1

Accepted in IEEE Signal Processing Letters

Improved Eigenfilter Design Method for Channel Shortening Equalizers for DMT Systems Toufiqul Islam, Member, IEEE, M. Shafi Al Bashar, Satya Prasad Majumder and Md. Kamrul Hasan

Abstract— In this paper, we propose an improved eigenfilter design method for time domain equalizer (TEQ) design for discrete multitone (DMT) systems. In conventional DMT systems, TEQs used tend to introduce spectral nulls which degrade the achievable signal-to-noise ratio at the corresponding subcarriers. As a result, bandwidth efficiency decreases. Tkacenko and Vaidyanathan recently proposed a low-complexity eigenfilter method which though performs nearly optimally in terms of observed bit rate but suffers performance loss due to spectral attenuation. We present a joint cost function which, along with good channel shortening, avoids spectral nulls in the useful signal band. Simulation results show that our method allows more subcarriers to carry bits and thus outperforms conventional eigenfilter method in terms of achievable bit rate. We also propose a heuristic choice of optimal transmission delay which yields profitable bit rate performance. Index Terms— channel shortening, time domain equalizers, discrete multitone, cyclic prefix, eigenfilter.

I. I NTRODUCTION

D

MT is a multicarrier modulation technique used for data transmission over twisted-pair (TP) lines, as in asymmetric digital subscriber line (ADSL) and very-high-datarate digital subscriber line (VDSL) systems. In multicarrier systems, intersymbol interference (ISI) and intercarrier interference (ICI) can be avoided by inserting cyclic prefix (CP) between consecutive symbols. However, the CP length ν needs to be no less than the considerable length of the TP channels, thus significantly reducing the bandwidth efficiency by a factor of N/(N + ν), where N denotes the FFT size. An attractive technique to combat the interference is the use of a TEQ in the receiver front end to shorten the channel, which in turn allows the CP to be confined to a predefined length. It was reported in [1] that two constraints need to be met by TEQs for good throughput performance: 1) the amplitude response associated with the shortened impulse response (SIR) shall have no spectral nulls (if original channel is free of spectral nulls); and 2) the SIR memory need to be shorter than a predefined CP length. Several methods proposed so far for the design of TEQs fall under the category of eigenfilter design method [1]–[7]. Minimum mean-squared error (MMSE) TEQ design method proposed in [2] minimizes the time-domain error between the TEQ output and a desired output, i.e. the output of an FIR filter of order ν. Another design technique called the T. Islam and S. P. Majumder are with the Department of Electrical and Electronic Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, M. S. Al Bashar is with the Department of Electrical and Computer Engineering, University of California, Davis, USA and M. K. Hasan is with the East West University, Dhaka, Bangladesh.

MSSNR method [3] attempts to maximize the shortening signal-to-noise ratio (SSNR), where the SSNR is defined as the ratio of energy inside a target window to the energy outside the window. The MMSE and MSSNR method exhibit equally spaced spectral nulls in the frequency domain, and will likely clobber a good subchannel [5]. Min-ISI method proposed in [4] partitions the SIR into signal path, ISI path and noise path. The MMSE, MSSNR and Min-ISI method require on the same order of multiplication and accumulation (MAC) operations [8], but Min-ISI method needs subchannel SNR information for weighting ISI in the frequency domain [4]. In [6], a computationally less extensive technique called minimum delay spread (MDS) method is presented which minimizes the delay spread of the effective channel. In [7], a low complexity eigenfilter method is presented which modified the MDS method incorporating cyclic prefix length and effects due to noise. The eigenfilter method proposed in [7] is shown to achieve near optimal bit rate performance. Though [7] requires Cholesky factor computation only once, this method also generates spectral nulls which hinders some potential subcarriers to carry bits. In this paper, we modify the eigenfiler method presented in [7]. We propose a new objective function which explicitly attempts to remove the spectral nulls in the frequency response of the shortened channel. Apart from minimizing residual power of SIR and filtered noise power, delay spread minimization of the desired portion of SIR with respect to a suitable time reference is incorporated to ensure that frequency response flattens to remove spectral nulls and hence allows to transmit more data over the useful signal spectrum. Delay dependent matrices in the proposed method can be updated for each transmission delay ∆ following the guidelines of [9] which further reduces computational burden. We also propose a heuristic choice of optimum transmission delay which not only allows solving generalized eigenvector (EV) problem only once for TEQ design but also yields profitable bit rate performance. II. R EVIEW OF E IGENFILTER M ETHOD Consider the SISO channel/equalizer model of Fig. 1. We assume that the input x(k) is zero mean and white with variance σx2 , noise u(k) is zero mean wide-sense stationary (WSS) random process with autocorrelation matrix Ru , and the processes x(k) and u(k) are uncorrelated. The channel h , [h(0), h(1), . . . , h(Lh − 1)]T and the TEQ w , [w(0), w(1), . . . , w(Lw − 1)]T are known as finite impulse response (FIR) filters of length Lh and Lw , respectively. In vector form, the effective channel impulse response c (of

