Turbo per tone equalization for ADSL systems 1

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1 Departement Elektrotechniek ESAT-SISTA/TR a Turbo per tone equalization for ADSL systems 1 Hilde Vanhaute and Marc Moonen 2 3 January 2004 Accepted for publication in the Proceedings of the IEEE International Conference on Communications, Paris, France, June 20-24, This report is available by anonymous ftp from ftp.esat.kuleuven.ac.be in the directory pub/sista/hvanhaut/reports/03-186a.ps.gz 2 K.U.Leuven, Dept. of Electrical Engineering (ESAT), Research group SCD, Kasteelpark Arenberg 10, 3001 Leuven, Belgium, Tel. 32/16/ , Fax 32/16/ , WWW: hilde.vanhaute@esat.kuleuven.ac.be. Hilde Vanhaute is a Research Assistant supported by I.W.T. (Flemish Institute for Scientific and Technological Research in Industry). 3 This research work was carried out at the ESAT laboratory of the Katholieke Universiteit Leuven, in the frame of the Belgian State, Prime Minister s Office - Federal Office for Scientific, Technical and Cultural Affairs - Interuniversity Poles of Attraction Programme ( ) - IUAP P5/22 ( Dynamical Systems and Control: Computation, Identification and Modelling ) and P5/11 ( Mobile multimedia communication systems and networks ), the Concerted Research Action GOA-MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology) of the Flemish Government, Research Project FWO nr.g ( Design of efficient communication techniques for wireless time-dispersive multi-user MIMO systems ) and was partially sponsored by Alcatel-Bell and Alcatel-MicroElectronics. The scientific responsibility is assumed by its authors.

2 Turbo per tone equalization for ADSL systems Hilde Vanhaute and Marc Moonen K.U.Leuven/ESAT-SCD, Kasteelpark Arenberg 10, 3001 Leuven-Heverlee, Belgium Abstract In this paper, we study the equalization procedure in discrete multitone (DMT)-based systems, in particular in DMTbased ADSL systems. Traditionally, equalization is performed in the time domain by means of a channel shortening filter. Shifting the equalization operations to the frequency domain, as is done in per tone equalization [1], increases the achieved bitrate with 5-10%. We show that the application of the turbo principle in this per tone equalization can provide significant additional gains. In the proposed receiver structure, equalization and decoding are performed in an iterative fashion. Optimal equalization methods for minimizing the bit error rate use maximum a posteriori (MAP) estimation algorithms, which suffer from high computational complexity for large signal constellations and long channel impulse responses. In this approach, the MAP equalizer is replaced by a linear equalizer, based on the minimum mean squared error (MMSE) criterion. We give a description of an efficient implementation of such an equalizer in the per tone structure. Simulations show that we obtain a bitrate increase of 14-17% compared to the original per tone equalization based receiver structure. I. INTRODUCTION Discrete multitone (DMT) modulation has become an important transmission method, for instance, for asymmetric digital subscriber line (ADSL), which provides a high bitrate downstream channel and a lower bitrate upstream channel over twisted pair copper wire. DMT divides the available bandwidth into parallel subchannels or tones, which are QAM-modulated by the incoming bitstream. After modulation with an inverse fast Fourier transform (IFFT), a cyclic prefix is added to each symbol. If the channel impulse response (CIR) order is less than or equal to the cyclic prefix length, demodulation can be implemented by means of an FFT, followed by a (complex) 1- tap frequency domain equalizer (FEQ) per tone to compensate for the channel amplitude and phase effects. A long prefix however results in a large overhead with respect to the data rate. An existing solution for this problem, currently used in ADSL, is to insert a (real) T -tap time domain equalizer (TEQ) before demodulation, to shorten the channel impulse response. Many algorithms have been developed to initialize the TEQ (e.g. [2]). However a general disadvantage is that the TEQ equalizes all tones simultaneously and as a result limits the performance. As an alternative to time domain equalization, per tone equalization (PTEQ) is proposed in [1]. The equalization is now carried out in the frequency domain with a (complex) multitap FEQ for each tone. This receiver scheme always results in a better performance while keeping complexity during transmission at the same level. In this paper, we apply the turbo principle in the per tone