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

Abstract

Abstract-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 = 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 [13] 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-thanGray 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.

Quadrature partial response signaling based on Alamouti Code

“…Thus , it is nec essary to define the following log a po ster iori prob ability ratio (LAPPR) LAPPR(i k ) = log (pr(~k = l l r)) (5) Pr(lk = Ojr)…”

“…In this section, we are going to derive the BER of QPRS Alamouti code with M-QAM using the results in [4] and [5]. As mentioned before, the optimum decision boundary for minimizing the bit error probability of the QPRS M -QAM must be determined.…”

“…Kim et al [3] derived the exact BER of various orthogonal STBC using the result in [4]. In [5], the exact pairwise error probability (PEP) using maximum a posterior probability (MAP) decoding in viewpoint of minimizing symbol error rate (SER) for orthogonal STBC was derived.…”

Pairwise optimization of modulation constellations for non-uniform sources

“…Compression will often have residual redundancy in the output due to non-ideal coding methods [3]. Rather than using traditional separate source and channel coding (which may be sensitive to noise-related errors in decoding if optimal variablelength source coding is used, as we later illustrate in Section VI), we can choose instead to directly exploit the non-uniformity of the source via the modulation scheme, while gaining noise-resiliency in many cases and significantly reducing system complexity and delay [3]. Such an approach, which can be characterized as a joint source-channel coding approach, is quite attractive for complexity-constrained and delaysensitive applications such as wireless sensor networks.…”

“…As noted in [13], however, there are many real-world examples of data sources which are highly nonuniform, such as text (email and instant/short messages), medical images and encoded voice data [2]. Compression will often have residual redundancy in the output due to non-ideal coding methods [3]. Rather than using traditional separate source and channel coding (which may be sensitive to noise-related errors in decoding if optimal variablelength source coding is used, as we later illustrate in Section VI), we can choose instead to directly exploit the non-uniformity of the source via the modulation scheme, while gaining noise-resiliency in many cases and significantly reducing system complexity and delay [3].…”