a frequency-domain training approach for equalization and noise suppression in discrete multitone...

Signal Processing 84 (2004) 327–339www.elsevier.com/locate/sigpro

A frequency-domain training approach for equalization andnoise suppression in discrete multitone systems�

Bo Wang, T+ulay Adal,∗

Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, Baltimore, MD 21250, USA

Received 17 December 2001; received in revised form 12 September 2003

Abstract

In discrete multitone (DMT) transceivers, a time-domain equalizer (TEQ) is used to shorten the e6ective channel impulseresponse such that a shorter cyclic pre7x can be used. In this paper, we pose the TEQ design problem completely in thefrequency domain by de7ning a frequency-domain weighted least-squares cost function. The de7nition of the frequency-domainTEQ design criterion, we show, allows for the introduction of a weighting function to control the spectral shape of the TEQ,and facilitates important extensions particularly useful for the DMT system. The TEQ can be used to suppress the noiseand interference in the DMT system, a feature especially useful in the frequency division multiplexing-based DMT system.Also, in the echo-cancellation-based DMT system, the TEQ can be used to jointly shorten the echo response thus reducingthe complexity of the echo canceler. We present corresponding algorithms for these two objectives and simulation results,which show that the weighted frequency-domain least-squares algorithm and the two extensions we derive can achieve theshortening (channel or channel and echo) as well as noise suppression objectives e6ectively.? 2003 Elsevier B.V. All rights reserved.

Keywords: Discrete multitone system; Time-domain equalizer; Noise and interference suppression; Joint response shortening

1. Introduction

Multi-carrier modulation (MCM) has been pro-posed for parallel communication in the late 1950s[8] based on the concept of creating multiple orthog-onal subchannels over which several data streams canbe sent without intersymbol interference (ISI). Thismodulation scheme provides Aexibility for adapting

� Research supported in part by Maryland Industrial Part-nerships and Nortel Networks under Grants MIPS-2218.12 andMIPS-2218.24.

∗ Corresponding author. Tel.: +1-410-455-3521; fax: +1-410-455-3969.

E-mail address: [email protected] (T. Adal,).

to di6erent channel environments by adjusting theenergy and constellation size of each carrier. Oneimplementation of MCM is the discrete multitone(DMT) system which uses the discrete Fourier trans-form (DFT) for modulation [17,23]. DMT is chosenas the industry modulation standard for asymmetricaldigital subscriber line (ADSL) modems and is alsoa candidate modulation scheme for very-high-speeddigital subscriber line (VDSL) systems.In the DMT system, a cyclic pre7x (CP) of length �

provides a guard time between transmitted symbols. Ifthe channel response is of length �+1 or shorter, DMTsymbols can be transmitted free of ISI. Using large �values to compensate for the length of the channel re-sponse, however, decreases the eLciency (introduces

0165-1684/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.sigpro.2003.10.017

mailto:[email protected]

328 B. Wang, T. Adal' / Signal Processing 84 (2004) 327–339

an overhead of �=(N+�) where N is the DMT symbollength). A time-domain equalizer (TEQ) to shorten thee6ective channel impulse response has been the mostpopular equalization approach for the DMT systems[2,6,12,15,16,20–22]. In addition, the TEQ is consid-ered to be part of the overall channel response, henceits spectral shape plays an important role for maxi-mizing the bit-rate over the channel [9,11].In this paper, we pose the TEQ design problem

completely in the frequency domain, introduce theweighted frequency-domain least-squares (WFD-LS)approach and derive the corresponding algorithmby minimizing a squared frequency-domain costfunction. We show that by introducing a weightingfunction, we can control the TEQ spectral shapeand improve the overall performance of the DMTsystem. We compare the performance of WFD-LSalgorithm to the time-domain optimal (TD-OP) andtime-domain least-squares (TD-LS) algorithms givenin [15] and show that di6erent weighting functionscan lead to di6erent TEQ spectral shapes for theWFD-LS algorithm, which cannot be realized easilyby the current TEQ methods that primarily work inthe time domain.We also show two important extensions of our TEQ

design approach to further improve the DMT systemperformance. First, we demonstrate that we can mod-ify the WFD-LS cost function and apply it to jointlyshorten the channel impulse response and suppress thenoise and interference in the DMT systems. We applythe stopband WFD-LS algorithm to the frequency di-vision duplex (FDD) DMT-ADSL systems to jointlyshorten the channel response and suppress a “virtual”noise as in [11], where the stopband is de7ned as theunallocated or highly noisy subchannels. It is shownby simulations that the stopband WFD-LS algorithmcan e6ectively serve the two purposes of shorten-ing the channel impulse response and suppressing thenoise and interference.The second extension we present is based on the ob-

servation that in echo-cancellation (EC)-based DMTsystem, the TEQ is placed prior to the echo canceler,hence it can a6ect the e6ective echo impulse responseas well [10,24]. We incorporate the joint shorteningidea of [15] into our TEQ design method and show thatwe can de7ne a composite squared cost function in thefrequency domain to account for these two shorteningpurposes. Simulation results show that both channel

and echo responses are e6ectively shortened by ap-plying the joint WFD-LS algorithm.The rest of the paper is organized as follows. In

Section 2, we 7rst present an overview of the TEQtraining approach for DMT systems. We then formu-late the TEQ design problem in the frequency domainand derive the WFD-LS algorithm to obtain the TEQcoeLcients. We present simulations to show that thisTEQ design approach is e6ective in shortening thechannel response and a weighting function can be usedto control the resultant TEQ spectral shape. Then inSection 3, the WFD-LS algorithm is extended suchthat the noise and interference can also be suppressed.Finally, we apply the WFD-LS algorithm to jointlyshorten channel and echo responses in Section 4. Sim-ulation results are presented for each case, and Section5 includes the discussions.

