Spectral factorization using FFTs for large‐scale problems

Abstract

A method is provided for scalar systems, which uses FFTs and provides spectral factorization directly from the periodogram. The method is block recursive, providing better estimates as time progresses with more data available. Although spectral factorization is a mature technique, the current methods available are generally too slow to cope with acoustic problems of the scale discussed here. SPECTRAL FACTORIZATION USING FFTs 955 problem, some simpler than others. For example, the earliest discrete-time attempt was due to Kolmogorov [3] and later a computational method by Wilson [4]. Since then, there has been a good number of approaches mostly summarized here [5]. The particular approach that concerns this paper is the feedback method proposed by Moir [6] and used for system identification [7]. In fact, the spectral factorization problem appears in all classes of frequency-domain problems where a quadratic cost function is involved. So besides the optimal filtering problem [2], it also has application in optimal control [8-10]. There is another method available for spectral factorization using FFTs, however. For example the cepstrum or kepstrum approach [11][12][13] can be used, but these approaches, when used on real data, need a great deal of smoothing to achieve accurate results. Although such a method could in principle be used for our application, the two methods are totally unrelated. We also note here that the estimation of the polynomial a.´ 1 / is not unique for the more general case when the spectrum has non-minimum-phase zeros. For second-order statistics, it is well established that it is only possible to estimate the minimum-phase equivalent (i.e. the zeros are reflected back within the unit circle of the´-plane). Although this can be a disadvantage, we can also use this as an advantage to obtain stable polynomials as shown in a later section applied to acoustics. Whilst it is possible to apply a similar approach to higher-order statistics, it is known that such an approach can lead to an error surface that is not convex and has local minima.rf .v 0 ; v 1 ; : : :be zero. 956 T. J. MOIR Now, as " k D L V k , we find by differentiating again P j D0w i w i Cj ; i D 0; 1 : : : n. Note that this operation is equivalent in the´-domain to the operation a.´ 1 /a.´/ D ja.´/j 2 . What we are effectively doing is re-forming the Laurent series based on the estimated spectral factor and creating SPECTRAL FACTORIZATION USING FFTs 959 frequency-domain method, which requires a primary and reference input [14]. Although other blind methods exist, for example, [15][16][17][18][19][20], these methods require special input signals associated with communication systems, for example, code division multiple access, or alternatively require signals that have higher-order statistics [21]. The possible exception to this is the Kepstrum method [11,12,22], which has found application in adaptive filtering [23]. It should be made clear however that no method that has Gaussian noise as the driving signal can reconstruct accurate...

On the Factorization of Rational Discrete-Time Spectral Densities

“…To this aim, we can follow the same lines of the proof of point 7) of Theorem 2. In fact, we can define Ap,1 := Ap \ ({ z ∈ C : |z| = 1 } ∪ {0, ∞}) and partition C0 as C0 = { z ∈ C : 1/z ∈ Ap,1 } ∪ { z ∈ C : |z| = 1 } ∪ Ap,1 and replace equation (35) with the more general expression for the degree of the pole pi of Φ(z)…”

“…October 6, 2018 DRAFT DRAFT and replace equation (35) with the more general expression for the degree of the pole p i of Φ(z)…”

“…Since the pioneering works of Kolmogorov and Wiener in the forties, a variety of methods have been proposed for the analysis and solution of this problem under different assumptions and in different settings, see e.g., [3], [27], [39], [45], [8], [49], [33], [34], [35], to cite but a few. We also refer to the relatively recent survey [42] that contains many other references and different points of view on this problem.…”