V. CONCLUSIONS
A general linear relationship between the coefficients of two dif-ferent subblock transformations was developed. This relationship holds for any mix of linear, invertible transforms and separable subblock transform geometries (as illustrated in Fig. 1). The relationship can be found by simply precomputing the result of an inverse transform ma-trix multiplied by a differing forward transform mama-trix.
This resultis a generalization of previous work by Jiang and Feng [2]. In that paper, it was also shown that the matrix giving their linear relation is sparse (which results in reduced computational load). This property holds for the DCT case when the subblocks areA : 1 ratios of the larger blocks. In general (i.e., for other transforms and subblock ra-tios), this sparseness may not be present. This means that the reduction in computational load may not be as great. Whether it is efficient to re-late transform coefficients by the method developed here will depend on the particular scenario as well as the applicability of any fast algo-rithms for the transforms of interest (these algoalgo-rithms may make re-lating the coefficients through an inverse transform operation followed by a forward transform operation more desirable).
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A Low-Complexity Adaptive Echo Canceller for xDSL Applications
Shou-Sheu Lin and Wen-Rong Wu
Abstract—A finite impulse response (FIR)-based adaptive filter struc-ture is proposed for echo cancellation in xDSL applications. The proposed algorithm consists of an FIR filter, a cascaded interpolated FIR filter, and a tap-weight overlapping and nulling scheme. This filter requires low com-putational complexity and inherits the stable characteristics of the conven-tional FIR filter. Simulations show that the proposed echo canceller can ef-fectively cancel the echo up to 73.4 dB [for a single-pair high-speed digital subscriber line (SHDSL)system]. About 55% complexity reduction can be achieved compared with a conventional FIR filter.
Index Terms—Adaptive filter, DSL, echo cancellation, interpolated FIR filter.
I. INTRODUCTION
In a digital subscriber loop (DSL) environment, full duplex trans-mission via a single twisted pair can be achieved using a hybrid circuit. Due to the impedance mismatch problem, the hybrid circuit will intro-duce echoes. A typical echo response, shown in Fig. 1, consists of a shortand rapidly changing head echo and a long and slowly decaying tail echo. Conventionally, an adaptive transversal FIR filter [1] is used to synthesize and cancel the echo. For high-speed applications such as HDSL [2], HDSL2 [3], and single-pair high-speed digital subscriber line (SHDSL) [4], the echo response is usually very long. The conven-tional FIR echo canceller may require hundreds of tap weights, and the computational complexity becomes very high.
In order to reduce the computational complexity, some researchers tried to use an adaptive infinite impulse response (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 [5], August et al. collected some echo responses for the European subscriber loops and used a cri-terion to determine the feedback filter optimally. In [6], Gordon et al. considered echo cancellation as a series expansion problem. They used a 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, since the existing loop re-sponses 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 [7]. The canceller is cascaded from 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 [7] has an uncontrollable transient 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,
Manuscriptreceived May 30, 2003; revised June 23, 2003. The associate ed-itor coordinating the review of this manuscript and approving it for publication was Prof. Xiaodong Wang.
The authors are with the Department of Communication Engineering, National Chiao-Tung University, HsinChu 30050, Taiwan, R.O.C. (e-mail: gii.cm84g@nctu.edu.tw; wrwu@cc.nctu.edu.tw).
Digital Object Identifier 10.1109/TSP.2004.826155 1053-587X/04$20.00 © 2004 IEEE
Fig. 1. Typical echo response.
it was overlooked in [7]. The authors only measured the cancellation performance of the tail echo and assumed that the remaining echo was cancelled perfectly.
In this paper, we propose a new adaptive echo canceller to remedy the problem mentioned above. In our work, the FIR and the IFIR filters are overlapped instead of being directly cascaded. Due to this over-lapping operation, some echo responses will be simultaneously can-celled by the FIR and IFIR filters. Although this will not affect the final performance, it will slow down the convergence. In order to solve the problem, some of tap-weights used in the FIR filter coefficients are nulled. We call this scheme coefficient nulling. After properly nulling certain tap weights of the FIR filter, we obtain a low-complexity yet high-performance echo canceller. We also derive the corresponding Wiener solutions, the minimum mean square error (MMSE), and the error return loss enhancement (ERLE).
This paper is organized as follows. In Section II, we describe the proposed adaptive echo canceller structure and analyze the complexity in detail. In Section III, we derive the Wiener solutions, MMSE, and ERLE of the proposed algorithm. In Section IV, we show the simulation results and evaluate the validity of the derived expressions. Finally, we draw our conclusions in Section V.