equalization procedure to further improve the performance. Turbo techniques have gained a lot of interest since the introduction of the successful turbo codes in 1993 [3]. The underlying iterative receiver scheme, originally developed for parallel concatenated convolutional codes, is now adopted in various communication problems, such as trellis-coded modulation (TCM) [4], code division multiple access (CDMA) [5], turbo equalization [6] [8]. In each of these systems, suboptimal joint detection and decoding is performed through the iterative exchange of soft information between Soft-Input/Soft- Output (SISO) components. This paper is organized as follows. We start with a description of the data model for the DMT system in section II. In section III we give a short review of per tone equalization in DMT-ADSL. Section IV describes turbo equalization in a single carrier system. In section V turbo per tone equalization is derived. An approximate implementation is presented in section VI and simulation results are given in section VII. II. DATA MODEL The following notation is adopted in the description of the DMT system. N is the symbol size expressed in number of samples, k the time index of a symbol, X n (k) is a complex subsymbol for tone n (n = 1 N) to be transmitted at symbol period k and vector X (k) denotes [X(k) 1 X (k) N ]T. Y (k) is the demodulated output for tone n (after the FFT) and n ˆX n (k) the symbol estimate (after frequency-domain equalization). Note that X n (k) = X (k) N (n 2), n = 2 (N/2) and that similar equations hold for Y n (k). In what follows, the index N (n 2) will be denoted as n. Further, ν is the length of the cyclic prefix and s = N + ν the length of a symbol including prefix.finally, n l is additive channel noise and y l is the received (time domain) signal with l the sample index. To describe the data model, we consider three successive symbols X (t) to be transmitted at t = k 1,k,k + 1, respectively. The kth symbol is the symbol of interest, the previous and the next symbol are used to include interferences from neighboring symbols in our model. T is the equalizer length. The received signal may then be specified as follows: y ks+ν T+2. = H X (k) + y (k+1) s X (k+1) or y = HX + n, X (k 1) n ks+ν T+2. n (k+1) s where H (N+T 1) 3N includes modulation with IFFT, adding of prefix and channel convolution. The channel is assumed to be known at the receiver.

3 III. PER TONE EQUALIZATION REVISITED Per tone equalization is based on transferring the TEQoperations to the frequency domain (i.e. after the FFTdemodulation) [1]. For a single tone n, we can write this as ˆX (k) n = D n F N (n,:) (Y w) = row n (F N Y) w D n. (1) }{{}}{{} T FFTs w n with D n the 1-tap FEQ, F N (n,:) the nth row of the N N DFT-matrix F N, w the T -tap TEQ and Y an N T Toeplitz matrix constructed with the received signal samples, with [y ks+ν+1 y ks+ν T+2 ] and [y ks+ν+1 y (k+1) s ] T as its first row and first column. w n can be considered as a complex multitap FEQ on tone n.it is proved in [1] that (1) can be modified into the following expression: ˆX (k) n = vn H z n (2) [ ] [ ] y IT with z n = Y n (k) = 1 0 I T 1 y. (3) 0 F N (n,:) }{{} F n and with v n the T -tap per tone equalizer for tone n. As such, for each tone n, the equalizer input consists of T 1 (real) difference terms y ks+ν (m 1) y (k+1) s (m 1) (m = 1 T 1), and the nth output Y n of the FFT.The first block row in F n is seen to extract these difference terms, while the last row corresponds to the single FFT. We can rewrite this input z n as z n = F n (HX + n) = G n X + N n, (4) with G n a (T 3N)-matrix. The vector v n in (2) can then be optimized by solving a least squares problem for each tone separately, hence the term per tone equalization. For more details, the reader is referred to [1]. A. General description IV. TURBO EQUALIZATION A (single carrier) transmission system using turbo equalization at the receiver, is depicted in Fig. 1. A rate r = k 0 /n 0 convolutional encoder is used to encode the data symbols u k,l (l = 1 k 0 ) into the bits c k,i (i = 1 n 0 ). The coded bits are sent through an interleaver, partitioned into groups of Q bits c n,j (j = 1 Q), which are finally mapped onto complex symbols x n from the 2 Q -ary symbol alphabet S = {α 1,α 2 α M } (M = 2 Q ). We require that the alphabet has zero mean M i=1 α i = 0 and unit energy ( M i=1 α i 2 )/M = 1. The symbols x n are transmitted over an ISI channel with channel coefficients h j (j = 0 L) and we assume that AWGN is present. The received symbols z n can then be expressed as z n L h j x n j + n n, j=0 or