A FREQUENCY-DOMAIN EIGENFILTER APPROACH FOR EQUALIZATION IN DISCRETE MULTITONE SYSTEMS

Transcription

1 A FREQUENCY-DOMAIN EIGENFILTER APPROACH FOR EQUALIZATION IN DISCRETE MULTITONE SYSTEMS Bo Wang and Tulay Adala Department of Computer Science and Electrical Engineering University of Maryland, Baltimore County, Baltimore, MD USA (buangl, ada1i)oengr. umbc. edu ABSTRACT The Discrete Multitone (DMT) modulation has been chosen as the industry standard for Asymmetrical Digital Subscriber Loop (ADSL) modems and is also a candidate scheme for very-high-speed digital subscriber line (VDSL) systems. The DMT system works on the condition that the length of the channel response is not longer than that of the guard band known as cyclic prefix to avoid,the intersymbol interference (ISI). A timedomain equalizer (TEQ) is utilized in the DMT system to shorten the effective channel impulse response such that the data rate loss due to the use of cyclic prefix is reduced. In this paper, we propose a new frequencydomain eigenfilter approach to train the TEQ. Simulation results indicate that this algorithm can effectively shorten the channel response. Furthermore, we show that this frequency-domain eigenfilter method provides different TEQ solutions by computing the eigenvectors corresponding to several smallest eigenvalues of a performance matrix. 1. INTRODUCTION Multi-carrier modulation (MCM) has been proposed for parallel communication in the late 1950s [l] based on the concept of creating multiple orthogonal subchannels over which several data streams can be sent without intersymbol interference (ISI). This modulation scheme provides flexibility for adapting to different channel environments by adjusting the energy and constellation size of each carrier. As a subclass of MCM, the Discrete Multitone (DMT) system utilizes the inverse fast Fourier transform (IFFT) and the fast Fourier transform (FFT) for implementing the modulation and demodulation respectively [2], [3], resulting in significant reduction of the system complexity. DMT has recently been chosen as the industry mod- Research supported in part by Maryland Industrial Partnerships and Nortel Networks under grant number ulation standard for Asymmetrical Digital Subscriber Loop (ADSL) modems, which offer powerful and flexible transmission capability enabling delivery of a variety of multimedia services over the existing telephone networks. DMT is also a candidate modulation scheme for very-high-speed digital subscriber line (VDSL) systems which can provide much higher bit rates over shorter loops [4]. Figure 1: Block diagram of the DMT system Fig. 1 shows the block diagram of the DMT system. The input data stream is coded by forward-errorcorrecting (FEC) and/or trellis coding schemes. The outputs of the encoder are grouped into QAM subsymbok. A complex-to-real IFFT transform used for modulation is performed to convert these Q AM subsymbols in the frequency domain to real data in the time d e main. Then the last 7 samples of each real-valued data vector are copied and prefixed to the data vector. At the receiver, the channel outputs are passed through the time domain equalizer (TEQ). After removing the samples corresponding to the cyclic prefix, the outputs of the TEQ are processed by the FFT transform which acts as the demodulation operation to convert the timedomain data back to frequency-domain QAM subsymbok. Then the transmitted data can be recovered by /99/$ IEEE 1058

