej e) | L = sup — — - ^ E amd | | ^ | | is the operator norm.

Hence, (5) is equivalent to

jy = Y( e J9) = H ( e J9) X ( ej e) = H ( ej e) J x (8) H(e^e) is called the transferfunction. If H(e"e) is rational, then the m-port is said

to be finite. Notice that the transferfunction possesses a stability property i n that it transfers I2 bounded input sequences into I2 bounded output sequences.

1.2. Scattering matrices and chain scattering matrices

If the bounded linear shift-invariant m-port is causal, that is J f : / ^ ( I+) * 'm( I+) and contractive, that is pf\\ < 1, then it is passive and hence 9t= . / . B y causality, S(z) ^ is analytic on IE, and via (6), contractivity implies (9).

S(z)S(z) < Im on IE ( 9 ) t

A (square) matrix S(z) analytic on IE and obeying (9) is commonly thought of as, and will be called, a scattering matrix ( S M ) , mapping incident waves A ( z ) = fa onto reflected waves B(z) = jb. This point of view is represented i n fig. 2a. Definition 2

A n SM S(z) will be said to be strictly contractive on E or strictly bounded (by I)

on E whenever strict inequality holds i n (9).

A square matrix S(z) which is analytic on ID and contractive there w i l l be called

a conjugate scattering matrix. • Fig. 2b. represents the case when the transfermatrix of an (n + m)-port as i n fig. l b .

is an S M 2 ( z ) = [ 2 L ( z ) ] . j= 1 2. If n = m , then the £fj ( z ) are o f dimension m .

*) Notice that definition (7a) implies that for x e l x e 'm( I _ ) l . x(z) is a series in z"1 I in z I. It is natural, then, to take z e IE [z e ID].

However, in all cases, the Zee-transform of x is denoted by X(z).

t ) The unit matrix of dimension m will be denoted by Im or 1 . It will also be designated I when m is unambiguous.

CHAPTER I 17

(a) (b)

Fig. 2. Representation of m-channel scattering systems. Double arrows indicate 'waves': incoming arrows for incident waves and outgoing arrows for (a) reflected waves (b) transmitted and reflected waves.

n m S ( x ) = T B . C z ) -= 2(z) " A j ( z ) -22 2( z ) J m ' i vz> - - B 2( z ) . (10a) S ( z ) S ( z ) < I in (z) > 1 (10b) Obviously, the 2.j(z) are scatttering matrices and it is quite natural to call 2u( z ) and S2 2( z ) reflection (scattering) matrices, £1 2( z ) and 22 1( z ) transmission (scattering) matrices.

We shall use the lower asterix (t) to indicate the para-Hermitian conjugate. Thus i f f(z) is any function of z, then ft( z ) is its para-Hermitian conjugate and is defined by

f , ( z ) = f ( l / z ) .

Of particular interest are square scattering matrices which are unitary on T . Thus i f ||-/|| = 1, then

S ( ej e) S ( ej e) = I = S ( ej e) S ( ej e) a.e. on T (11a)* S*(z) is analytic on ID. If an S M S(z) a n a l y t i c on IE and contractive t h e r e -is such that I - S ( l / z ) S ( z ) van-ishes a.e. on T, then it must van-ish for allmost all z. Indeed, i f S(z) has a meromorpfic extension to ID, then S«(z) has a meromorphic extension to IE and I — S„(z)S(z) is analytic a.e. and must vanish a.e. since it does so i n an infinit number of points (a.e. on T ) .

Hence

S.(z)S(z) I = S(z)S,(z) a.e. ( l i b ) If S(z) has no meromorphic extension to ID, then it has a pseudo-meromorphic ex-tension to ID [ 1 7 ] . That is, since S(z) is analytic on E and invertible a.e. on E , on account of (11a), then the function T(z) — [ S ( l / z ) ]_ 1 is meromorphic on ID. T(z)T(z) > I on ID, by (9). O n T , S(z) = T(z) = S ;1 (z), and s ; ' ( z ) is called the (unique) pseudomeromorphic extension to D of S(z). We are again let to ( l i b ) . A scattering matrix (conjugate scattering matrix) obeying ( l i b ) is said to be

para-unitary. A para-unitary S M is often called lossless.

In chapter II, we will be concerned with the problem of constructing realizations of rational para-unitary SM's. The motivation is that such realizations have nice numerical properties and that a rational transfer function of a passive m-port can be realized as a submatrix of a para-unitary S M E(z) which is, then, usually called

18 CHAPTER I

an embedding. Realizations of embeddings bearing the 'nice properties' will be called lossless (X) filters. The precise definition of such filters is deferred to section 1.6. We shall come back to the embedding approach i n chapter III and we shall then tackle the problem of constructing partial or approximating embeddings via recursive 2-filter constructions.

