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 $f$.

(c) $ (10\%) $ Derive the average probability of error based on $ \eta_{\mathrm{MAP}} $ in part b). (Hint: $ Q(x)\triangleq \int_{x}^{\infty}\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}\mathrm{d}y $ and $$Q(-x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \mathrm{d}y.$$

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題組內容

(c) \( (10\%) \) Derive the average probability of error based on \( \eta_{\mathrm{MAP}} \) in part b). (Hint: \( Q(x)\triangleq \int_{x}^{\infty}\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}\mathrm{d}y \) and $$Q(-x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \mathrm{d}y.$$

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