國 立 交 通 大 學
電信工程學系
博 士 論 文
內插信號處理: 回音消除, 預先編碼, 與渦輪等化
Interpolated Signal Processing: Echo Cancellation,
Precoding, and Turbo Equalization
研 究 生 ：林 壽 煦
指導教授 ：吳 文 榕
Precoding, and Turbo Equalization
研 究 生: 林 壽 煦
Student: Shou-Sheu Lin
指導教授: 吳 文 榕 博士
Advisor: Dr. Wen-Rong Wu
國 立 交 通 大 學
電信工程學系
博 士 論 文
A Dissertation
Submitted to Department of Communication Engineering College of Electrical Engineering and Computer Science
National Chiao Tung University in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
in
Communication Engineering June 2005
Hsinchu, Taiwan, Republic of China
內插信號處理: 回音消除, 預先編碼, 與渦輪等化
研究生: 林壽煦 指導教授: 吳文榕
國立交通大學電信工程學系
摘要
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 i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。 ii
Interpolated Signal Processing: Echo Cancellation,
Precoding, and Turbo Equalization
Student: Shou-Sheu Lin
Advisor: Dr. Wen-Rong Wu
Department of Communication Engineering
National Chiao Tung University
Abstract
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%.
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.
Acknowledgements
First of all, I would like to thank my advisor, Professor Wen-Rong Wu for his extensive guid-ance, valuable suggestions, and constructive discussions to make this dissertation possible. I really appreciate his time and efforts in improving my papers, thesis and writing skills. I did learn a lot from him not only in the academic works but also his attitude to everything. I consider myself to be truly fortunate to have had the opportunity to work with such a knowledgeable and friendly person.
I am extremely grateful to all the members of my PhD dissertation committee for their thoughtful comments and valuable suggestions. With their comments, the content of disserta-tion is further improved and more direcdisserta-tion about future works is inspired.
I wish to thank Jeff Lee, Hua-Lung Yang and Yinman Lee for their discussion and help. I am extremely grateful to Chao-Yuan Hsu for his extensive support. His in-depth discussion and help saves me much times for documentation preparation and presentation.
I would like to special thanks to my elder brother Jun-Fu Lin for his inspiration to undertake the Ph.D. studies. Without his motivation, I will never think about pursuing a doctoral degree after graduate.
I am deeply indebted to my beloved wife, Sun-Ting Lin, for her support and understanding. I would like to thank my lovely daughter, Rui-Yu Lin, for her understanding. I express my deep gratitude to my mother-in-law, Fu-Hei Chen, and aunt, Siu-Er Dai, and her family for taking care of my daughter. I wish to thank to my grandma, Moon Hsu, for taking care of my daily
his grandson. There are so many peoples, I should thank. Finally, I would like to thank to all relatives and friends who ever encourage or help me.
Contents
Abstract iii
Acknowledgements v
Contents vii
List of Tables xi
List of Figures xiii
Glossary xvii
1 Introduction 1
2 Digital Subscriber Loop Environment 5
3 Interpolated Echo Canceller 11
3.1 The IFIR Echo Canceller . . . 14
3.2 Theoretical Analysis . . . 18
3.3 Optimal Interpolation Filter Design . . . 21
3.4 Simulation Results . . . 26
3.4.1 Loop Characteristics and Topologies . . . 26
3.4.2 The Cutting Point and the Interpolation Factor . . . 29 vii
3.4.4 Discussions . . . 33
3.5 Conclusions . . . 38
4 Interpolated Decision Feedback Equalizer and Precoder 39 4.1 System Model and Conventional DFE . . . 42
4.2 The Proposed Interpolated DFE . . . 45
4.2.1 Interpolated DFE . . . 45
4.2.2 Adaptive IDFE . . . 51
A) Convergence in the Mean . . . 51
B) Convergence in the MSE . . . 52
4.3 Simulation Results . . . 55
4.3.1 Channel CSA No. 3 . . . 55
4.3.2 Different Loop Topologies . . . 56
4.3.3 The Adaptive IDFE . . . 57
4.3.4 Symbol Error Rate vs. SNR (for DFE and THP) . . . 58
4.3.5 Discussions . . . 59
4.4 Conclusions . . . 61
5 Fast Interpolated Turbo Equalizer 63 5.1 System Model and Notations . . . 67
5.2 Summary of Previous Works . . . 70
5.2.1 The BCJR Equalizer/Decoder . . . 70
A) Trellis Diagram Representation of an ISI Channel . . . 70
B) Trellis Diagram Representation of a Convolutional Code . . . 71
5.2.2 The BCJR Equalizer . . . 73
5.2.3 The BCJR Decoder . . . 75
CONTENTS ix
5.2.5 Filter-based MMSE SISO Equalizer . . . 80
A) OSL Equalizer . . . 81
B) LSL Equalizer . . . 88
C) LSLN, LSLP, and LSLH Equalizers . . . 90
5.3 The Proposed Fast Interpolated Turbo Equalizer . . . 94
5.3.1 FSISL Equalizer . . . 94
5.3.2 FDISL Equalizer . . . 102
5.3.3 Complexity Analysis . . . 106
5.4 Simulation Results . . . 109
5.4.1 Parameters of Reference Filters . . . 112
5.4.2 Parameters for IR Interpolation . . . 113
5.4.3 BER Simulations . . . 115
A) Low-ISI Proakis A Channel . . . 115
B) Severe-ISI Proakis C Channel . . . 116
C) Long ISI CSA #6 Channel . . . 117
5.4.4 Discussions . . . 121
5.5 Conclusions . . . 122
6 Conclusions and Future Works 125 A Derivation of the Correlation Matrices for DFE 129 B Derivation of the Correlation Matrices for IDFE 133 C Derivation of the BCJR Algorithm for Optimal Equalization 139 C.1 The BCJR Algorithm (Probabilistic Form) . . . 144
C.2 The BCJR Algorithm (Logarithmic Form or Log-MAP) . . . 147
D.1 2-PAM (BPSK) . . . 151 D.2 4-PAM . . . 152
E Complexity Analysis 155
List of Tables
3.1 Computational complexity comparison for the FIR and IFIR echo cancellers . . 18
3.2 ERLE performance vs. various interpolation filters (LIN: linear, TRS: truncated sinc, HWS: Hanning-windowed sinc, CWS: Chebyshev-windowed sinc, OPS: optimal-symmetric, OPT: optimal) . . . 25
3.3 Computational complexity ratio for different interpolation factor (N1 = 50, S=2) 31 3.4 Echo return loss and path loss for CSA loops . . . 34
3.5 SNR with or without echo interference for CSA #1 at CPE side . . . 37
4.1 Computational complexity analysis . . . 53
5.1 Soft-input and soft-output conversion formulae for M-PAM . . . 86
5.2 Complexity of SISO equalizers without SISO conversion . . . 107
5.3 Channel models for performance evaluation . . . 111
D.1 Parameters for calculating the soft-input (4-PAM) . . . 153
D.2 Parameters for calculating the extrinsic information (4-PAM) . . . 154
E.1 Complexity of FDISL equalizer . . . 156
E.2 Complexity of LSL equalizer . . . 157
E.3 Complexity of BCJR equalizer . . . 158
E.4 Complexity of BCJR decoder . . . 159
E.5 Complexity summary of the turbo equalizers . . . 159 xi
List of Figures
2.1 The network architecture of digital subscriber loop - the last mile. . . 6
2.2 A typical DSL loop with different size of twisted-pair wires and bridge taps. . . 7
2.3 Full-duplexing transmission of a DSL channel. . . 8
2.4 The echo caused by the impedance mismatch of a hybrid circuit. . . 9
3.1 A typical echo response. . . 12
3.2 The adaptive FIR echo canceller. . . 12
3.3 The adaptive IFIR echo canceller. . . 15
3.4 The filter responses of the IFIR echo canceller. . . 17
3.5 Illustration of finding optimal interpolation filter. . . 22
3.6 Interpolation filters (M=4, S=2). . . 26
3.7 SHDSL echo responses for CSA loops at CO side. . . 27
3.8 SHDSL echo responses for CSA loops at CPE side. . . 28
3.9 Overlapping of FIR and IFIR filters (M=4, S=2, N1=50). . . 29
3.10 ERLE performance for different CSA loops. . . 30
3.11 ERLE versus the cutting point and the interpolation factor. . . 32
3.12 Simulated ERLE performance at CO side. . . 33
3.13 Simulated ERLE performance at CPE side. . . 34
3.14 Total echo return loss at CPE side. . . 35
3.15 The power spectral densities of echo, residual echo, and composite noises. . . . 36 xiii
4.1 Typical channel impulse response in digital subscriber loop application. . . 41 4.2 Conventional adaptive DFE structure. . . 43 4.3 (a) Optimal feedback filter response of conventional DFE. (b) Interpolated
feed-back filter response of proposed IDFE. . . 47 4.4 Proposed low-complexity adaptive IDFE structure. . . 48 4.5 Complexity ratio with respect to a conventional DFE,(Nf, Nb)=(16,180). The
minimum complexity ratio (C.R.) is 24% when the interpolation factor equals 8. 54 4.6 (a) Optimal feedforward filter coefficients for IDFE and DFE. (b) Optimal
feed-backward filter coefficients for IDFE and DFE. . . 56 4.7 (a) Performance of DFE and IDFE for eight CSA loops at received SNR=40 dB.
