2. An angle-modulated signal is described by

$$x_c(t) = 10 \cos[2\pi(10^6)t + 0.1 \sin(10^3)\pi t]$$

(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 \(x_{c}(t)\).

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

4. Consider a baseband communication system with $s_1(t)$ and $s_2(t)$ depicted in Fig. 1. (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 $s_2(t)$.

114年 - 114 國立臺北大學_碩士班一般入學考試試題_民俗藝術與文化資產研究所：語文論述能力(民俗與文化)#147240

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