Exact and approximate polynomial decomposition methods for signal processing applications

Signal processing is a discipline in which functional composition and decomposition can potentially be utilized in a variety of creative ways. From an analysis point of view, further insight can be gained into existing signal processing systems and techniques by reinterpreting them in terms of functional composition. From a synthesis point of view, functional composition offers new algorithms and techniques with modular structure. Moreover, computations can be performed more efficiently and data can be represented more compactly in information systems represented in the context of a compositional structure. Polynomials are ubiquitous in signal processing in the form of z-transforms. In this paper, we summarize the fundamentals of functional composition and decomposition for polynomials from the perspective of exploiting them in signal processing. We compare exact polynomial decomposition algorithms for sequences that are exactly decomposable when expressed as a polynomial, and approximate decomposition algorithms for those that are not exactly decomposable. Furthermore, we identify efficiencies in using exact decomposition techniques in the context of signal processing and introduce a new approximate polynomial decomposition technique based on the use of Structured Total Least Norm (STLN) formulation.Texas Instruments Leadership University Consortium ProgramBose (Firm

Structured total least norm and approximate GCDs of inexact polynomials

The determination of an approximate greatest common divisor (GCD) of two inexact polynomials f=f(y) and g=g(y) arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation S*(f,g) of the Sylvester resultant matrix S(f,g). In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of S(f,g), and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix S*(f,g), and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented

Decoupling Multivariate Polynomials Using First-Order Information

We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the first-order information of the polynomials in a set of operating points, which is captured by the Jacobian matrix evaluated at the operating points. The polyadic canonical decomposition of the three-way tensor of Jacobian matrices directly returns the unknown linear relations, as well as the necessary information to reconstruct the univariate polynomials. The conditions under which this decoupling procedure works are discussed, and the method is illustrated on several numerical examples