(b) Performance of DFE and IDFE for eight CSA loops at received SNR=20 dB. 57 4.8 Learning curves of adaptive IDFE and DFE with step size 0.000638.
Theo-retical steady-state MSE (J∞), MMSE (Jmin), and misadjustment (ψ) are also
shown. . . 59 4.9 Low-complexity interpolated THP (associated with adaptive IDFE) . . . 60 4.10 Symbol error rate comparison for adaptive IDFE plus ITHP and DFE plus THP
(without error propagation), and for adaptive IDFE and DFE (with error propa-gation). . . 61
5.1 The system model of turbo equalization for coded data transmitted over ISI channel. . . 68 5.2 A discrete-equivalent ISI channel with memory length µ . . . 71 5.3 The state transition diagram for the ISI channel shown in Fig. 5.2, where Q =
Mµis total numbers of states for the channel. . . 72
LIST OF FIGURES xv 5.5 The state transition diagram of the convolutional encoder shown in Fig. 5.4,
where P = 2η is total numbers of states for the code. . . 73
5.6 A generic SISO processing unit. . . 77
5.7 The block diagram of the turbo equalizer. . . 77
5.8 A filter-based MMSE SISO linear equalizer. . . 81
5.9 The optimal block time-invariant filter vs. the iteration and block number. . . . 95
5.10 The average reliability function vs. the iteration and block numbers. . . 96
5.11 Optimal block time-invariant filter vs. the reliability function (Channel: Proakis B channel with 4-PAM at SNR=18.0 dB). . . 97
5.12 Optimal block time-invariant filter vs. the reliability function (Channel: Proakis B channel with 4-PAM at SNR=18.0 dB). . . 98
5.13 The detailed filter structure of the LSL. . . 103
5.14 The reference optimal filters for the SHDSL CSA #6 channel at SNR=18.0 dB. 105 5.15 The IR complexity ratio vs. interpolation factor for the SHDSL CSA #6 channel. 109 5.16 The computational complexity of proposed fast interpolated SISO linear equal-izers (FSISL and FDISL) and LSL equalizer. . . 110
5.17 The complexity ratio of proposed turbo equalizers (block length: 512 symbols). 111 5.18 The channel response of CSA #6 channel for SHDSL application. . . 112
5.19 The zoom-in view of the reference optimal filters (Z = 15, λ=8, SHDSL CSA #6 channel, and SNR=18.0 dB). . . 114
5.20 The NIE performance of reference optimal filters in Fig. 5.18, where the worst one is -39.7 dB and the average is -44.3 dB. . . 115
5.21 The IR interpolation for the CSA #6 channel response and the reference optimal filters in Fig. 5.18. . . 116
5.22 The interpolation filter used for IR interpolation in Fig. 5.20. . . 117
5.23 BER comparison for the proposed FSISL (dashed-x) and LSL (solid) turbo equalizers over Proakis A channel . . . 118
equalizers over Proakis C channel. . . 119 5.25 BER comparison for the proposed FSISL (dashed-x), FDISL (dotted-point), and
LSL (solid) turbo equalizers over SHDSL CSA #6 channel . . . 120 5.26 BER comparison for the proposed FSISL (dashed-x), FDISL (dotted-point), and
Glossary
ADSL Asymmetric Digital Subscriber Line
ADSL2 Asymmetric Digital Subscriber Line - Second Generation ADSL2+ Extended Bandwidth ADSL2
ANSI American National Standards Institute
APP A Posteriori Probability
ARMA Autoregressive and Moving Average AWGN Additive White Gaussian Noise
BCJR The Optimal MAP Algorithm by L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv
BICM Bit-Interleaved Coded Modulation
BER Bit-Error Rate
BPSK Binary Phase Shift Keying
CDMA Code-Division Multiple Access CO Central Office
CPE Customer Premise Equipment
DFE Decision Feedback Equalizer
DMT Discrete Multi-Tone
DS Downstream
DSL Digital Subscriber Line
ERL Echo Return Loss
ERLE Echo Return Loss Enhancement
ETSI European Telecommunications Standards Institute
EXIT EXtrinsic Information Transfer
FBF Feedback Filter
FDD Frequency Division Duplex
FFF Feedforward Filter
FFT Fast Fourier Transform
FIR Finite Impulse Response
FS Fractionally-Spaced
FSISL Fast Singly Interpolated SISO Linear
FDISL Fast Doubly Interpolated SISO Linear
GSM Global System for Mobile Communications
xix
HDSL2 High Bit Rate Digital Subscriber Line - 2nd Generation IC Interference Cancellation
IDFE Interpolated Decision-Feedback Equalizer
IFIR Interpolated Finite Impulse Response
i.i.d. Independent and Identically Distributed
IIR Infinite Impulse Response
IR Individual Response
ISDN Integrated Service Digital Network
ITHP Interpolated Tomlinson-Harashima precoder
ITU International Telecom Union
ISI Intersymbol Interference
LDPC Low-Density Parity-Check Codes
LLR Log-Likelihood Ratio
LMS Least Mean Square
LS Least-Squares
LSL Low-complexity SISO Linear
LSLH Low-complexity SISO Linear, Hybrid
LSLN Low-complexity SISO Linear, No a priori
MFB Matched-Filter Bound
ML Maximum Likelihood
MMSE Minimum Mean Square Error MSE Mean Square Error
NIE Normalized Interpolated Error
NSC Non-Systematic Convolutional Code
OFDM Orthogonal Frequency Division Multiplexing
OSL Optimal SISO Linear
PAM Pulse Amplitude Modulation
pdf Probability Density Function
PL Path Loss
PMP Pontryagin’s Maximum Principle
POTS Plain Old Telephone Service
PSD Power Spectral Density
PSK Phase Shift Keying
PSTN Public Switched Telephone Network
QAM Quadrature Amplitude Modulation
xxi
SC-FDE Single-Carrier Frequency-Domain Equalization
SDSL Symmetrical Single Pair High Bitrate Digital Subscriber Line
SHDSL Single-Pair High-Speed Digital Subscriber Line SISO Soft-Input Soft-Output
SNR Signal to Noise Ratio
SOVA Soft Output Viterbi Algorithm
T1 ANSI 1.544 Mbps time-division multiplex service
TDD Time Division Duplex
telco Telecom Company
TERL Total Echo Return Loss
THP Tomlinson-Harashima Precoder
US Upstream
VA Viterbi Algorithm
VDSL Very-High-Bit-Rate Digital Subscriber Line
VDSL2 Very-High-Bit-Rate Digital Subscriber Line - Second Generation WR Whole Response
Chapter 1
Introduction
S
Ofar, the twisted-pair copper wire is known to be the most popular broadband access mediain the world. Several digital subscriber line (DSL) technologies have been proposed and successfully commercialized. With these technologies, the broadband internet access era has been becoming reality. While the DSL transmission rate is greatly enhanced, it is still far from Shannon’s limit. Sophisticated communication systems and advanced signal processing schemes are still continuously developed.
