Maximum likelihood detection

Before talking about decoding, we should introduce a probabilistic criterion for decoding, and show that it is equivalent to finding the closest codeword. Given a received vector $\rbf$, the decision rule that minimizes the probability of error is to find that codeword $\cbf_i$ which maximizes $P(\cbf=\cbf_i\vert\rbf)$. This is called the maximum a posteriori decision rule. (Proof that this minimizes probability of error is shown in the communications class.) We note by Bayes rule that

$$P(\cbf\vert\rbf) = \frac{P(\cbf) P(\rbf\vert\cbf)}{P(\rbf)},$$

where, for example, $P(\rbf)$ is the probability of observing the vector $\rbf$. Now, since $P(\rbf)$ is independent of $\cbf$, maximizing $P(\cbf\vert\rbf)$ is equivalent to maximizing

$$P(\cbf) P(\rbf\vert\cbf).$$

If we now assume that each codeword is chosen with equal probability, then maximizing $P(\cbf) P(\rbf\vert\cbf)$ is equivalent to maximizing

$$P(\rbf\vert\cbf).$$

A codeword which is selected on the basis of maximizing $P(\rbf\vert\cbf)$ is said to be selected according to the maximum likelihood criterion. We shall assume throughout the text a maximum likelihood criterion.

Let us see what this means for us.

$$P(\rbf\vert\cbf) = \prod_{i=1}^n P(r_i\vert c_i)$$

Assuming a BSC channel with crossover probability $p$, we have

$$P(r_i\vert c_i) = \begin{cases} 1-p & \text{ if} c_i=r_i \\ p & \text{ if} c_i \neq r_i \end{cases}$$

Then

$$\begin{aligned} P(\rbf\vert\cbf) &= \prod_{i=1}^n P(r_i\vert... ...1-p)^n \left(\frac{p}{1-p}\right)^{d(\cbf,\rbf)}. \end{aligned}$$

Then if we want to maximize $P(\rbf\vert\cbf)$, we should choose that $\cbf$ which is closest to $\rbf$, since $0 \leq (p/(1-p)) \leq 1$. Thus, under our assumptions, the ML criterion is the minimum distance criterion. In every case, we should choose the error vector of lowest weight.