MAP decoding for multi-antenna systems with non-uniform sources: exact pairwise error probability and applications

We study the maximum a posteriori (MAP) decoding of memoryless non-uniform sources over multiple-antenna channels. Our model is general enough to include space-time coding, BLAST architectures, and single-transmit multi-receive antenna systems which employ any type of channel coding. We derive a closed-form expression for the codeword pairwise error probability (PEP) of general multi-antenna codes using moment generating function and Laplace transform arguments. We then consider space-time orthogonal block (STOB) coding and prove that, similar to the maximum likelihood (ML) decoding case, detection of symbols is decoupled in MAP decoding. We also derive the symbol PEP in closed-form for STOB codes. We apply these results in several scenarios. First, we design a binary antipodal signaling scheme which minimizes the system bit error rate (BER) under STOB coding. At a BER of 10^-6, this constellation has a channel signal-to-noise ratio (CSNR) gain of 4.7 dB over conventional BPSK signaling for a binary nonuniform source with p₀ Delta₌ P(0) = 0.9. We next design space-time linear dispersion (LD) codes which are optimized for the source distribution under the criterion of minimizing the union upper bound on the frame error rate (FER). Two codes are given here: one outperforms V-BLAST by 3.5 dB and Alamouti's code by 12.3 dB at an FER of 10^-2 for a binary source with p₀ = 0.9, and the other outperforms V-BLAST by 4.2 dB at an FER of 10^-3 for a uniform source. These codes also outperform the LD codes of constructed under a different criteria. Finally, the problem of bit-to-signal mapping is studied. It is shown that for a binary source with p₀ = 0.9, 64-QAM signaling, and SER = 10^-3, a gain of 3.7 dB can be achieved using a better-than-Gray mapping. For a system with one transmit and two receive antennas that uses trellis coding with 16-QAM signaling, a 1.8 dB gain over quasi-Gray mapping and ML decoding is observed when MAP decoding is used for binary sources with p₀ = 0.9.

I. Introduction

Ideally, a lossless or lossy source coder would compress data into an independent, identically distributed (i.i.d.) nearly uniform bit-stream (for sufficiently long blocklengths). However, most practical source coding methods are not ideal; hence there exists a residual redundancy (in the form of non-uniform distribution and/or memory) at their output which will be present at the input of the channel encoder. For example, the line spectral parameters at the output of codebook-excited linear predictive (CELP) speech vocoders may contain up to 42% of (residual) redundancy due to non-uniformity and memory (see, e.g., [3]). Another example is the bit-stream at the output of vector quantizers with moderate blocklengths. Furthermore, natural data sources, which in certain complexity-constrained applications (e.g., wireless sensor networks) are transmitted uncompressed over the channel, exhibit even higher amounts of redundancy. For example, binary images may contain as much as 80% of redundancy due to non-uniformity; this translates into a probability as high as 97% for having a “0” (as opposed to a “1”) in the image bit-stream (see, e.g., [34] and the references therein).