2. Impulse response shortening for DMT systems

2.1. Background

TEQ design approaches typically approximate thechannel transfer function in DMT systems by an au-toregressive moving average (ARMA) model

H (z−1) =B(z−1)A(z−1)

=z−d ∑L−1

i=0 biz−i∑M−1i=0 aiz−i

: (1)

A TEQ whose transfer function is equal to A(z−1)can be introduced at the receiver side, such that thecascade of the channel H (z−1) and the TEQ approx-imate a suLciently short target response B(z−1). InEq. (1), L is the length of the target response, M is thelength of the TEQ 7lter, and d is the delay of the tar-get response. Note that L should be less than or equalto �+ 1, where � is the length of the cyclic pre7x, asthis implies ISI-free transmission. Usually, the chan-nel impulse response cannot be perfectly shortened bythe TEQ such that some residual energy of the short-ened impulse response will lie outside the �+ 1 con-secutive taps with the highest total energy. We can usethe shortening signal-to-noise ratio (SSNR) [15] tomeasure the shortening capability of the TEQ, whichis de7ned as the ratio of the energy in the largest con-secutive �+1 taps to the energy in the remaining tapsof the shortened response.

B. Wang, T. Adal' / Signal Processing 84 (2004) 327–339 329

Most TEQ design methods, such as [2,15,16], usethe time-domain squared-error at the TEQ outputas the design criterion and hence cannot control theTEQ spectral shape easily. Chow et al. [6] propose afrequency-domain algorithm for TEQ training, whichis an adaptive solution to the TEQ design problem.The algorithm minimizes the mean-squared error ofthe equalized response by using frequency-domainleast mean squares (LMS) for the adaptation andwindowing in the time domain. However, the con-vergence of Chow et al.’s algorithm is slow [16],which is also con7rmed by our simulation results in[21]. For the DMT system that naturally operates inthe frequency domain, TEQ design in the frequencydomain might o6er additional advantages such asthe ability to control the TEQ spectral shape. In thispaper, we introduce the weighted frequency-domainleast-squares approach for TEQ design that mini-mizes a squared frequency-domain cost function. Weshow that by introducing a weighting function, wecan control the TEQ spectral shape and improve theoverall performance of the DMT system.

2.2. Weighted frequency-domain least-squaresalgorithm

During the initialization phase in aDMT transceiver,a pseudo-random sequence, which is known to thetransmitter as well as the receiver, is transmitted re-peatedly over the channel to form a periodic signal ofperiod N [4]. Suppose X (e−j!k ) and Y (e−j!k ) are theDFTs of the training and the received signals, then thechannel frequency response can be estimated as

H (e−j!k ) =Y (e−j!k )X (e−j!k )

≡ B(e−j!k )A(e−j!k )

;

k = 0; : : : ; N − 1: (2)

We would like to model the channel as the ratio ofB(e−j!k ) and A(e−j!k ), where A(e−j!k ) and B(e−j!k )are the DFTs of the TEQ and the target responsewith delay d, respectively. If there is no error in themodeling, we achieve

B(e−j!k )X (e−j!k ) = A(e−j!k )Y (e−j!k );

k = 0; : : : ; N − 1 (3)

or

B(e−j!k ) = A(e−j!k )H (e−j!k );

k = 0; : : : ; N − 1: (4)

The coeLcients of the channel response h, theTEQ, a, and the target response, b, are assumed to bereal, which makes their frequency responses conju-gate symmetric. Hence we can introduce the follow-ing frequency-domain least-squares cost function forthe TEQ design:

Els(�) =N=2∑k=0

W (k)|H (e−j!k )A(e−j!k )− B(e−j!k )|2;

(5)

where �T=[a1; : : : ; aM−1; b0; : : : ; bL−1] is the vector ofall the parameters of the ARMAmodel to be estimated(note that a0 is assumed to be 1), N is the fast Fouriertransform (FFT) length (hence is even) as well as theDMT symbol length, k is the index of the subchannelfrequencies, and W (k) is a non-negative weightingfunction that can be used to control the TEQ spectralshape.If we di6erentiate Els(�) with respect to the para-

meter vector �, and set the result to zero, we obtain

��= �; (6)

where

�=

[�11 �12

�21 �22

];

�11 =

�0 �1 : : : �M−2

�1 �0 : : : �M−3

......

�M−2 �M−3 : : : �0

;

�12 =

−pd−1 −pd : : : −pd+L−2

−pd−2 −pd−1 : : : −pd+L−3

......

−pd−M+1 −pd−M+2 : : : −pd+L−M

;


+

hn+

+ TEQ

an

d

-

x(n)

y(n)

u(n)

e(n)

Response Target

b n

Channel

Delay

Fig. 1. Time-domain equalizer in DMT system.

�21 =�T12;

�22 =

�0 �1 : : : �L−1

�1 �0 : : : �L−2

......