In the DSL environment, there exists a couple of major impairments that impose challenges for receiver signal processing. This includes echo cancellation, channel equalization, and pre-coding. To achieve full duplex transmission over a single pair of wire, a hybrid circuit is employed in the transmitter. However, echoes are produced due to the impedance mismatch problem. The echo will significantly interfere the receive signal. In many cases, the echo inter-ference is so strong and the receiver becomes impossible to work properly. A common remedy to this problem is to use an echo canceller. The echo channel response is usually very long and this puts a computational complexity burden for the receiver. Originally, the twisted-pair channel is used for voice transmission and the bandwidth is narrow. It will introduce a severe intersymbol interference (ISI) for high speed data transmission. In DSL, it is not uncommon to observe a channel with hundreds taps. Channel equalization and precoding are the techniques
devices are very high. Though above techniques are suggested in standards and popular in the off-the-shelf DSL related products, the computational complexity due to the long channel response remain an issue, specially when the transmission rate is further increased.
In this dissertation, we will focus on the complexity reduction issue in echo cancellation, channel equalization, and precoding. We explore the DSL channel characteristics and propose effective signal processing schemes. Our goal is to develop algorithms with implementable complexity for the next generation DSL applications. There are two distinct properties in the DSL environment. The first one is that the channel and echo responses are almost static. This means that many signal processing structures and parameters can remain the same for a long period of time and this is a great design advantage. The other property is that the channel and echo responses exhibits low-pass characteristics. This indicates that even though the channel or echo response is long, we can use a low-order system to model it. Exploiting these two proper-ties, we develop our low-complexity algorithms through a general framework of interpolation. To guarantee the stability, we use the finite impulse response (FIR) structure for the interpolated filters. With our interpolated signal processing schemes, the computational complexity can be significantly lower than the conventional approaches.
This dissertation is organized as follows. The DSL environments and impairments are briefed in Chapter 2. In Chapter 3, we investigate the echo cancellation problem and pro-pose a low-complexity adaptive interpolated FIR (IFIR) echo canceller. Theoretical Wiener solution, minimum mean square error (MMSE) and convergence behaviors are analyzed in de-tail. We also propose a least-squares method to design the optimal interpolation filter. These results were verified with extensive simulations with versatile loop topologies and noise envi-ronments. Base on the similar IFIR filtering structure, we then extend the interpolation scheme to channel equalization and precoding. This is described in Chapter 4 in which we propose a low-complexity adaptive interpolated decision feedback equalizer (DFE) and a low-complexity interpolated Tomlinson-Harashima precoder. Theoretical Wiener solutions, minimum mean
3 square errors (MMSEs) and convergence behaviors are also analyzed in details. In Chapter 5, we study the turbo equalization algorithm. In this chapter, we propose fast interpolated turbo equalizers with complexity an order of magnitude less than the conventional approach. An ac-tual DSL channel is used for simulations and the results show that the performance is similar to the conventional approach. Finally, we draw our conclusions in Chapter 6.
Chapter 2
Digital Subscriber Loop Environment
The invention of telephone is a remarkable milestone in the contemporary history of human beings. With the telephone network, people can communicate each other without any distance limitation. Lately, the modem was introduced and digital data can be transmitted/received over the network. With digital processing technology, voice, text, music, picture, and video can all be converted into their digital forms. In addition to voice, today’s telephone network extends its application to multimedia communication.
As shown in Fig. 2.1, the telephone network consists of a core network and an access net-work. The core network, shown on the left-side of the figure, relays information from the central office (CO) and then conducts it to the different service networks such as circuit-switched pub-lic switched telephone network (PSTN) for voice communication or packet-routed data network for Internet access. The access network, shown on the right-hand side, transports information at the customer premise (CPE) side to the CO side. The distance between the CPE and the CO is usually around one mile. Thus, we also call it as the last mile. For the PSTN, any user within the telephone network needs a dedicated physical link. The media cost becomes an important issue when the access network is built. For low-cost consideration, the twisted-pair wire was chosen as the transmission media for the last mile.
The telephone network was initially designed for the voice communication only, i.e., the so-5
PSTN
Core network Access network
PSTN
Core network Access network
Figure 2.1: The network architecture of digital subscriber loop - the last mile.
called plain old telephone service (POTS). The voice signal is transmitted directly on a single pair of twisted-pair wire in its analog form. The bandwidth requirement for POTS is set as 4 KHz. The signal transmission between two end users during a phone call can be described as follows. A 4 KHz analog voice signal is first transmitted from a CPE to a CO. Then, it is sampled and digitized at 8 KHz rate before entering a class 5 switch (a voice switch) [1]. After the class 5 switch, the voice signal is transported and switched digitally within the core network of the PSTN and finally converted back to the original analog waveform in a remote class 5 switch near the called user. Due to the sampling operation, the class 5 switch limits the bandwidth utilization of a twisted-pair wire. As we can see, the signal transmission path can be divided into three sections. They are a twisted-pair wire section from the calling user to the first class 5 switch, a digital link section between the first and the last class 5 switch, and finally a twisted-pair wire section from the last class 5 switch to the called user. Note that the bandwidth of the twisted-pair wire is wider than 4 kHz and is capable of carrying more information. The digital section is the throughput/bandwidth bottleneck in a telephone network.
Though the original purpose of the twisted-pair wire is for voice communication, it’s possi-ble to extend its use for other types of application. For example, fax uses the digital modulation technology such as quadrature-amplitude-modulation (QAM) to transmit digital data. However,
7 due to the sampling constraint in the class 5 switch, the bandwidth utilization of a twisted-pair wire for digital communication is still limited to 4 KHz. For higher speed communication re-quirement such as Internet access, the bandwidth is not sufficient. To solve the problem, the class 5 switch is bypassed and transmission characteristic of the twisted-pair wire above 4 KHz was exploited. This results in the development of new systems such as xDSLs (ISDN, HDSL, SHDSL, ADSL and VDSL) pumping more throughput with wider bandwidth utilization. Note that a new type of switch is required at CO side to handle the new applications. Using this approach, the bottleneck in the digital section is resolved.
