108 年 國立中正大學通訊工程學系碩士班通訊甲組《通訊原理》

1. To combat the effects of aliasing in practice, which of the following methods is correct?
   (a) Use a high-pass filter to attenuate the in-band noise.
   (b) The signal is sampled at a rate slightly lower than the Nyquist rate.
   (c) Reduce the signal amplitude so that aliasing effect can be reduced.
   (d) Use a low-pass filter to attenuate those high frequency components of the signal before sampling.
   (e) None of the above.

2. A PAM system produces the signal $s(t) = \sum_{n=-\infty}^{\infty} m(nT_s)h(t – nT_s)$, where $T_s$ is the sampling period and $m(t)$ is the message signal. Let $H(f)$ be the Fourier transform of $h(t)$ and $M(f)$ be the Fourier transform of $m(t)$. Determine the Fourier transform of $s(t)$.
   (a) $S(f) = \sum_{k=-\infty}^{\infty} M\left(f - \frac{k}{T_s}\right) H\left(f - \frac{k}{T_s}\right)$
   (b) $S(f) = \sum_{k=-\infty}^{\infty} M\left(\frac{k}{T_s}\right) H\left(f - \frac{k}{T_s}\right)$
   (c) $S(f) = \sum_{k=-\infty}^{\infty} M\left(f - \frac{k}{T_s}\right) H(f)$
   (d) $S(f) = \sum_{k=-\infty}^{\infty} M(f) H(f) \exp(j2\pi k f T_s)$
   (e) None of the above.

3. Let $m(t)$ be the message signal and $W$ be the bandwidth of $m(t)$. The double-sideband suppressed carrier (DSB-SC) amplitude modulation has the modulated signal $u(t) = A_c m(t) \cos(2\pi f_c t)$. Let the received signal be $r(t) = u(t)$. Suppose we demodulate the received signal by first multiplying $r(t)$ by a locally generated signal $\cos(2\pi f_c t + \phi)$, and then passing the product through an ideal low pass filter with bandwidth $W$. What will happen when $\phi = \pi/2$?
   (a) The demodulated signal is zero.
   (b) The demodulated signal has the maximum signal strength.
   (c) The demodulated signal has the best signal-to-noise ratio.
   (d) The demodulated signal has large frequency components.
   (e) None of the above.

4. For both FM and PM with sinusoidal message signal, we have $u(t) = A_c \cos(2\pi f_c t + \beta \sin(2\pi f_m t))$. Let $J_n(\beta)$ be the Bessel function of the first kind of order $n$. What is the Fourier transform of $u(t)$?
   (a) $\sum_{n=-\infty}^{\infty} j^n A_c J_n(\beta) \left[ \delta\left(f - f_c - n f_m\right) + \delta\left(f + f_c + n f_m\right) \right]$
   (b) $\sum_{n=-\infty}^{\infty} A_c J_n(\beta) \cos(2\pi (f_c+n f_m)t)$
   (c) $\sum_{n=-\infty}^{\infty} j^n A_c J_n(\beta) \left[ \delta\left(f - f_c - n f_m\right) - \delta\left(f + f_c + n f_m\right) \right]$
   (d) $\sum_{n=-\infty}^{\infty} A_c J_n(\beta) \sin(2\pi (f_c+n f_m)t)$
   (e) None of the above.

5. Determine the power spectral density of $m(t) = A \cos(2\pi f_m t)$.
   (a) $\frac{A^2}{2} [\delta(f - f_m) + \delta(f + f_m)]$
   (b) $\frac{A^2}{4} [\delta(f - f_m) + \delta(f + f_m)]$
   (c) $A^2 [\delta(f - f_m) + \delta(f + f_m)]$
   (d) $A^2 [\delta(f - f_m) - \delta(f + f_m)]$
   (e) None of the above.

6. A digital communication system generates the transmitted signal $u(t) = \sum_{n=-\infty}^{\infty} a_n h(t – nT_b)$, where $a_n$ is an i.i.d. random process with $P\{a_n = +1\} = P\{a_n = -1\} = 1/2$, and $T_b$ is the symbol period. Define $H(f)$ as the Fourier transform of $h(t)$. Determine the power spectral density of $u(t)$.
   (a) $\frac{1}{T_b} H(f)$
   (b) $|H(f)|^2$
   (c) $T_b |H(f)|^2$
   (d) $T_b H(f)$
   (e) None of the above.

7. Let $X(f)$ be the Fourier transform of $x(t)$. Which of the following is a necessary and sufficient condition for $x(t)$ to satisfy
   $x(nT) = \begin{cases} 1 & n=0 \\ 0 & n \neq 0 \end{cases}$
   (a) $\sum_{m=-\infty}^{\infty} X\left(f + \frac{m}{T}\right) = T$
   (b) $\sum_{m=-\infty}^{\infty} X\left(f - \frac{m}{T}\right) = T$
   (c) $\sum_{m=-\infty}^{\infty} X\left(f + mT\right) = T$
   (d) $\sum_{m=-\infty}^{\infty} X\left(f - mT\right) = T$
   (e) None of the above.

