Behaviour of the spectral factorization for continuous spectral densities

A Control Theoretical Approach to the Polynomial Spectral-Factorization Problem

“…Sometimes it is convenient to normalize the first coefficient of the spectral factor polynomial such that a(0) = 1 in which case L(z) = c.a z −1 a(z) (3) where c = a 2 0 is a constant whose meaning varies depending on the application. The problem of spectral factorization has been around since the earliest days of optimal filtering and was found independently by Wiener [26] and Kolmogorov [11].…”

“…For example Youla [29,30], Wilson [28] and Sayed and Kailath [22]. Furthermore, see [3,4,19,20] for other methods for both the scalar and multivariable cases, and [14,24] for some state-space approaches to the problem.…”

Uniform FIR approximation of causal Wiener filters, with applications to causal coherence

“…This forms the basis for a common practical algorithm for computing spectral factors F (z) which is outlined in [12]. A finite number of Fourier coefficients of log S xx (Ω) are computed and then the frequency response (52) can be computed at a finite number of values z = e jΩ (and then, if needed, a time-domain representation can be recovered).…”

“…In the Fourier analytic method only finitely many Fourier coefficients are used, whereas the Wilson-Burg method can only be applied to PSDs from from finite-length autocorrelation functions. The question of how these finite order approximations affect the resulting minimum phase factor F (Ω) was addressed partially in [12]. It turns out that the spectral factorization mapping S xx → F is not a continuous mapping on the space of continuous …”

Quantitative results on continuity of the spectral factorization mapping in the scalar case

“…[8]). The spectral factor can also be expressed in terms of zeros of polynomial (5), and therefore the map (4) is continuous on P N , the set of all functions of the form (5). Papers [6], [7] are devoted to estimating the constant C N in the inequality…”