基於高階統計量之韋納（Wiener）濾波處理

Abstract

本篇論文提出了一個用高階統計量之平均平方誤差準則來設計韋納濾波器 的方法；此方法可適用於當廣義穩態隨機訊號x(n)及期望之訊號d(n)都是 非高斯分佈且受高斯雜訊污染時。在本文中，我們理論上證明根據所提出 的準則設計之韋納濾波器與當訊號未受污染時依據二階統計量所設計之傳 統韋納濾波器完全相同；而且就像後者一樣，前者亦可以由解一個基於 高階統計量的）垂直原理所得之Wiener-Hopf方程式來獲得。此外，本文 更進一步地提出了一個基於高階統計量之廣泛投影定理；此定理包括本文 所提之準則以及Delopoulos與Giannakis所提基於高階統計量之平均平方 誤差準則相關的投影原理為特例，其中相關的投影原理為將高階統計量投 影至二階統計量。再者，當統計量階數M=3時，本文所提之準則會等效於 Delopoulos與Giannakis所提之準則。本文中亦將所提的基於高階統計量 之韋納濾波器應用於時間延遲估計與系統鑑別，並且提供一些相關的模擬 結果來驗證本文所提方法之優越性能。最後，我們並做了一些討論及結論 。 This thesis proposes a cumulant (higher-order statistics) based mean-square-error (MSE) criterion for the design of Wiener filters when both the given wide-sense stationary random signal x(n) and the desired signal d(n) are non-Gaussian and contami- nated by Gaussian noise sources. It is shown that the designed cumulant based Wiener filter associated with the proposed cri- terion is identical to the conventional correlation (second- order statistics) based Wiener filter as if both x(n) and d(n) were noise-free measurements. As the latter, the former can be also obtained by solving a cumulant based Wiener-Hopf equation associated with a (cumulant based) orthogonality principle. Then a generalized cumulant based projection theorem is proposed which includes the projection of cumulants to correlations as- sociated with the proposed cumulant based MSE criterion and that associated with Delopoulos and Giannakis' cumulant based MSE criterion as special cases. Moreover, the proposed cumulant based criterion and Delopoulos and Giannakis' cumulant based MSE criterion are equivalent for cumulant order M = 3. The proposed cumulant based Wiener filter is then applied to time delay esti- mation and system identification. Some simulation results are then provided to demonstrate the good performance of the pro- posed cumulant based Wiener filter. Finally, we draw some con- clusions.