5800, 26 150, 24 1200, 26 1200, 26 300, 24 300, 26 feet, AWG CO side CPE side
Figure 2.2: A typical DSL loop with different size of twisted-pair wires and bridge taps.
A DSL loop may consist of twisted-wire pairs with different sizes. A typical DSL loop is shown in Fig. 2.2, in which each segment of the twisted-pair wire is labeled with a length in feet and a diameter in gauge number. A smaller gauge number stands for a wire with larger diameter. Since the telephone company built the network before user subscription, it doesn’t know who will connect to a particular DSL loop in advance. The bridge taps provide access points for potential users in that service area. Whenever the wire gauge is changed or a bridge tap is placed, impedance mismatch will occur and this will result in signal reflection. When the signal is transmitted from one side to the other, it will be self-interfered with the reflection. The destructive reflection may cause spectral nulls at some frequencies. For a given segment of a twisted-pair wire, it is well known that the transfer function of twisted-pair wire is non-flat and the attenuation is proportional to the square root of frequency [1]. Thus, the overall transfer function of a typical DSL loop in Fig. 2.2 is frequency selective. The corresponding impulse
rectangular pulse transmitted from the CO side may result in a wide-spread distorted pulse at the CPE side. Thus, an effective equalizer is needed to compensate the ISI effect at receive side.
Transmit Transmit Receive Receive Hybrid Echo Echo Signal Tx filter Rx filter Hybrid Rx filter Tx filter Channel
Figure 2.3: Full-duplexing transmission of a DSL channel.
In the telephone network, a special design was used to achieve full-duplex transmission over a single twisted pair of wire. Fig. 2.3 shows the architecture. In the figure, we can see a device called hybrid performing the duplexing operation. A simplified hybrid circuit [1] is shown in Fig. 2.4. The circuitry was developed according to the Wheatstone bridge principle. In fact, a hybrid circuit is an impedance bridge consisting of two voltage dividers. When the bridge is balanced, i.e., R1 Zi = R2 Zb , (2.1)
where Zi defined as the impedance measured at the primary side of the coupling transformer.
Thus, a voltage applied at transmit terminals causes zero voltage difference between receive terminals. In the hybrid circuit, a coupling transformer shown on the bottom-left side is usually required to couple the signal between the transceiver and the twisted-pair wire and block the undesired DC signal (due to the unequal ground level between the CO and CPE sides). The
9 loop impedance represents the impedance of the cascaded twisted-pair wire and the loading in the receiver side. Obviously, Zi will depends on the loop to which the hybrid connects. The
balance condition in (3.1) is then difficult to hold accurately. Thus, the transmit signal will partially return to the receive terminals and this is called echo. The echo signal will interfere the receive signal seriously. Thus, a short rectangular pulse transmitted from CO side will return a dispersed received pulse (echo) at the same side. An echo canceller is the commonly device used to eliminate the unwanted echo signal. Except for ISI and echo, there are other kinds of impairments in a telephone network such as crosstalk noise, impulsive noise, radio frequency interference (RFI) noise, and thermal noise.
Loop Impedance Zi 1 R R2 b
Z
Transmit Tx Rx (Echo)Figure 2.4: The echo caused by the impedance mismatch of a hybrid circuit.
A transceiver must be designed to cope with the impairments mentioned above so that in-formation can be transmitted and received properly. A transceiver usually consists of an analog front-end and a digital processing unit. The functionality of a typical signal processing unit (for receiver) includes error-correcting, echo cancellation, equalization, and detection. The analog front-end is used to transform the digital signal into proper analog waveform for signal trans-mission, and the operation is reversed for signal reception. The analog front-end includes a transmit filter, a receive filter, an analog to digital signal conversion circuit (ADC), and digital to analog signal conversion circuit (DAC). The transmit filter is designed to shaping the trans-mit pulse in time domain for transmission or the power spectral density of the transtrans-mit signal
anti-aliasing and filtering noises outside the signal band. ADC and DAC perform the analog and digital signal conversion, respectively.
To increase the throughput, a wider transmission bandwidth is required and a higher clock rate for a transceiver is necessary, too. Like the proverb said - “No pains, no gains”. A higher clock rate will result in longer (more severe) echo and ISI responses. For low speed transmission such as analog modem and ISDN, the response length of these impairments is not too long. The computational complexity for the echo canceller and the equalizer is moderate. For high speed transmission such as ADSL, SHDSL, and next generation xDSL, however, the response length will be much longer. The computational complexity will become the main obstacle in the transceiver design.
Chapter 3
Interpolated Echo Canceller
In a DSL environment, full duplex transmission via a single twisted pair can be achieved using a hybrid circuit. Due to the impedance mismatch problem, the hybrid circuit will introduce echoes. In an analog telephone, a hybrid circuit with 10-20 dB echo return loss is good enough for voice conversation. For high-speed digital transmission, the required echo return loss, how-ever, is much higher (50-70 dB). In versatile subscriber loop environments, it is difficult to design a hybrid circuit achieving a high echo return loss, even when using an adaptive hybrid circuit. Thus, an additional adaptive digital echo canceller [2, 3] is required.
A subscriber loop is composed of telephone wires with different gauges and lengthes [1], and has many possible topologies. A typical echo response, shown in Fig. 3.1, usually consists of a short and rapidly changing head echo, and a long and slowly decaying tail echo. Since the subscriber loop between a central office and a customer premise is fixed, its characteristics will not vary with time (except for temperature variation which is very slow). As a result, the echo response is usually considered as time-invariant. Conventionally, an adaptive transversal FIR filter, shown in Fig. 3.2, is used to synthesize and cancel the echo. For a lower speed appli-cation such as ISDN, the echo canceller has about 50 taps, and the computational complexity of the transversal filter structure is moderate. However, in higher speed applications such as HDSL [4–6], HDSL2 [7], and SHDSL [8–10], the echo response is usually much longer. The
transversal FIR filter approach becomes very high. In order to reduce the computational com-h h 1 α− 0 Nh−1 t h
Figure 3.1: A typical echo response.
Noise k x k n k y h dk Σ Σ Echo path Echo canceller + − 1 w k e k y
Figure 3.2: The adaptive FIR echo canceller.
plexity, some researchers tried to use an adaptive IIR filter to cancel the tail echo. However, the adaptive IIR filtering suffers from the local minima and stability problems. Since an IIR filter usually consists of a feedforward and a feedback filter, a compromising approach is to let the feedforward filter be adaptive only. In [11], August et al. collected some echo responses for the European subscriber loops, and used a criterion to determine the feedback filter optimally. In [12], Gordon et al. considered echo cancellation as a series expansion problem. They used a
13 set of IIR orthonormal functions to expand the echo response and let the expanding coefficients be adaptive. The orthonormal responses were obtained using a set of predetermined cascaded feedback filters. If only a small number of loops are considered, good performance can be obtained using these methods. However, the existing loop responses are versatile, it will be difficult to find a feedback filter that will always yields the optimal performance.