�L−1 �L−2 : : : �0

;

� = [− �1 − �2 : : : − �M−1 pd pd+1 : : : pd+L−1]T

and

�i =N=2∑k=0

|H (e−j!k )|2 cos (i!k)W (k);

i = 0; : : : ; M − 1;

pi =N=2∑k=0

[Rk cos(i!k)− Ik sin(i!k)]W (k);

i = d−M + 1; : : : ; d+ L− 1;

�i =N=2∑k=0

cos(i!k)W (k);

i = 0; : : : ; L− 1;

Rk ≡ Re{H (e−j!k )}; Ik ≡ Im{H (e−j!k )};k = 0; : : : ; N=2:

Hence solution of the set of linear equations shownin Eq. (6) yields the optimal parameter vector � suchthat the frequency-domain squared-error in Eq. (5) isminimized. It is assumed that the set of linear equa-tions in Eq. (6) has a unique solution, i.e., � is of fullrank. This will be the case for most cases of practical

interest as the entries of � will typically result in dis-tinct columns/rows for the matrix. The likelihood of�being rank de7cient increases when a large number ofthe weights, W (k)’s are set to zero, a situation unde-sirable in the design anyway as it implies discountingerror over a large portion of the frequency band.The resulting set of equations given in Eq. (6)

yields the weighted frequency-domain least-squares(WFD-LS) algorithm. An unweighted version ofEls(�) is used in [14] for continuous-time systemmodeling.Our frequency-domain approach is closely related

to the time-domain TEQ design methods as expected.To observe the relationship of the two, note that theerror sequence between the TEQ output and the targetresponse shown in Fig. 1 can be expressed as

e(n) = y(n) ∗ an − x(n) ∗ bn−d; (7)

where ∗ denotes linear convolution, x(n) is the input(training sequence), y(n) is the channel output, an arethe coeLcients of the TEQ, bn and d are the coeL-cients and delay of the target response, respectively.Note that during the initial start-up phase, typicallymultiple blocks of data are averaged, hence decreas-ing the e6ect of additive noise, which is assumed to bezero mean. If we thus assume that the additive chan-nel noise u(n) shown in Fig. 1 is negligible, then y(n)is also periodic when x(n) with length N is transmit-ted repeatedly through the channel whose length is as-sumed to be less than or equal to N . Hence the errorsequence in Eq. (7) can be rewritten as

e(n) = y(n)˜ an − x(n)˜ bn−d; (8)

where ˜ represents circular convolution. After theDFT (with size equal to DMT symbol length N ),


we obtain

E(e−j!k ) = Y (e−j!k )A(e−j!k )− X (e−j!k )B(e−j!k );

k = 0; : : : ; N − 1: (9)

The two-error sequences e(n) and E(e−j!k ) are nowrelated through the Parseval’s relation for the DFT:N−1∑n=0

|e(n)|2 = 1N

N−1∑k=0

|E(e−j!k )|2

=1N

N−1∑k=0

|X (e−j!k )|2

×|H (e−j!k )A(e−j!k )− B(e−j!k )|2; (10)where the receiver noise is not taken into account.During the initialization of the DMT transceiver,

the complex pseudo-random sequence X (e−j!k ) is 7rstconverted to a real sequence x(n) through the inversediscrete Fourier transform (IDFT) and then transmit-ted through the channel repeatedly. In the ADSL stan-dard [4], X (e−j!k ) has Aat energy over all subchannels.Comparing Eqs. (5) and (10), we note that W (k) de-7ned in the frequency-domain cost function Els(�) isequivalent to |X (e−j!k )|2 in Eq. (10). So we can thinkthat, in WFD-LS algorithm, a “virtual” input data x(n)with varying spectral density |X (e−j!k )|2 (orW (k)) isused, thus generalizing the time-domain TEQ designmethod in which only training sequence with 7xedspectral density is utilized. In the simulation part, weshow that the weighting function W (k) can be usedto control the spectral shape of the TEQ.

2.3. Simulation results for WFD-LS algorithm

We compare the performance of the WFD-LS algo-rithmwith the twomethods given in [15]: time-domainoptimal and time-domain least-squares algorithms.Also, we investigate the e6ect of di6erent weightingfunctions on the resultant TEQ spectral shapes andSSNR values of the shortened channel response.Fig. 2 shows the con7gurations for CSA loops 4

and 6. Their squared transfer functions when cas-caded with a 30–1000 kHz bandpass 7lter are given inFig. 3. In this simulation section, we only considerthe downstream case, that is, N = 512 and � = 32,and do not consider the additive noise e6ect in the

400’

26 AWG

800’ 550’

26 AWG

CSA loop 6

CSA loop 4

9000’

800’26 AWG

26 AWG

6250’

26 AWG

26 AWG

Fig. 2. Con7gurations of CSA loops 4 and 6.

0 50 100 150 200 250

− 140

− 120

− 100

− 80

− 60

− 40

− 20Squared Transfer Function

Frequency bins

Mag

nitu

de (

dB)

Solid: CSA loop 4

Dashed: CSA loop 6

Fig. 3. Channel transfer function for CSA loops 4 and 6.

channel due to the averaging e6ect during initializa-tion described in Section 2.2.First, two di6erent weighting functions Wa(k) and

Wb(k) shown in Fig. 4(a) are chosen for the WFD-LSalgorithm. These weights are determined by the mag-nitude response of a Butterworth 7lter. We applythree TEQ training algorithms (TD-OP, TD-LS, andWFD-LS) to CSA loop 4 with TEQ length 20. Theoptimal delays of the target response for these fourcases are determined in terms of maximum SSNRvalues. The second column of Table 1 shows the re-sultant SSNR values for the four cases for CSA loop4. We can see that TD-OP algorithm achieves the bestshortening performance because it explicitly works tomaximize the SSNR [15]. The SSNR values obtainedby TD-LS and WFD-LS with Wa(k) weighting func-tion are almost the same. For the WFD-LS algorithmwith Wb(k) weighting function, the resultant SSNR isless compared to that obtained by TD-OP but betterthan those obtained by the other two.


