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

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=(m₁,m ₂)^T such that m=Dn, where n=(n₁,n₂ )^T is an arbitrary integer vector. One out of every |det(D)| 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 |det(D)|=N (maximally decimated). For a matrix B, [B]_(i,j) denotes the (i,j) element of B, diag(a₀,a₁,…,a_N-1) denotes an N×N diagonal matrix whose (i,i) element is a_i-1