To retain the FIR structure of the echo canceller, and to reduce the complexity, an interesting echo canceller structure was proposed in [13, 14]. The canceller is cascaded of an adaptive FIR head echo canceller and an adaptive interpolated FIR (IFIR) tail echo canceller. Since the tail echo always decays smoothly, an IFIR filter with a small number of coefficients can effectively cancel the echo. Unfortunately, the IFIR filter proposed in [13, 14] has an uncontrollable tran-sient response, and the direct cascade of an FIR and an IFIR filter will leave a certain period of the echo response uncancelled. Although this problem is critical, it was overlooked in [14]. Recently, a new interpolated FIR echo canceller was proposed [15] to solve the problem. In this work, the FIR and the IFIR filters are overlapped instead of being directly cascaded. Due to this overlapping operation, some echo responses will be simultaneously cancelled by the FIR and IFIR filters. Although this will not affect the final performance, it will slow down the conver-gence. In order to solve the problem, some of tap-weights used in the FIR filter coefficients are nulled. This results in a high performance yet low-complexity echo canceller.
An IFIR filter consists of an interpolation filter and an upsampled FIR filter. It is known that the interpolation filter for an IFIR filter has great impact on the interpolated result. Many interpolation filters have been proposed in [16]. These filters are general in the sense that they are independent to the echoes being interpolated. Since they are general, they cannot yield best performance for all types of echoes. If we know the characteristics of the signal which we will interpolate, we can design a better interpolation filter. The purpose of this thesis is to enhance the performance of the IFIR echo canceller in [15] via the interpolation filter design. We propose a least-squares method to obtain the optimal interpolation filter for a single or multiple DSL loops. Simulations show that the optimal interpolation filter can have much better performance
This chapter is organized as follows. In Section 3.1, we describe the IFIR echo canceller structure in [15] and review its properties. In Section 3.3, we discuss the proposed least-squares interpolation filter design. In Section 3.4, we show the simulation results and discuss the com-plexity reduction issues. Finally, we draw our conclusions in Section 3.5.
§ 3.1 The IFIR Echo Canceller
Let the length of an echo response be Nh and its response be h = [h0 h1 ... hNh−1]
T
. The received echo signal can be described as follows:
yk = hTxk+ nk (3.1)
where xk = [xk xk−1 ... xk−Nh+1]
T is the transmitted signal, n
k is a zero-mean additive noise
which may be the additive white Gaussian noise (AWGN), the near-end crosstalk (NEXT) noise, or both combined.
From Fig. 3.1, we can clearly see that a typical echo response has a fast changing head echo and a slowly varying tail echo. Thus, we can select a cutting point to segment these two portions. Let hh = [h0 ... hα−1]T, xh = [xk ... xk−α+1]T, ht = [hα ... hNh−1]
T, and x t =
[xk−α... xk−Nh+1]
T. Then, the echo response can be re-expressed as:
yk = hThxh+ hTtxt+ nk (3.2)
In absence of noise, yk can be synthesized and cancelled by an Nh-tap FIR filter. This filter,
having h as its response, can be decomposed to an α-tap and an (Nh − α)-tap FIR filter; one
cancels the head echo and the other cancels the tail echo. In general, (Nh− α) is much larger
than α. As a result, the tail echo canceller will dominate the overall computational complexity. Since the tail echo is slowly varying, we can use a filter with a lower complexity to approximate ht. The idea is to use an IFIR filter, which is an interpolation filter cascaded by a filter with an
3.1. THE IFIR ECHO CANCELLER 15 3.3. The FIR filter w1 is used to model the head echo response hh and the IFIR filter g ∗ wU2,
where ‘∗’ denotes the convolution operation, is used to model ht. Note that w2Uis an upsampled
version of a filter w2. Basically, w2 tries to model the downsampled version of ht, and g is a
FIR filter that interpolates w2. Let the downsampling factor be M, and the FIR filter and the
IFIR filter be overlapped for No= Ng− M taps. The head echo canceller length is extended to
N1 = α + No instead of just α. As Fig. 3.3 shows, xk is the input to w1 and ˜xk is that to wU2.
The output of the IFIR echo canceller in Fig. 3.3 can be expressed as: ˜
yk = wT1x1,k + wT2˜x2,k (3.3)
where w1 = [w1,0 w1,1 · · · w1,N1−1]
Tis the N
1-tap head echo canceller, x1,k = [xk xk−1 · · ·
xk−N1+1]
T is its input vector, w
2 = [w2,0 w2,1 · · · w2,N2−1]
T is the N
2-tap tail echo canceller,
and ˜x2,k =
˜
xk−αx˜k−α−M · · · ˜xk−α−(N2−1)M
T
is its input vector. In terms of z-transform representation, we have wU 2(z) = w2(z−M). Rewriting (3.3), we have Noise k x k n k y h dk Σ Σ Echo path
Interpolated echo canceller
2 U w + − g 1 w Σ k e k y k x −α xk−α
Figure 3.3: The adaptive IFIR echo canceller.
˜ yk = h wT1 wT2 i x1,k ˜ x2,k = wTx˜k. (3.4)
where S be the number of w2 tap-weights involved in calculating an interpolated value for a
single side span. Then, the interpolator output can be expressed as follows:
˜ xk = Ng−1 X i=0 gixk−i. (3.5)
Generally, the impulse response of the interpolation filter g is peaking at the center, slowly decaying to its two sides and is symmetric around the center. The simplest response of g is a tri-angular window function with 2M-1 taps, which gives a linear interpolation result. The design of g is important to the IFIR filter and will be discussed later. Since the impulse response of IFIR filter is the convolution of g and wU
2, it exhibits two transient responses; each one decaying
to zero (each with Nosamples), one in the front end of g∗wU2 and the other in the tail end. Since
the tail end of htalways decays to zero, there is no problem with that transient response in the
tail. However, the head portion of ht has an abrupt rising edge. As a consequence, the front
end transient response of g ∗ wU
2 cannot model that of ht. A simple way to solve this problem
is to increase the length of w1, and overlap w1 with the front end transient response of g ∗ wU2.
The IFIR echo canceller overlap the last No taps of w1with the first Notaps of g ∗ w2Uto cover
the full front-end transient response. Fig. 3.4 shows how the FIR and IFIR responses are over-lapped. Note that in the structure, there are (S−1) echo samples being cancelled simultaneously by two filters; w1 and g ∗ w2U. As shown in [15], these taps are redundant and they will slow
down the convergence rate of the IFIR echo canceller. An easy and efficient way to overcome this problem is to null w1; we can let the coefficients
w1,N1−(S−1)M · · · w1,N1−2M w1,N1−M
all be zeros. This nulling scheme removes the redundant taps and accelerates the convergence rate. To obtain the tap weights of w1 and w2, an adaptive algorithm is applied. From (3.4), we
can see that the IFIR echo cancellation filter, similar to a conventional FIR filter, has a linear structure. As a result, adaptive algorithms developed for the conventional FIR filter can be di-rectly applied here. For the complexity consideration, the simplest adaptive algorithm, namely
3.1. THE IFIR ECHO CANCELLER 17 2 U w g Overlapped Interpolated Nulled tap t h 1 α− h h α Nh−1 0 2 U ∗ g w 1 w ∗ (Avoid underdetermined problem)
Figure 3.4: The filter responses of the IFIR echo canceller.
the least mean square (LMS), is employed. The LMS algorithm is given by [19]
wk+1 = wk+ µekx˜k (3.6)
where µ is the step size controlling the convergence rate, and ek = yk− ˜ykis the error signal.