II. PROPOSEDECHOCANCELLER
Let the length of an echo response beNhand its response beh = [h0h1. . . hN 01]T. The received echo signal can be described as fol-lows:
yk= hTxk+ nk (1)
wherexk = [xk xk01. . . xk0N +1]T is the transmitted signal, and nkis a zero-mean additive white Gaussian noise (AWGN).
From Fig. 1, we can clearly see that a typical echo response has a fastchanging head echo and a slowly varying tail echo. Thus, we can select a cutting point to segment these two portions. Lethh = [h0. . . h 01]T,xh= [xk. . . xk0 +1]T,ht= [h . . . hN 01]T, and xt = [xk0 . . . xk0N +1]T. Then, the echo response can be re-ex-pressed as
yk= hThxh+ hTtxt+ nk: (2) In absence of noise,yk can be synthesized and cancelled by an Nh-tap FIR filter. This filter, havingh as its response, can be decom-posed to an -tap and an (Nh0 )-tap FIR filter, where one cancels the head echo, and the other cancels the tail echo. In general,(Nh0 ) is much larger than . 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 upsampled response [8]. The detailed structure of the proposed algorithm is shown in Fig. 2. The FIR filter w1 is used to model the head echo responsehh and the IFIR filter g 3 wU
2, where “3” denotes the convolution operation and is used
Fig. 2. Adaptive FIFIR echo canceller.
to modelht. Note thatwU2 is an upsampled version of a filterw2. Basically,w2tries to model the downsampled version ofht, andg is a FIR filter that interpolatesw2. Let the downsampling factor beM and the FIR filter and the IFIR filter be overlapped forNo= Ng0 M taps. The head echo canceller length is extended toN1 = + No instead of just . As Fig. 2 shows, xkis the input tow1, and ~xkis that towU2. The output of the proposed echo canceller in Fig. 2 can be expressed as
~yk= wT1x1;k+ wT2~x2;k (3) wherew1 = [w1;0 w1;1 1 1 1 w1;N 01]T is theN1-tap head echo canceller, x1;k = [xk xk01 1 1 1 xk0N +1]T is its input vector, w2 = [w2;0 w2;1 1 1 1 w2;N 01]T is theN2-tap tail echo canceller, and~x2;k= [~xk0 ~xk0 0M 1 1 1 ~xk0 0(N 01)M]T is its input vector. In terms ofz-transform representation, we have wU2(z) = w2(z0M).
Rewriting (3), we have
~yk= wT1 wT2 x~x1;k 2;k
= wT~xk: (4)
Letg = [g0 g1 1 1 1 gN 01]T be an Ng-tap interpolation filter; its length equals2SM 0 1, where S is the number of w2tap-weights in-volved in calculating an interpolated value for a single side span. Then, the interpolator output can be expressed as follows:
~xk= N 01
i=0
gixk0 i: (5)
Generally, the impulse response of the interpolation filter g is peaking at the center, slowly decaying to its two sides, and is sym-metric around the center. The simplest response ofg is a triangular window function with2M 0 1 taps, which gives a linear interpolation result. 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 withNo samples), with one in the front end ofg 3 wU2 and the other in the tail end. Since the tail end ofht always decays to zero, there is no problem with that transient response in the tail. However, the head portion of ht has an abruptrising edge. As a consequence, the front-end transient response of g 3 wU2 cannot model that ofht. A simple way to solve this problem is to increase the length ofw1and overlapw1with the front-end transient response ofg 3 wU2. Here, we overlap the lastNo taps ofw1 with the first No taps ofg 3 w2U to cover the full front-end transient response. Fig. 3 shows how the FIR and IFIR responses are overlapped. Note that in the structure, there are(S 0 1) echo samples being cancelled
Fig. 3. Filter responses of the proposed FIFIR echo canceller. TABLE I
COMPUTATIONALCOMPLEXITYCOMPARISON FOR THEFIRANDFIFIR FILTERS
simultaneously by two filters:w1andg 3 wU2. It seems that there are (S 0 1) redundant taps. As we will show later, this is indeed the case, and the(S 0 1) redundant taps will slow down the convergence rate of the proposed echo canceller. An easy and efficient way to overcome this problem is to null the redundant taps ofw1; we let the coefficients [w1;N 0(S01)M 1 1 1 w1;N 02Mw1;N 0M] all be zeros. This nulling scheme removes the redundant taps and accelerates the convergence rate.
