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.