The convergence behavior of the adaptive IFIR echo canceller and the upper bound of µ for convergence will be discussed later.
The computational complexity of the adaptive IFIR echo canceller can be easily evaluated. Table 3.1 summarizes the numbers of additions and multiplications required in the echo can-cellation, for an IFIR and a conventional FIR echo canceller. As we can see, the complexity reduction for the IFIR echo canceller comes from the IFIR filter. The computational complexity of w2 is only one M-th of that of the corresponding FIR filter.
Echo emulation Taps weight update
Operation + × + ×
FIR Nh− 1 Nh Nh Nh
IFIR N1+ N2+ Ng− 2 N1+ N2+ Ng N1+ N2 N1+ N2
§ 3.2 Theoretical Analysis
The Wiener solution, minimum mean squared error (MMSE), and error return loss enhancement (ERLE) of the IFIR echo canceller have been derived in [15]. Here, we only give the final results. These formulas will be used for performance evaluation. Let R = E˜xk˜xTk be the
input correlation matrix (without nulling), and p = E [˜xkyk]a cross correlation vector. The
Wiener solution with coefficient nulling is ˆ
wo = ˆR−1ˆp (3.7)
where ˆR and ˆp is the correlation matrix and vector for the nulled filter, respectively. ˆR and ˆ
pare obtained by eliminating the i-th row and i-th column of R, and the i-th row of p, where i ∈ {(N1−(S−1)M), · · · , (N1−2M), (N1−M)}, respectively. By doing so, the corresponding
weights in w1 will be all zeros, i.e., [w1,N1−(S−1)M · · · w1,N1−2M w1,N1−M] = 01×(S−1).
From definition, the correlation matrix without nulling is
R = Rx1x1 Rx1˜x2 RTx1˜x2 R˜x2˜x2 . (3.8)
Assume that the transmitted signal xkis white. The correlation matrix of x1,k is
Rx1x1 = σ
2
xIN1×N1 (3.9)
where σ2
xis the transmitted signal variance. The correlation matrix of ˜x2,k is
Rx˜2˜x2 = σ
2
3.2. THEORETICAL ANALYSIS 19 where M= gT, 0, ..., 0 | {z } (N2−1)M 0, ..., 0 | {z } M , gT, 0, ..., 0 | {z } (N2−2)M ... 0, ..., 0 | {z } (N2−1)M , gT (3.11)
is an N2-by-(Nh−α) interpolation matrix. The cross correlation matrix of x1,k and ˜x2,k is given
by Rx1x˜2 = σ 2 x 0α×No 0α×(Nh−N1) INo×No 0No×(Nh−N1) MT. (3.12)
If we assume that noise nkis independent of the transmitted signal xk, then, the cross correlation
vector is given by p = σx2 h(0 : N1− 1) Mh(α : Nh− 1) (3.13)
where the notation h(i : j) denote a vector whose elements consisting of the i-th to the j-th component of h.
The residual echo response is
∆h = h − ( ˆwo,1+ g ∗ ˆwUo,2) (3.14)
where ˆwo,1and ˆwo,2are the optimal weights for w1and w2, respectively, and ˆwUo,2is an
upsam-pled version of ˆwo,2. The MMSE is then equal to the summation of the residual echo power and
the noise variance.
MMSE = (∆hT∆h)σ2
x+ σ2n (3.15)
where σ2
nis the noise variance. The theoretical ERLE then equals
ERLE = 10 · log10
hTh
that the update equation for the adaptive IFIR filter is identical to that of a standard adaptive FIR filter, except for the input vectors. Thus, the existing results for the adaptive FIR filter can be applied. Using the independence theory [19], we assume that {˜xk} be a sequence of statistically
independent vectors. Defining the weight-error vector as εk = ˆwk− ˆwo, we then have [19]
E [εk+1] =
I− µ ˆRE [εk] . (3.17)
Thus, if the step size satisfies the following condition, the mean of εkwill converge to zero as k
approaches infinity:
0 < µ < 2 λmax
(3.18) where λmaxis the largest eigenvalue of the correlation matrix ˆR.
Let the MSE of the adaptive IFIR filter be denoted as Jk = E [e2k]. The MSE, Jk, will
converge to a steady-state value equal to J∞if, and only if, the step-size µ satisfies the following
two conditions [19]: 0 < µ < 2 λmax , N1+N2−S+1 X n=1 µλn/2 (1 − µλn) < 1 (3.19)
where λnis the n-th eigenvalue of the correlation matrix ˆR, and N1+ N2− S + 1 is the number
of adjustable tap weights in the proposed IFIR echo canceller. The MSE value in the steady state is given by J∞ = Jmin 1 −N1+NP2−S+1 n=1 µλn/2 (1 − µλn) (3.20)
where Jminis the MMSE shown in (3.15). The misadjustment ψ can then be expressed as
ψ = N1+NP2−S+1 n=1 µλn/2 (1 − µλn) 1 −N1+NP2−S+1 n=1 µλn/2 (1 − µλn) . (3.21)
From simulations we found that the convergence speed of the proposed adaptive IFIR filter is somewhat slower than that of the conventional LMS FIR filter. This is due to the fact that the input correlation matrix for the IFIR filter is not truly diagonal.
3.3. OPTIMAL INTERPOLATION FILTER DESIGN 21
§ 3.3 Optimal Interpolation Filter Design
Given a value for the interpolation factor M, the optimal interpolation filter is the ideal lowpass filter with bandwidth π/M; however, the corresponding impulse response is an unrealizable sinc function. Therefore, many suboptimal interpolation filters have been proposed [16]. These filters are general in the sense that they are independent to the signal being interpolated. If we know the characteristics of the signals for which we will be interpolating, we can design a better interpolation filter. In this section, we propose a least-squares method to obtain the optimal interpolation filter for a set of given tail echo responses.
We first consider the interpolation filter design for a single tail echo response. Let
f = f0 f1 ... f(N2−1)M
T
= hα hα+1 ... hα+(N2−1)M
T
(3.22)
be a tail echo response. Here, we take ((N2 − 1)M + 1)-sample from the original tail echo. If
the original tail echo is not long enough, we can pad zeros. Let
ˆf = hfˆ0 fˆ1 ... ˆf(N 2−1)M iT = [f0 0, · · · , 0 | {z } M −1 fM 0, · · · , 0 | {z } M −1 · · · f(N2−1)M] T (3.23)
be the corresponding downsampled and then upsampled response used in interpolation, g be the interpolation filter response, and
˜f =hf˜0 f˜1 ... ˜f(N
2−1)M
iT
(3.24)
be the interpolated tail response. Note that
˜ fk= Ng−1 X i=0 gifˆk−i. (3.25)
h
Interpolated echo tail f ∗ ˆf g 0 1 I I2 ˆ ∗ g f
Minimize sum of the error square in this interval f
What is optimal g?
Figure 3.5: Illustration of finding optimal interpolation filter.
