Noise Optimized Minimum Delay Spread Equalizer Design for DMT Transceivers Toufiqul Islam 1 , Satya Prasad Majumder 1 and Md. Kamrul Hasan 1,2 1 Department of Electrical and Electronic Engineering Bangladesh University of Engineering and Technology, Dhaka, Bangladesh. 2 Department of Electrical and Electronic Engineering East West University, Dhaka, Bangladesh. E-mail: {toufiq56, khasan, spmajumder}@eee.buet.ac.bd Abstract— Time-domain equalizer (TEQ) design for multicarrier transceivers has recently received much attention. In this paper, we consider generalization of one such design method which takes into account the noise observed in discrete multitone (DMT) systems. We propose an iterative TEQ design method which jointly minimize delay spread of the channel and filtered noise at the output of the equalizer. Experimental results show that our method can successfully minimize delay spread and noise when compared to other reported techniques.

I. Introduction Channel shortening, a generalization of equalization, has recently become necessary in receivers employing multicarrier modulation (MCM). MCM techniques like orthogonal frequency division multiplexing (OFDM) and DMT have been deployed in applications such as the wireless LAN standards IEEE 802.11a and HIPERLAN/2, digital audio broadcast (DAB) and digital video broadcast (DVB) in Europe, and asymmetric and very-high-speed digital subscriber loops (ADSL, VDSL). MCM is attractive because intercarrier interference (ICI) and intersymbol interference (ISI) can be avoided by inserting a cyclic prefix (CP) between consecutive symbols. If the channel impulse response (CIR) spans no more than ν + 1 samples, where ν is the CP length, then linear convolution with channel turns into circular convolution, enabling one-tap frequency domain equalization (FDE). However, the CP reduces the bandwidth efficiency by the factor N/(N + ν), where N is the length of the DMT symbol. Hence, ν cannot be high as it in turn reduces bit rate. A well-known technique to combat the ICI/ISI caused by the inadequate CP length is the use of a time-domain equalizer (TEQ) in the receiver front end. The TEQ is a finite impulse response filter that shortens the channel so that the delay spread of the combined channel-equalizer effective impulse response (EIR) is not longer than the CP length. Several TEQ designs have been proposed [1],[2],[3],[4]. One of the drawback of these methods is the need to search over the range of admissible transmission delays, ∆ in order to design optimum TEQ. In [5], a unique TEQ design method is proposed which claims to minimize the delay spread of the overall channel impulse response (called MDS method). MDS method is computationally less intensive as it does not require to search for optimum ∆. This approach is independent of the CP length and attempts to

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squeeze the EIR as much as possible. This is advantageous since EIR squeezing allows further to reduce CP length to increase data rate and provides additional robustness to synchronization offset. Recently, in [6], MDS method is modified to account for the true time reference about which delay spread needs to be minimized. Unfortunately, both these methods, [5] and [6], do not account for any knowledge of noise statistics. Due to noise source models for DMT systems, such as near-end crosstalk (NEXT) and farend crosstalk (FEXT) [7], it is only natural to exploit such knowledge to obtain a more robust equalizer. In this paper, we generalize the method of [6] to some aspects. First, we modify the cost function to account for minimization of noise. Secondly, a trade-off parameter is also introduced to set appropriate weight for minimizing delay spread and noise. Thirdly, we propose an iterative algorithm for TEQ update where trade-off parameter is adaptively adjusted at each iteration under a specific criteria. Simulation results provided show that our method performs comparatively well in terms of delay spread minimization and filtered noise suppression.

II. System model The channel/equalizer model is shown in Fig. 1 and the notation is summarized in Table 1. We make the following assumptions here. The channel, h and the TEQ, w are finite impulse response (FIR) filters. 2 • The input xk is zero mean and white with variance σx . • The noise uk is zero mean wide-sense stationary (WSS) random process with covariance matrix Ru . • The processes xk and uk are uncorrelated. • The channel, h is known at the receiver. In practice, channel state information is obtained via training sequence. •

The effective channel is given by ck = hk ∗ wk where ‘∗’ denotes linear convolution. Then the output yk can be written as yk

= = =

rk ∗ wk xk ∗ hk ∗ wk + uk ∗ wk xk ∗ ck + qk

(1)

The MDS design of [5] chooses w in order to minimize the following cost function: 1 X (k − kref )2 c2k (6) Jds = Ec

uk xk

rk

h

yk

w

k

where Ec is the energy of the effective channel. Jds is, in fact, the square of the delay spread. Hence, [5] actually attempts to minimize the delay spread of the effective channel. This method assumes centroid of the original channel, kh , as the time reference, kref . Now, we show that choice of [5] for kref is not optimum. From (6), after some modification, we get that X X 1 X 2 2 2 kc2k + kref k ck − 2kref gk2 } (7) { Jds = Ec

c

Fig. 1. System Model with channel/equalizer.