0 50 100 150 200 250− 45

− 40

− 35

− 30

− 25

− 20

− 15

− 10

− 5

0

5

0

5

10

Frequency bins

−10*

log(

wei

ghtin

g fu

nctio

n) (

dB)

− 45

− 40

− 35

− 30

− 25

− 20

− 15

− 10

− 5

10

−10*

log(

wei

ghtin

g fu

nctio

n) (

dB)

(a)

Solid: Wa(k)

Dashed: Wb(k)

Dashdot: Wc(k)

Dotted: Wd(k)

0 50 100 150 200 250

Frequency bins(b)

Solid: Wa(k)

Dashed: We(k)

Dashdot: Wf(k)

Dotted: Wg(k)

Fig. 4. (a) First set of weighting functions; (b) second set of weighting functions.

Table 1SSNRs for di6erent algorithms and di6erent loops

Channel CSA loop 4 (dB) CSA loop 6 (dB)

TD-OP 58.19 64.87TD-LS 55.39 61.07WFD-LS (Wa(k)) 55.37 61.83WFD-LS (Wb(k)) 56.72 63.08

Since the ultimate goal in the TEQ design is toachieve the maximum bit-rate, besides the SSNRmea-sure, the TEQ spectral shape plays an important role

0 100 200 300− 70

− 60

− 50

− 40

− 30

− 20

− 10

0

10

20

30

(a) Frequency bins

Mag

nitu

de (

dB)

− 60

− 50

− 40

− 30

− 20

− 10

0

10

20

Mag

nitu

de (

dB)

Solid: TD−OP

Dashed: TD−L

Dashdot: WFD−LS with Wa(k)

Dotted: WFD−LS with Wb(k)

0 100 200 300

(b) Frequency bins

Solid: TD−OP

Dashed: TD−L

Dashdot: WFD−LS with Wa(k)

Dotted: WFD−LS with Wb(k)

Fig. 5. (a) TEQ spectral shapes for CSA loop 4; (b) TEQ spectral shapes for CSA loop 6.

for maximization of the channel throughput as it can beconsidered to be part of the overall channel response[11,9]. Next, we discuss the resultant TEQ spectralshapes obtained by these three methods. We note thatthe TEQ spectral shape obtained by TD-OP results indeep nulls (see Fig. 5(a)) although this method is op-timal in terms of SSNR. All subchannels that coincidewith these nulls or are near can only be assigned noneor small number of bits, resulting in an overall TEQfrequency response that is undesirable [9]. The TEQshapes obtained by TD-LS and WFD-LS with Wa(k)weighting function for CSA loop 4 are very similar as


− 60

− 50

− 40

− 30

− 20

− 10

0

10

20

30

− 60

− 70

− 80

− 50

− 40

− 30

− 20

− 10

0

10

20

Mag

nitu

de (

dB)

(a) (b)0 100 200 300

Frequency bins0 100 200 300

Frequency bins

Mag

nitu

de (

dB)

Solid: Wa(k)

Dashed: Wb(k)

Dashdot: Wc(k)

Dotted: Wd(k)

Solid: Wa(k)

Dashed: We(k)

Dashdot: Wf(k)

Dotted: Wg(k)

Fig. 6. (a) TEQ spectral shapes for the 7rst set of weighting functions for CSA loop 6; (b) TEQ spectral shapes for the second set ofweighting functions for CSA loop 6.

shown in Fig. 5(a). WhenWb(k) is used as the weight-ing function for WFD-LS algorithm, the TEQ gainnear the high frequency end is suppressed as wouldbe expected by observing the characteristics shown inFig. 4(a). We also apply these three methods to CSAloop 6 with TEQ length 14, and the results obtainedare consistent with those obtained for CSA loop 4 asshown in Table 1 and Fig. 5(b).Hence as shown in the simulations, WFD-LS

allows for e6ective control of the TEQ shape byselecting proper weighting functions. The TD-OPalgorithm is the most computationally complex onesince it is based on eigenvector computation [15].The computational requirements for the time-domainand frequency-domain LS methods on the other handare similar because they both solve a set of linearequations to obtain the coeLcients of the TEQ.Next, we show the relationship among di6erent

weighting functions, TEQ spectral shapes and the re-sultant SSNR values for the WFD-LS algorithm. TheTEQ length is chosen as 14 and the delay of the targetresponse as 22 for all the following cases. We use theset of weighting functions shown in Fig. 4 for CSAloop 6. As seen in Fig. 6, the TEQ shape follows theweighting function trends in low or high-frequencybins. What is more important to note is the changein the SSNR with di6erent weighting functions (seeTable 2). Higher suppression of the low frequencies

Table 2SSNRs by WFD-LS algorithm for di6erent weighting functionsfor CSA loop 6

Channel SSNR (dB)

Wa(k) 61.83Wb(k) 63.08Wc(k) 63.44Wd(k) 64.03We(k) 60.95Wf(k) 58.99Wg(k) 57.34

leads to decrease in SSNR values and reverse forthe high-frequency bins. However, we note that theSSNR, the index for shortening channel response,may not be the best measure for quantifying optimalperformance.

3. Joint impulse response shortening and noisesuppression for DMT systems

A major advantage of the DMT system is that itsmodulation and demodulation can be implementatedby an eLcient algorithm, the inverse FFT(IFFT)/FFT.However, due to the 7nite length of the FFT operationin the DMT receiver, the neighboring subchannels


interfere with each other, creating leakage e6ect.Hence the narrowband noise and interference can bespread outside the band deteriorating the neighbor-ing subchannel SNRs [7,11]. In [11], Kerckhove andSpruyt propose a constrained mean square error crite-rion for the FDD DMT system to reduce the leakageof stopband noise and interference into the passband.In the following section, we extend this idea by con-sidering suppression of all the noise and interferencewithin the transmission band and incorporating theseinto our least squares cost function de7ned in Eq. (5).