2-filter realization of unitary embeddings is primarily based on the closely related theory o f J-unitary embeddings and the problem of factoring J-unitary

matrices [ 5 ] . These matrices are called chain scattering matrices and we introduce them now.

Given a 2m-port characterized by an S M 2(z), relating port signals as i n (10a) with n = m. Suppose the transmission scattering matrix 22 1( z ) is not identically singular. We n o w characterize the 2m-port equivalently by a matrix ©(z) which, by defini-tion, relates the port signals as follows:

A , ( z ) B2( z )

= 0(z)

It is not too difficult to show that 0(z) allways exists under the assumptions stated* It will be called the chain scattering matrix ( C S M ) originating from the correponding ing S M i n (lOab). It is, then, readily verified that 0(z) i n (12a) is J-expansive on IE, that is, introducing the signature matrix J m i n (12b),

J = 1 + ( - 1 )

m m m (12b)t

we have

0(z)J 0(z) J > 0

m on IE (12c)

A n y matrix 0(z) satisfying (12c) will be called a C S M . O f course, 0(z) is rational when 2(z) is such, but it is not i n general analytic on IE. In case £ ( z ) is unitary on T , the C S M will be J-unitary, that is,

0(ej e)J0(ej e) = J a.e. T (12d)

Similarly, the para-unitary property of E ( z ) translates to a para J-unitary property of 0(z):

0„,(z)J0(z) = J = 0(z)J0»(z) a.e. (12e)

A para J-unitary C S M is often called J-lossless.

Conversely, if 0(z) satisfies (12c), then it is related via (12a) and (10a) to a unique para-unitary S M S ( z ) . It is readily verified that a 2 m x 2 m para J-unitary C S M and the corresponding 2 m x 2 m para-unitary S M are related as follows:

*) Extensions to the case n < m will be given in chapter III.

CHAPTER I 19

-2

(z)

-S

(z)

A C S M is not i n general a stable transfermatrix. It will be such whenever ^ ' ( z ) and 2 ~2^ ( z ) are stable transferfunctions. More about C S M ' s and their relationship to SM's will be the subject of section 1.4.

We now first turn to the notion of the degree of a rational matrix and its factori-zation.

1.3. Degree and factorization of rational matrices

In this section we collect some standard definitions and relevant theorems concer-ning the degree (a measure of complexity) of a rational matrix. We shall consider the problem of degree reduction or factorization of a rational matrix and we shall concentrate on chain scattering matrices. The basic result [7] is that a rational para J-unitary C S M factors out, without adding complexity, into a finite product of para J-unitary C S M ' s which are themselves of simplest complexity.

A n y p x q rational matrix R(z) has a partial fractions decomposition as follows: R(z) = 2 R (z) + R (z)

ai

where R J z ) is a p x q matrix polynomial, and for

(14)

R (z) = S R . ( z - a . r1

aj i = l _1 >

(15)

is the principal or polar part of R(z) at the finite pole a. e (£. The R _ . are constant p x q matrices. Let for / < 0 and some fixed i; e (p, R(z) = 2 R.(z — £)'

i=l 1

be the Laurent series of R(z) at £. O n the set of matrix coefficients R. so defined, one can construct a triangular block Toeplitz-matrix (16).

T

(R,|) =

Whereby rank Tk( R , | ) < r, with r = m i n ( k - p , k - q ) . Definition 3a.

A finite pole a, € (t of a rational p x q matrix R(z) is said to be of order L, with as in (15). It is said to be of degree 5. = rank T, (R,a.). The order / and the j J ij i

20 CHAPTER I

the matrix R ' ( z ) = R „ ( V z ) = .2 { R . z ' + RQ with R J z ) as i n (14). T h e degree

5 of R(z) is the sum o f its unipolar degrees. • Definition 3b.

A point £ G (p is said to be a zero of a regular rational p x p matrix R ( z ) i f and only i f £ is a pole o f R_ 1( z ) . The order and the degree of the zero £, is by

defini-tion the order and the degree o f the pole £ of R "1 (z). • A recursive algorithm performing Toeplitz rank search can be found i n [ 8 ] .

Anticipating the next chapter, we n o w introduce elementary degree reducing

factors and we summarise criteria for degree reduction using such factors.

A n y p x p nonsingular matrix w i t h pole at z = £Q of order 1 and degree 1 has u v

the form R„ r - , some p-vectors u and v, and some p x p constant R„

° z " £o 0

of full rank. Definition 4.