Table 1 Channel shortening notation Notation Meaning length of CP ν FFT size N delay of effective channel 4 Lh × 1 CIR vector h Lw × 1 TEQ impulse response vector w Lc × 1 EIR vector c Lc × Lw channel convolution matrix H Lw × Lw noise covariance matrix Ru n × n identity matrix In transpose AT diagonal matrix with entries diag[ ] in the main diagonal

kref = kc

Jds =

where Λkref

(2)

where H is the Toeplitz convolution matrix of the unequalized channel, h. H is configured as h0

  h1   ..  .  H =  hL −1 h   0   ..  . 0

0

···

0 .. . .. .

h0 .. . hLh −2 hLh −1 .. .

··· .. . ··· ··· .. .

hLh −Lw hLh −Lw +1 .. .

0

···

hLh −1

X

gk2

(energy)

= diag[0, 1, . . . , Lc − 1] − kref ILc (10)

Jds =

(11) wT HT Hw Noise and filtered signal power at the output of the TEQ can be written respectively as σq2

Jn

Dg

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wT Ru w

(12)

=

σx2 wT HT Hw

(13)

=

σq2 σx2f

=

wT Ru w σx2 wT HT Hw

J , αJds + (1 − α)Jn ,

(14)

0≤α≤1

(15)

Hence, the problem is to find optimum TEQ wopt by minimizing J in (15): wopt

(3)

1 X 2 (4) kgk (centroid) = Eg k s 1 X (k − kg )2 gk2 (delay spread) (5) = Eg k

=

Finally, the objective function can be formulated by coupling the two cost functions of (11) and (14) via a trade off parameter, α :

k

kg

wT HT Λ2kref Hw

The cost function to minimize noise power is defined as ratio of filtered noise power and filtered signal power:

We restate the definition of the following fundamental quantities from [6] for a given impulse response {gk , −∞ ≤ k ≤ ∞ }: =

(9)

Now using (2), we get

            

cT Λ2kref c

cT c is a diagonal weighting matrix defined as

σx2f

III. Problem Formulation

Eg

(8)

where kc is the centroid of EIR. It reveals that we need to minimize (6) using kc , not kh . The cost function of (6) can be written as

where qk is the filtered noise. In vector form, the effective channel impulse response, c can be written as



k

Now, using (3),(4) and setting ∂Jds /∂kref equal to zero, we obtain

Λkref

c = Hw

k

k

= arg min J w

wT Xw (16) w wT Yw (1 − α) Ru (17) = αHT Λ2kref H + σx2

= arg min where

X Y

= HT H

(18)

IV. Optimum TEQ Design 2.8

From (4), it is obvious that we need to know EIR, c, to find kc which in turn require to solve (16) first. So we proceed to iteratively minimize the (15) using centroid obtained at the previous iteration as time reference for the next [6]. At ith iteration, optimum w is obtained as

2.6

J (dB)

2.4 2.2 2 1.8

wi = arg min J |kref (i−1 )

(19)

w

1.6 1

Note that at ith iteration, Λkref in (10) present inside matrix X in (17) is formed using kref (i − 1). Finally, kref for ith iteration is set as the centroid of the effective channel: kref (i) =

1 X 2 kck Ec k

= = where Υ

=

cT Υc cT c wi T HT ΥHwi wi T HT Hwi diag[0, 1, . . . , Lc − 1]

α(i) where

=

Z =

wi

H Λ2kref (i) Hwi wi T Zwi

(20)

T

HT Λ2kref (i) H +

(21)

1 Ru σx2

The resulting algorithm is as follows • Precompute Y of (18) and initialize kref (0) = kh i.e as centroid of the CIR and α(0) = 0.5 (arbitrary choice). • for i=1, 2, . . . do the following 1. Compute weighting matrix Λkref of (10) using kref (i− 1). 2. Compute matrix X of (17) and obtain optimum TEQ for ith iteration, wi solving (19). 3. Compute kref (i) using (20). 4. Compute α(i) using (21). Here, Λkref is calculated using kref (i) as it is available. Within few iterations, kref becomes fixed and J ceases to be minimized more. Then iteration is stopped and optimum TEQ, wopt is obtained. Fig. (2) shows that J cannot increase from iteration to iteration. It is obvious because when kref is set to kc (see (8)), Jds will minimize which effectively minimizes J as true time reference is gradually reached with iterations.

V. Simulation Results We now proceed to analyze how our design works. The channels used are eight standard downstream carrier service area (CSA) loops combined with a plain old telephone service (POTS) splitter and a twelfth-order Chebyshev bandpass filter for the 30-1000 kHz frequency band, and truncated to 512 samples. Data for the channel and noise was obtained from the Matlab DMTTEQ Toolbox [9]. We used the following asymmetric digital subscriber line (ADSL) input parameters:

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3

4 5 iteration index

6

7

8

Fig. 2. Cost function J gradually stabilizes with iterations (for CSA loop 8).

