A necessary condition for linear phase in two-dimensional perfect reconstruction QMF banks

Abstract

One dimensional (1-D) perfect reconstruction (PR) QMF banks have been studied extensively. If all the analysis filters are linear phase in a PR QMF bank, we call it a 1-D linear phase PR QMF bank. Nguyen and Vaidyanathan showed a necessary condition for 1-D linear phase PRQMF banks [1989]. Recently, two dimensional (2-D) PR QMF banks have been studied. This paper shows a necessary condition for 2-D linear phase PR and MF banks. Our result is easily generalized to M dimensional linear phase PR QMF banks. In a 2-D system, subsampling is defined by a subsampling matrix D, where D is a 2*2 nonsingular matrix of integers. The sampling retains only samples at points m=(m/sub 1/,m/sub 2/)/sup T/ such that m=Dn, where n=(n/sub 1/,n/sub 2/)/sup T/ is an arbitrary integer vector. One out of every mod det(D) mod samples of the sequence is retained. A 2-D N channel analysis/synthesis filter bank is shown. We assume that all channels share the same subsampling matrix D such that mod det(D) mod =N (maximally decimated). For a matrix B, (B)/sub (i,j/) denotes the (i,j) element of B, diag(a/sub 0/,a/sub 1/,...,a/sub N-1/) denotes an N*/spl times/N diagonal matrix whose (i,i) element is a/sub i-1/. >