題組內容

3. Consider the BPSK system whose transmitted signal is given by $$x(t) = \begin{cases} A \cos(2\pi f_c t), & m = 0, \\ -A \cos(2\pi f_c t), & m = 1, \quad 0 \le t \le T, \quad f_c >> \frac{1}{T}, \end{cases}$$ The message $m$ has the distribution of $\mathbb{P}(m=0)=2/9$ and $\mathbb{P}(m=1)=7/9$. 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 $f$.

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