2

Accepted in IEEE Signal Processing Letters

A. Development of a new objective function

u(k) x(k)

r(k)

h

y(k)

w

c Fig. 1.

System model with channel/equalizer.

length Lc = Lh + Lw − 1) can be written as c = Hw, where c = [c(0), c(1), . . . , c(Lc − 1)]T , and H is the tall convolution matrix of the unequalized channel h, which is Lc × Lw Toeplitz. The output y(k) is given by y(k) = x(k) ∗ c(k) + u(k) ∗ w(k) = xf (k) + q(k) (1) where xf (k) and q(k) are filtered signal and noise processes, respectively. The method in [7] attempts to optimize the TEQ to shorten the effective channel c(k) and minimizes the filtered noise power σq2 with respect to the filtered signal power σx2f . The objective function is given by Jeig , αJshort + (1 − α)Jnoise ,

0≤α≤1

where Jshort and Jnoise are defined as follows: P (k − ∆)|c(k)|2 k fP Jshort , 2 k |c(k)| 2 σq2 σq P = Jnoise , σx2 k |c(k)|2 σx2f

(2)

w

(7)

D(∆) , ILc − G(∆)

(8)

and The residual and desired portion of the channel can be expressed, respectively, as cres

=

D(∆)Hw

(9)

cdes

=

G(∆)Hw

(10)

and (4)

The penalty function penalizes uniformly samples outside k ∈ [∆, ∆+ν] where ∆ ∈ [0, Lc −ν −1]. For each ∆, the optimum TEQ by eigenfilter method is obtained as = argmin Jeig

G(∆) , diag[01×∆ , 11×(ν+1) , 01×(Lc −∆−ν−1) ]

(3)

Here Jshort and Jnoise are, respectively, the channel shortening and noise suppression objective functions, and α is a trade-off parameter. The penalty function f (k) is formulated as ½ 0, 0 ≤ k ≤ ν f (k) , (5) 1, otherwise.

weig,∆

Similar to approaches in [3] and [4], here the effective channel response is divided into two parts– desired portion cdes and residue portion cres . The proposed objective function comprises the following goals: 1) Minimize the energy of the residual portion cres to minimize ISI. 2) Minimize the delay spread of the desired portion cdes with respect to a time reference so that it approaches a delta function. This is to ensure flat frequency response and thus to avoid encountering nulls. 3) Minimize noise power σq2 . Now, we will proceed to develop the proposed objective function. We define two diagonal window matrices, each of size Lc × Lc , to separate the desired and residue portion of the effective channel c for a particular ∆ as

Then the cost function for minimizing the residual energy can be written as wT HT D2 (∆)Hw cres T cres (11) = wT HT Hw cT c To achieve goal 2), we introduce the following cost function for minimizing delay spread within the window with respect to a time reference kn . This will penalize the desired portion cdes if it deviates from the shape of the delta function.