A regular degree 1 p x p matrix [ RQ u v ] is said to be a left elementary

degree reducing factor of a rational p x q R(z) i f R(z) has a pole at £Q and the factorization R(z) = [ R „ u v. ] R . ( z ) leaves a remainder R , ( z ) which has

0 z - £ 0 1 1

degree 1 less, i.e., 6Q( R1) = 5Q( R ) - 1 and 5 ( Rt) = 5 ( R ) - 1. a If R(z) is of degree 1 with pole at | , i f A and B are nonsingular and constant,

then A R ( z ) B is of degree 1 and has its pole at % . In a product of elementary degree 1 factors, constant multipliers can be shifted along at will. Hence they can be choosen so that A R ( z ) B is standard.

Definition 5.

If R(z) is of degree 1 with pole at £0, then it w i l l be said to be a standard degree 1 matrix or a standard elementary factor (with pole at £ ) i f for any | 1: R ( l ) = I, and for £Q = 1: R( 1) = I.

Criteria for degree reduction by factorization using (standard) elementary factors are summarized i n the next, well-known [ 6 ] , [ 7 ] , theorem.

Theorem 1.

If R(z) is a p x q rational matrix, then i n the factorization R ( z ) : the elementary factor is degree reducing if and only if

1. the p /Q vector [ op ... . . o v] belongs to the column range space o f (the space spanned by the columns of) T; ( R , z0) (thus zQ is a pole o f R(z)),

2. i f zf = zQ + uv, for any p-vector w such that the p(/j + 1) vector [ o . . . o w] ~ belongs to the column range space of T; + 1( R , z1) , we have u w = 0 ('thus

Zj = zQ + uv is a zero of R(z)),

3. i n case z0 = , there exists a p-vector x such that the p(/Q + 1) vector

[ op . . . Op x ] belongs to the column range space of J[ + 1( R , zQ) and ux = —1. A n alternative set o f conditions is the following. There exists a vector g(z) * analitic at zQ such that

1. l i m g(z) = 0 and l i m R(z)g(z) = v,

z->zQ z*zo

2. for any vector h(z) analitic at zi = zQ + uv such that l i m R(z)h(z) = w,

CHAPTER I 21

E ( Z )

3. in case z„ = z , , one has u [ l i m R(z) ] = — 1.

0 1 z*zQ z-zo D

Elementary para unitary S M ' s and para J-unitary C S M ' s will play a crucial role in chapters to follow. We now introduce their standard forms.

Proposition 1.

If v,i\ e C , i f M < 1 and r? =fc 1, if u, v are complex m-vectors such that uu = 1 — \v\2 and vJv = | T ? |2 — 1, then

0 V M * l « - ( i z - m - v ) ™ (17a)

is a standard degree 1 para unitary S M . irfz) — det U(z,j») = -—= - — — and I — » z — ^ ^ ( z ) i ^ ( z ) = 1,

i i } J ( 2 ^ - ! m + V V~J ( 1 7 b )

is a standard degree 1 para J-unitary C S M , iMz) - det J(z,r?) = -—W - — — and 1-7? z - 7? 0 , ( z ) 0 ( z ) = 1,

i i i ) w i t h J(z,7?) as in (ii), J(— z,+r?)| __j is a standard degree 1 para J-unitary C S M with pole at 1.

Proof

We shall only show (ii), the proof of (i) and (iii) is similar.

Let a e C\{1} be the pole of an elementary m x m matrix A ( z ) normalized to I z — 1

at z = 1: A ( z ) = I + — — — x y , x and y m-vectors. (1 ~a)(z — a)

case 1: \a\ = 1. A(( z ) J A ( z ) - J = 0 iff (1 - a ) ( x y - J y x J ) + J y x J x y = 0 and (a - l ) ( x y - J x y J ) + J y x J x y = 0, hence x J x = 0 and y = e x j , with e = ± 1 . Next, A ( z ) J A ( z ) - J > 0 in |z| > 1 iff e J x x J , 22 ~J =s > 0 in |z| > 1.

( z - a ) ( z - a ) Hence y = x j .

case 2: | a | ¥= 1. Observe that any a, |a| > 1 (|a| < 1) can be put at infinity (zero)

by means of the fractional conformal transformation £(z) = 1 —az 1 ~3 ( f_ 1( z ) ) z —a 1 - a

whereby £(J£) = IE ( |_ 1( I D ) = ID) and |£| = 1. Hence £(z) preserves properties. J — 1

N o w in the transformed domain, i f |a| > 1, then A(£) = I + 5-^5 x y , and i f

t _ 1 „ |a| - 1

| a | < 1, then A ( f ) = I + ^ x y .

