Bayes Rule in Digital Communication with BPSK modulation
Because of the stochastic nature wireless communication, Bayes rule is very useful in optimizing symbol detection rule for various modern digital communication systems. The idea is that based on the known information about the distribution of possible transmitted symbols, we can use Bayes rule to help minimize the bit error-rate (BER) in signal transmission. To highlight the use of Bayes rule in signal detection, I am going to detail a simple application in digital signal transmission using BPSK modulation.
To start easy, we consider a simple signal model, in which the symbols transited are selected from the set S = {-sa, sa}. Binary phase shifting keying, or BPSK, is a set of transmit symbols which is able to represent exactly one bit. A possible bit-to-symbol mapping using BPSK modulation is that s= sa maps to bit 0 and s= -sa maps to bit 1. Thus, if we want to transmit a bit 0, we transmit the symbol sa, and if we want to transmit a bit 1, we transmit the symbol –sa. The received signal is represented as r= s+w, where s is the symbol transmitted and w is the noise (considered to be zero-mean Gaussian distributed in this case).
To detect the symbol transmitted, we need to use a detection rule. The optimal detection rule that minimizes the error probability is called the Maximum A-Posteriori (MAP) detection. The MAP detection rule is defined as:
However, the PMF of s given r is not known. Since r is given by r=s+w, and w is a zero-mean Gaussian random variable, we can compute the probability of r given s and use Bayes rule. Because by Bayes rule:
The MAP detector is then equivalent to:
(We assume p(-sa)=p(sa)=0.5 because of the stochastic nature of transmitted bits)
Intuitively, this gives us that: if the received signal r is greater than 0, then the MAP detector gives us s= sa; if the received signal r is less than 0, then the MAP detector gives us s= -sa. There is a detailed proof on why MAP detector is the optimal detector for BPSK modulation, which is not in the scope of this course. But this example should illustrate a simple application of Bayesian inference in signal transmission and gives us a good hint about how Bayes rule can be suited to different modulations in modern digital communication systems.
Link: https://jcis.sbrt.org.br/jcis/article/view/394