3.1. Stopband WFD-LS algorithm

During the initialization phase, the channel trans-fer function is 7rst identi7ed, then the channel noisepower spectral density is estimated [19]. SupposeU (e−j!k ); k = 0; : : : ; N − 1, is the DFT of the chan-nel noise, then the noise frequency response at theTEQ output is U (e−j!k )A(e−j!k ) where A(e−j!k ) isthe DFT of the TEQ. Thus, we can de7ne a stopbandleast-squares cost function which accounts for the ISIand the noise at the TEQ output:

Esls(�) =N=2∑k=0

W (k)|H (e−j!k )A(e−j!k )− B(e−j!k )|2

+ $N=2∑k=0

Su(k)|A(e−j!k )|2; (11)

where Su(k) ≡ |U (e−j!k )|2 is the power spectrumof the channel noise, $ is the normalized coeLcientto balance between the dual objectives of channelresponse shortening and noise suppression. Hence,the cost function de7ned in Eq. (11) can be usedwhen channel shortening and colored noise suppres-sion need to be jointly achieved. Di6erentiating Esls(�)with respect to each unknown coeLcient ai and bi, andsetting the results to zero, yields

�′�= �′; (12)

where

�′ =

[�′

11 �′12

�′21 �′

22

];

�′11 =

�′0 �′

1 : : : �′M−2

�′1 �′

0 : : : �′M−3

......

�′M−2 �′

M−3 : : : �′0

and �′12, �

′21, and �′

22 are equal to �12, �21 and�22 given in Eq. (6), and

�′ = [− �′1 − �′

2 : : : − �′M−1 pd pd+1 : : : pd+L−1]T;

�′i = �i + $

N=2∑k=0

cos (i!k)Su(k);

≡ �i + $�i; i = 0; : : : ; M − 1;

$ ≡ SC�0

�0:

The variable SC is the suppression coeLcient.The optimal parameters ai (i = 1; : : : ; M − 1) andbi (i = 0; : : : ; L − 1), which can minimize the stop-band least-squares error in Eq. (11), can be obtainedby solving the linear equations in Eq. (12). The re-sulting algorithm is called the stopband WFD-LS(SWFD-LS) algorithm.

3.2. Simulation results for stopband WFD-LSalgorithm

In this section, we apply the stopband WFD-LSalgorithm to jointly shorten the channel impulse re-sponse and suppress noise in the FDD DMT system.We assume that frequency bins 0–45 are used forstopband and subchannels 46–256 for the passband.The CSA loop 6 cascaded with a highpass 7lter isused in the simulations, whose squared transfer func-tion is shown in Fig. 7. The stopband WFD-LS algo-rithm is implemented with N = 512; L= 33 and M =20. The weighting function W (k) assumes its defaultvalue 1 here. The optimal delay d for the target re-sponse is selected for channel shortening only (SC=0)such that the SSNR is maximized, and then the samedelay value is used for the joint channel shorteningand noise suppression case.Suppose we have a “virtual” noise (which is also

used for simulations in [11]) whose power spectraldensity is Aat in subchannels 0 up to 39. In our for-mulation, we let Su(k)= 1 for k =0; : : : ; 39, and 0 for


50 100 150 200 250− 100

− 90

− 80

− 70

− 60

− 50

− 40

Frequency Subchannels

Squ

ared

Cha

nnel

Tra

nsfe

r F

unct

ion

Mag

nitu

de (

dB)

Fig. 7. Channel transfer function for CSA loop 6 cascaded with ahighpass 7lter.

0 50 100 150 200 250 300− 50

− 40

− 30

− 20

− 10

0

10

20

Frequency Subchannels

Squ

ared

TE

Q T

rans

fer

Fun

ctio

n M

agni

tude

(dB

)

TEQ Spectral Responses for Different SC Values

Solid: SC equal to 0.0

Dashed: SC equal to 0.1

Fig. 8. TEQ spectral responses with and without noise suppression.

the remaining frequency bins. Usually, it is impossibleto completely suppress the noise because we cannotachieve zero gain at these noisy frequency bins for theresultant TEQs. In this case, because the inserted“virtual” noise lies in the stopband of the FDD DMTsystem, we de7ne the TEQ’s passband energy toTEQ’s stopband energy ratio (PESER) as themeasure for the noise and interference suppressionfor this case.We run the stopband WFD-LS algorithm to shorten

the channel impulse response with suppression coef-7cient SC equal to 0,0.05, and 0.1, respectively. Notethat when SC is equal to 0, this case corresponds to thedesign for the objective of channel shortening only.Fig. 8 shows the squared transfer functions of the two

Table 3SSNRs and PESERs for di6erent SC values

SC SSNR (dB) PESER (dB)

0 57.92 3.910.05 53.30 57.390.1 53.15 59.03

resultant TEQs by using the stopband WFD-LS algo-rithm for SC equal to 0 and 0.1, respectively. We canobserve in Fig. 8 that when we introduce noise sup-pression for the frequency bins 0–39, the gain of theTEQ at these noisy subchannels decreases substan-tially. Table 3 shows the resultant SSNR and PESERvalues for the three di6erent SC values. As seen in Ta-ble 3, the joint shortening and noise suppression causesfew dB loss in the SSNR but provides very signi7-cant energy suppression of the TEQ in the stopband,which may signi7cantly improve the performance ofthe DMT receiver [11]. Note that the TEQ shape inthe passband is minimally a6ected but its gain is in-creased as the noise suppression term moves mostlythe zeros in the vicinity of the suppression region tothe low frequency end.