Jres

,

Jdes

,

(6)

Delay parameter ∆ is varied over the useful range and the optimum delay is chosen for the best bit rate. III. A NALYSIS The aforementioned method has been proposed to lower the computational cost, especially for the Cholesky factor computation. But unfortunately [7] does not have any control over the frequency response of the TEQ. Optimal TEQ obtained by this method has deep nulls in the frequency domain. Those subchannels with deep nulls become useless. As a result, the presence of these nulls in the magnitude response of the shortened channel reduces the total achievable bit rate of the DMT systems. If the equalizer design problem can incorporate a technique for eliminating these nulls in addition to the channel shortening, it would be possible to achieve much higher bit-rate. In this section, a proposal is presented to achieve this goal.

=

∆+ν 1 X (k − kn )2 |cdes (k)|2 Ec

1 Ec

k=∆ ∆+ν X

(k − kn )2 |c(k)|2

(12)

k=∆

where Ec is the energy of effective channel. Here, kn is taken as the mid-position of the window. Hence, from (12) we get Jdes =

wT HT Λ2 (∆)Hw cT Λ2 (∆)c = wT HT Hw cT c

where

(13)

Lc elements

{ }| z Λ(∆) = diag[0, . . . , 0, ∆, ∆ + 1, . . . , ∆ + ν, 0, . . . , 0] | {z } ∆ zeros

−kn G(∆) (14) The cost function to minimize noise power at the output of the TEQ is defined as Jnoise ,

wT Ru w σx2 wT HT Hw

(15)

3

Accepted in IEEE Signal Processing Letters

Apparently, it might seem that if goal 2) is satisfied then goal 1) would be automatically ensured. However, a closer inspection would suggest that satisfying goal 2) only may not necessarily minimize the residual energy– which is the main target of channel shortening. Clearly, trade-off parameters among these objectives are required for optimum solution. Defining α and β as two trade-off parameters, final objective function can be written as J , βJres + (1 − α − β)Jdes + αJnoise

(16)

1.5 1

Original channel Shortened channel

k =30 ref

0.5 0 −0.5

∆

=23

opt

−1

Boundary Indicates desired portion

∆heu=26

−1.5 −2 −2.5 0

B. Optimum TEQ design For each ∆, optimum TEQ for the proposed method can be obtained by minimizing J, i.e., wT Xw (17) w∆ = argmin J = argmin T w w w Yw where α X = βHT D2 (∆)H + (1 − α − β)HT Λ2 (∆)H + 2 Ru σx Y = HT H Here Y is independent of delay and Cholesky factorization for Y has to be calculated only once over the possible range of ∆. w∆ will be the generalized eigenvector corresponding to the smallest generalized eigenvalue of the matrix pair (X, Y) for the particular ∆. Delay parameter ∆ is varied and final TEQ wopt is obtained for ∆opt which yields the maximum bit rate. Fig. 2 shows the original channel and SIR by the proposed method for ∆opt = 23, Lw = 17 , ν = 32, α = 0.399 and β = 0.6 (up to 200 samples shown). Higher value of β is used to set higher priority for suppression of residual portion. Now, residual and desired portion of H, Hres = D(∆)H and Hdes = G(∆)H, respectively, can be updated for each ∆ easily by following [9] which will significantly reduce the complexity of solving generalized EV problem for all ∆ values. For example, computing A(∆) = HTres Hres from A(∆−1) requires only Lw (Lw +ν) MACs against L2w (Lh −ν) MACs required to compute A(∆) each time. C. Heuristic choice of optimum ∆ How to find the optimum ∆ without tracing its whole range is still an open problem for delay optimized TEQ design methods. Many heuristic choices have been proposed for the optimum delay. One of the choices is the length of the cyclic prefix. Here, we propose a criteria for choosing a heuristic ∆ for which bit rate performance is roughly between that of [7] and the proposed method. The heuristic choice is based on the proportion of energy distribution on either side of a particular time reference kref of the original channel. In this letter, we consider index of the maximum value of channel h as Pkref −1vector the time reference kref . Defining El , k=0 h2 (k) and PLh −1 Er , k=k h2 (k), heuristic ∆ can be obtained as (see ref +1 Fig. 2) ¸ · El (ν + 1) (18) ∆heu = kref − round El + Er Performance for this choice of ∆ is examined in Section IV.