for a block of N samples z n = Hx n + n n, where z n = [z n N2 z n+n1 ] T, n n = [n n N2 n n+n1 ] T and x n = [x n N2 L x n+n1 ] T. H is the N (N + L) channel convolution matrix, with N = N 1 + N At the receiver the SISO equalizer and SISO decoder both produce soft information about the transmitted bits c n,j (or c k,i ), which is exchanged as Log Likelihood Ratios (LLR), defined as L(c n,j ) = log P(c n,j = 1) P(c n,j = 0). A fundamental property of a SISO component is that the calculated a posteriori LLR L p can always be split up into an a priori term L a and extrinsic information L e [9]. For both components this results in L equ p L dec p (c n,j ) = L equ e (c n,j ) + L equ a (c n,j ) (c k,i ) = L dec e (c k,i ) + L dec a (c k,i ) The extrinsic LLR can be considered as an update of the available a priori information on the bit c n,j (resp. c k,i ), obtained through equalization (resp. decoding). This extrinsic information, delivered by one component, is used as a priori information by the other component, after (de-)interleaving, as can be seen in Fig. 1. The SISO decoder uses the optimal (Log-)MAP (maximum a posteriori) algorithm, or a suboptimal version of it (Max- Log-MAP or SOVA) [10]. The SISO equalizer, as it was first proposed by Douillard et al. [6], also applies the MAP algorithm to the underlying trellis of the channel convolution. However, for long channel impulse responses and/or large symbol alphabets, this MAP-based equalization suffers from impractically high computational complexity. A suboptimal, reduced-complexity solution is to replace the MAP equalizer by linear processing of the received signal, in the presence of a priori information about the transmitted data. Several algorithms can be found, such as linear equalization based on the minimum mean squared error (MMSE) criterion [11], soft intersymbol interference (ISI) cancellation [7] [12] or MMSE decision feedback equalization [8]. In this paper, we focus on linear MMSE equalization using a priori information. B. Linear MMSE equalization using a priori information Fig. 2 shows a SISO equalizer based on MMSE equalization. First, the mean E{x p } x p and variance Cov(x p,x p ) v p of the transmitted symbol x p are calculated (p = n N 2 L n + N 1 ), given the a priori information L a (c p,j ), j = 1 Q. Then the equalizer estimates ˆx n using the observation z n as follows [11] ˆx n = w H n(z n HE{x n } + h n x n ) with w n [w n,n2 w n, N1 ] T the equalizer taps and h n the (N 2 +L+1)th column of H. w n can be found by minimizing the following cost function min w n J(w n ) = min w n E { ˆx n x n 2}

4 n n transmitter L equ a u k,l c k,i interleaver mapping channel SISO L equ e deinterc n,j symbol x n ISI z n encoder L dec û k,l a SISO equalizer leaver decoder interleaver L dec e receiver Fig. 1. Transmission system with a receiver performing turbo equalization. z n linear symbol bit ˆx n p(ˆx n x n) L e(c n,j ) MMSE extrinsic extrinsic L a(c n,j ) estimator x n equalizer prob. LLR x n, v n v n estimator estimator Fig. 2. A SISO equalizer based on MMSE equalization. resulting in w n = Cov(z n,z n ) 1 Cov(z n,x n ). Instead of calculating the inverse of the covariance matrix, the equalizer taps can also be found by solving a least squares problem, for instance by a QR decomposition [13]. After MMSE equalization, we assume that the pdfs p(ˆx n x n = α i ),α i S are Gaussian so the parameters µ n,i E{ˆx n x n = α i } and σ 2 n,i Cov(ˆx n, ˆx n x n = α i ) can be easily calculated [5]. This assumption drastically simplifies the computation of the SISO equalizer output LLR L e (c n,j ). A. General description V. TURBO PER TONE EQUALIZATION The exchange of soft information between equalizer and decoder, which is essential in turbo equalization, is difficult to realize in a time domain equalization (TEQ) based DMT receiver. Since the output signal of the TEQ is a time domain signal which does not have a finite alphabet, it is not possible to express LLRs based on the outputs. On the other hand, in a per tone equalization based receiver the equalization is carried out in the frequency domain based on (distorted) QAM symbols. A symbol mapping expresses the relation between the QAM symbols and the coded bits, thus LLRs can be easily deduced. Per tone equalization is thus more suited for the introduction of turbo techniques in the equalization procedure. Including a priori information L a (X (t) p ) = [L a (c (t) p,1 ) L a(c (t) p,q p )] T with Q p the number of bits on tone p (p = 1 N) for symbols t = k 1,k and k+1, in the expression for linear MMSE equalization (2) and taking