在高斯/非高斯雜訊下之適應式自迴歸模型建立:演算法及應用

Abstract

適應式自迴歸模型建立已廣泛地應用在訊號處理的領域上。自 迴歸 模型的係數可以使用最小平均平方差(LMS)預測誤差濾波器求 出已廣為 人知。然而當輸入訊號受相加性白雜訊干擾時，此濾波器 將求出有偏差 的值。在本論文中我們提出一類新的適應式濾波器來 解決這一個問題。 這類濾波器名為ρ-LMS濾波器，是由估計理論導 出。主要的方法是對輸 入及誤差訊號加以濾波，再將結果輸入LMS 濾波器。由於雜訊減小，偏 差問題也相對地減小。ρ-LMS濾波器在 高斯雜訊下是線性的。而在非高 斯的雜訊下是呈非線性。由於廣泛 性的非高斯濾波問題非常困難，因此 我們只考慮機率分佈可被高斯 和逼近的雜訊。線性及非線性ρ-LMS濾波 器的一階及二階統計特性 的理論值亦在本論文中導出。一個由牛頓法導 出的快速演算法用來 加速ρ-LMS濾波器的收斂速度，因而導出了快速牛 頓ρ-LMS濾波 器。另外ρ-LMS濾波器亦能輸出濾波後的結果，這個重 要性質是其 他LMS型式的濾波器所沒有的。換言之自迴歸模型建立及濾 波可以 整合在一濾波器□。當訊號為高斯時，線性的ρ-LMS濾波行為 像卡 門濾波器。而當訊號為非高斯時，非線性的ρ-LMS濾波器像馬氏濾 波器。本論文提出的濾波器亦應用在弦波加強及主動式迴音消除上，並有 令人滿意的結果。至於在濾波方面，我們有其他方法處理偏差 問題。我 們可以使用有偏差的驅動雜訊變異數去補償偏差係數所造 成的影響。當 自迴歸階數不高時，此法可以有不錯的次佳結果。我 們將此結果應用在 語音加強上。有雜訊的語音首先分解成次頻訊號，然後次頻訊號被模擬成 低階自迴歸模型，因此低階的卡門濾波器可 用來加強這些訊號，加強後 的次頻訊號最後被合成加強的全頻語音。在自迴歸係數的處理上，只使用 一般的預測誤差濾波器來估計。卡 門濾波器在偏差係數下的工作效能亦 在本文中有分析。此法不只降 低了運算量亦有不錯的效能。 Adaptive autoregressive (AR) modeling is widely used in signal processingIt is well-known that the coefficients of an AR model can be easily obtained using an LMS prediction error filter. However, this filter givesa biased solution when the input signal is corrupted by additive white noise.In this thesis, we propose a new class of adaptive filter to solve the problem. This class of filters, known as the ρ-LMS filter, is derived fromthe estimation theory. The main idea is to filter the noisy input and error signal and then use the results in the LMS algorithm. Since the noise level is reduced, the bias problem will consequently be reduced. The ρ-LMS filter is linear when the noise is Gaussian, however, it becomes nonlinear when the noise is non-Gaussian. Since the general non-Gaussian filtering is difficult, we only consider the case where the noise distribution can be approximated by a Gaussian sum. Theoretical analysis of the first- and second-order statistics for the linear and nonlinear ρ-LMS filters are also presented. A fast algorithm derived from the Newton method is developed to accelerate the convergence rate of ρ-LMS filters. This leads to the fast ρ-LMS-Newton filters. Asa by- product, the ρ-LMS filter can output filtered results. This is animportant property not shared by other LMS-type filters. In other words,AR modeling and filtering are combined in a single filter. The linear ρ-LMS filter can act like a Kalman filter when noise is Gaussian, and the nonlinear ρ-LMS filter can act like a Masreliez's filter when noise isnon-Gaussian. The proposed filters are then applied in line enhancement and active noise cancellation and satisfactory results are observed. As far asfiltering concern, there is another way to deal with the bias problem. We canuse a biased driving noise variance to compensate for the effect caused by the biased AR coefficients. When the AR order is low, this approach can yield a good suboptimal result. We use this method in speech enhancement.Noisy speech is first decomposed into subbbands. Subband signals are then modeled as low-order AR process, such that low-order Kalman filter can be applied to enhance subband signals. Enhanced subband signals are finallycombined to form the enhanced fullband speech. To identify AR coefficients,the conventional prediction-error filters are used. The performance ofthe Kalman filter with biased parameters is analyzed. This approach not only greatly reduces the computational complexity, but also achieves good performance.