Mixed norm low order multirate filterbank design: Relaxed commutant lifting approach

Abstract

In this paper, we reformulate the design of IIR synthesis filters in classical multirate systems as an optimization problem involving a new norm called P m-norm where m is any positive integer. That optimization problem can be solved using a recent generalization of the commutant lifting techniques in operator theory. The introduced norm is actually a trade-off in handling energy distortion and error peak distortion. Our development allows the designer to select from a family of filters the one which is best suited for specific applications. The wellknown H 2 and H∞ designs then can be viewed as special cases when m = 1 and m → ∞ respectively. The computation relies mainly on FFT technique and a finite section of certain Toeplitz matrices. The obtained filters are of low order and attractive for practical implementation. Moreover, the proposed approach works for non-rational transfer functions. A new method for inner outer factorization of a rational matrix-valued function is also developed.

Redheffer Representations and Relaxed Commutant Lifting

“…See [11] for an application in filterbank design. In [14] a particular (central) solution is given; while descriptions of all solutions are obtained in [16,17,20,22,25].…”