Spectral factorization of the autocorrelation sequence of an IIR filter using patterns of three functions (Q1775980)

Consider an infinite impulse response (IIR) filter, i.e., a linear time-invariant system whose input \(x_n\) and output \(y_n\) satisfy a difference equation of the form: \[ y_n=\sum _{k=1}^p b_ky_{n-k} +\sum _{j=o} ^q a_j x_{n-j}. \] Denote by \(\{\rho_i\}\) the autocorrelation sequence of the mentioned IIR filter and let \(H(z)=\alpha(z)/\beta(z)\), where \(\alpha(z)=\sum_{j=1}^q a_jz^{-j}\), \(\beta(z)=1-\sum_{k=1}^p b_k z^{-k}\). The paper presents a 3-step method of seeking \(H(z)\) from \(\{\rho_i\}\) by utilizing the 3-pattern identification method of an autoregressive moving-average process.