A] t( ? ) J A ( g ) - J = 0 iff : J x y ( | a |2 - 1) = y ( x J x ) y and y x j ( | a |2 - 1) = y ( x J x ) y , all a, |a| # 1. Hence y = e x j , e = ± 1 , and x J x = | a |2 — 1.

A © J A(£) - J > 0 in H I > 1 iff : for | a | > 1, e ^ " J - J x x J > 0 in | | | > 1 and for | a | < 1, e - ^T,}^ J x x J > 0 i n |f| > 1.

(1 - \a\2M

Hence e = +1 and we are led to (17b). The rest of the proof is omitted. Q Elementary factors J(z,7?) are pseudo commutative i n the following sense. Proposition 2.

If J J ( Z , T ? ) and J2( z , i / ) are elementary para J-unitary C S M ' s , with r\i=v, then there always exist elementary para J-unitary C S M ' s J3(z,i>) and J4( Z , T ? ) such that J j t z . T j J J ^ z . i O = J3(z,^)J4(z,7?).

22 CHAPTER I

Thus an exchange of poles is always possible, as follows:

[I + Z ~ l ruuJ] [! + - - — - _ " 1 , v v J ) = [1+ —A^ - x x J ] [1 +

( l - r j K z - 1 7 ) ( l - v ) ( z - v ) ( l - j > ) ( z - p ) 1 1 (1 -7?)(z-17) where

i) uJu = fjr? — 1 = y j y and vJv = vt> — 1 = x j x , ii) x and y are computable from u and v and vice versa. Proof.

We have

? ? - l x j y ~ ,17-1 uJv ~ ~ u = a(y + = x) and a(u + - — v) = y

\-v t\ — v \ - v r\-v , v~ 1 uJv ~ ~ v-\ x j y ~

v H u = px and pv = x + y

1-17^-17 l - T J f - 1 7

A n d (i) will be satisfied i f |j3|2 = = 1 + j "J V* , , = 1 + ,l x J y l„ .

\aV \v-n\ k-T7l

The following is a fundamental result. n Theorem 2. [ 4 ] , [ 7 ] .

If 0 ( z ) is a rational m x m para J-unitary C S M with pole at z = r? (17 # 1) then i) there always exists an m-vector g(z) analitic at 17, lim g(z) = 0, and an m-vector

v, v = lim 0 ( z ) g ( z ) such that vJv = IT?!2 — 1. Moreover, when |rj| = 1, ^[lim 0 (Z) - M ^ - ] = - 1 .

Z-»-T) Z~V

ii) if the vector v in the elementary factor J(z,T?) in (17b) is as in (i) and if this

factor J(z,ij) is used to extract the pole r? of 0 ( z ) , then and only then, all conditions of theorem 1 are satisfied.

iii) The matrix 0 j ( z ) = J- 1( z , T 7 ) 0 ( z ) with J ( Z , T 0 from (ii), is also a para J-unitary

C S M . • Notice that, when 17 is a pole of 0 ( z ) , there always exists a vector g(z) analytic

at 17, with l i m g(z) = 0 and such that l i m 0 ( z ) g ( z ) = x is non-zero. The fact that we also have x j x with the proper sign following from (12cd), since

lim g ( z ) [ 0 ( z ) J 0 ( z ) - J] g(z) = x j x . Z>TJ

When using the factor (17b) as degree reducing factor, we have with the notations of theorem 1, z = tyfj and v j [ l i m _ 0 ( z ) h ( z ) ] = [lim gt( z ) ©t( z ) ] J [lim _ 0 ( z ) h ( z ) ] where h(z) is a vector analytic at z = Z, with l i m 0 ( z ) h ( z ) = w.

z->z.

Hence (ii) follows from (12e). 1

The proof of (iii) is more involved and will not be repeated here. We shall come back to it in Chapter III.

Theorem 2 says that up to a constant J-unitary multiplier, any rational para J-unitary C S M 0 ( z ) consists of and can be factored into a finite product of elementary standard factors J.(z,i7.). Each factor contains a pole of 0 ( z ) and the number of elementary factors in the product form equals the number of

CHAPTER I 23

poles* of 0 ( z ) . The degree of the factors add up to the degree of 0 ( z ) . Such a factorization will be called minimal and irreducible. Minimal because of the property that 0 ( z ) factors out without adding complexity, and irreducible because of the fact that the factors i n the product form are of lowest degree. Note that proposition 2 implies that in a minimal irreducible factorization o f a para J-unitary C S M , the sequence of poles can be chosen at will.

