On the ill conditioning of locating transmission zeros in least squares ARMA filtering (Q1059386)

To generate autoregressive-moving average (ARMA) filters the Nevanlinna- Pick algorithm is commonly used. This algorithm solves the problem to minimize \[ S[f_ n]=\frac{1}{2\pi}\int^{\pi}_{-\pi}| \frac{1}{\sigma (e^{i\theta})}-f_ n(e^{i\theta})|^ 2\omega (e^{i\theta})d\theta \] where \(\sigma\) is an (unknown) spectral factor of the spectral density \(\omega\) for a real stationary zero mean scalar stochastic sequence, \(f_ n\) ranges over \(L_ n=span\{U_ 0(z),U_ 1(z),...,U_ n(z)\}\) where \(U_ j(z)\) \((j=0,1,...,n)\) form a basis of the space of rational functions with poles \(1/\alpha_ k\), \(k=1,2,...,n\). The complex numbers \(\alpha_ k\) belong to the open unit disc and are called ''transmission zeros'' of the filter. The minimum of S is evidently a function of these transmission zeros. To obtain an optimal least-squares rational approximation of 1/\(\sigma\) one must optimally choose the set of \(\alpha_ k\) (they are then interpolation points for the Nevanlinna-Pick algorithm). The author shows by some simple examples that the problem of finding genuine transmission zeros of a given filter is a very ill-conditioned one. Thus, it means that if the chosen transmission zeros are far away from the exact ones, the Nevanlinna-Pick algorithm can still give an ARMA filter which is a very good least squares approximant.