Lossless Filter Examples
The simplest lossless filter is a unit-modulus gain
where can be any phase value. In the real case can only be 0 or , hence .
A lossless FIR filter can only consist of a single nonzero tap:
for some fixed integer , where is again some constant phase, constrained to be 0 or in the real-filter case. We consider only causal filters here, so .
where , , and . The polynomial can be obtained by reversing the order of the coefficients in , conjugating them, and multiplying by . (The factor above serves to restore negative powers of and hence causality.) Such filters are generally called allpass filters.
Properties of Paraunitary Filter Banks