50

100 k

150

200

Fig. 2. Original and equalized channel impulse responses for CSA loop # 1. Location of kref and ∆heu are shown for that channel.

IV. E XPERIMENTAL R ESULTS We now proceed to analyze how our design method compares with [7]. As the literature appears to be moving towards the goal of perpetually increasing the bit rate, we opted to compare on the basis of achievable bit rate. The channels used are eight standard downstream CSA loops commonly used in ADSL system simulation (obtained from [10]). We add a fifthorder Chebyshev highpass filter with cutoff frequency of 4.5 kHz and passband ripple of 0.5 dB to each CSA loop to take into account the effect of the splitter at the transmitter. The DC channel, channels 1-3, and the Nyquist channel are not used. ADSL system simulation parameters are listed below. −7 • Desired probability of error is 10 . • DFT size, N = 512 and sampling frequency, fs = 2.208MHz. • ν = 32, Lw = 16 and Lh = 512. • Input signal x(k) consists of QAM symbols. 2 • Input power, σx = 21 dBm, SNR gap, Γ = 9.8 dB (For uncoded QAM constellations, Γ = 9.8 dB for a symbol error probability of 10−7 ) • Input noise consists of near-end crosstalk (NEXT) noise plus additive white noise with power density -110 dBm/Hz. As the input consists of two-dimensional QAM symbols, the number of bits to allocate in the ith subchannel is, º ¹ SNRi ) , 0≤i≤N −1 (19) bi = log2 (1 + Γ with Γ = 9.8 dB here. Here, b.c indicates floor operation. We assume that the subchannels are mutually isolated from each other and sufficiently narrowband. SNRi is taken as the ratio of desired signal power to residual plus noise power on subcarrier i [4]. From this, the bit rate Rb was calculated using, fs X bi bps (20) Rb = N +ν i In Figs. 3(a) and 3(b), original and equalized channel frequency response and the corresponding bit allocation into different subcarriers are shown, respectively. From Fig. 3(a), it is clear that channel equalized by eigenfilter method in [7] contains several nulls in the useful signal band whereas the proposed method eliminates those nulls. Thus the proposed method increases the possibility of achieving higher bit rate

4

Accepted in IEEE Signal Processing Letters 3

bit rate (Mb/s)

2.5 2 1.5 1 0.5 0

1

2

3

4 5 CSA Loop Index

6

7

8

Fig. 4. Observed bit rate for CSA loop # 1-8 using various TEQ design methods. From left to right, height of the bars for each CSA loop denote bit rate obtained by MMSE-UEC [2], MSSNR [3], MDS [6], eigenapproach [1], Min-ISI [4], eigenfilter [7], proposed method with and without ∆heu , respectively.

Magnitude (dB)

0 (a) 10 20 h proposed eigenfilter

30 40 0

0.2

0.4 0.6 Normalized frequency

Number of bits

8 (b)

6 4

0.8

1

eigenfilter proposed without β proposed with ∆heu

2 0

0

50

100 150 Subcarrier Index

200

250

Fig. 3. Frequency response of original and equalized channel in (a) and bit allocation over subcarriers in (b) for TEQ designed by eigenfilter method [7] and the proposed method, respectively, for CSA loop # 1.

