A Simple and Explicit Formulation of Non-Unique Wiener Filters for Linear Predictor with Rank-Deficient Autocorrelation Matrix

Shunsuke KOSHITA  Masahide ABE  Masayuki KAWAMATA  Takaaki OHNARI  Tomoyuki KAWASAKI  Shogo MIURA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E99-A   No.8   pp.1614-1617
Publication Date: 2016/08/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E99.A.1614
Type of Manuscript: LETTER
Category: Digital Signal Processing
Wiener filter,  linear predictor,  noise-free input,  rank-deficient input autocorrelation matrix,  

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This letter presents a simple and explicit formulation of non-unique Wiener filters associated with the linear predictor for processing of sinusoids. It was shown in the literature that, if the input signal consists of only sinusoids and does not include a white noise, the input autocorrelation matrix in the Wiener-Hopf equation becomes rank-deficient and thus the Wiener filter is not uniquely determined. In this letter we deal with this rank-deficient problem and present a mathematical description of non-unique Wiener filters in a simple and explicit form. This description is directly obtained from the tap number, the frequency of sinusoid, and the delay parameter. We derive this result by means of the elementary row operations on the augmented matrix given by the Wiener-Hopf equation. We also show that the conventional Wiener filter for noisy input signal is included as a special case of our description.