Depth-L Nyquist (M) Filters and Biorthogonal Partners

In the past, Nyquist filters have been studied by many researchers. A pair of filters are said to be biorthogonal partners of each other if their cascade forms a Nyquist filter. The theory and design of traditional biorthogonal partners have been studied extensively. In this paper, we extend these results to depth-L Nyquist filters and biorthogonal partners. Two potential applications of them are discussed in fractionally spaced equalizer (FSE) and filter bank multicarrier (FBMC) systems. Moreover, the necessary and sufficient condition is derived for the existence of an finite impulse response (FIR) depth-L biorthogonal partner. We find out that the existence depends on the cardinality of the largest so-called congruous-zero set. In addition, we will show how to design these filters by using the eigenfilter method. Finally, performance comparisons are carried out to demonstrate the advantage of depth-L biorthogonal partners. It is shown that the depth-L version is more robust against inter-symbol interference (ISI) and timing synchronization error compared to the traditional biorthogonal partner.