Let the interpolation error response be vi = fi− ˜fi. We can then define a cost function using
the error response as
ξ(g) =
I2
X
i=I1
v2i (3.26)
where I1 = Ng − 1 and I2 = (N2 − 1)M. Note that the cost function does not include the
interpolation errors for the first Ng− 1 and last Ng− 1 transient responses of ˜fk, as shown in
Fig. 3.5. Obviously, the optimization problem is a classic least-squares problem. Define an error vector as v = [vI1 vI1+1 · · · vI2]
T. We can rewrite the cost function in (3.26) using a
vector form.
ξ(g) = vTv
= (fs− ˜fs)T(fs− ˜fs) (3.27)
where fs = f (I1 : I2)is the reduced-length tail response, ˜fs = ˜f(I1 : I2)is the corresponding
interpolated response. If we expressed ˜fsin the following matrix form,
˜f
3.3. OPTIMAL INTERPOLATION FILTER DESIGN 23 where F= ˆ fI1 fˆI1−1 · · · ˆfI1−(Ng−1) ˆ fI1+1 fˆI1 · · · ˆfI1−(Ng−2) ... ... ... ... ˆ fI2 fˆI2−1 · · · ˆfI2−(Ng−1) (3.29)
is a (I2− I1+ 1)-by-NgToeplitz matrix, then the cost function in (3.27) can be re-expressed as:
ξ(g) = (fs− Fg)T(fs− Fg). (3.30)
Taking the derivative with respect to g in (3.30) and set the result to zero, we can obtain the least-squares solution as [20]
ˆ
g = (FTF)−1FTfs. (3.31)
Extending the idea developed above, we can find a single optimal interpolation filter for a set of echo loops. Let fs,ibe the reduced-length tail response for the i-th loop, N be the number
of loops considered, and ˜fs,i the corresponding interpolated response. Now we augment the
interpolated response in (3.28) as follows: ˜f s,1 ˜f s,2 ... ˜f s,N = F1 F2 ... FN g. (3.32)
where Fi is the matrix in (3.29) for the i-th loop. Let ¯fs =
h ˜f s,1 ˜fs,2 · · · ˜fs,N iT , and ¯ F=h F1 F2 · · · FN iT
. We can then define a cost function similar to (3.30) as ¯
ξ(g) = (¯fs− ¯Fg)T(¯fs− ¯Fg) (3.33)
The cost function in (3.33) is equivalent to ¯ ξ(g) = N X i=1 ξi(g) = N X i=1 I2 X j=I1 |vij|2. (3.34)
as
˜
g= (¯FTF)¯ −1F¯T¯fs. (3.35)
and the sum of averaged squared errors as
¯ ξmin =
¯fs− ¯F˜g 2
= ¯fsT I− ¯F ¯FTF¯−1F¯T¯fs. (3.36) Assuming that the head echo, including the overlapped portion is cancelled perfectly, we have the theoretical ERLE for loop i as
ERLE = 10 · log10kh ik2
ξmin,i (3.37) where hiis the i-th echo response vector, and ξmin,i= kfs,i− Fi˜gk2is the sum of the minimum
squared errors for that loop. Note that the optimal interpolation filter is usually not symmet-ric. The computational complexity for the interpolation operations, using the optimal filter, is higher than that when using the conventional symmetric ones. We can solve the problem by constraining the filter response to be symmetric, i.e., gi = gNg−1−i, i = 0, 1, ...(SM − 2). The
optimal solution for a single echo loop is identical to that in (3.31) except for the matrix F. The matrix now becomes:
F= ˆ fI1 + ˆfI1−(Ng−1) · · · ˆfI1−(SM−1) ˆ fI1+1+ ˆfI1−(Ng−2) · · · fˆI1−SM ... ... ... ˆ fI2 + ˆfI2−(Ng−1) · · · ˆfI2−(SM−1) . (3.38)
Compared to (3.29), the column dimension of (3.38) is reduced by half. The (Ng − 1 − i)-th
column of F in (3.29) is added to the i-th column, i = 0, 1, ...(SM − 2) except for the middle column corresponding to the central tap of g. That means the middle column ofF is unchanged
3.3. OPTIMAL INTERPOLATION FILTER DESIGN 25 Using a similar method, we can find the optimal symmetric interpolation filter for a single or for multiple loops via (3.31) or (3.35).
In this paragraph, we compare the performance of some existing interpolation filters with proposed ones. Those we considered include the linear, the truncated-sinc, the Hanning-windowed sinc, and the Chebyshev-windowed sinc filters. The responses of the Chebyshev-windowed sinc, the optimal, and the optimal symmetric interpolation filters are shown in Fig. 3.6 (S = 2 and M=4), and the theoretical ERLEs for the CSA loop #1 [7] are shown in Table 3.2. Note that the optimal interpolation filters are optimized for eight CSA loops [7] then apply to CSA loop #1. As we can see, for the case of S=1 only the optimal interpolation filter has ERLE higher than 70-dB. For the case of S=2, the proposed interpolation filters outperform other fil-ters by an amount of more than 10 dB. For the case of S=3, the proposed filter still has the best performance, however, the performance difference is reduced. Specifically the performance of the Chebyshev-windowed sinc filter is close to that of the proposed ones. It seems that the performance bound has been reached, and that there is no need to consider a span of more than three. We conclude that the proposed method provides a systematic rather than a heuristic or a trial-and-error way to find the interpolation filter. From the results shown in Table 3.2, we can see that the two-span symmetric interpolator seems a good choice for the IFIR echo canceller.
Table 3.2: ERLE performance vs. various interpolation filters (LIN: linear, TRS: truncated sinc, HWS: Hanning-windowed sinc, CWS: Chebyshev-windowed sinc, OPS: optimal-symmetric, OPT: optimal)
Filter LIN TRS HWS CWS OPS OPT S=1 62.4 32.3 31.5 28.5 68.9 70.1 S=2 - 39.1 55.1 57.8 73.2 73.3 S=3 - 43.4 66.4 75.0 75.9 76.3
0 2 4 6 8 10 12 14 -0.2 0 0.2 0.4 0.6 0.8 1 M ag in ut ud e Sample Chebyshev Optimal (symmetric) Optimal
Figure 3.6: Interpolation filters (M=4, S=2).
§ 3.4 Simulation Results
In this section, we report some computer simulation results to demonstrate the effectiveness of the optimal IFIR echo canceller. Specifically, we have taken SHDSL as the application example and evaluated the echo cancellation performance under scenarios with different loop topologies, echo cutting points, interpolation factors, and noise environments.
