Factor graph based receiver design for PCC-coded BPSK transmissions over a frequency-selective fading channel

This paper presents an iterative receiver to detect and decode parallel concatenated convolutional (PCC) coded BPSK transmissions over a frequency selective Rayleigh fading channel. The scheme generalizes work by Valenti and Woerner to frequency selective channels. A detailed derivation of the receiver from factor graph theory is presented. For the purposes of channel estimation, a Wiener filter based estimator is derived, and the interaction between the channel estimation procedure and the message passing algorithm is illustrated. In particular it is shown that, in contrast to Valenti and Woerner's scheme, the pilot symbols appear as variables in the factor graph, leading to the insertion and removal of "pilot symbol LLRs" in the receiver. Finally, the BER performance of the system is demonstrated through computer simulations. I. INTRODUCTION The sum-product algorithm, which operates in a factor graph (1), may be used to design iterative receivers for finite-state transmitter-channel combinations (2). It has also been shown to give rise to the decoding algorithms for the best known classes of error correcting codes, namely PCC and LDPC codes. When the channel is unknown, it is desirable to incorporate channel estimation into the iterative algorithm in a manner which combines somewhat naturally with the sum-product algorithm. One method of achieving this for flat-fading channels is to approximate the channel as finite-state in order to facil- itate application of the sum-product algorithm (3), (4). However for a frequency selective channel, this method has high complexity as the number of channel states is exponential in the channel length. This paper derives a receiver algorithm for PCC coded transmissions over a frequency selective Rayleigh fading channel based on message passing and the sum-product algorithm. A channel estimator is also derived based on optimum Wiener filtering, which has low complexity and is decoupled from the equalizer module. The channel estimator is shown to combine with the sum-product algorithm as follows. First, the sum-product algorithm operates in the same way as if the channel parameters were known exactly, i.e. the state space of the L-tap channel is not expanded beyond 2 L. The effect of this is that a subset F of the pendant factor nodes in the graph require knowledge of the channel; channel estimates are substituted for the actual values. Then, after each iteration of message passing, summaries are computed for all bi- nary variables, and these summaries are used to update the channel estimates. The subsequent iteration of message passing then begins with the re-sending of messages from all nodes in F with the updated channel estimates in place of the actual values. The scheme may be regarded as a generalization of the work of (5) to frequency selective channels. The proposed technique is in contrast to existing schemes for joint channel estimation and turbo equalization for frequency selective fast fading channels, which not only incorporate channel estimation into an expanded equalizer soft-input soft-output (SISO) module, but also approximate the channel as Gauss-Markov, thus inhibiting accurate modelling of the Jakes spectrum (6), (7), (8).