F r o m proposition 1 and theorem 2, we deduce the following. Corollary 2.

If 0 ( z ) is a para J-unitary C S M with poles at 17., j = 1, . . . , N (multiplicity counted), if 9(z) = d e t 0 ( z ) , then

i) 0(z) = c l l -T?'Z t |c| = 1 hence is para unitary, not necessarily contractive i n E , 1=1 z - %

ii) 5 ( 0 ) = 8(9) iff for all i = 1, . . . , N 1/n. # 1?. j = I , . . , , N . 'i J

In particular, i f 0 ( z ) is a stable transfermatrix, then I77.I < 1, all i , 9{z) is

an all-pass function and 5 ( 0 ) = 8(9). n

It is not to difficult to derive similar factorization results for para unitary SM's 2 ( z ) as well. Thus if 5 ( 2 ) = N , then we can factor 2 ( z ) to 2N( z ) . . . 2 , ( z ) 20 whereby the 2.(z) are degree 1 para unitary standard SM's and 2Q is constant unitary. In this case, the poles extracted i n the elementary factors all belong to E> and hence det 2(z) is an all-pass function whose degree is N [ 1 ] .

1.4. Positive Hermitian Matrices, Spectral factoring and Minimal

Embedding.

In section 2, we have introduced para J-unitary C S M ' s as equivalent representations of para unitary SM's 2 ( z ) = [ 2 . . ( z ) | . .= 1 2, with 2..(z) s q u a r e T h u s let «|ffl be the set of 2m x 2m rational para unitary SM's 2 ( z ) with invertible transmission matrix 22 1( z ) (dimension m) and let 0". be the set of 2m x 2m para J-unitary CSM's. We define a map P: n&f0 •* .f which assigns to each 2 (z) i n ^ 0 its corresponding C S M 0 ( z ) i n P is one-one and onto. Hence (13) can be written as

0 ( z ) = P 2 ( z ) .

A l s o , i f ©5( z ) . . . ©1( z ) 0( ) is a minimal irreducible factorization of 0 ( z ) e ifQ,

then P- 10 ( z ) = 2 ( z ) e V /Q and 11

2 ( z ) = P -1 { P 2s( z ) . P 25_1( z ) . . . P 2f l} *

£ 26( z ) * 25_1( z ) * . . . * 20

*) These poles are counted according to their multiplicity, a pole of degree m is counted as m poles.

t) A product of factors 8-—— with |a| < 1 and |fl| = 1 is often called an all-pass function. z - a

It is also called a Blashke product or a (rational) inner function. | ) An extension to the defectuous case is defered to chapter III.

^|) Although the 8 E.(z) are elementary factors, they are not 'standard' in the sense of our previous definitions of standard para unitary SM factors.

24 CHAPTER I

P and * are often called the Redheffer operator and Redheffer star product respectively [ 9 ] .

A n S M S(z) in <3/Q is called a minimal embedding o f one o f its m x m submatrices, Sk ;( z ) say, when 5 ( 2 ) = 5 ( 2k ;) . Fundamental results o n minimal embeddings [1] are summarized in theorem 3.1 o f chapter III. Hereafter we shall confine ourselves to SM's S(z) for which [I + S(z)] is invertible a.e. SM's which do not obey this property can be reduced according to the following lemma.

Lemma 1. [ 1 ].

If SQ( z ) is an m x m Scattering matrix, then there exists a constant unitary trans-formation U such that

U Sn( z ) U = S„ (z) + 1 . . + [-1 , ]

0V 0 m—[rl_ m—|r]+

where

i) [r]_= rank [I - S0( z ) ] in |z| > 1, [ r ]+= rank [I + SQ( z ) ] i n | z | > 1, ii) SQ( z ) is an S M o f dimension [ r )+ + [r]_— m ,

iii) [I - SQ( z ) ] and [I + SQ( z ) ] are o f rank [ r ]+ + [ r ] _ - m in |z| > 1. •

N o w let 2 ( z ) = [2..(z)].j =]2 belong to WQ and put 22 2( z ) = SQ( z ) . Clearly, I — S0( z ) SQ( z ) is not identically singular on T and is strictly positive o n IE, by the maximum modulus theorem. [I — SQ( z ) ] and [I + SQ( z ) ] have their poles i n IE and have strictly positive Hermitian parts there. Indeed, He[I — SQ( z ) ] — | [(I - SQ(z)) + (I - S0( z ) ) ] = ~ [(I - S0(z))(I - S0(zV) + I - SQ( z ) S0( z ) ] > 0 on IE. A n d similarly for [I + SQ( z ) ] . Moreover [I - SQ( z ) ]_ 1 and [I + S0( z ) ]_ 1 exist, by lemma 1, they are analytic o n IE and have strictly positive Hermitian parts there. Put

Zf l( z ) ^ t I - S0( ^ [ I + S0( z ) ] (18a) Then

H e [ ZQ( z ) ] = | [ Zo( z ) + Z0( z ) ] > 0 o n E (18b)

A matrix Z(z) analytic on IE and with positive-definite (non-negative) Hermitian part in IE will be called positive Hermitian* ( o n IE). It will be called positive real when Z(z) is, moreover, real, that is when Z(z) = Z(z).

