A decoder for low-complexity equalization of coded ...

A DECODER FOR LOW-COMPLEXITY EQUALIZATION OF CODED FREQUENCY SELECTIVE MIMO CHANNELS Noura Sellami




, Inbar Fijalkow



France T´el´ecom R&D, 38-40 rue du G´e n´eral Leclerc,92794 Issy les Moulineaux, France, [email protected] Equipe de Traitement des Images et du Signal ENSEA-Universit´e de Cergy Pontoise, 6 av. du Ponceau, 95014 Cergy-Pontoise, France, [email protected]

Abstract - In this paper, we propose a decoder for low-complexity equalization of coded frequency selective multiple-input multiple-output channels. The decoder combines the Parallel Decision Feedback Decoding (PDFD) and the List-type approach. This alternative achieves a good trade-off between performance and complexity even when the channel is not minimum phase. Keywords - Space-time coded MIMO channel, joint equalization and decoding, reduced-state algorithms, per-survivor processing, List-type approach. I. I NTRODUCTION The growing demand for new services at high data rates indicates the need for new techniques to increase channel capacity. Foschini and Gans [1] have demonstrated the enormous capacity potential gain of systems with antenna arrays at both transmitter and receiver. In order to achieve the promised high data rates over coded frequency selective multiple-input multiple-output (MIMO) channels, an equalizer has to be applied to reduce the channel time dispersion due to multipath propagation at high data rates. In terms of performance, a good solution for the equalization part is to use a Maximum A Posteriori (MAP) [2] or Viterbi equalizer. However, the complexity of these algorithms is proportional to the number of states of the trellis which grows exponentially with the product of the channel memory and the number of transmit antennas [3]. When the channel memory becomes large and high-order constellations are used, the algorithm becomes impractical. Therefore, a reduced complexity approach is needed. Some suboptimum solutions have been proposed among them shortening the memory of the channel [4] and using a whitened matched filter in order to make the channel minimum phase [5][6]. In the case of trellis encoded signals transmitted over frequency selective channels, the code can be optimally decoded by maximum likelihood joint equalization and decoding (supertrellis). Unfortunately, the state complexity of the trellis grows exponentially with the channel memory and the constellation size. In [7], a reduced-state Viterbi decoder operating on a subset of combined intersymbol interference (ISI) and code states is used for single-input single-output (SISO) frequency selective coded channels. The trellis has a reduced number of states taking into account a reduced number of

0-7803-7589-0/02/$17.00 ©2002 IEEE

taps of the channel. The remaining ISI terms, not represented by the reduced-state trellis, are cancelled by an internal persurvivor processing (PSP) (see [8][9] and references therein). It has been shown in [7][10] that even the least complex Viterbi algorithm that operates on the code trellis, referred to as Parallel Decision Feedback Decoding (PDFD), achieves a good performance. However, this receiver suffers from error propagation, especially when the channel is not minimum phase. In [11][12], a low-complexity receiver, called Generalized Parallel Decision Feedback Decoder (GPDFD), using PDFD and the List-type approach [13][14] is proposed. The List-type approach was first presented for hard-output Viterbi algorithms. The principle is to retain more than one survivor  per state (a list of survivors) in order to avoid error propagation. Simulation results in [12] show that for reasonable  values of , the optimal performance of the supertrellis can be attained. In this paper, as shown in Fig.1, we use a convolutional code before the channel without interleaving. We propose to generalize the joint equalization and decoding process presented in [11][12] to the coded frequency selective MIMO channels. We get also some insight on the justification of the proposed scheme good behavior. II. S YSTEM MODEL We consider a frequency selective fading MIMO channel with  transmitting antennas and  receiving antennas. The channel between each transmit antenna and each receive antenna is modeled by a Rayleigh fading with a memory of  symbols. As shown in Fig.1, the input information bit sequence is first encoded with a rate    convolutional encoder. The output of the encoder is mapped to PSK/QAM symbols and demultiplexed into  streams that are transmitted simultaneously by the  transmitting antennas. The constellation size is  . Thus,    coded bits are mapped into one  !#" symbol. We assume that transmissions are organized into bursts of $ symbols. For the sake of simplicity, the channels are supposed to be invariant during a burst and to change independently from burst to burst. The received baseband signal sampled at the symbol rate at antenna % at time  is a linear combination of the  transmitted signals perturbed with noise,

PIMRC 2002

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In this expression,/ >)  ? are modeled as independent samples of a zero mean5 white complex Gaussian noise with vari/ ance @ A and
) 76  is the 6CBED tap gain5 from transmit antenna F to receive antenna % . The tap gains
) E6  are modeled 4 as independent complex Gaussian random variables 132   with zero mean and variance @ D 76 *G We assume that H 0 @ D 76 I KJG The problem we address is then to recover the information bits from the noisy observations G III. A DECODING ALGORITHM FOR EQUALIZATION For coded frequency selective channels with interleaving, an equalizer has to be designed independently of the decoder. If the channel memory is large and high order constellations are used, the complexity of optimal solutions (MAP and Viterbi equalizer) becomes prohibitive. In order to decrease the complexity of the receiver, the use of the system presented in Fig.1 seems to be very interesting. In fact, the equalization and the decoding can be performed jointly using only the code trellis. In the following, we will present a lowcomplexity receiver which is a state-reduction variant of the supertrellis, using the List-type approach. A. The decoder structure We assume in the sequel of the paper that the modulation used is BPSK and the code rate  is equal to J*   LJ and   MN . The generalization to other constellations and code rates is straightforward. We present here a generalization of the algorithm proposed in [11][12] to our MIMO system. As explained in [11][12], the Viterbi algorithm operates directly on the code trellis. The receiver is then a generalized version of the Viterbi decoder and it has the following features : O the number  of survivors at each state can be set to an arbitrary positive integer [13][14]. O the branch transition metric calculation takes into account the intersymbol interference due to the past symbols through the use of a per-survivor processing. The decoder takes as observable the sequence of samples and TUQVWQX$YAJZG We assume that *)  P for JRQA%SQM the decoder trellis starts from the all zero state. Let [  P be the trellis state at time  . The branches between states [  \]J# and [  P are labeled by

 P_^`8   P*^bacacab^`8 -  P*G

At time  , a state [d [  P is as  sociated to a survivor list e