To obtain the tap weights ofw1 andw2, an adaptive algorithm is applied. From (4), we can see that the proposed 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 directly applied here. For the complexity consideration, the simplest adaptive algorithm, namely, the least mean square (LMS), is employed. The LMS algorithm is given by [9]
wk+1= wk+ ek~xk (6)
where is the step size controlling the convergence rate, and ek = yk0 ~ykis the error signal. In spite of the joint input vector, the above equation is identical to a typical LMS weights update equation for a transversal FIR filter. Hereafter, for the sake of convenience, we will call the proposed echo canceller the FIFIR echo canceller.
The computational complexity of the adaptive FIFIR echo canceller can be easily evaluated. Table I summarizes the numbers of additions and multiplications required in the echo cancellation for an FIFIR and a conventional FIR filter. As we can see, the complexity reduction for the proposed structure comes from the IFIR filter. The computational complexity ofw2 is only oneMth of that of the corresponding FIR filter. Consider the SHDSL application studied in Section IV. The echo response length is 250, and the cutting point is set as 31. For interpola-tion factor 2, 4, and 8, the complexity reducinterpola-tion ratios over a conven-tional FIR filter are 63%, 45%, and 40%, respectively.
Given and M, the optimal g is an ideal lowpass filter with band-width=M; however, its impulse response is an unrealizable infinite sinc function. A simple remedy is to multiply the sinc function by a finite length widnow. From extensive simulations, we found that a tunnable stopband attenuation window, such as a Chebyshev window, gives satisfactory results.
III. THEORETICALANALYSIS
In this section, we consider some theoretical aspects of the proposed FIFIR filter. First, we derive the optimal Wiener solution and the corre-sponding MMSE. Using these results, we calculate the ERLE bounds for the adaptive FIFIR echo canceller.
Definex2;k = [xk0 xk0( +1) 1 1 1 xk0N +1]T as the input data for the interpolator. Then,~x2;kcan be expressed as
~x2;k= Mx2;k (7) where M = gT; 0; . . . ; 0 (N 01)M 0; . . . ; 0 M ; gT; 0; . . . ; 0 (N 02)M .. . 0; . . . ; 0 (N 01)M ; gT (8)
is anN2-by-(Nh0 ) matrix. Without loss of generality, we can always pad zeros in the original echo so that the length ofx2;kequals[Ng+ (N20 1)M]. From (3) and (7), we then have
~yk= wT1x1;k+ wT2Mx2;k: (9) The input vector of the FIFIR filter consists ofx1;kand ~x2;k. The vectorx1;kis typically white; however,~x2;kis not. The signal~xk is the output from the interpolation filterg. To simplify our analysis, we assume thatg is an ideal lowpass filter. In other words, the frequency response ofg is flatin the desired passband =M. In this case, ~x2;k is a white vector. With this assumption, the FIFIR echo canceller is identical to a conventional FIR echo canceller. The mean-squared error (MSE) criterion is then given as
J = E (yk0 ~yk)2 : (10)
If we take the derivative with respect tow and set the result to zero, we can obtain the Wiener solution of the FIFIR filterwoas
wo= R01p (11)
whereR = E[~xk~xTk] is the input correlation matrix, and p = E[~xkyk] is a cross correlation vector.
Next, we find closed-form expressions forR and p. The correlation matrix can be rewritten as
R = E ~xk~xTk = RRx xT Rx ~x
x ~x Rx ~~ x : (12)
Note thatxkis usually white. Thus
Rx x = E x1;kxT1;k
= 2xIN 2N (13)
where2xis the transmitted signal variance. From (7), the correlation matrix is
R~x ~x = E ~x2;k~xT2;k = E Mx2;kxT2;kMT = 2
whereM is the interpolation matrix in (8). The cross correlation matrix is Rx ~x = E x1;k~xT2;k = E x1;kxT2;kMT = 2 x I0 2N 0 2(N 0N ) N 2N 0N 2(N 0N ) M T: (15) If we assume that noisenkis independent of the transmitted signalxk, then the cross correlation vector is
p = Ef~xkykg = E x1;k Mx2;k (h Tx k+ nk) = 2 x Mh( : Nh(0 : N10 1) h0 1) (16)
where the notationh(i : j) denotes a vector whose elements consisting of theith to the jth component of h.