§ 3.4.1 Loop Characteristics and Topologies
To test the robustness of the optimal IFIR echo canceller, we used eight CSA loops in [7] for simulations. The method to model echo responses was described in [1, 21]. The echo path con-tained a transmit shaping filter, a transmit differential hybrid circuit (with a 135Ω termination
3.4. SIMULATION RESULTS 27 impedance), a CSA loop, a receive differential hybrid circuit, and a receive filter. The trans-mit/receive filter was modelled as a 6-th order Butterworth lowpass filter with a 3-dB cutoff frequency at 775 KHz. The primary inductance of the transformer in the hybrid circuits was 3 mH. For the SHDSL application, the sampling rate was as high as 775 KHz. The simulated echo responses at the central office side (CO) and the customer premise side (CPE) are shown in Fig. 3.7 and Fig. 3.8, respectively. The responses exhibited a short and rapidly changing head portion and a long and slowly decaying tail portion, as was expected. The line code of
0 50 100 150 200 250 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 M ag itu de (V ) Sample 1 2 3 4 5 6 7 8
Figure 3.7: SHDSL echo responses for CSA loops at CO side.
the transmit signal was 16-PAM. Here, AWGN with -140 dBm/Hz was used to contaminate the received signal. The cutting point α was set as 39 and the interpolation factor M was set as 4. The optimal symmetric interpolation filter (S=2) obtained by using the eight CSA loops, was applied. We then have N1 = 50. During the training period, the far-end transmit signal was
0 50 100 150 200 250 -0.15 -0.1 -0.05 0 0.05 M ag ni tu de (V ) Sample 1 2 3 4 5 6 7 8
Figure 3.8: SHDSL echo responses for CSA loops at CPE side.
turned off. After that, the transceiver was operated in a full duplex data transmission mode. For a faster convergence, the step size was varied using the following scheme. The training period was divided into five stages and the overall period was 12,000 samples. In each stage, the step size was simply reduced by a factor of two. The step size was initialized as 1/Nh = 0.004. The
emulated echo response, which was an overlapped combination of the FIR and IFIR responses, is shown in Fig. 3.9. Note that there was one nulled tap (zero weight) located in the tail end of w1. As we can see, the tail response was modelled accurately using the IFIR filter except
for the transient response in the beginning, however, w1 compensated for that effectively. All
the eight CSA test loops both at the CO and the CPE side were simulated. The resultant ERLE performances are shown in Fig. 3.10. As the figure shows, the ERLE was between 73.0 and 77.1 dB. The averaged ERLE was around 74.0 dB at the CO side, and 74.5 dB at the CPE side.
3.4. SIMULATION RESULTS 29 35 40 45 50 55 -1 0 1 2 3 4 5 6 7 x 10-3 M ag ni tu de Sample h w1 g*w2U Nulled tap
Figure 3.9: Overlapping of FIR and IFIR filters (M=4, S=2, N1=50).
The low sensitivity of the optimal IFIR echo canceller to different topologies and loop char-acteristics exhibited its feasibility to real-world applications. Generally speaking, theoretical ERLE predictions, which are also shown in Fig. 3.10, were accurate. The higher the ERLE, the larger the difference between the theoretical and empirical ERLEs. This was because if the ERLE was higher, a smaller step size was required to hold the independence theory. However, we used the same step size for all cases.
§ 3.4.2 The Cutting Point and the Interpolation Factor
In this set of simulations, all the parameters and the interpolation filters were identical to those in the previous one. However, only CSA loop #1 was used. To show the influence of the cutting point and the interpolation factor, we give theoretical ERLE in a two-dimensional plot
1 2 3 4 5 6 7 8 72 74 76 78 80 82 E R LE (d B )
CSA Loop No. Theoretical @ CO Simulated @ CO Theoretical @ CPE Simumalted @ CPE
Figure 3.10: ERLE performance for different CSA loops.
in Fig. 3.11. In the figure, the cutting point varies from 30 to 70 and the interpolation factor from two to eight. Note that the performance surface is monotonically increasing with respect to the increasing of the cutting point position and the decreasing with the interpolation factor. The ERLE is always higher than 67.4 dB (76.9 dB in average). For an interpolation factor less than (or equal to) 4 and the cutting point greater than 30, the ERLE performance excesses 70 dB. A larger cutting point or a smaller interpolation factor implies higher computational complexity. Thus, there must be a compromise between performance and complexity. The computational complexity comparison for the IFIR and the conventional FIR echo cancellers is shown in Table 3.3. The complexity ratio in Table 3.3 is defined as the ratio of the computational complexity of the IFIR canceller and that of the FIR canceller. Here, the cutting point is 39 and the interpolation filter spans 4 samples. For the interpolation factor of 2, 4, and 8, computational
3.4. SIMULATION RESULTS 31 complexity for the IFIR canceller is (in terms of additions and multiplications) 62%, 43%, and 36% of that for the conventional FIR canceller, respectively. Using the interpolation factor of 4, we can achieve more than 70-dB ERLE while reduce 57% complexity.
Table 3.3: Computational complexity ratio for different interpolation factor (N1 = 50, S=2)
Echo emulation Taps weight update Complexity ratios
×, + + × + × + ×
M=2 156 158 151 151 62% 62%
M=4 114 116 101 101 43% 43%
M=8 105 107 76 76 36% 36%
§ 3.4.3 Noise Environments: AWGN and NEXT
In this subsection, we evaluate the performance of the optimal IFIR echo canceller under differ-ent noise environmdiffer-ents. The scenario includes pure AWGN and a composite noise consisting of AWGN and NEXT noise. The power spectral density (PSD) of the coupled NEXT can be evaluated using the formula shown below [21, 22].
PSDN EXT =PSDDisturber· xnf3/2
0 ≤ f < ∞, n < 50, xn= 0.8536 · 10−14· n0.6
(3.39)
where the PSDDisturberis the PSD of a disturbing xDSL line code such as 16-PAM, DMT, 2B1Q,
etc. The parameter xnis a coupling coefficient and its value depends on the number of disturbers
n. Both self-NEXT as well as foreign-NEXT were considered. For the self-NEXT case, NEXT noise was contributed by 10-SHDSL disturbers. For the foreign-NEXT case, NEXT noise was contributed by 24-ISDN disturbers, 10-HDSL disturbers, non-collocation1 24-T1 disturbers, or
1Contrary to [8], here, we assume that the T1 and SHDSL terminals are non-collocation, a factor of 15.5 dB downward of interferer PSD is included for modelling the adjacent binder effect [22, 23].
2 3 4 5 6 7 8 40 45 50 55 60 65 70 75 80 60 65 70 75 80 85 Cutting point (α) Interpolation factor (M) E R LE ( dB )
Figure 3.11: ERLE versus the cutting point and the interpolation factor.
10-downstream/upstream-ADSL disturbers. All NEXTs were from pairs in the same binder group except T1 crosstalk that was from pairs in an adjacent binder group. Finally, NEXT and AWGN (-140 dBm/Hz) were then summed together to form a composite noise. The ERLE performance at the CO and the CPE side for different CSA loop under the composite noise en-vironments were simulated and the results are shown in Fig. 3.12 and Fig. 3.13, respectively. In either case, the ERLE under AWGN is the highest. The ERLE is lower in the NEXT environ-ments and inversely proportional to the NEXT power. Note that the step sizes used here were all the same. From (3.8) and (3.13), we can see that the Wiener solution is independent of the noise power. The noise power only affects the misadjustment. It is well known that the smaller the step size, the smaller the misadjustment [19]. This is to say that if we could reduce the step size, all the ERLEs would be as high as that in the AWGN case. However, as we will discuss
3.4. SIMULATION RESULTS 33 1 2 3 4 5 6 7 8 40 45 50 55 60 65 70 75 80 E R LE (d B )
CSA Loop No. AWGN ISDN HDSL T1 ADSL SHDSL
Figure 3.12: Simulated ERLE performance at CO side.
below, this might not be necessary.
§ 3.4.4 Discussions
The mission of an echo canceller is to cancel the echo signal produced by the transmission symbols. It can do nothing about noise. Thus, if the level of residual echo is below that of noise, echo is no longer the main factor limiting the system performance. So, lets define the total echo return loss (TERL) as
TERL = (ERL + ERLE), (3.40)
where ERL is the echo return loss of the hybrid circuit. The value of ERL usually depends on the loop topology, loop characteristics as well as the receiver location. For convenience, we
