Matrix Extension with Symmetry and Its Application to Filter Banks

Let P be an r×smatrix of Laurent polynomials with symmetry such that P(z)P∗(z) = Ir for all z ∈ C\{0} and the symmetry of P is compatible. The matrix extension problem with symmetry is to find an s × s square matrix Pe of Laurent polynomials with symmetry such that [Ir,0]Pe = P (that is, the submatrix of the first r rows of Pe is the given matrix P), Pe is paraunitary satisfying Pe(z)Pe(z) = Is for all z ∈ C\{0}, and the symmetry of Pe is compatible. Moreover, it is highly desirable in many applications that the support of the coefficient sequence of Pe can be controlled by that of P. In this paper, we completely solve the matrix extension problem with symmetry and provide a step-by-step algorithm to construct such a desired matrix Pe from a given matrix P. Furthermore, using a cascade structure, we obtain a complete representation of any r × s paraunitary matrix P having compatible symmetry, which in turn leads to an algorithm for deriving a desired matrix Pe from a given matrix P. Matrix extension plays an important role in many areas such as electronic engineering, system sciences, applied mathematics, and pure mathematics. As an application of our general results on matrix extension with symmetry, we obtain a satisfactory algorithm for constructing symmetric paraunitary filter banks and symmetric orthonormal multiwavelets by deriving high-pass filters with symmetry from any given low-pass filters with symmetry. Several examples are provided to illustrate the proposed algorithms and results in this paper.