A positive Hermitian matrix is commonly thought of, and will be called, a (passive)

impedance matrix (IM). The adjective 'passive' refers to the passivity property of

the S M S(z) induced by a positive Hermitian Z(z), via a relation inverse to (18a). This inverse relation always exists. Hence Z(z) is an I M iff:

| [Z(z) + Z(z)] > 0 on IE (18c)

If Z(z) is analytic on IE and has no poles on T , then (18c) is implied by (18d).

*) A positive Hermitian matrix is often called a positive real matrix since the real part of xZ(z)x, any x 6 ( tm, is X^ (Z(z) + Z(z)]x.

CHAPTER I 25

W ( ej e) = | [ Z ( ej e) + Z ( ej e) ] > 0 a.e. on T (18d)

Z(z) may have poles o n T, but for (18c) to be true, such poles, e-"e° say, must be o f order 1 and the residue H° at e^e° must be such that H ^ e-^0 is Hermitian and positive. This is an easy extension to the matrix case of a well k n o w n scalar result on positive real functions [ 1 ].

The para-Hermitian part of Z(z) is the matrix

W(z) = i [ Z (z) + Z(z)] (19a)

l *

A n I M Z(z) is said to be skew para-Hermitian, i f W(z) = 0 a.e. It is then often called lossless, since the S M induced by a skew para-Hermitian IM is para-unitary. Recalling that SQ( z ) i n (18a) is strictly bounded (by I) a.e. on T , and observing that

WQ( z ) = [I - S ^ z ) ] "1 [I - S0( z ) S0/ z ) ] [I - S0 t( z ) ] "1 =

= [I - S0 t( z ) ] - ' [I - S0 t( z ) S0( z ) ] [I - S ^ z ) ] -1 (19b)

it follows that for all x e <tm, we have x W0( z ) x j= 0 a.e. on T . However, the degree of x WQ( z ) x , hence of WQ( z ) , may be less than twice the degree o f x ZQ( z ) x , hence of ZQ( z ) , because a para-Hermitian portion may cancel between x ZQ( z ) x and x ZQ^ ( z ) x .

ZQ( z ) (in 18a) being analytic on IE, it is harmonic on IE and H e [ ZQ( z ) ] is a non-negative (actually a strictly positive) harmonic function on IE. This is (18b). If ZQ( z ) is analytic on IE (has no poles on T ) , then H e [ ZQ( z ) ] is continuous on IE and hence [8] it is the Poisson integral of its restriction to T :

H e [ Z0( z ) ] = i f W0( e ^ ) d v 3 , z e IE (20a)

Moreover,

i n 7 + Jf

ZAz) = X + — ƒ - — S _ w „ ( eJ ¥' ) d ^ , z 6 IE (20b) 0 0 27T _„ z „ e1^ 0

where XQ is constant skew-Hermitian. The positive Hermitian ZQ( z ) generated via (20b) defines an S M

SQ( z ) = [ ZQ( z ) + I ] "1[ ZQ( z ) - I] (20c)

which is strictly bounded by I a.e. o n T . A more general result due to Herglotz [2] is the following.

F o r any positive Hermitian Z(z) (on IE) there is a positive matrix valued measure w(i/>) such that

1 T Z + ew

Z(z) = X + - ƒ =- d w ( ^ ) , z e E (20d) z v -7T z — e1^

26 CHAPTER I

dw(i/>) = W(eJ^) dtp + dws(y?) whereby Wie1^) is non-negative integrable on TT and

ws(i^) is a positive singular measure (constant a.e. on T ) . Moreover, [ 1 8 ] ,

W ( ej v) = l i m H e Z ( r e ^ ) a.e. and Z(z) defines an S M S(z) ^ [Z(z) + I ] "1 [Z(z) - I], r + l

Thus for the rational ZQ( z ) in (18a), d w ^ ) , if non-zero, will consist of a number of Dirac impulses on T , located at the positions of the poles of ZQ( z ) on T . We shall now give a particular form of para J-unitary C S M ' s in .fQ, which we

assume, without loss of generality, to originate from minimal embeddings S ( z ) = [ 2 . . ( z ) ] .j = 1 > 2 in V /Qo f 22 2( z ) .

