Synthesis of spectral densities using finite automata

A method for designing a finite automaton whose output exhibits a given rational power spectral density R(z) belonging to a particular class, is presented. If the automaton input is composed of independent and identically distributed symbols, its state process is a homogeneous Markov chain whose transition probability matrix π may by obtained from the input probability mass function and the state transition function. Since the poles of R(z) only depend on π whereas its zeros depend on the matrix A specifying the output function, a matrix π with the desired eigenvalues (and perhaps additional ones) is first derived and, the matrix A is determined so as to ensure the realization of the desired zeros (as well as the cancellation of the additional poles possibly introduced in the first step). The method exploits the properties of circulant matrices; in particular, a sufficient condition is provided under which a circular matrix with given eigenvalues (ordered in a Hermitian sequence) is stochastic