115 年 國立中山大學電波聯合碩士班《通訊理論》

1. (5%) Suppose a communication channel has bandwidth W. Let the symbol (sampling) interval be
   $t_s = \frac{1}{W}$. Which of the following pulse waveforms does not satisfy the ISI-free (Nyquist) criterion?
   (A) $p_1(t) = 2W \text{sinc}^2(2Wt) + W\text{sinc}(Wt)$
   (B) $p_2(t) = 4W\text{sinc}(4Wt) - W\text{sinc}^2(Wt)$
   (C) $p_3(t) = 2W\text{sinc}(Wt) \cos(2\pi Wt)$
   (D) $p_4(t) = 2W\text{sinc}^2(2Wt) - W\text{sinc}^2(Wt)$
   (E) None of the above

2. (5%) An M-ary communication system transmits at a rate of 2000 symbols per second. What is the equivalent bit rate in bits per second for M = 16?
   (A) 4000
   (B) 6000
   (C) 8000
   (D) 10000
   (E) None of the above

3. (5%) Consider a 3-ary communication system in which each transmitted message is chosen from one of three symbols, $m_{-1}, m_0$, and $m_1$. These symbols are transmitted using the waveforms $-p(t), 0$, and $p(t)$, respectively, where the pulse $p(t)$ has duration $T_M$. At the receiver, a matched filter matched to $p(t)$ is employed. Let $r$ denote the output of the matched filter sampled at time $T_M$. Assume that the messages are quiprobable, i.e., $P(m_{-1}) = P(m_0) = P(m_1) = 1/3$. The energy of the pulse $p(t)$ is $E_p$ and the channel noise is additive white Gaussian noise (AWGN) with two-sided power spectral density $N_0/2$. Determine the symbol error probability $P_e$ using the optimum decision thresholds. (The Q-function is defined as $Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-z^2/2} dz$.)
   (A) $P_e = Q\left(\frac{E_p}{2N_0}\right)$
   (B) $P_e = Q\left(\frac{E_p}{3N_0}\right)$
   (C) $P_e = Q\left(\sqrt{\frac{3E_p}{2N_0}}\right)$
   (D) $P_e = Q\left(\frac{E_p}{N_0}\right)$
   (E) None of the above

4. (5%) Let $g(t)$ be a signal with Fourier transform $G(f)$. Define $G_1(t)$ as the inverse Fourier transform of the frequency-domain function $g(af - f_0)$. Which of the following expressions correctly gives $G_1(t)$?
   (A) If $a > 0$, $G_1(t) = \frac{1}{a} G(-t/a) e^{j2\pi f_0 t/a}$
   (B) If $a < 0$, $G_1(t) = \frac{1}{a} G(t/a) e^{j2\pi f_0 t/a}$
   (C) If $a = 0$, $G_1(t) = g(f_0) \delta(t)$

5. (5%) Consider a narrowband FM signal that can be approximated by
   $s(t) \approx A_c \cos(2\pi f_c t) - \beta A_c \sin(2\pi f_m t) \sin(2\pi f_c t)$,
   where $A_c$ is the carrier amplitude, $f_c$ is the carrier frequency, $f_m$ is the modulating frequency, and $\beta << 1$ is the modulation index. Which of the following statements is correct?
   (A) The minimum value of the signal envelope is $A_c\sqrt{1 - \beta^2}$
   (B) The maximum value of the signal envelope is $A_c\sqrt{1 + \beta^2}$
   (C) The total average power of the FM is $\frac{(1+\beta^2)A_c^2}{2}$
   (D) The total average power of the FM is $\frac{(1+\beta^2)A_c^2}{4}$
   (E) None of the above

Consider a signal

with its Fourier transform $X(f)$. Which of the following statements is wrong?

(A) The imaginary part of $X(f)$ is zero.

(B) The value of $\int_{-\infty}^{\infty}X(f)\,df$ is 9.

(C) The value of $X(0)$ is 36.

(D) The value of $\int_{-\infty}^{\infty}|X(f)|^2\,df$ is 324.

(E) The value of $\int_{-\infty}^{\infty}f\cdot X(f)\,df$ is 0.

The signals $x_i(t)$ undergo sampling with sampling period $T_s=0.02$ second. Which of the following signals can be recovered perfectly from the sampled signal?