4. Joint channel and echo response shortening forDMT systems

In the ADSL standard, two types of DMT systemsare de7ned for full-duplex operation. One is the FDDDMT, where frequency bandwidth is split into up-stream and downstream bands. The other version ofthe DMT system is called EC-based DMT, where thefrequency bands for upstream and downstream datatransmission are overlapped. The lowest subchannelsare not used to avoid the interference with the plainold telephone service (POTS). The frequency rangefrom 30 to 138 kHz is used for upstream data trans-mission, while 30–1104 kHz is used for downstreamdata communication, thus the frequency band 30–138 kHz is overlapped [18]. The EC-based DMT sys-tem can make more eLcient use of the bandwidth atthe cost of more complex modem design, sensitivityto near-end crosstalk and the need for analog circuitwith greater dynamic range. In this transceiver, echocancellation separates two signals traveling simultane-ously in opposite directions in the same channel over


the overlapped frequency bands. Shortening the echoresponse can reduce the complexity of the echo can-celer whose computational requirement is proportionalto the length of the echo impulse response [10,15].

4.1. Joint WFD-LS algorithm

We can incorporate the joint channel and echoresponse shortening idea of [15] into our WFD-LSalgorithm by modifying the least-squares cost func-tion in Eq. (5), to obtain a “joint” least-squares costfunction [21]:

Ejls(�′) =N=2∑k=0

W (k)|H (e−j!k )A(e−j!k )− B(e−j!k )|2

+&N=2∑k=0

W ′(k)|G(e−j!k )A(e−j!k )

−C(e−j!k )|2; (13)

where echo transfer function G(e−j!k ) is approxi-mated by an ARMA model C(e−j!k )=A(e−j!k ), whilethe channel response is assumed to be, as before, givenby B(e−j!k )=A(e−j!k ). Here we assume that the delayand the length of the target echo response c or its fre-quency response C(e−j!k ) are de and S, respectively.Hence �′ = [a1; : : : ; aM−1; b0; : : : ; bL−1; c0; : : : ; cS−1]T

(again a0 is assumed to be 1). This pole-zero-zeromodel assumption is valid for practical cases and isalso used in [15]. In Eq. (13), & is de7ned as thenormalized coeLcient, which can balance the twoshortening purposes.If we di6erentiate Ejls(�′) with respect to each un-

known coeLcient ai; bi and ci, and set the results tozero, we obtain another set of linear equations, thistime given by

�′′�′ = �′′; (14)

where

�′′ =

�′′

11 �′′12 �′′

13

�′′21 �′′

22 0

�′′31 0 �′′

33

;

�′′11 =

�′′0 �′′

1 : : : �′′M−2

�′′1 �′′

0 : : : �′′M−3

......

�′′M−2 �′′

M−3 : : : �′′0

;

�′′13 =

−&p′′de−1 −&p′′

de: : : −&p′′

de+S−2

−&p′′de−2 −&p′′

de−1 : : : −&p′′de+S−3

......

−&p′′de−M+1 −&p′′

de−M+2 : : : −&p′′de+S−M

;

�′′31 =

−p′′de−1 −p′′

de−2 : : : −p′′de−M+1

−p′′de

−p′′de−1 : : : −p′′

de−M+2

......

−p′′de+S−2 −p′′

de+S−3 : : : −p′′de+S−M

;

�′′33 =

�′′0 �′′

1 : : : �′′S−1

�′′1 �′′

0 : : : �′′S−2

......

�′′S−1 �′′

S−2 : : : �′′0

:

The submatrices �′′12, �

′′21 �

′′22 are equal to �12, �21

and �22 given in Eq. (6), and

�′′ = [− �′′1 − �′′

2 : : : − �′′M−1 pd pd+1

: : : pd+L−1 p′′de

p′′de+1 : : : p′′

de+S−1]T;

where

�′′i = �i + &

N=2∑k=0

|G(e−j!k )|2 cos(i!k)W ′(k)

≡ �i + &)i; i = 0; : : : ; M − 1;

&= JSC�0

)0;

p′′i =

N=2∑k=0

[R′k cos(i!k)− I ′k sin(i!k)]W ′(k);

i = de −M + 1; : : : ; de + S − 1;

�′′i =

N=2∑k=0

cos(i!k)W ′(k); i = 0; : : : ; S − 1


0 20 40 60 80 100 120 140 160 180 200− 1

− 0.5

0

0.5

1

(a)

0 20 40 60 80 100 120 140 160 180 200− 1

− 0.5

0

0.5

1

(b)

0 20 40 60 80 100 120 140 160 180 200− 1

− 0.5

0

0.5

1

(c) Sample Number

Fig. 9. (a) Original channel response; (b) shortened channel re-sponse by WFD-LS algorithm; (c) shortened channel response byJWFD-LS algorithm (JSC=0.1).

and

R′k = Re{G(e−j!k )}; I ′k = Im{G(e−j!k )};k = 0; : : : ; N=2:

Here JSC is de7ned as the joint shortening coeL-cient. Hence solution of the set of linear equationsshown in Eq. (14) yields the optimal parametersai (i=1; : : : ; M −1), bi (i=0; : : : ; L−1) and ci (i=0;: : : ; S−1), such that the “joint” least squares cost func-tion in Eq. (13) is minimized. The resulting algorithmis called the joint WFD-LS (JWFD-LS) algorithm.

4.2. Simulation results for joint WFD-LS algorithm

We show the impact of the TEQ on joint shorteningof channel and echo response by the following simu-lations. The original channel and echo responses usedin the simulations are shown in Figs. 9(a) and 10(a),respectively. The curve shown in Fig. 9(a) is the nor-malized channel response of CSA loop 6 cascadedwith a 30–1000 kHz bandpass 7lter, whose squaredtransfer function before normalization is shown in Fig.3. The curve in Fig. 10(a) shows the normalized echoresponse based on the simpli7ed echo path model in[5], and the bandwidth used is 30–135 kHz. We useWFD-LS algorithm for channel only shortening, and

0 50 100 150− 1

− 0.5

0

0.5

1

(a)

0 50 100 150− 0.2

− 0.15− 0.1

− 0.050

0.050.1

0.15

(b)

(c)0 50 100 150

− 0.15

− 0.1

− 0.05

0

0.05

0.1

Sample Number

Fig. 10. (a) Original echo response; (b) shortened echo responseby WFD-LS algorithm; (c) shortened echo response by JWFD-LSalgorithm (JSC=0.1).

