Polyphase Realization of VSB Filter Banks for Spectrally Efficient Transmitter Multiplexers and Receiver Demultiplexers Using Spectral Factorization

Various Nyquist polynomials and their spectral factorizations to obtain suitable finite impulse response (FIR) vestigial sideband (VSB) filters are summarized here. The filters of interest have transfer functions that are polynomials with complex coefficients which are symmetric, due to Lawton, and polynomials whose coefficients are conjugate symmetric or conjugate antisymmetric, which are also linear phase. The resulting polyphase implementations using polynomials with complex coefficients for the VSB filters result in computational savings of 25% over linear phase prototype VSB filters with real coefficients, and 50% over VSB filters obtained by factorization into minimum and maximum phase polynomials. Relationships to symmetries of the complex scaling functions and wavelets obtained from the usual two-scale difference equations are described.