(A) $x_1(t), x_3(t)$

(B) $x_1(t), x_5(t)$

(C) $x_2(t), x_4(t)$

(D) $x_1(t), x_3(t), x_5(t)$

(E) $x_3(t), x_4(t), x_5(t)$

Let $X(t)=A\cdot\cos(2\pi f_c t+\Theta)$ be a real-valued random process, where $A$ is a constant and $\Theta$ is a random variable uniformly distributed between $0$ and $2\pi$. Which of the following is wrong?

(A) $X(t)$ is wide-sense stationary (WSS).

(B) The mean value of $X(t)$ is zero.

(C) The autocorrelation function of $X(t)$ is $R_X(\tau)=\frac{A^2}{2}\cos(2\pi f_c\tau)$.

(D) The power spectral density function of $X(t)$ is $S_X(f)=\frac{A^2}{4}\left[\delta(f-f_c)+\delta(f+f_c)\right]$.

(E) The average power of $X(t)$ is $A^2$.

Consider an amplitude-modulated signal expressed by

Which of the following is wrong?

(A) The amplitude sensitivity $k_a$ can be any positive real number.

(B) The message $m(t)$ can be recovered from an envelope detector.

(C) We may apply DSB-SC (double sideband-suppress carrier) to reduce transmission power.

(D) We may apply SSB (single sideband) scheme to reduce required transmission bandwidth.

(E) The transmission bandwidth of VSB (vestigial-sideband) is between those of SSB and DSB-SC.

Consider a frequency-modulated signal expressed by

where the message $m(t)$ has the maximum amplitude of $A_m$ and the highest frequency $F_m$. Which of the following is wrong?

(A) The instantaneous frequency of $s(t)$ is $f_c+k_fm(t)$.

(B) The transmission power of $s(t)$ is $\frac{A_c^2}{2}$.

(C) If we connect an FM modulator with an integrator, we can generate a phase-modulated signal.

(D) The maximum frequency deviation is $k_f A_m$.

(E) The transmission bandwidth of $s(t)$ is approximately $2k_f A_m+2F_m$.

Consider the following baseband signal $g(t)$:

(A) Determine the impulse response of a filter $h(t)$ matched to the signal.

(B) Plot the matched filter output $g_0(t)=g(t)*h(t)$ in the absence of noise.

Consider the frequency demodulation scheme shown in Fig. 1, where the incoming FM signal is given by

The delay line produces a delayed signal $s(t-T)$ and leads to a phase shift of $\pi/2$ radians at the carrier frequency $f_c$, i.e., $2\pi f_cT=\pi/2$. Assume that $\beta<1$ and $T$ is sufficiently small, such that

Find the output signal of the demodulator.

🖼️【此處有附圖，請對照原卷】

Consider antipodal signaling with amplitude imbalance. That is, a logic-1 symbol is transmitted as a rectangular pulse of amplitude $A_1$ and duration $T$, and a logic-0 symbol is transmitted as a rectangular pulse of amplitude $-A_2$, where $A_1\geq A_2>0$. The receiver employs a fixed decision threshold at zero. Define the ratio $\rho=A_2/A_1$, and note that the average signal energy, for equally likely 1s and 0s, is

The channel noise is additive white Gaussian noise (AWGN) with two-sided power spectral density $N_0/2$. Show that the resulting probability of bit error can be expressed as

Consider a real-valued message signal $m(t)$ whose spectrum is nonzero over the frequency range 200 Hz to 3 kHz, as illustrated in Fig. 2. This message is SSB modulated to produce the transmitted signal

where $\hat{m}(t)$ denotes the Hilbert transform of $m(t)$. At the receiver, $s(t)$ is demodulated using a carrier of the form $\cos\left(2\pi(f_c+\Delta f)t\right)$. Assume that the lowpass filter is ideal with unity gain and extends over $[-4k,4k]$ Hz, and $f_c=20$ kHz.

(A) Derive the expression for the demodulated signal at the output of the lowpass filter.

(B) Plot the spectrum of the demodulated signal when $\Delta f=20$ Hz.

(C) Plot the spectrum of the demodulated signal when $\Delta f=-10$ Hz.

🖼️【此處有附圖，請對照原卷】