8. A binary communication system has the received signal $r(t) = s(t) + n(t)$ for $0 \le t \le T$, where $s(t)$ is the transmitted signal with $P\{s(t) = s_A(t)\} = P\{s(t) = s_B(t)\} = 1/2$, $n(t)$ is the AWGN with power spectral density $N_0/2$, and $T$ is the symbol period. Assume that the ML receiver is employed. Determine the bit error rate.
   (a) $Q\left(\frac{\sqrt{d}}{\sqrt{2N_0}}\right)$, where $d = \int_0^T |s_A(t) - s_B(t)|^2 dt$
   (b) $Q\left(\frac{d}{\sqrt{2N_0}}\right)$, where $d = \int_0^T |s_A(t) - s_B(t)|^2 dt$
   (c) $Q\left(\frac{d}{\sqrt{N_0}}\right)$, where $d = \int_0^T |s_A(t) - s_B(t)|^2 dt$
   (d) $Q\left(\frac{d^2}{\sqrt{N_0}}\right)$, where $d = \int_0^T |s_A(t) - s_B(t)|^2 dt$
   (e) None of the above.

9. Two signal vectors are employed for a binary communication system which are given by
   $s_1 = (+A, +A)$
   $s_2 = (+A, -A)$
   The received signal can be represented as $r = s + w$, where $P\{s = s_1\} = P\{s = s_2\} = 1/2$ and $w = [w_1, w_2]$ is a 2-dimensional zero-mean Gaussian noise vector with $E[w_i^2] = N_0/2$ for $i = 1, 2$ and $E[w_1w_2] = 0$. What is the bit error rate for the optimum detector?
   (a) $Q\left(\frac{A}{\sqrt{N_0}}\right)$
   (b) $Q\left(\frac{\sqrt{A^2}}{\sqrt{2N_0}}\right)$
   (c) $Q\left(\frac{A^2}{\sqrt{N_0}}\right)$
   (d) $Q\left(\frac{A}{\sqrt{2N_0}}\right)$
   (e) None of the above.

10. Under the same signal-to-noise ratio, which of the following binary modulation techniques has the best performance.
    (a) Binary pulse position modulation.
    (b) Binary frequency-shift keying.
    (c) On-off keying.
    (d) Binary phase-shift keying.
    (e) None of the above.

1. (25%) Consider the case of binary PAM signal over the AWGN channel with received signal $r = s + w$.
   where $s$ is the transmitted signal and $w$ is the AWGN with variance $N_0/2$. The two
   possible signal points are $s_1 = -\sqrt{E_b}$ and $s_2 = \sqrt{E_b}$, where $E_b$ is the energy per bit. The prior
   probabilities are $P[s = s_1] = p$ and $P[s = s_2] = 1 - p$.

(a) (10%) For the optimum detector, a threshold $\lambda$ can be used to make decision on the
transmitted signal. If $r > \lambda$, the decision $s = s_1$ is made and if $r < \lambda$, the decision
$s = s_2$ is made. Determine the threshold $\lambda$.
(b) (15%) Given the optimum detector, what is the bit error probability?

2. (25%) The discrete sequence $r_k = \sqrt{E_b}c_k + n_k$, for $k = 1, 2, ..., n$ represents the output sequence of samples from a demodulator, where $c_k = \pm 1$ are elements of one of two possible codewords $c_1 = [1, 1,..., 1]$ and $c_2 = [1, 1,..., 1, -1, -1, . . ., -1]$. The codeword $c_1$ is an all-one vector and $c_2$ has $w$ elements which are $+1$ and $n-w$ elements which are $-1$, where $w$ is some positive integer. The noise sequence $\{n_k\}$ is white Gaussian with variance $\sigma^2$.
   (a) (10%) What is the maximum-likelihood detector for the two possible transmitted signals?
   (b) (10%) Determine the error probability as a function of the parameter $(\sigma^2, E_b, w)$.
   (c) (5%) What is the value of $w$ that minimizes the error probability?

1. (25%) Consider the case of binary PAM signal over the AWGN channel with received
   signal
   $r = s + w$.
   where $s$ is the transmitted signal and $w$ is the AWGN with variance $N_0/2$. The two
   possible signal points are $s_1 = -\sqrt{E_b}$ and $s_2 = \sqrt{E_b}$, where $E_b$ is the energy per bit. The prior
   probabilities are $P[s = s_1] = p$ and $P[s = s_2] = 1 - p$.

(a) (10%) For the optimum detector, a threshold $\lambda$ can be used to make decision on the
transmitted signal. If $r > \lambda$, the decision $s = s_1$ is made and if $r < \lambda$, the decision
$s = s_2$ is made. Determine the threshold $\lambda$.
(b) (15%) Given the optimum detector, what is the bit error probability?