2 using the adaptive frequency domain equalizer (FEQ) followed by a decoder. In the DMT system, the cyclic prefix, whose length is longer than that of the channel response, is inserted between transmitted symbols such that the linear convolution of the data and the channel impulse response becomes a circular one corresponding to term-by-term product in the frequency domain. However, on many practical channels such as digital subscriber loop, 7 should be large to compensate for the length of the channel response, resulting in a manifest decrease of the data rate (y/(n + 7)) for practical DMT systems with finite FFT length N. A time domain equalizer to shorten the effective channel impulse response has been the most popular equalization approach in the DMT system [5] - [lo]. Figure 2: Time domain equalizer in DMT system In Fig. 2, a FIR filter a with M taps is used as the TEQ such that the linear convolution of the channel response h and the TEQ a approximates the target response b with L taps and delay d. Here L should be less than or equal to Usually, the channel impulse response can not be perfectly shortened by the TEQ such that some residual energy of the shortened impulse response will lie outside the consecutive taps with the highest total energy. We can use the shortening signal to noise ratio (SSNR) [8] to evaluate the performance of the TEQ. This SSNR is defined as the ratio of the energy in the largest consecutive taps to the energy in the remaining taps of the shortened response. The TEQ methods given in [6] - [8] use cost functions that minimize the residual error in the time d e main. These methods require the computation of the autocorrelation and crosscorrelation coefficients of the transmitted and the received signals. Because of severe frequency selective channel distortion, the input to the equalizer at the receiver will be highly correlated causing the conventional time-domain least mean squares (LMS) type algorithms to suffer from slow con- vergence. The group at Amati Communication Corp. (Chow et 01. [5]) proposed an algorithm which minimizes the mean squared error of the equalized response by using frequency-domain LMS for the adaptation and windowing in the time domain. However, the convergence of Amati's algorithm is slow [?I, which is also confirmed by our simulation results in [9]. In [8], the authors proposed a time-domain alge rithm for optimal shortening of the channel response in terms of SSNR. This method utilizes eigenanalysis to generate the coefficients of the TEQ [8]. In [6], the authors developed an algorithm by imposing a unitenergy constraint (UEC) on the target channel b as shown in Fig. 2 such that the b vector obtained from minimum mean square error (MMSE) approach is the eigenvector corresponding to the minimum eigenvalue of a matrix. It is also shown in that paper that the MMSE obtained by applying UEC is lower than that obtained by applying unit-tap constraint (UTC) to the target channel b [6]. In this paper, we pose the TEQ problem completely in the frequency domain by defining a squared cost function in the frequency domain. Instead of applying the UTC to the first tap of the TEQ in [9], [lo], the UEC is used in this paper, leading to an eigenfilter method [ll]. The coefficients of the TEQ can be obtained by computing the eigenvector corresponding to the minimum eigenvalue of a performance matrix. Because the ultimate goal in the TEQ design is to maximize the channel throughput which is determined by the overall SNRs at the output of the FFT block, the minimization of the error at the TEQ output will not necessarily result in channel capacity maximization. Hence the TEQ solution, which is obtained by the eigenvector corresponding to the minimum eigenvalue of the performance matrix, we note, may not be the optimal one in terms of system performance such as the channel throughput. In the simulation part, we show that the TEQ solutions obtained from the eigenvectors corresponding to several smallest eigenvalues (including the minimum one) can shorten the channel response effectively. 2. WEIGHTED FREQUENCY-DOMAIN EIGENFILTER APPROACH As shown in Fig. 2, the error sequence between the TEQ output and that of the target response can be expressed as: where * denotes linear convolution. If the additive channel noise un shown in Fig. 2 is averaged out or is negligible, then yn is also periodic when zn with length N is transmitted repeatedly through the channel whose length is assumed to be less than or equal to N. Hence 1059

3 .. Po the error sequence in Eqn. (1) can be rewritten as: e, = - bn-d (2) represents circular convolution. After DFT transformation (with size equal to DMT symbol length N), we get:,iqe-jwh) = y(e-jwh)a(e-jwh) - X(e-jwh ) B( e-jwh) (3) where k = 0,...,N - 1. Amati's algorithm [5] minimizes the mean squared error of the equalized response by using frequency-domain LMS for the adaptation (the error signal for the frequency-domain LMS algorithm is: B(e-Jwh))X(e-jwh) - A(e-J"h)Y(e-jwh)) and windowing in the time domain. Instead, we introduce the following cost function: and *22 = 921 = 2 : * [ PL-1 PL-2 0 "1 Pl Po a -. PL-1 where H(e-jwh) can be obtained by Yfe-jwh))lX(e-jWh) during the initializationphase [12],&= (UO,...,UM-l, bo,..., b ~ - ~ is ) the parameter vector to be estimated. The coefficients of the channel response h, the TEQ a and the target channel b are assumed to be real, which makes their frequency responses conjugate symmetric, that is: H(e-jWw-w) = H*(e-jwh), A(e-jw(N-k) )= K(e-jWh) and B(e-JW(N-h)) = B*(e-jWh). N is the FFT length, IC is the index of the subchannel frequencies, and W(k) is a non-negative weighting function which can be used to control the TEQ spectral shape [14]. And the performance matrix Q can be expressed as: *11 = *= [ x: x: ] Po p1... PM po... PM-2 1(M-1 1(M-2... PO (5) Thus the TEQ problem can be posed as choosing 8 to minimize E(6) in Eqn. (4) while satisfying the constraint BT8 = 1. Under the UEC, we can use the method of Lagrange multipliers [13]. Defining: L(6, A) = BT*6 + X(1-6%) (10) Differentiating L(6, A) with respect to 8 yields: *B,j, = X6,i, (11) Hence the Oeig vector is an eigenvector of the symmetric positive-definite matrix 9, and X is the corresponding eigenvalue. In this case, the TEQ problem becomes a filter design problem based on the eigenfilter method [ll]. The error E(8) can be rewritten as: ~(6) = e;,wejg = 6~,X6,jg = A (12) 912 = -Pd+l -Pd+L-l -Pd - 1 -Pd Pd+L-2 -Pd-M+l -Pd-M+2. -Pd+L-M Hence the minimum value of E(6) equals to the minimum eigenvalue of the performance matrix *. We call the resulting algorithm the weighted frequencydomain eigenfilter (WFD-EF) algorithm. I060

