The discrete-time bounded-real lemma in digital filtering

The Lossless Bounded-Real lemma is developed in the discrete-time domain, based only on energy balance arguments. The results are used to prove a discrete-time version of the general Bounded-Real lemma, based on a matrix spectral-factorization result that permits a transfer matrix embedding process. Some applications of the results in digital filter theory are finally outlined

Factorability of lossless time-varying filters and filter banks

We study the factorability of linear time-varying (LTV) lossless filters and filter banks. We give a complete characterization of all, degree-one lossless LTV systems and show that all degree-one lossless systems can be decomposed into a time-dependent unitary matrix followed by a lossless dyadic-based LTV system. The lossless dyadic-based system has several properties that make it useful in the factorization of lossless LTV systems. The traditional lapped orthogonal transform (LOT) is also generalized to the LTV case. We identify two classes of TVLOTs, namely, the invertible inverse lossless (IIL) and noninvertible inverse lossless (NIL) TVLOTs. The minimum number of delays required to implement a TVLOT is shown to be a nondecreasing function of time, and it is a constant if and only if the TVLOT is IIL. We also show that all IIL TVLOTs can be factorized uniquely into the proposed degree-one lossless building block. The factorization is minimal in terms of the delay elements. For NIL TVLOTs, there are factorable and unfactorable examples. Both necessary and sufficient conditions for the factorability of lossless LTV systems are given. We also introduce the concept of strong eternal reachability (SER) and strong eternal observability (SEO) of LTV systems. The SER and SEO of an implementation of LTV systems imply the minimality of the structure. Using these concepts, we are able to show that the cascade structure for a factorable IIL LTV system is minimal. That implies that if a IIL LTV system is factorable in terms of the lossless dyadic-based building blocks, the factorization is minimal in terms of delays as well as the number of building blocks. We also prove the BIBO stability of the LTV normalized IIR lattice

The role of lossless systems in modern digital signal processing: a tutorial

A self-contained discussion of discrete-time lossless systems and their properties and relevance in digital signal processing is presented. The basic concept of losslessness is introduced, and several algebraic properties of lossless systems are studied. An understanding of these properties is crucial in order to exploit the rich usefulness of lossless systems in digital signal processing. Since lossless systems typically have many input and output terminals, a brief review of multiinput multioutput systems is included. The most general form of a rational lossless transfer matrix is presented along with synthesis procedures for the FIR (finite impulse response) case. Some applications of lossless systems in signal processing are presented

Circulant and skew-circulant matrices as new normal-form realization of IIR digital filters

Normal-form fixed-point state-space realization of IIR (infinite-impulse response) filters are known to be free from both overflow oscillations and roundoff limit cycles, provided magnitude truncation arithmetic is used together with two's-complement overflow features. Two normal-form realizations are derived that utilize circulant and skew-circulant matrices as their state transition matrices. The advantage of these realizations is that the A-matrix has only N (rather than N2) distinct elements and is amenable to efficient memory-oriented implementation. The problem of scaling the internal signals in these structures is addressed, and it is shown that an approximate solution can be obtained through a numerical optimization method. Several numerical examples are included

Minimal structures for the implementation of digital rational lossless systems

Digital lossless transfer matrices and vectors (power-complementary vectors) are discussed for applications in digital filter bank systems, both single rate and multirate. Two structures for the implementation of rational lossless systems are presented. The first structure represents a characterization of single-input, multioutput lossless systems in terms of complex planar rotations, whereas the second structure offers a representation of M-input, M-output lossless systems in terms of unit-norm vectors. This property makes the second structure desirable in applications that involve optimization of the parameters. Modifications of the second structure for implementing single-input, multioutput, and lossless bounded real (LBR) systems are also included. The main importance of the structures is that they are completely general, i.e. they span the entire set of M×1 and M×M lossless systems. This is demonstrated by showing that any such system can be synthesized using these structures. The structures are also minimal in the sense that they use the smallest number of scalar delays and parameters to implement a lossless system of given degree and dimensions. A design example to demonstrate the main results is included

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

On designing H∞ filters with circular pole and error variance constraints

Copyright [2003] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we deal with the problem of designing a H∞ filter for discrete-time systems subject to error variance and circular pole constraints. Specifically, we aim to design a filter such that the H∞ norm of the filtering error-transfer function is not less than a given upper bound, while the poles of the filtering matrix are assigned within a prespecified circular region, and the steady-state error variance for each state is not more than the individual prespecified value. The filter design problem is formulated as an auxiliary matrix assignment problem. Both the existence condition and the explicit expression of the desired filters are then derived by using an algebraic matrix inequality approach. The proposed design algorithm is illustrated by a numerical example

Optimal Controller and Filter Realisations using Finite-precision, Floating- point Arithmetic.

The problem of reducing the fragility of digital controllers and filters implemented using finite-precision, floating-point arithmetic is considered. Floating-point arithmetic parameter uncertainty is multiplicative, unlike parameter uncertainty resulting from fixed-point arithmetic. Based on first- order eigenvalue sensitivity analysis, an upper bound on the eigenvalue perturbations is derived. Consequently, open-loop and closed-loop eigenvalue sensitivity measures are proposed. These measures are dependent upon the filter/ controller realization. Problems of obtaining the optimal realization with respect to both the open-loop and the closed-loop eigenvalue sensitivity measures are posed. The problem for the open-loop case is completely solved. Solutions for the closed-loop case are obtained using non-linear programming. The problems are illustrated with a numerical example

An improved sufficient condition for absence of limit cycles in digital filters

It is known that if the state transition matrix A of a digital filter structure is such that D - A^{dagger}DA is positive definite for some diagonal matrix D of positive elements, then all zero-input limit cycles can be suppressed. This paper shows that positive semidefiniteness of D - A^{dagger}DA is in fact sufficient. As a result, it is now possible to explain the absence of limit cycles in Gray-Markel lattice structures based only on the state-space viewpoint

Review of Unbiased FIR Filters, Smoothers, and Predictors for Polynomial Signals

Extracting an estimate of a slowly varying signal corrupted by noise is a common task. Examples can be found in industrial, scientific and biomedical instrumentation. Depending on the nature of the application the signal estimate is allowed to be a delayed estimate of the original signal or, in the other extreme, no delay is tolerated. These cases are commonly referred to as filtering, prediction, and smoothing depending on the amount of advance or lag between the input data set and the output data set. In this review paper we provide a comprehensive set of design and analysis tools for designing unbiased FIR filters, predictors, and smoothers for slowly varying signals, i.e. signals that can be modeled by low order polynomials. Explicit expressions of parameters needed in practical implementations are given. Real life examples are provided including cases where the method is extended to signals that are piecewise slowly varying. A critical view on recursive implementations of the algorithms is provided