Memory- $m$ Antipodal Processes: Spectral Analysis and Synthesis

Discrete-time two-valued processes are of paramount importance in Information Engineering and their analysis is usually addressed assuming they have at most the memory of one step in the past. Since this is a quite limiting assumption, a general analytic formula is given here for the spectrum of stochastic antipodal processes with finite memory m ges 1. The formula is derived within a generalized Markov framework and depends on the eigenstructure of a suitably defined transition matrix that is also exploited to give ergodicity conditions. The complexity of the overall analysis depends on the size of such matrix which is exponential in m. Within that framework a slightly less general but more tractable scheme for the generation of antipodal processes with prescribed spectral profile is introduced leveraging on a linear probability feedback. For such a scheme, whose complexity is linear in m, an alternative spectrum formula is derived as well as a synthesis procedure going from spectrum specification to feedback filter design. Both the general case and the linear probability feedback scheme are exemplified in some special cases.