Proposition 3.

Suppose: (1) SQ( z ) is a rational S M , of dimension m, which is strictly bounded (by I) a.e. on T ,

(2) ZQ( z ) is a rational I M , of dimension m, with Hermitian part strictly positive a.e. on T ,

(3) SQ( z ) and ZQ( z ) are related to each other via (20c) or (18a),

(4) S(z) = [E..(z)]. is a minimal embedding in y /Q of ^2 2(z) ~ s 0^z^

l2 Put: (1) Tf( z ) — [ i ( Z0( z ) + I ) ] S2 1( z ) , T ( z ) à 21 2( z ) [ I ( Zo( z ) + I)], (2) V(z) ^ 2n 1 t( z ) [ I - S ^ ( z ) ] [ I - Sr i( z ) ] -12 : „ ( z ) . o 1 21 (21a) (21b) (21c) Then Tf( z ) and Tr( z ) are m x m matrices, with poles i n ID and invertible a.e., such that

i) Tf( z ) Tf, ( z ) = WQ( z ) = Tr. ( z ) Tr( z ) (22a)

ii) Tf( z ) = Tf. ( z ) V ( z ) with (22b)

(a) V ( z ) from (21c) a para unitary S M of dimension m, (b) T (z) and V (z) right co-prime,*

(c) S ( Tf) = S(V).*

iii) 0 ( z ) = P 2 ( z ) in is of the form (23) and is minimal, that is 5 ( 0 ) = 5 ( Sn) .

0 ( z ) T - V z ) T " ' ( z ) ^ [ Z0( z ) + I] • M Z „ ( z ) - I ] J L 2 0* f I Z0( z ) - I ] i-[Zn (z) + I] 2 o* (23) Proof.

This proposition is actually a direct consequence of the existance of the minimal embedding 2 ( z ) (theorem 3.1), which in turn follows from the factorability [10] of [I - S0( z ) S J z ) ] , to 22 1( z ) 22 1 t( z ) , hence of [I - S0. ( z ) S0( z ) ] to S1 2, ( z ) 21 2( z ) . This implies the factorability of WQ( z ) = - [ ZQ( z ) + ZQ^ ( z ) ] , on account of (19b) and (20c), whence (22a) via (21ab). Next, 22(b) follows from (21c) and (21ab) with (a) via (22a), and (b,c) via (21c), the minimality of S ( z ) and the properties of [I — S0( z ) ]_ 1. Finally, (23) follows via (13), and the minimality of 0 ( z ) is

implied by the minimality of 2 ( z ) , see also chapter III. •

C H A P T E R I 27

A n y 0 ( z ) in ,'T can be represented by (23), on account of (13) and definitions of Tf( z ) , Tf( z ) and ZQ(z) as in (21a), (21b) and (18a). Tf( z ) and Tr( z ) are (non-unique) left and right spectral factors of WQ( z ) , which are minimal when S2 1( z ) and 2J 2( z ) are such.

In the minimal case, E2 1( z ) , hence Tf( z ) , can always be chosen to be minimum

phase (maximum phase), that is to have an analytic inverse on IE (on ID). E1 2( z ) , hence Tf( z ) , is then almost always automatically m a x i m u m phase ( m i n i m u m phase).

A n y S ( z ) i n ?/„ with 5 ( E ) = 5 ( E , J and with either E "^(z) and E "1 (z) or 2i 2' ( z ) and S ~*(z) analytic o n IE will be called a phase restricted minimal

embedding (of EJ 2( z ) ) i n WQ. The collection o f all phase restricted minimal

embeddings i n V /0 for which E2 1( z ) is m i n i m u m phase (maximum phase) will be designated H ('•#,•). The collection o f all minimal 0 ( z ) in for which Tf _ 1( z ) and T ' ^ z ) i n the representation (23) are both analytic o n IE (on D ) will be denoted by if (tf,). The map P : 4SC * & is one-one and onto. In chapter III, we will be concerned with E-filter realizations o f elements E ( z ) in V / j , embedding some S M SQ( z ) induced by some I M ZQ( z ) that originates from string data ('wave data'). Since elements in <%/ have equivalent representations of form (23) i n £ , it is o f interest to obtain para J-unitary C S M ' s in ,f constructed on the Laurent expansion o f positive Hermitian matrices.

