題組內容

4.Consider the BPSK system whose transmitted signal is given by \[ x(t) = \begin{cases} A \cos(2\pi f_c t), & m = 0, \\ -A \cos(2\pi f_c t), & m = 1, \end{cases} \quad 0 \le t \le T, \quad f_c \gg \frac{1}{T}, \] 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 \( f \).

iii) (10pt) Derive the average probability of error based on \( \eta_{\text{MAP}} \) in part ii). Hint: \( Q(x) \triangleq \int_{x}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} dy \) and \( Q(-x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} dy \).

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題組內容

iii) (10pt) Derive the average probability of error based on \( \eta_{\text{MAP}} \) in part ii). Hint: \( Q(x) \triangleq \int_{x}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} dy \) and \( Q(-x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} dy \).

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