into account that X(k) n ˆX (k) n = w H n n n = v(k) n, leads to ( ) z n G n E{X} + g X(k) n n + g n X(k) n (k) = ( X ) and v (k) and w n = Cov(z n,z n ) 1 Cov(z n,x (k) n ). (5) w n can be calculated as [11] w n = [ G n R XX G H n + (1 v (k) n )(g n g H n + g n g H n ) + E{N n N H n } ] 1 gn, with R XX = Cov(X,X). From the independence of the bits c (t) p,j, it follows that the symbols X(t) p are independent and that the covariance matrix R XX is a 3N 3N diagonal matrix with the variances v p (t) = Cov(X p (t),x p (t) ) on its diagonal (t = k 1,k,k+1; p = 1 N). Further, g n is the (N +n)th column of G n, and E{N n N H n } = F n E{nn H } F H n = F n σn 2 I N+T 1 F H n [ 2 = σn 2 IT 1 f n fn H 1 with f n = FN H (n,n T + 2 : N). B. Complexity reduction For the computation of the equalizer coefficients on tone n, the covariance matrix Cov(z n,z n ) has to be inverted, see (5). This leads to a high computational complexity, which, however, can be drastically reduced if we exploit the specific PTEQ structure. As can be seen by combining (3) and (4), the first T 1 rows of G n are common for every tone. In what follows, we denote the first T 1 rows with the subscript diff, since they act on or refer to the difference terms. The covariance matrix can be split into submatrices, according to the structure of G n and E{N n Nn H } [ ] Dn d Cov(z n,z n ) = n where D n = D + (1 v (k) n )(g diff,n g H diff,n + g diff,n g H diff,n ) = D + 2(1 v (k) n )R{g diff,n g H diff,n} with D = Cov(z diff,z diff ) d H n u n = G diff R XX G H diff + 2σ 2 NI T 1. ],

5 TABLE I COMPLEXITY OF COMPUTATION OF EQUALIZER COEFFICIENTS operation per iteration initialization G diff,n X O(N ut) (for all tones) FFT O(N u log(n u)) Cov(z n,z n) O(N ut(t + N u)) D 1 O(T 3 ) per tone O(T 2 ) TOTAL (per DMT-symbol) O(N ut(t + N u)) R{ } selects the real part of the argument. D is a (real) symmetric tone-independent matrix and D n is a rank-2 modification on D to eliminate a priori information on tone n and tone n. The third equality follows from the fact that g diff,n = gdiff,n. The inverse of the covariance matrix can also be split into submatrices: [ [Cov(z n,z n )] 1 Bn b = n By expressing that the product Cov(z n,z n ) [Cov(z n,z n )] 1 should be equal to the identity matrix, the submatrices B n, b n and v n can be found as follows: b H n p n = D 1 n d n 1 v n = u n d H np n b n = p n v n B n = D 1 n b n p H n. In this computation, D 1 n is needed. This inverse can be calculated in an efficient way. To this aim, write D n as v n D n = D + a n a H n + (a n a H n) T with a n = 1 v n g diff,n, and define q n = D 1 a n c n = a T nq n = q T na n C d n = a H nq n = q H na n R. By applying twice the matrix inversion lemma 1, it can be shown that the inverse is equal to ]. D 1 n = D 1 2(1 + d n)r{q n q H n } 2R{c nq n q T n } (1 + d n ) 2 c n 2. The D 1 should be calculated only once. This reduces the complexity of inverting D n for all tones together from O(N u T 3 ) to O(T 3 + N u T 2 ), with N u the number of used tones. The complexity associated with the computation of the equalizer coefficients is summarized in Table I. Typical values for downstream transmission are N u NFFT 2 = 256 and T = 16, leading to a total complexity of O(N u T(T + N u )). 1 More specifically: if A = B+cc H, then A 1 = B 1 B 1 cc H B 1 1+c H B 1 c. VI. APPROXIMATE IMPLEMENTATION The equalizer filter coefficients have to be updated for every tone and for every iteration, based on the available a priori information. To reduce this computational burden one can make some approximations by assuming that 1) the previous symbol X (k 1) is perfectly known from the previous equalization and decoding step, which gives a zero variance for all the tones of the previous symbol; 2) from the second iteration, the variance of the symbol of interest is also set to zero for all tones. This is true for perfect a priori information. Moreover, there is no a priori information available about the next symbol X (k+1), leading to unit variance for all tones in all iterations. In this way, the filter coefficients are fixed for all DMT symbols, but the coefficients of the first iteration differ from those of further iterations. All the coefficients can be initialized before transmission. Only the mean of the transmitted symbols has to be computed. The initialization complexity is given in Table I, but this should be done only once. Simulations have shown that these approximations do not introduce a large performance degradation (see section VII). VII. SIMULATION RESULTS Time-domain simulations were performed for a downstream ADSL