運用代數化簡及係數外差之時域最小均方差適應性濾波演算法

Abstract

本論文主要提出兩種新的最小均方差適應性濾波演算法。第一種是由 代數化簡的適應性濾波，第二種是由適應性濾波器係數的外插。在重新組 合過濾波式子後，第一種新的"代數化簡最小均方差(ARLMS)適應性濾波演 算法"較一般的最小均方差(DLMS)適應性濾波演算法減少將近百分之五十 的乘法，但加法增加比百分之五十多一些，此外這新的演算法多引進一額 外的更新變數。另一方面，第二種是"係數外插最小均方差適應性濾波( ELMS)演算法"，它經由係數外插而得到一接近收斂值的解，因而使得大部 份的係數可提早收斂，因此使得整個系統至收斂的更新次數減少。 首先我們推導出代數化簡最小均方差適應性濾波架構的最佳係數解和最小 平均平方誤差(MMSE)。當系統滿足某特殊條件下，代數化簡最小均方差適 應性濾波架構跟Wiener濾波器的最佳係數解和MMSE會一樣，不然則會不同 ，且代數化簡最小均方差適應性濾波架構的MMSE會比Wiener濾波器的MMSE 小，此特殊條件是：理想訊號的期望值等於輸入訊號的期望值乘上最佳係 數和的共軛值。 接著我們介紹ARLMS演算法和它的一些特性，由於新引進一額外的更新變 數，使得 ARLMS演算法之均方差的式子和LMS演算法的形式有些不同， 因而我們重新探討係數更新式子，並推導出ARLMS演算法的係數更新式子 。由模擬顯示，若系統是在那種特殊條件且有 一適當的步級狀況下，則 ARLMS演算法的穩態平均平方誤差會比DLMS演算法的小。另外當 輸入訊號 和理想訊號的期望值都是零的情形下，我們推導出ARLMS演算法在平均和 平均平 方的收斂條件，這收斂條件是受DLMS演算法的步級μ和額外係數 更新式子之步級α之影響。另外我們推導出一完整的穩態平均平方誤差， 這穩態平均平方誤差是μ和α的函數式，且這ARLMS演算法的穩態平均平 方誤差會比DLMS演算法的稍大一點。若適當的選擇μ和α ，這ARLMS演算 法的表現則跟DLMS演算法相當。接著一簡單的使用法則被提出,這法則規 範了μ和α的範圍使得ARLMS演算法和DLMS演算法有相近的表現。此外， 我們也將這ARLMS演算法運用在高畫質電視和傳統電視的鬼影消除器、迴 聲消除器和HDSL等化器設計上。 再來我們將ARLMS演算法與SA, SRA, SSA和FELMS這些方法結合，以減少在係數更新部份的運算量，若適 當的選擇μ和α，由模擬結果可發現這些結合後的方法和原本的方法有相 近的表現。而ARLMS演算法也被應用至二維的適性濾波器，由影像雜訊消 除的模擬，我 們可發現二維的ARLMS演算法和傳統的二維DLMS演算法有相 近的表現。 最後我們介紹ELMS演算法的幾種不同 的類型，由模擬結果發現，ELMS演算法對步級和外插係數的時間點非常敏 感，為減少ELMS演算法可能發生的不合適之估計，我們提出一些預防措施 ，以使得這演算法有較好的結果。 In this thesis, two new types of LMS adaptive filtering algorithmsare proposed. The first one is based on algebraic reduction of adaptive filtering equation, while the second one is based on extrapolation of the filter coefficients. By rearranging the filtering equation, the algebraic-reduction LMS (ARLMS) adaptive filtering algorithm costs 50% fewer multiplications at the expense of 50% more additions than the direct-form LMS (DLMS) algorithm. In addition, the new algorithm introduces an extra adaptation parameter other than the filter coefficients. On the other hand, the extrapolated LMS (ELMS) algorithm extrapolates the adaptation coefficients to values close to the converged ones. As a result, considerable adaptation iterations are saved. The thesis first derives the optimal coefficient solution and minimal mean square error (MMSE) of the algebraic-reduction adaptive filtering (ARAF) structure. From the derivations, the optimal coefficient solutions and MMSEs for ARAF structure and direct-form Wiener filter are the same when system satisfies a specific condition. Otherwise, their optimal coefficient solutionsand MMSEs are different and the ARAF structure has a smaller MMSE than that ofthe direct-form Wiener filter. The specific condition is that the mean of input signal multiplied by the conjugated summation of the adaptive coefficient equals the mean of desired signal. Secondly, we introduce the ARLMS algorithm and explore its properties. FromARAF structure, an ARLMS algorithm is proposed, in contrast to the DLMS algorithm which is a realization of Wiener adaptive filter. The mean square errors of ARLMS and DLMS algorithms are different, because the ARLMS algorithmintroduces an extra adaptation parameter. Simulations show that the ARLMS algorithm has a smaller steady-state MSE than that of the DLMS algorithm when the step sizes are properly chosen and system is not under the mentioned specific condition. The conditions of convergence in mean and mean square for for the ARLMS algorithm are derived under the conditions of zero-mean input and desired signals. Accordingly, closed-form steady-state mean square error (MSE) as a function of the LMS algorithm step size μ and an extra compensation step size α are derived, which is slightly larger than that of the DLMS algorithm. Meanwhile, convergence bounds for μ and α are also derived. It is shown that, when μ is small enough and α is properly chosen, the ARLMS algorithm has comparable performance to that of the DLMS algorithm. Correspondingly, simple working rules and ranges for α and μ to make such comparability are provided. For verification, the ARLMS algorithm has been applied to HDTV and NTSC multi-path equalizations, Echo cancellation and HDSL equalization. Thirdly, the combined ARLMS algorithms with the signed LMS algorithm (SA), signed regressor algorithm (SRA), signed-signed (or signed product) algorithm (SSA) and the Duhamel's fast exact least mean square (FELMS) adaptive algorithm are simulated to converge as fast as their counter DLMS variants. They maintain comparable performances to those of the DLMS variants when α isproperly chosen. Further, the ARLMS adaptive algorithm is extended to two- dimensional adaptive filtering. Simulations on image noise reduction also showthat the 2-D algorithm has comparable performance to that of the conventional 2-D LMS algorithm when a is properly chosen. Finally, the ELMS algorithms in various forms are detailed. Simulations show that the ELMS algorithm is very sensitive to step size, and the time instances that the extrapolation is made. To avoid wide and/or incorrect extrapolation, pre- cautious scheme is included, which improves the extrapolation accuracy noticeably.