Wisuwat Bhosri

An Operator Perspective on Signals and Systems

Preface.- I Basic Operator Theory.- 1 The Wold Decomposition.- 2 Toeplitz and Laurent Operators.- 3 Inner and Outer Functions.- 4 Rational Inner and Outer Functions.- 5 The Naimark Representation.- 6 The Rational Case.- II Finite Section Techniques.- 7 The Levinson Algorithm and Factorization.- 8 Isometric Representations and Factorization.- 9 Signal Processing.- III Riccati Methods.- 10 Riccati Equations and Factorization.- 11 Kalman and Wiener Filtering.- IV Interpolation Theory.- 13 Contractive Nevanlinna-Pick Interpolation.- V Appendices.- 14 A Review of State Space.- 15 The Levinson Algorithm.- Index.

The Wold Decomposition

In this chapter we will introduce the classical Wold decomposition for an isometry U on K. The Wold decomposition was initially used to decompose a wide sense stationary random process into its deterministic and purely nondeterministic parts. We will also study unilateral and bilateral shifts, and present an introduction to Toeplitz operators. Finally, the Wold decomposition along with Toeplitz and shift operators will play a fundamental role in our approach to signal processing and factorization theory.

Multirate Filterbank Design: A Relaxed Commutant Lifting Approach

In this paper, we reformulate the design of the IIR synthesis filters in classical multirate systems as an interpolation problem involving a norm called the <i>P</i> _m norm where <i>m</i> is any positive integer. This interpolation problem can be solved using relaxed commutant lifting techniques in operator theory. The <i>P</i> _m norm is actually a tradeoff in handling energy distortion and error peak distortion. Our development allows the designer to select from a family of filters the one which is best suited for a specific application. The well-known <i>H</i> ² and <i>H</i> ^¿ design methods can be viewed as special cases when <i>m</i> = 1 and <i>m</i> ¿ ¿ respectively. The computation relies mainly on FFT techniques and a finite section of certain Toeplitz matrices. The resulting filters are given in state space form and maybe useful for practical implementation.

Mixed norm low order multirate filterbank design: Relaxed commutant lifting approach

In this paper, we reformulate the design of IIR synthesis filters in classical multirate systems as an optimization problem involving a new norm called Pm-norm where m is any positive integer. That optimization problem can be solved using a recent generalization of the commutant lifting techniques in operator theory. The introduced norm is actually a trade-off in handling energy distortion and error peak distortion. Our development allows the designer to select from a family of filters the one which is best suited for specific applications. The well-known H2 and Hinfin designs then can be viewed as special cases when m=1 and mrarrinfin respectively. The computation relies mainly on FFT technique and a finite section of certain Toeplitz matrices. The obtained filters are of low order and attractive for practical implementation. Moreover, the proposed approach works for non-rational transfer functions. A new method for inner outer factorization of a rational matrix-valued function is also developed.

Toeplitz and Laurent Operators

Toeplitz and Laurent operators play a basic role in systems theory. For example, a lower triangular Toeplitz matrix can be viewed as an input-output map for a linear causal time invariant system. First we introduce the Fourier transform. Then we will study Toeplitz and Laurent operators. The Fourier transform will be used to turn Laurent operators into multiplication operators and visa versa.

The Naimark Representation

This chapter is devoted to the Naimark representation theorem and its consequences. The Naimark dilation allows us to use geometric methods to compute inner-outer factorizations and solve signal processing problems. Let A be any matrix whose entries A jk are operators mapping a Hilbert space E into Y. Then A# denotes the matrix obtained by taking the adjoint of the entries of A and then transposing this matrix, that is, the entries of A# are given by (A#) jk =A* kj . If A defines an operator mapping ⊕ 0 n e into ⊕ 0 m Y, then A#=A* is the adjoint of A. Throughout l+ c (e) denotes the set of all vectors in l+2(e) with compact support. Finally, recall that the controllability matrix W determined by the pair of operators {A on χ,B where B maps e into χ is given by

Isometric Representations and Factorization

W = [B AB A^2 B \cdots ].

The Rational Case

(5.0.1) In general, W is not necessarily an operator mapping l+2(e) into ⊕. However, W is a well-defined linear map from l+ c (e) into ⊕.