109年 - 109 國立臺北大學_碩士班一般入學考試試題_通訊工程學系：通訊原理#147253

(a) \( (10\%) \) Determine \( A \) and \( B \).

(b) \( (10\%) \) Determine the modulation index.

(a) \( (10\%) \) Consider \( x_{c}(t) \) as a PM signal with its phase deviation constant \( k_{p} = 10 \). Find modulating signal \( m(t) \).

(b) \( (10\%) \) Consider \( x_{c}(t) \) as an FM signal with its frequency deviation constant \( k_{f} = 10\pi \). Find modulating signal \( m(t) \).

(c) \( (10\%) \) Estimate the bandwidth of \( x_{c}(t) \).

(a) \( (10\%) \) Use coherent demodulation to demodulate the received signal and formulate the detection problem as hypothesis testing. You must derive the probability density function for each hypothesis.

(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.

(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 $$Q(-x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \mathrm{d}y.$$

(a) \( (7\%) \) Determine the corresponding optimal receive filter \( h^{\star}(t) \)

(b) \( (7\%) \) Evaluate \( s_1(t) * h^*(t) \) and \( s_2(t) * h^*(t) \) at \( t = T \) and compute the optimal ML detection threshold.

(c) (6%) Suppose now \( s_1(t) \) is fixed as that in Fig. 1 but we are allowed to design \( s_2(t) \). Find the optimal \( s_2(t) \) such that the probability of error is minimized. Fig. 1 $s_1(t)$ and $s_2(t)$.