內插信號處理: 回音消除, 預先編碼, 與渦輪等化

Abstract

在有線通訊中，回音消除、預先編碼、與等化是常用的信號處理運算。對於如數位用戶迴路(DSL)等之高速通訊系統，通道響應通常是非常的長，因此前述的信號處理計算複雜度需求也是相當的高。在本論文中，我們提出了一種基於內插濾波的高效率演算法以降低複雜度運算量。內插濾波是由一個有限脈衝響應(FIR)濾波器、一個內插有限脈衝響應濾波(IFIR)與一種將述兩個濾波器係數重疊的方法串聯實作而成。內插濾波方法可有效的降低計算複雜度同時保有傳統有限脈衝響應濾波器的數值穩定性優點與性能。 基於所提出的內插濾波器架構，我們將其廣泛運用於回音消除、決策回授等化 (DFE)、與Tomlinson-Harashima預先編碼(THP)等信號處理功能。在論文中，我們對所提濾波器架構做了完整的理論分析，相關的理論公式如最佳解與輸出訊號雜訊比(SNR)等也提供了詳細的推導。爲了適應通道變化與降低實作複雜度，我們使用最小平均平方(LMS)法作為適應性演算法。對於所提之適應性演算法，其收斂行為與理論效能，我們也做了完整的理論分析與驗證。論文所提出的適應性內插演算法，可以較低的計算複雜度達到傳統演算法相同的效能。對內插回音消除而言，模擬結果顯示在各種單迴路高速用戶迴路(SHDSL)拓撲下，可消除73.0分貝(dB)的回音同時可節省57%的複雜度。對內插決策回授等化、與內插預先編碼，複雜度節省則可高達76%。 渦輪等化是一種結合等化與錯誤更正解碼的重複等化方法，其效能遠超越傳統的分離式等化與錯誤更正解碼方法。然而，前者的複雜度卻遠超過後者。以往的研究主要集中在無線方面的運用。因無線通道的響應長度是屬於較短或稀疏的，以格子(trellis)為基礎的演算法如軟輸出Viterbi演算法或BCJR演算法可有效的運用於無線系統。很不幸地，此類演算法的複雜度與通道的響應長度成指數成長。針對此一問題，Tüchler等人於西元2002年提出了以濾波器為架構的低複雜度渦輪等化器。雖然複雜度可以大幅度的降低，但當通道的響應長度很長時，複雜度還是相當的高。由於數位用戶迴路的通道的響應長度通常長達數百，到目前為止，尚無可行的渦輪等化器可供使用。 基於內插濾波的概念，我們提出一個可應用於長通道的快速渦輪等化器。該快速等化器可將複雜度降低十倍以上。在Tüchler的渦輪等化器中，最佳濾波器係數運算與等化濾波運算的計算量相當的大。我們以理論證明，不同的最佳濾波器是可以被內插的。我們僅需事先計算好少數的最佳濾波器係數，之後便可以內插的方式來快速計算出最佳濾波器係數。因此，複雜度便可大幅度的降低。如果最佳濾波器響應或通道響應本身也是可以被內插的，則我們可運用IFIR方式再次減低複雜度。就我們所知，我們所提出的快速渦輪等化器是目前唯一可應用於SHDSL系統也是世界上複雜度最低的渦輪等化器。在數位位元錯誤率(BER)得等於10-5時，我們可用四次重複等化方式得到8.8分貝的效能改進，而複雜度僅需Tüchler渦輪等化器的3.7%。就整體複雜度而言，所提出的快速渦輪等化器大約為傳統分離式等化與錯誤更正解碼方法的三倍，但這已代表所提的方法已可達到實際應用的目標。

Echo cancellation, precoding, and equalization are common signal processing operations performed in wireline communications. For high-speed systems such as digital subscriber line (DSL), the channel response is usually very long and the computational complexity requirement for those operations is very high. In this thesis, efficient algorithms based on interpolated filtering are developed to solve the problem. The idea of interpolated filtering is realized with a cascade of an FIR and an interpolated FIR (IFIR) filter, and with a tap-weight overlapping method. The interpolated filtering scheme can effectively reduce the computational complexity while inherits all the numerical stability advantages of the conventional FIR filter. The interpolated filtering framework is then applied to echo cancellation, decision feedback equalization (DFE), and Tomlinson-Harashima precoding (THP). Performance is theoretically analyzed and close-form solutions such as optimum solutions and output signal to noise ratio (SNR) are derived also. For accommodating the channel variation and reducing implementation complexity, adaptive algorithms based on the least-mean-squared (LMS) algorithm are then considered. Convergence behavior is analyzed and the performance is theoretically evaluated. While proposed adaptive interpolated algorithms can achieve similar performance as conventional algorithms, the computational complexity is much lower. For echo cancellation, simulations with a wide variety of loop topologies show that the interpolated echo canceller can effectively cancel the echo up to 73.0 dB and achieve 57% complexity saving (in SHDSL applications). For DFE and THP, the computational saving can be as high as 76%. It is well known that a turbo equalizer, a joint iterative equalization and decoding scheme, can significantly outperform a conventional receiver performing equalization and decoding separately. However, the complexity is much higher than the conventional receiver. Previous works mainly focus on the wireless applications in which the channel length is short or sparse. This enables the use of the trellis-based algorithms such as soft-output Viterbi algorithm (SOVA) and BCJR. Unfortunately, the computational complexity of these algorithms grows exponentially with the channel length. In 2002, Tüchler et al. proposed a low-complexity filter-based turbo equalizer reducing the complexity dramatically. Even so, the computational complexity remains tremendous if the channel length is long. So far, there is no turbo equalizer with reasonable complexity designed for a channel with hundreds of taps, which is common in DSL applications. With the interpolated filtering approach, a fast turbo equalizer with complexity an order of magnitude less is proposed. The most computationally intensive operations in Tüchler’s equalizer are the calculation of optimal filter coefficients and the filtering operation of the equalizer. The relationship between optimal filter coefficients and reliability information is exploited and a fast algorithm, which calculates the current optimal coefficients by interpolating two pre-calculated known optimal coefficients, is proposed. If the channel response has a smooth shape, the interpolated equalizer can be applied to reduce the complexity further. Closed-form expressions for interpolated optimal filter coefficients are also derived. To the best of our knowledge, the computational complexity of the proposed turbo equalizer (for SHDSL application) is the lowest in the world. The performance gain at BER 10-5 is about 8.8 dB (with four iterations) and the complexity is only 3.7% of the conventional turbo equalization. Also, the complexity is less than three times of the conventional un-turbo equalizer scheme. This indicates that the complexity of the proposed turbo equalizer is lower enough such that real-world implementation becomes feasible.