JWFD-LS algorithm for joint channel and echo re-sponse shortening. We chooseM=16, L=33 for bothchannel only and joint shortenings; S =33 and de =0for the JWFD-LS algorithm. The weighting functionsW (k) and W ′(k) assume their default value 1. Thejoint shortening coeLcient (JSC) is chosen as 0.1, 0.5and 1.0 for the JWFD-LS algorithm. In WFD-LS andJWFD-LS algorithms, unsuitable choice of the delayd of the target channel response could cause a perfor-mance degradation. We observed that the optimum de-lays for channel only and joint shortening algorithmsusually are di6erent. To ensure a fair comparison of thetwo algorithms mentioned, we search for the optimaldelay of the target channel response to obtain the bestperformances in terms of shortened channel SSNR forboth channel only and joint shortening cases.Fig. 9 shows the original and shortened channel re-

sponses for both channel shortening only and jointshortening with JSC equal to 0.1. Fig. 10 shows thecorresponding original and shortened echo responsesfor WFD-LS and JWFD-LS algorithms. The reducedmaximum magnitude of the shortened echo responseis due to the low gain of the resultant TEQ at thefrequency range 30–135 kHz where most echo signalenergy lies. The joint shortening results for JSC equal


Table 4Channel and echo SSNR values for WFD-LS and JWFD-LS al-gorithms

Channel EchoSSNR (dB) SSNR (dB)

WFD-LS 61.92 23.89JWFD-LS (JSC = 0:1) 60.44 47.99JWFD-LS (JSC = 0:5) 60.38 50.78JWFD-LS (JSC = 1:0) 60.31 51.45

to 0.5 and 1.0 are not shown in Figs. 9 and 10 becausethey are very similar to those when JSC is 0.1. Wenote that both WFD-LS and JWFD-LS can shorten theecho response, but the JWFD-LS can achieve a muchbetter result at the expense of small loss in the chan-nel shortening. Table 4 shows the resultant channeland echo SSNR values for WFD-LS and JWFD-LS al-gorithms. We can see that JWFD-LS algorithm losesfew dBs in the SSNR for channel response shorten-ing compared to WFD-LS algorithm but provides asigni7cant gain in the echo response shortening.We also apply the time-domain joint LS shorten-

ing method [15] to the above normalized channel andecho responses shown in Figs. 9(a) and 10(a), similarresults are obtained compared to those by JWFD-LSalgorithm with weighting functions equal to 1.

5. Conclusions

We introduce a frequency-domain approach for de-signing the time-domain equalizer for the DMT sys-tem. The least-squares cost de7ned in the frequencydomain allows for control of the TEQ magnitude re-sponse by a weighting function. The performance ofthe algorithm derived, WFD-LS, is comparable to thatof time-domain least squares TEQ design when noweighting is used. As the simulation results suggest,by appropriate choice of the weighting function, theresulting TEQ shape can be controlled with the objec-tive of improving the channel throughput. The TEQshape also a6ects the SSNR values achieved, e.g., sup-pression of the higher frequency bands improving theSSNR, approaching the values obtained by the optimaltime-domain design approach, TD-OP, for the caseswe studied. When noting the importance of the result-ing TEQ shape on the overall DMT performance, it

is also important to emphasize the presence of deepnulls in the spectrum achieved by the TD-OP leadingto an undesirable overall TEQ response even thoughit is the best in terms of maximizing the SSNR, theobjective function used in its design.The ultimate goal in the TEQ design is to maxi-

mize the channel throughput that is determined by theoverall SNRs at the output of the FFT block. Refs.[1,3,9,13] address this problem by de7ning a criterionfor bit rate maximization, the geometric SNR. Thecriterion is de7ned at the TEQ output hence cannotaccount for the leakage e6ect introduced by the FFT7lter-bank in the DMT receiver. Our TEQ schemeprovides the added Aexibility of controlling the TEQspectral shape in the presence of colored noise. How-ever, the choice of the optimal TEQ spectral shapeand the achievable maximum channel throughput forthe DMT system with 7nite-length FFT operation arestill open issues. Besides the weighting choice, theparameter choice, e.g. the 7lter order and the bestdelay values are important topics that need furtherinvestigation.We also noted two other important extensions with

the TEQ: echo and channel response shortening forthe echo-cancellation-based DMT systems for reduc-ing the complexity of the echo canceler and joint noisesuppression and channel response shortening for im-proving the overall DMT performance. For both cases,we can easily de7ne the appropriate objectives directlyin the frequency domain. The stopband WFD-LS al-gorithm designed for noise suppression and channelresponse shortening can create deep notches in thetransfer function of the TEQ very e6ectively, whilethe weighting function in the WFD-LS algorithm canproduce smooth change of the TEQ spectral shape.Besides o6ering the ability to control the TEQ

spectrum e6ectively and suggesting additional advan-tages in implementation, e.g., by relaxing the require-ments for synchronization, a frequency-domain TEQapproach also o6ers the added advantage of enablingderivation of adaptive approaches for tracking chan-nel variations. Noise sources in DMT systems, suchas crosstalk, can vary rapidly as service is added ordeleted on one or more of the loops in the bundledcable. Also, AM radio stations are known to changetheir power levels during the day, constituting anothersource of time-varying noise [19]. All these changesrequire the update of the TEQ for best performance


during data transmission. In [20], we introduce anadaptive weighted frequency-domain least-squaresapproach to adapt the TEQ to changing channel en-vironment such as the noise spectrum. The adaptiveWFD-LS algorithm solves a set of linear equationsrecursively and hence allows adaptive control of theTEQ shape during data transmission.