by allowing more subcarriers to carry bits by flattening the frequency response in the useful signal band. From Fig. 3(b), we notice that shortened channel by eigenfilter method cannot carry data bits in potential subcarriers like 63,109-112 etc. due to attenuation. Moreover, proposed method not only makes those subcarriers useful but also lets more subcarriers (up to 154, which is 134 for [7]) to carry bits. In Fig. 3(b), we see that the proposed method with ∆heu achieves better performance as well. To emphasize the importance of using two trade-off parameters, we also plotted bit allocation for shortened channel obtained by optimizing J = α(Jres + Jdes ) + (1 − α)Jnoise instead of (16) (i.e., without using β). As different weighting is performed for desired and residual portion, single tradeoff parameter cannot successfully adjust suitable weighting between noise and shortening objective functions. Without using β, bit rate comes down as low as 0.65 Mbps for CSA loop # 1. Generally, α ∈ [0.39, 0.399] and β ∈ [0.588, 0.6] yield good results for all the test channels. In Fig. 4, we have shown comparative achievable bit rate performance of the proposed method with some state-of-the-art methods for CSA loop # 1-8. Achievable bit rates for each method is computed using (20). For each method considered except MDS method [6], we varied the delay parameter ∆ and chose the value that yielded the best bit rate. From Fig. 4, it is clear that the proposed method gets very close to Min-ISI method and

achieves higher bit rate than the other methods. Like MDS and eigenfilter method, the proposed method also requires only one Cholesky factor computation (for Y in (17)) for all ∆ values. Note that the proposed method using ∆heu requires generalized EV problem solving only once. With that significant computational advantage, bit rate performance for ∆heu manifested in Fig. 4 can be considered profitable. V. C ONCLUSION In this letter, we have modified the objective function of [7] to account for the spectral nulls. The weighting matrices can be updated for each delay to further reduce the computational complexity. We have also proposed a heuristic method for computing optimum value of the delay parameter. The experimental results have demonstrated that the proposed method achieves bit rate very close to that of Min-ISI method and higher than that of the other methods for all the CSA loops considered. R EFERENCES [1] B. Farhang-Bouroujeny and M. Ding, “Design methods for time-domain equalizers in DMT transceivers,” IEEE Trans. Commun., vol. 12, pp. 1450–1455, 2000. [2] N. A. Dhahir and J. M. Cioffi, “Effficiently computed reduced-parameter input-aided MMSE equalizers for ML detection ,” IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 903–915, May 1996. [3] P. J. Melsa, R. C. Younce, and C. E. Rohrs, “Impulse response shortening for discrete multitone transceivers,” IEEE Trans. Commun., vol. 44, pp. 1662–1672, Dec. 1996. [4] G. Arslan, B. L. Evans, and S. Kiaei, “Equalization for discrete multitone transceivers to maximize bit rate,” IEEE Trans. Signal Processing, vol. 49, no. 12, pp. 3123–3135, Dec. 2001. [5] R. K. Martin, M. Ding, B. L. Evans, and C. R. Johnson, Jr, “Infinite Length Results and Design Implications for Time-Domain Equalizers”, IEEE Transactions on Signal Processing, vol. 52, no. 1, pp. 297-301, Jan. 2004. [6] R. Schur and J. Speidel, “An efficient equalization method to minimize delay spread in OFDM/DMT systems,” in Proc.IEEE Int. Conf. Commun., Helsinki, Finland, June 2001, vol. 1, pp. 1–5. [7] A. Tkacenko and P. P. Vaidyanathan, “A low-complexity eigenfilter design method for channel shortening equalizers for DMT systems,” IEEE Trans. Commun., vol. 51, no. 7, pp. 1069–1072, July 2003. [8] R. K. Martin, K. Vanbleu, M. Ding, G. Ysebaert, M. Milosevic, B. L. Evans, M. Moonen, and C. R. Johnson. Jr., “Implementation complexity and communication performance tradeoffs in discrete multitone modulation equalizers,” IEEE Trans. Signal Processing, vol. 54, no. 8, pp. 3216–3230, Aug. 2006. [9] R. K. Martin, M. Ding, B. L. Evans and C. R. Johnson. Jr., “Efficient Channel Shortening Equalizer Design,” EURASIP Journal on Applied Signal Processing, vol. 2003, no. 13, pp. 1279–1290, 2003. [10] G. Arslan, M. Ding, B. Lu, M. Milosevic, Z. Shen and B. L. Evans , “MATLAB DMTTEQ Toolbox 3.1,” The University of Texas at Austin, May 10, 2003.