The system transfer function for a QMF bank is For perfect reconstruction, we need Tz=z-l T z z l for some l∈ℕ l , which implies the equivalent conditions 4z-1 P 0 z2 P 1 z2=z-l 4 z -1 P 0 z 2 P 1 z 2 z l P 0 z2 P 1 z2=14z-l-1 P 0 z 2 P 1 z 2 1 4 z l 1 P 0 z P 1 z=14z-l-12 P 0 z P 1 z 1 4 z l 1 2 For FIR polyphase filters, this can only be satisfied by P 0 z= β 0 z- n 0 P 0 z β 0 z n 0 P 1 z= β 1 z- n 1 P 1 z β 1 z n 1 where we have n 0 + n 1 =l-12 n 0 n 1 l 1 2 and β 0 β 1 =14 β 0 β 1 1 4 .

In other words, the polyphase filters are trivial, so that the prototype filter H 0 z H 0 z has a two-tap response. With only two taps, H 0 z H 0 z cannot be a very good lowpass filter, meaning that the sub-band signals will not be spectrally well-separated. From this we conclude that two-channel 1 perfect reconstruction QMF banks exist but are not very useful.