ARMA filter and method for designing the same

摘要

A near minimum order ARMA type recursive filter with guaranteed stability and convergence is provided together with a method for obtaining the parameters of such filter. The amplitude/frequency response of the filter approximates an arbitrarily selected frequency spectrum of amplitude, and the phase response approximates a substantially linear function of frequency with an arbitrarily selected slope because the parameters are identified, off-line, using a minimization process that minimizes an integral error norm. The first step involves performing an inverse discrete Fourier transform of the arbitrarily selected frequency spectrum of amplitude to obtain a truncated sequence of coefficients of a stable, pure moving-average filter model, i.e., the parameters of a non-recursive filter model. The truncated sequence of coefficients, which has N+1 terms, is then convolved with a random sequence to obtain an output sequence associated with the random sequence. A time-domain, convergent parameter identification is then performed, in a manner that minimizes an integral error function norm, to obtain the near minimum order parameters alpha i and beta j of the model having the desired amplitude- and phase-frequency responses, the parameters satisfying the relationship: <IMAGE> where: alpha i is the ith auto-regressive parameter, and beta j is the jth moving-average parameter, respectively, of an ARMA-type recursive filter; n and m denote the order of the auto-regressive and the moving-average parts of the ARMA model, respectively; yk and uk are associated elements of the kth element of the output sequence and the random sequence, respectively; k is an integer; and v is a shift integer selected to provide the desired slope of the phase response given by 2 pi (v-N/2).