4 3. SIMULATION RESULTS In this section, we apply the WFD-EF algorithm to shorten the channel impulse response of CSA loop 6 cascaded with a bandpass filter. Figs. 3 and 4 show the channel response and squared transfer function of CSA loop 6. We choose N = 512, the length of cyclic prefix, 7 = 32, L = 33 and the TEQ length M = 14. The weighting function W(k) assumes its default value 1. The delay of the target response is chosen as 18, 22, and 26 respectively. resultant SSNR values by these TEQ solutions. From Table 1 we note that, in most cases, the SSNR vales for XZ to for the three different delays are bigger than those for XI. For the channel response and the parameter set used in this simulation, the TEQ solutions corresponding to the eigenvalues bigger than X8 have very low shortening capability. Figs. 5 and 6 show the TEQ spectral shapes corresponding to XI to X8 with delay d equal to 22. From the figures, we can see that the resultant TEQs by frequency-domain eigenfilter method have some deep nulls in the passband. However; the frequency-domain eigenfilter method provides several TEQ solutions with different TEQ shapes and shortening capabilities. The best solution can be picked up by checking the corresponding achievable data rates. Table 1: SSNRs for different delays and eigenvalue selections I Delay I d=18 I d=22 I d=26 1 A db I db Xz db I db db db Figure 3: Channel impulse response of CSA loop 6 Xi i db i db db I db I 64.67dB I db Figure 4: Squared channel transfer function for CSA loop 6 After setting the above parameters for the WFD- EF algorithm, we compute the eight eigenvectors corresponding to the smallest eight eigenvalues XI to of the performance matrix + in Eqn. (5). The eigenvalues A1 to A8 are in ascending order. The resultant TEQ coefficients are obtained from these eight eigenvectors for three different delays respectively. Table 1 shows the Figure 5: TEQ spectral shapes for eigenvalues XI to A4 4. CONCLUSIONS In this paper, we propose a weighted frequency-domain eigenfilter (UEC is used) method to estimate the codcients of the time domain equalizer of the DMT system 1061

5 lhns. Inform. Theory, vol. 42, no. 3, pp , May [7] M. Nde, and A. Gather, Time-domain equalizer training for ADSL, in Pmc. Int. Conf. on Communications, pp , (Montreal, Canada), June [8] P. J. W. Melsa, R. C. Younce, and C. E. Rohrs, Joint impulse response shortening for discrete multitone transceivers, IEEE Bans. Communications, vol. 44, no. 12, pp , Dec Figure 6: TEQ spectral shapes for eigenvalues Xg to X8 such that the channel impulse response is shortened. We find that the eigenvectors corresponding to several smallest eigenvalues of the performance matrix result in different TEQ solutions. The best TEQ coefficients can be obtained by testing the corresponding system performances. 5. REFERENCES [l] M. L. Doelz, E. T. Heald, and D. L. Martin, Binary data transmission techniques for linear systems, Pmc. IRE., vol. 45, pp , May [2] S. B. Weinstein, and P. M. Ebert, Data transmission by frequency division multiplexing using the discrete fourier transform, IEEE Ipnms. Communication TechnoL, vol. COM-19, no. 5, pp , Oct [3] A. Peled, and A. Rub, Fkequency domain data transmission using reduced computational complexity algorithms, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp , (Denver, CO), Apr [4] J. M. Cioffi, V. Oksman, J. J. Werner, T. Pollet, P. M. P. Spruyt, J. S. Chow and K. S. Jacobsen, Very-high-speed digital subscriber lines, in IEEE Communications Magazine, pp , Apr [9] B. Wang, and T. Adali, An efficient training method for equalization of discrete multitone transceivers, in Pmc. IEEE/IEE International Conference on Telecommunication (ICT), Korea, vol. 2, pp , June [lo] B. Wang, and T. Adali, Joint impulse responses shortening for discrete multitone systems, to appear in Proc. IEEE Globecom Symposium on Communication Theory, Rio de Janeiro, Brazil, Dec [ll] P. P. Vaidyanathan, and T. Q. Nguyen, Eigenfilters: A new approach to least-squares FIR filter design and applications including Nyquist filters, IEEE Bans. Circuits and Systems, vol. CAS-34, no. 1, pp , Jan [12] J. A. C. Bingham, and F. van der Putten, T1.413 Issue 2: Standards Project for Interfaces Relating to Carrier to Customer Connection of Asymmetrical Digital Subscriber Line (ADSL) Equipment, ANSI Document, No. TlE1.4/97-007R6, Sept [13] G. Golub, and C. F. Van Loan, Matrix Computations. 2nd Ed., Johns Hopkins University Press, [14] B. Wang, and T. Adali, A weighted frequencydomain least squares approach for equalization and noise suppression in discrete multitone systems, submitted to IEEE Int. Conf. on Communications, New Orleans, LA, June [5] J. S. Chow, J. M. Cioffi, and J. A. C. Bingham, Equalizer training algorithms for multicarrier modulation systems, in Proc. Int. Conf. on Communications, pp , (Geneva, Switzerland), May [6] N. Al-Dhahir, and J. M. Cioffi, Efficiently computed reduced-parameter input-aided MMSE equalizers for ML detection: A unified approach, IEEE 1062