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

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 well-known 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.