115 年 國立臺灣大學電信工程學研究所乙組《通信原理》

(1) Please illustrate the following terms:
(a) Gibb's phenomenon;
(b) Hilbert transform

(2) Please determine
(a) $\int_{-\infty}^{\infty} \frac{\sin(2\pi t)}{t} dt$
(b) $\int_{-\infty}^{\infty} \frac{\sin^2(\pi t)}{\pi t^2} dt$
(c) $\int_{-\infty}^{\infty} \sin(30\pi\tau)\cos(6\pi(t-\tau))d\tau$
(d) $\int_{-\infty}^{\infty} \sin(30\pi\tau)\cos(6\pi(t-\tau))d\tau$

(3) Please determine the inverse Z transforms of the following functions:
(It is OK to show only the causal solution).
(a) $X(z) = \frac{1+z^{-1}}{4-3z^{-1}-z^{-2}}$
(b) $X(z) = \log(1+2z^{-2})$

(4) Suppose that $X(j\omega)$ is the continuous Fourier transform of $x(t)$.
(a) What is the Fourier transform of $\text{Re}\{x(3t+6)\}$? (where $\text{Re}\{\cdot\}$ means taking the real part)
(b) What is the Fourier transform of $X(jt)$?
(c) Suppose that the bandwidth of $x(t)$ is $W$. Please estimate the bandwidth of $|x(t+1)|^2$.
(d) Continue in (c), how do we sample $|x(t+1)|^2$ without aliasing effect?

Part 2: Principles of Communication
(Please simplify your answers as much as possible and write down detailed steps)
(5) In a fading channel the received signal is $y = hx + w$, where $x$ is a quadrature-amplitude modulation (QAM) signal and both the fading coefficient $h$ and noise $w$ are zero-mean complex-Gaussian distributed with variance $E[|h|^2] = E[|w|^2] = 1$.
(a) Please find the variance of the real part of $h$.
(6%)
(b) Assume that $x \in \{\sqrt{P}(m_1 + jm_2)|m_1, m_2 = -1, 0, 1, 2\}$. Also, the point $x = \sqrt{P}(1-j)$ is sent. The receiver knows the realization of $h$. Please find the set of QAM constellation points having the largest pairwise error probability to $x$ under the maximum likelihood (ML) rule.
(6%)

(c) Please use union-bound and Q-function to find the upper-bound of the symbol error probability of ML demodulator given realization $h = h'$. You can simplify the answer assuming P is high.
(d) Using the Q-function upper-bound $Q(t) \le (1/2)e^{-t^2/2}$, based on your simplified answer in (c), please find the upper-bound of the symbol error probability when $|h|^2$ is Chi-squared distributed with probability density function (PDF) $f(t) = e^{-t}, t \ge 0$.

(6) Consider coding in a binary finite field. A 1-bit coded packet $[W_1 W_2]c_t$ is sent at discrete time index $t$, where message vectors $W_1$ and $W_2$ are $4 \times 1$ and $4 \times 1$ binary row vectors respectively while $c_t$ is binary $1 \times 8$ coding vector. Let spaces $\Omega_1$ be span$\{g_j: j = 1 \dots 4\}$ and $\Omega_2$ be span$\{g_j: j = 1 \dots 8\}$ where $g_j$ is $8 \times 1$ binary elementary basis with the $j$-th element being 1. In noiseless channel, the receiver has knowledge space $S(t)$ being span$\{c_\tau: \tau = 1 \dots t\}$
(a) Let $c_1 = [0,1,0,0,0,1,0,0]^T$ and $c_2 = [0,1,0,0,0,0,0,0]^T$ where $T$ is the transpose. Find the rank $\gamma_t = \text{rank}(S(t) \cap \Omega_2)$ at $t =1$ and $2$ where $\cap$ is set intersection.
(b) Continue from (a), find $\gamma_t = \text{rank}(S(t) \oplus \Omega_1) - \text{rank}(\Omega_1)$ at $t =1$ and $2$, where the linear sum space $S(t) \oplus \Omega_1$ is the span of all vectors belong to set union $S(t) \cup \Omega_1$.