Some Properties of Laurent Polynomial Matrices

In the context of multivariate signal processing, factorizations involving so-called para-unitary matrices are relevant as well demonstrated in the book of Vaidyanathan [11], or [4, 1] and more recently in a series of papers by McWhirter and co-authors [5, 6]. However, known factorizations of matrix polynomials, such as the Smith form [10], involve unimodular matrices but usual factorizations such as QR, eigenvalue or singular value decompositions, have not been proved to exist for polynomial matrices, if defined with para-unitary matrices, except for very restrictive matrices [2]. It is clear that Cholesky factorization requires square roots, and that EVD and SVD require roots of higher degree polynomials. But one can ask oneself whether the closure of the field of polynomial coefficients is enough or not. It turns out that it is not. Nevertheless, density arguments allow to approximate any polynomial matrix by SVD factorization involving paraunitary polynomial matrices. With that goal, we define the appropriate framework for Laurent polynomial matrices, that is, polynomial matrices with both positive and negative powers in a single variable, particularly the notion of ordrer and degree. We introduce a Smith form for these matrices involving “L-unimodular” matrices which are matrices with a monomial non-zero determinant. The ‘Elementary Polynomial Givens Rotations’ of [6] are of that kind. 1 Notations and Definitions Throughout the paper, vectors and matrices are denoted with underline lowercase and bold uppercase letters respectively. I and 0 denote identity and zero matrices. The entries of a matrix M are denoted mij , where subscript ij denotes the i-th row and the j-th column of M . (∗) stands for complex conjugation, () for conjugate transposition. Let Z be the set of integers, N the subset of positive integers, R the field of real numbers, C the field of complex numbers, C∗ = C\{0} and C be the unit circle. C(A) denotes the set of continuous functions from C to A. 1Laboratoire I3S, CNRS UMR7271, Université de Nice Sophia Antipolis, 2000 route des Lucioles, BP 121, 06903 Sophia Antipolis Cedex, France (e-mail: [email protected]). 2GIPSA-Lab, CNRS UMR5216, 11 rue des Mathematiques, Grenoble Campus, BP 46, 38402 St Martin d’Heres cedex, France. Let C[z] be the ring of polynomials with complex coefficients: p(z) = ∑n i=0 piz i with n ∈ N, pi ∈ C. If pn 6= 0, then the polynomial degree of p is deg(p) = n, if pn = 1, p is said to be monic. Moreover, if pn = 1 and p0 6= 0, p is said to be L-monic. Let C[z, z−1] be the ring of Laurent polynomials: p(z) = ∑n i=m piz i with m,n ∈ Z,m ≤ n, pi ∈ C. Let us suppose that pmpn 6= 0, then one can always factorize p(z) = pnz π(z) with π ∈ C[z] and Lmonic. The L-degree of p is defined as the degree of π and is denoted as d(p) = n−m. According to this L-degree, C[z, z−1] is an Euclidean ring, so a Principal Ideal Domain (PID). The invertible elements of C[z, z−1] are non-zero L-monomials: p(z) = az with a ∈ C∗ and α ∈ Z. As greatest common divisors (gcd) are defined up to an invertible element, one can set for uniqueness purposes, the gcd to be a L-monic polynomial of C[z]. Let Cn×n[z−1] be the ring of n×n matrices corresponding to Finite Impulse Response (FIR) sytems: M(z) = ∑l k=0 Mkz −k. The order of M is the greatest index such that Mk 6= 0. Let Cn×n[z] be the ring of n× n matrices with polynomials entries, the order is defined in the same way. The units of Cn×n[z] are matrices whose determinant is constant and are called unimodular matrices [10, 11]. Let Cn×n[z, z−1] be the ring of matrices with Lpolynomials entries, also called L-polynomial matrices. The units of Cn×n[z, z−1] are matrices whose determinant is a non-zero L-monomial and are called L-unimodular matrices. Let tilde denote the paraconjugation: M̃(z) = M( 1 z∗ ),∀z ∈ C ∗. On the unit circle, one has M̃ = M . One says H is parahermitian iff H = H̃, and U is paraunitary iff UŨ = I. Let us define a sesquilinear form according to paraconjugation for two vectors of Cn×1[z, z−1] by 〈u, v〉 = ũ v. As paraconjugation is an involution, 〈, 〉 is a sesquilinear parahermitian form, and one has 〈̃u, v〉 = 〈v, u〉. Moreover, let H ∈ Cn×n[z, z−1], then 〈Hu, v〉 = 〈u, H̃v〉. If H is parahermitian, we then have 〈Hu, v〉 = 〈u,Hv〉. Let Cn×n(z) be the ring of n × n matrices whose entries are rational functions. Let H be a proper rational matrix, the degree or the McMillan degree of H is defined as the sum of the degrees of the denominator polynomials in its Smith-McMillan form. It appears to be the minimal number of delays to 1 ha l-0 07 47 25 3, v er si on 1 30 O ct 2 01 2 Author manuscript, published in "9th IMA International Conference on Mathematics in Signal Processing, Birmingham : United Kingdom (2012)"