Signal detection

Let $r[n]$ be a noisy signal that is either:

$$ H_0: R[n] = W[n] $$

$$ H_1: R[n] = s[n] + W[n] $$

where $s[n]$ is the signal that we are trying to detect, and $W[n]$ is an i.i.d. zero-mean Gaussian process with variance $\sigma^2$.

The maximum a posteriori rule can be written as:

$$ \frac{f(r[0], r[1], \dots, r[L-1] | H_1)}{f(r[0], r[1], \dots, r[L-1] | H_0)} \overbrace{\gt}^{'H_1'} \underbrace{\lt}_{'H_0'} \frac{p_0}{p_1}$$

Given that $W[n]$ is Gaussian, this can be rewritten as:

$$ \frac{\prod_{n = 0}^{L - 1} \left( \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(r[n] - s[n])^2}{2 \sigma^2}} \right)}{\prod_{n = 0}^{L - 1} \left( \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(r[n])^2}{2 \sigma^2}} \right)} $$