Thus let Z(z) = 1 + 2 2 Z.z'1 be an m x m matrix series, analytic o n IE, which

originates from a positive measure W(I/J) on

T

ƒ dw(ifi) normalized to I. F o r simplicity, we assume that dw(i/>) = W (eJ l i !) dip

27T _t¡ u

with W0( eJ ¥' ) strictly positive a.e. on T. Hence, the series Z(z) = ZQ( z ) is the expansion o f the integral part in (20b) which is normalized such that

i - ƒ Wn( ej^ ) d ^ = I (24a)

277 _„ 0

1 It .,

We, therefore, have that Z = — ƒ e ^ W ^ e ^ ) d<p. W ^ e ^ ) being strictly

¿Ti _7r

positive a.e. on TT, it follows that d e t W0( eJ^ ) ¥= 0 a.e. whence log det WQ(e>ip)

It is k n o w n [ 1 1 ] , [12], then, that W0( eJ V' ) is factorable uniquely (modulo is integrable on T.*

It is k n o w n [11], [1 constant unitary factors) as

W0( ej ¥' ) = Tf( ej^ ) ff( e ^ ) (24b)

whereby both Tf( ew) and T~'(eJ I'') have analytic extensions to IE and 1 TT z + J v

d e t T . ( z ) = exp { - ƒ r log det W „ ( ew) d^} , |z| > l ( 2 4 c )

'} The condition of integrability of log det W0(eJ l^) is sometimes refered to as the Szegô-Wiener-Masani condition, [11], [12].

28 CHAPTER I

Tf( z ) is said to be a minimal m i n i m u m phase spectral factor. (Outer i n the non-rational case). This section is mainly concerned with rational IM's and we, therefore, have assumed that the series ZQ( z ) is the Laurent expansion of a rational I M . Notice, however,

that the above factorization result has a more general validity. N o w i f Tf( e ^e) i s m x m rational, we can always extract the poles of Tf( z ) minimally i n standard degree 1 para-unitary S M factors (17a), to obtain the co-prime factorization (25a).

Tf( z ) = Tr, ( z ) V ( z )

(25a)t-where Tf( z ) , T (z) and V ( z ) are as i n (22a), (22b) with Tf( z ) m i n i m u m phase, and V ( z ) a product of standard factors (17a). Then, the matrix i n (25b) is a para J-unitary C S M i n if . 0 ( z ) V ( z ) T " ' ( z ) r* pZ0(7.) + l] 2 ^ II - [ Z0( z )

I]

The latter property can be established by showing that P 1 0 ( z ) is a minimal embedding i n ** o f SQ( z ) = [ ZQ( z ) + I ] "1 [ ZQ( z ) - I] and with transmission scattering matrices as i n (21ab). Notice that since ^ [ ZQ( z ) + ZQ( z ) ] > Tf( z ) Tf( z ) on E , we can always assume that

Tf( ° ° ) > 0 (25c)

This normalization together with those i n (24a) and (25a) make 0 ( z ) unique. Of course, (25b) only establishes the existance of a rational embedding i n .yj of the (Laurent) series ZQ( z ) , hence an embedding i n $f o f the related SQ( z ) . The actual factorization of W( )(eJ''f') through degree reduction of ZQ( z ) , by factorization of 0 ( z ) , will be considered i n chapter III. There, we shall also elaborate o n the non-rational case for which we shall obtain partial realizations which will be shown to be approximations i n a least squares sense. We shall obtain recursive procedures to perform this task.

1.5. Inverse m-ports and Dual 2m-ports.

If 2 ( z ) is i n ••$/ with £2 2( z ) = SQ(z) of dimension m , and with 2)2 1(z) m i n i m u m phase, then it follows from

I - S0( z ) S „ , ( z ) = U + Z ^ z ) ] -1 [ Z0( z ) + Z0, ( z ) ] [I + Z ^ z ) ] "1 = 22 1( z ) S2 U( z )

that ^2 1( z ) is stable invertible i f and only if ZQ( z ) has no poles on TT and WQ( e ' '0) has full invariant rank o n T . In this case, 0 ( z ) = P £ ( z ) is not only an equivalent representation i n i'T^ of a 2m-port with input-output relations given i n (8a), it is itself a stable tranfermatrix of a 2m-port .9": x •* y relating via (8a), (11) and (23) input signals X ^ z ) = A2( z ) , X2( z ) = B2( z ) and output

t) In the non-rational case (25a): Tf(ej f l) = f ( eJ t ;) V ( eJ C' ) wiU define Tf(eJ") for all z iff f r(eJ l 7) is analytic on ID and the unitary (inner) V(e?6) is mxm and co-prime with f (ej 9). [13],[14].