New precoding for intersymbol interference cancellation using nonmaximally decimated multirate filterbanks with ideal FIR equalizers

In this paper, we propose a new precoding method for intersymbol interference (ISI) cancellation by using non-maximally decimated multirate filterbanks. Unlike the existing preceding methods, such as the TH and trellis precodings, the new preceding (i) may be independent of the ISI channel; (ii) is linear and does not have to implement any modulo operation; (iii) gives the ideal FIR equalization at the receiver for any FIR ISI channel including spectral-null channels; (iv) expands the transmission bandwidth in a minimum amount. The precoding is built on non-maximally decimated multirate filterbanks. Based on multirate filterbank theory, we present a necessary and sufficient condition on an FIR ISI transfer function in terms of its zero set such that there is a linear FIR N×M precoder so that an ideal FIR equalizer exists, where the integers K and N are arbitrarily fixed. The condition is easy to check. As a consequence of the condition, for any given FIR ISI transfer function (not identically 0), there always exist such linear FIR precoders. Moreover, for almost all given FIR ISI transfer functions, there exist linear FIR precoders with size N×(N-1), i,e., the bandwidth is expanded by 1/N. In addition to the conditions on the ISI transfer functions, a method for the design of the linear FIR precoders and the ideal FIR equalizers is also given. Numerical examples are presented to illustrate the theory