題組內容

4. Consider the BPSK system whose transmitted signal is given by

$$x(t) = \begin{cases} A \cos(2\pi fc t), & m = 0, \\ -A \cos(2\pi fc t), & m = 1, \quad 0 \le t \le T, \quad f_c >> \frac{1}{T}, \end{cases}$$

The message $m$ has the distribution of $\mathbb{P}(m=0) = 4/9$ and $\mathbb{P}(m=1) = 5/9$. The received signal is given by $y(t) = x(t) + n(t)$ where $n(t)$ is a zero mean white Gaussian noise with power spectral density $N_0/2$ for all $f$.

ii) (10pt) Derive the detection threshold \( \eta{\mathrm{ML}} \) of maximum likelihood detection for the hypothesis testing problem derived in part i). Also, derive the detection threshold \( \eta{\mathrm{MAP}} \) of maximum a posterior detection.