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Abstract

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..sub.i and .beta..sub.j of the model having the desired amplitude-and phase-frequency responses, the parameters satisfying the relationship: ##EQU1## where: .alpha..sub.i is the ith auto-regressive parameter, and .beta..sub.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; y.sub.k and u.sub.k 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).

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