Acknowledgements

The authors would like to thank AleksandarPurkovic for his comments and input.

References

[1] N. Al-Dhahir, J.M. CioL, Optimum 7nite-length equalizationfor multicarrier transceivers, IEEE Trans. Commun. 44 (1)(January 1996) 56–64.

[2] N. Al-Dhahir, J.M. CioL, ELciently computed reduced-parameter input-aided MMSE equalizers for ML detection: auni7ed approach, IEEE Trans. Inform. Theory 42 (3) (May1996) 903–915.

[3] N. Al-Dhahir, J.M. CioL, A bandwidth-optimizedreduced-complexity equalized multicarrier transceiver, IEEETrans. Commun. 45 (8) (August 1996) 948–956.

[4] J.A.C. Bingham, F. van der Putten, T1. 413 Issue 2: StandardsProject for Interfaces Relating to Carrier to CustomerConnection of Asymmetrical Digital Subscriber Line(ADSL) Equipment, ANSI Document, No. T1E1.4/97-007R6,September 1997.

[5] W.Y. Chen, DSL Simulation Techniques and StandardsDevelopment for Digital Subscriber Line Systems,Macmillian Technical Publishing, 1998.

[6] J.S. Chow, J.M. CioL, J.A.C. Bingham, Equalizer train-ing algorithms for multicarrier modulation systems, in:Proceedings of the International Conference on Communica-tions, Geneva, Switzerland, May 1993, pp. 761–765.

[7] J.M. CioL, V. Oksman, J.J. Werner, T. Pollet, P.M.P. Spruyt,J.S. Chow, K.S. Jacobsen, Very-high-speed digital subscriberlines, IEEE Commun. Mag. (April 1999) 72–79.

[8] M.L. Doelz, E.T. Heald, D.L. Martin, Binary datatransmission techniques for linear system, Proc. IRE 45 (May1957) 656–661.

[9] B. Farhang-Boroujeny, M. Ding, An eigen-approach to thedesign of near-optimum time domain equalizer for DMTtransceivers, in: Proceedings of the International Conferenceon Communications, Vancouver, Canada, June 1999.

[10] M. Ho, Multicarrier echo cancellation and multichannelequalization. Ph.D. Thesis, Stanford University, June 1995.

[11] J.V. Kerckhove, P. Spruyt, Adapted optimization criterionfor FDM-based DMT-ADSL equalization, in: Proceedings ofthe International Conference on Communications, Dallas, TX,June 1996, pp. 1328–1334.

[12] N. Lashkarian, S. Kiaei, Fast algorithm for 7nite-lengthMMSE equalizer with application to discrete multitonesystems, in: Proceedings of the IEEE International Conferenceon Acoustics, Speech, and Signal Processing, Phoenix, AZ,March 1999.

[13] N. Lashkarian, S. Kiaei, Optimum equalization of multicarriersystems via projection onto convex set, in: Proceedings ofthe International Conference on Communications, Vancouver,Canada, June 1999.

[14] E.C. Levy, Complex-curve 7tting, IRE Trans. Automat.Control AC-4, (May 1959) pp. 37–44.

[15] P.J.W. Melsa, R.C. Younce, C.E. Rohrs, Impulse responseshortening for discrete multitone transceivers, IEEE Trans.Commun. 44 (12) (December 1996) pp. 1662–1672.

[16] M. Na7e, A. Gatherer, Time-domain equalizer training forADSL, in: Proceedings of the International Conferenceon Communications, Montreal, Canada, June 1997,pp. 1085–1089.

[17] A. Peled, A. Ruiz, Frequency domain data transmissionusing reduced computational complexity algorithms, in:Proceedings of the IEEE International Conference onAcoustics, Speech, and Signal Processing, Denver, CO, April1980, pp. 964–967.

[18] D.J. Rauschmayer, ADSL/VDSL Principles: A Practical andPrecise Study of Asymmetric Digital Subscriber Lines andVery High Speed Digital Subscriber Lines, MacmillianTechnical Publishing, 1998.

[19] T. Starr, J.M. CioL, P.J. Silverman, Understanding DigitalSubscriber Line Technology, Prentice-Hall, Englewoods, NJ,1999.

[20] B. Wang, Frequency domain approaches for equalizer designfor discrete multitone systems, Ph.D. Dissertation, Universityof Maryland, Baltimore, May 2000.

[21] B. Wang, T. Adal,, An eLcient training methodfor equalization of discrete multitone transceivers, in:Proceedings of IEEE/IEE International Conference onTelecommunication, Vol. 2, Korea, June 1999, pp. 189–193.

[22] B. Wang, T. Adal,, Time-domain equalizer design for discretemultitone systems, in: Proceedings of the IEEE InternationalConference on Communications (ICC), New Orleans, LA,June 2000.

[23] S.B. Weinstein, P.M. Ebert, Data transmission by frequencydivision multiplexing using the discrete Fourier transform,IEEE Trans. Commun. 19 (5) (October 1971) 628–634.

[24] R.C. Younce, P.J.W. Melsa, S. Kapoor, Echo cancellationfor asymmetrical digital subscriber lines, Proceedings of theInternational Conference on Communications, New Orleans,LA, May 1994, pp. 301–306.

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