DFT size, N = 512 and sampling frequency, f s = 2.208 MHz. • ν = 32, Lw = 16 and Lh = 512. 2 • Input power, σx = 21 dBm. Input signal xk consists of QAM symbols. • Input noise consists of near-end crosstalk (NEXT) noise plus additive white noise with power density -110 dBm/Hz. •

α can be adapted based on proportionate ratio of delay spread and filtered noise power : T

2

Fig. (3) shows the original and equalized channel impulse responses upto 470 samples for CSA loop 8 designed using the proposed method . As we can see, our method shortened the channel quite well. Fig. (4) shows the corresponding magnitude responses of original and shortened channel. As the cost function jointly minimize delay spread and filtered noise, in Fig. (5) and Fig. (6), we have plotted delay spread and output SNR variation as function of TEQ length, Lw . Signal-to-noise ratio (SNR) at the output of the equalizer can be obtained using (12) and (13) as: SNR = =

σx2f σq2 σx2 wT HT Hw wT Ru w

(22)

As expected, increasing Lw results in improved performance (delay spread reduces, output SNR rises) which comes at the expense of the increased complexity in implementing the additional equalizer taps. It is also noticed from Fig. (5) and Fig. (6), delay spread and output SNR reaches a floor with increasing Lw . In Table 2, we compared our method with other existing methods on the basis of delay spread. Except [6], our method achieves delay spread which is lower than other methods. Minimum delay spread attained by modified MDS method in [6] is expected as it corrects the original time reference problem present in [5] and only targets to minimize the delay spread. Whereas in this paper, we perform a trade-off between minimizing delay spread and minimizing noise power. In Table 3, we compared output SNR for the methods which are derived from [5]. It is clear from Table 3 that our method achieves higher SNR than other methods. Incorporation of α allows appropriate weight for Jds and Jn so that both output noise and delay spread are minimized in optimum manner.

1

Table 2 Observed delay spread for various methods (averaged over eight CSA loops) Delay Spread (µs) Method 6.27 MDS Design [5] 7.61 Eigenfilter Method [4] 4.50 MSSNR Method [3] 4.40 Modified MDS Design [6] 4.64 MMSE Method [1] 4.42 Proposed

Original Channel Shortened Channel

0.5 Amplitude

0 −0.5 −1 −1.5 −2 −2.5 0

100

200

300

400

k

Fig. 3. Original and shortened channel impulse responses (for CSA loop 8). 10 Original Channel Shortened Channel Magnitude (dB)

0

−10

−20

−30 0

50

100 150 Subcarrier Index

200

Table 3 Output SNR (in dB) comparison of various delay spread minimizing methods MDS Eigenfilter Modified Proposed CSA MDS [6] [4] [5] loop no. 32.91 29.31 22.01 26.94 1 28.63 27.10 20.75 28.05 2 31.81 28.91 20.82 26.27 3 31.56 28.00 18.35 27.27 4 31.54 29.10 21.57 26.30 5 30.29 27.87 19.15 24.55 6 32.57 29.81 17.49 27.05 7 31.49 30.01 15.08 24.54 8

250

VI. Conclusion Fig. 4. Original and shortened channel frequency responses (for CSA loop 8).

Delay Spread ( µs)

6.5 6 5.5

In this paper, we have generalized the modified MDS method to account for noise encountered in the system. Trade off parameter involved in the cost function successfully adjusts weight for delay spread and noise minimization. Simulation results show that our method jointly achieves superior delay spread minimization and filtered noise suppression at the output of the equalizer. This method can be easily extended to incorporate multiantenna system following the guidelines of [8].

5 4.5

References [1]

5

10 TEQ Length

15

20 [2]

Fig. 5. Variation of delay spread as function of TEQ length (for CSA loop 1).

[3]

Output SNR (dB)

[4]

32

[5]

30 [6]

28 [7]

26 5

10 TEQ Length

15

20

[8]

[9]

Fig. 6. Equalizer output SNR as function of TEQ length (for CSA loop 1).

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N. A. Dhahir and J. M. Cioffi, “Effficiently computed reducedparameter input-aided MMSE equalizers for ML detection ,” IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 903–915, May 1996. 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. 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. 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. 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. R. L´opez-Valcarce, “Minimum delay spread TEQ design in multicarrier systems,” IEEE Signal Processing Letters, vol. 4, pp. 112– 114, 1997. T. Starr, J. M. Cioffi, and P. J. Silverman, Understanding Digital Subscriber Line Technology. Upper Saddle River, NJ: Prentice-Hall, 1999. A. Tkacenko and P. P. Vaidyanathan, “Eigenfilter design of MIMO equalizers for channel shortening,” in Proc. IEEE Int. Conf. Acoustic Speech Signal Processing., Orlando, FL, May 2002, vol. 3, pp 2361–2364. 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.