On the exact spectral factorization of rational matrix functions with applications to paraunitary filter banks (Q6596531)

Spectral factorization is a process by which a positive definite matrix function \(S\) on the unit circle \(\mathbb{T}\) is expressed in the form \(S(t)=S_+(t)S_+^*(t)\), where \(S_+\) can be extended analytically inside \(\mathbb{T}\) and \(S_+^*\) is its Hermitian conjugate. The main result of the paper says that if \(S\) is an \(r\times r\) polynomial matrix function which is positive definite on \(\mathbb{T}\) and \(S(z)=M(z)M^*(z)\) is its lower-upper factorization with \N\[ M(z)= \left(\begin{array}{ccccc} f_1^+(z) & 0 & \dots & 0 & 0 \\\N\xi_{21}(z) & f_2^*(z) & \dots & 0 & 0 \\\N\vdots & \vdots & \ddots & \vdots & \vdots \\\N\xi_{r-1,1} (z) & \xi_{r-1,2}(z) & \dots & \xi_{r-1}^+(z) & 0 \\\N\xi_{r1}(z) & \xi_{r2}(z) & \dots & \xi_{r,r-1}(z) & f_r^+(z) \end{array}\right). \] \NIf the entries of the matrix function \(M\) and the poles of the functions \(\xi_{ij}\) inside \(\mathbb{T}\) are known exactly, then the spectral factorization of \(S\) can also be found exactly. Some applications of the obtained results in signal processing are discussed.