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This short chapter presents characterizations of nonnegative univariate polynomials, with an emphasis on trigonometric polynomials. The basic result (the well known Riesz-Fejér theorem) is the existence of a spectral factorization for a globally nonnegative trigonometric polynomial.

There are several ways of characterizing nonnegative polynomials that may be interesting for a mathematician. However, not all of them are appropriate to computational purposes, by “computational” understanding primarily optimization methods.

Are the notions and results presented in the previous two chapters valid in the multivariate case? The answer is mostly yes, but with some limitations. The notion of Gram matrix is related directly only to sum-of-squares polynomials. Unlike the univariate case, multivariate nonnegative polynomials are not necessarily sum-of-squares. However, trigonometric polynomials are sum-of-squares, but the degrees of the sum-of-squares factors may be arbitrarily high, at least theoretically.

In Section 1.4, we have presented parameterizations of univariate polynomials that are positive on an interval.

Filter design is one of the perennial topics in signal processing. FIR filters are often preferred for their simple implementation and robustness and so they are an appropriate subject for this first chapter devoted to applications.

In this chapter, we explore the use of positive polynomials in the design of FIR filterbanks (FB). The study is confined to a single class, that of orthogonal FBs. Two-channel FBs are discussed first, as the simplest instance of the problem; naturally related with it are the design of compaction filters or of signal-adapted wavelets. We go then to DFT modulated FBs, with an arbitrary number of channels; similarly to the two-channel case, the free parameters of the whole FB are the coefficients of a single prototype filter.

Stability is a basic property of dynamic systems. In this chapter, we explore several issues related to the stability of multidimensional discrete-time systems. First come stability tests: given a system, we have to decide if it is stable or not. Then, we discuss a robust stability problem, for the case where the coefficients of the system depend polynomially on some bounded parameters. Finally, we show how to build a convex stability domain around a given stable system. For all these problems, the solutions we present are based on the use of positive polynomials.

IIR filters can give the same magnitude performance with fewer parameters than FIR filters. However, they cannot have exact linear phase. Their design is more complicated due to the difficulty in ensuring stability and to the nonconvexity of the optimization problems. In this short chapter, we give few guidelines for the optimization of IIR filters, insisting on algorithms that use positive polynomials.