loop of 4 km with additive Gaussian noise. The psd of the transmitted signal was -40 dbm/hz while the psd of the noise was -140 dbm/hz. For the encoding, we chose a high-rate R = 9/10 recursive systematic convolutional code of order 4, with an octal representation [ ], as described in [14]. Natural mapping was selected for both square and non-square constellations, since natural mapping has a better performance than Gray mapping in iterative schemes [15]. The number of equalization taps was set to T = 16. The MAP decoding is done using the dual code as described in [16]. Since the trellis is not terminated the last bits of the decoded sequence are more sensitive to errors. If we force the (de)interleaver to map well-conditioned bits onto the end of the coded sequence, we can reduce the BER at the end of the codeword. This kind of interleaver is further called an adjusted (random) interleaver. The comparison between the original per tone equalization and the turbo per tone equalization is based on equal target bit error rate for both schemes. The performance is then measured by the achievable capacity. The turbo scheme is initialized with a certain bit loading, which gives rise to a specific bit error rate (BER), whereas in the original per tone scheme the bit loading is calculated given the BER. For infinite granularity this bit loading is computed as follows 2 b = n b n = n log 2 ( 1 + SNR n γ c Γ ), with γ c the coding gain and Γ the SNR gap, which expresses the distance between the theoretical Shannon capacity and the 2 There is no noise margin included in this calculation.

6 TABLE II BER FOR TURBO PER TONE EQUALIZATION ( 10 6 ) interleaver iter.1 iter.2 iter.3 random adjusted TABLE III CAPACITY (MBPS) interleaver Turbo-PTEQ PTEQ (RS) PTEQ (RS+Wei) random adjusted practically achievable bitrate [17]. This SNR gap depends on the target BER, which is taken from the 3th iteration of the turbo per tone equalization scheme. The SNR is the same as the SNR after equalization in the first iteration of the turbo scheme, since in the absence of a priori information the two equalization procedures are equal. The ADSL standard provides Reed Solomon (RS) codes for the error correction with a coding gain of 3 db. The standard states that as an option a 4D 16-state trellis code (the Wei-code) can be concatenated with an RS code. This concatenated coding scheme results in a coding gain of 5.2 db. Table II shows the BER performance of the turbo per tone equalization. From simulations it could be seen that 3 iterations are sufficient for convergence. Making the above mentioned adjustments to the interleaver further decreases the bit error rate. In Table III the capacity obtained with the turbo-pteq based receiver is compared to the capacity obtained with the receiver based on the original PTEQ, with two different coding schemes. It can be seen that the turbo-pteq performs 14 to 17% better than the PTEQ scheme. The results for the approximate implementation are shown in Table IV. The approximations can be made either in the first or the second and third iteration, or in all iterations. The performance degradation is small, despite the fixed equalizer filters. This means that the approximations made are justified in this situation. VIII. CONCLUSIONS In this paper, we introduced the turbo principle in the per tone equalization for DMT-ADSL modems. A receiver scheme where equalization and decoding are performed in TABLE IV BER: APPROXIMATE IMPLEMENTATIONS ( 10 6 ) iter.1 iter.2&3 iter.1 iter.2 iter.3 MMSE MMSE approx approx. MMSE approx an iterative fashion, was presented. We have described an efficient implementation of the turbo per tone equalization, taking into account the structure of the per tone equalization. It is shown that the new scheme performs significantly better than the original per tone equalization based scheme. First approximations to reduce the complexity give almost the same results. However, further reduction of the complexity is still necessary and this will be the subject of further research. ACKNOWLEDGMENTS Hilde Vanhaute is a Research Assistant with the I.W.T.. This work was carried out at the SCD laboratory of the Katholieke Universiteit Leuven, in the frame of the Interuniversity Poles of Attraction Program ( ) - IUAP P5/22 and P5/11, the Concerted Research Action GOA-MEFISTO- 666, Research Project FWO nr.g REFERENCES [1] K. Van Acker, G. Leus, M. Moonen, O. van de Wiel, and T. Pollet, Per tone equalization for DMT-based systems, IEEE Trans. Commun., vol. 49, pp , Jan [2] N. 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