To obtain the Wiener solution without redundant taps inw1, we must eliminate theith row and ith column of R, as well as the ith row of p, where i 2 f(N1 0 (S 0 1)M); 1 1 1 ; (N1 0 2M); (N10 M)g. By doing so, the corresponding weights inw1 will be all zeros, i.e., [w1;N 0(S01)M 1 1 1 w1;N 02M w1;N 0M] = 012(S01). Then, the Wiener solution with coefficient nulling can be solved by
^
wo= ^R01^p (17)
where ^R and ^p are the correlation matrix and vector for the nulled filter, respectively. Using (11) or (17), we now are ready to derive the corresponding MMSE and ERLE.
The residual echo response is given by
1h = h 0 wo;1+ g 3 wo;2U (18)
wherewo;1andwo;2are the optimal weights forw1andw2, respec-tively, andwUo;2is an upsampled version ofwo;2. The MMSE is then equal to the summation of the residual echo power and the noise vari-ance.
MMSE= (1hT1h)2x+ 2n (19)
wheren2 is the noise variance. The theoretical ERLE equals ERLE= 10 1 log10 h
Th
1hT1h: (20)
So far, we have obtained two Wiener solutions for two adaptive FIFIR echo cancellers. We are then concerned with which one will be better. Although a theoretical comparison is not available, we have ob-tain the following results using extensive simulations. We found that the ERLEs for these two cancellers are almost the same; however, the eigenvalue spreads for two input correlation matrices are significantly different. The eigenvalue spread is defined asmax=min, wheremax andminare the maximum and the minimum eigenvalues of an input correlation matrix. The eigenvalue spread ofR is usually much larger than that of ^R. It is well known that the convergence rate of an adap-tive algorithm is inversely proportional to the eigenvalue spread, which means that the convergence rate of the adaptive echo canceller without nulling will be much slower. Thus, we will use the one with nulling as the proposed echo canceller.
IV. SIMULATIONRESULTS
To test the robustness of our FIFIR echo canceller, we used eight CSA loops in [3] for simulations. We considered an SHDSL
applica-Fig. 4. SHDSL echo responses for CSA loops atCO and CPE side.
Fig. 5. Overlapping of FIR and IFIR filters (M = 4, S = 3, N = 50).
tion where 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. 4. The line code of the transmit signal was 16-PAM. Here, AWGN with0140 dBm/Hz was used to contaminate the received signal. The cutting point was set at 31, and the inter-polation factorM was setat4. A Chebyshev windowed sinc function [11] with 23-tap was selected as the interpolation filter. We then have N1 = 50. During the training period, the far-end transmit signal was turned off. After that, the transceiver was operated in a full duplex data transmission mode. For a faster convergence, the LMS algorithm with a variable step size was applied. The training period was divided into five stages, and the overall period was 12 000 samples. In each stage, the step size was reduced by a factor of two. The emulated echo sponse, which was an overlapped combination of the FIR and IFIR re-sponses, is shown in Fig. 5 for CSA loop # 1. Note that there was two nulled taps (zero weights) located in the tail end ofw1. As we can see, the tail response was modeled accurately using the IFIR, except for the transient response in the beginning; however,w1compensated for that effectively.
All eight CSA test loops, both at the CO and the CPE side, were simulated. The resultant ERLE performances are shown in Fig. 6. As
Fig. 6. ERLE performance for differentCSA loops.
the figure shows, the ERLE was between 73.4 and 77.5 dB. The aver-aged ERLE was around 74.7 dB atthe CO side and 75.1 dB atthe CPE side. These ERLE results exceed the general requirement for a DSL echo canceller (60–70 dB). The low sensitivity of the proposed echo canceller to different topologies and loop characteristics exhibited its feasibility to real-world applications. Generally speaking, theoretical ERLE predictions, which are also shown in Fig. 6, 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.
V. CONCLUSIONS
A low-complexity, finite impulse response, and adaptive filter struc-ture is proposed for the echo cancellation application in high-speed
baseband xDSL systems. The proposed echo canceller is a general-ized adaptive interpolated FIR structure. It inherits all the numerical stability advantages of the conventional FIR filter while effectively re-ducing its computational complexity. Using the proposed tap-weight overlapping and nulling scheme, the performance loss due to the un-controllable transient response problem was avoided. The theoretical performance bounds for the proposed echo canceller were also derived and verified. Finally, simulations using standard test loops were con-ducted to demonstrate the effectiveness of the proposed echo canceller.
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