Distributed Source Coding

Introduction   DSC using Convolutional Codes   DSC Using Turbo Codes    Other DSC Work

(Last modified November 2001. For more recent work, refer to the Wyner-Ziv Coding of Video webpage)

DSC Using Turbo Codes

Simple convolutional coding is not enough to approach the results of the Slepian-Wolf distributed source coding theorem.Thus, the next logical step is to apply better channel coding techniques – codes that give lower probability of errors for the same encoding rate.Turbo coding, first introduced in 1993 by Berrou, et al. [13], have been demonstrated to be superior error correcting channel codes.Studies have shown that these codes allow systems to perform close to the Shannon channel capacity limit [13].

 
Convolutional Coder and Decoder
    A system based on turbo codes, composed of two parallel concatenated convolutional codes, was implemented (Fig. 9). The bit sequence X is fed into a 16-state, rate 4/5 systematic convolutional encoder of generator matrix h0 = 23, h1 = 35, h2 = 31, h3 = 37 and h4 = 27 [14].  In parallel, a randomly interleaved version of X is used as an input to another convolutional encoder of the same type.  At both encoder outputs, only the parity bits (1 parity bit for every 4 input bits), XP,enc1 and XP,enc2 are sent to the decoder resulting in Rx = 0.25+0.025 = 0.5.  The decoder is composed of two soft-input soft-output (SISO) constituent decoders.  Each SISO decoder uses a priori probabilities (Papriori) for X and the “channel” probabilities (Pchannel) calculated from side information Y and the corresponding parity bits XP,enci  to calculate extrinsic probabilities (Pextrinsic) and a posteriori probabilities (Papp) for X. The iterative decoding is executed by passing the extrinsic probability results of one SISO decoder as the a priori probabilities of the other SISO decoder.  Iterations between the two constituent decoders are performed until a satisfactory convergence is reached.  The final estimate X' is based on thresholding the a posteriori probabilities produced by the final iteration.

Simulation Results

    The turbo coded system was applied to bit sequences of length L = 104, 105, and 106and a maximum of 24 iterations were executed. The results are plotted in Figure 10.  Each point on the plot is generated from a total of 107 sample bits.  It can be observed from the plots that the probability of error, Pe, of the turbo-coded system has a very steep drop, similar to the “ideal” plot predicted by the Slepian-Wolf theorem.It is also very evident that the turbo-coded system produces acceptable probability of errors at H(X|Y) values much closer to the information-theoretic bound, when compared to the convolutional coded system.


Figure 10 – Pe plots for systems based on convolutional codes (CC) and turbo codes (TC).
Bit sequences of length N = 104, 105, and 106 were used.

    An important component of turbo coding is the “interleaver gain.”  It has been shown that the greater the interleaver length, the better the performance of the turbo code.  The results do show that as X was grouped into larger sequence lengths, the error-free H(X|Y) values moved closer to the ideal point.One way to understand this is that the longer sequences and more “random” interleaving bring the system closer to the random coding and asymptotic equipartition principles assumed by the proof of the Slepian-Wolf theorem.

Turbo Codes with Puncturing

    To be able to send X at lower rates, it is possible to apply the concepts of turbo code puncturing.  An alternative turbo-coded system was implemented wherein X was sent at a rate of RX = 0.25.This was done by puncturing half of the parity bits from each constituent convolutional encoder.  Ideally, the system should have close to zero errors at H(X|Y£0.25.The results of the simulations (107 sample bits per point) are shown in Figure 11.  Like the previous results, the punctured turbo code produces steep probability of error curves and larger sequence lengths shift the plot closer to the ideal curve.


Figure 11 - Pe results for punctured turbo code.


 

    Results in this study suggest that applying turbo codes for distributed source coding is very promising.  The simulation results show that acceptable probability of errors can be achieved at H(X|Y) values close to the Slepian-Wolf bound.  This method can easily be extended to other X-Y dependency structures by calculating the a priori and channel probabilities relative to the specific dependency.  Furthermore, turbo coding encoders are very simple, making them reasonable to use in such distributed system scenarios wherein there are low-complexity nodes communicating to a more complex central processing unit.