109 年 國立臺灣聯合大學系統（清華、政治、陽明交通、中央四校聯招）研究所電機類《通訊系統（通訊原理）》

(a) (4%) Explain why $R_X(\tau) = \sin(2\pi f_c \tau)$ cannot be the autocorrelation function of a wide-sense stationary (WSS) random process $X(t)$.

(b) (6%) Below is a list of the analog modulation schemes for the transmission of message $m(t)$, where the transmission signal is modeled as $s(t) = \text{Re}\{(s_1(t) + j s_q(t)) e^{j 2\pi f_c t}\}$. Provide what should be placed in the three blanks under $s_q(t)$ column below.

Note: DSB-C = Double-sideband with carrier, DSB-SC = Double-sideband with suppressed carrier, SSB = Single-sideband

(c) (6%) Suppose the oscillator at the receiver has a phase difference $\phi$ to the oscillator at the transmitter, and suppose the value of $\phi$ can be perfectly estimated via a separate low-power pilot tone. Show that we can demodulate $s_I(t)$ and remove $s_q(t)$ from $s(t)$ for all the above four modulations by two product modulators and two lowpass filters as shown below.

🖼️【此處有附圖，請對照原卷】
Note: You shall represent $s_I(t)$ as a function of $o_I(t)$, $o_Q(t)$ and $\phi$.

In the above diagram, we obtain that for sequence transmission $\{a_k\}$,
$y(iT_b) = \sum_{k=-\infty}^{\infty} a_k \cdot p((i - k)T_b) + w(t) * c(t)|_{t=iT_b} = n_i$
where $p(t) = g(t) * c(t)$, and "$\ast$" is the convolution operation. Denote by $G(f)$, $C(f)$ and $P(f)$ the Fourier transforms of $g(t)$, $c(t)$ and $p(t)$, respectively. The noise $w(t)$ is zero-mean and white with power spectra density $N_0/2$.

(a) (6%) If $n_i = 0$ and the function $p(t)$ satisfies
$p(kT_b) = \begin{cases} 1, & k = k_0; \\ 0, & k \ne k_0, \end{cases}$
determine $y(iT_b)$.

(b) (6%) Determine $\sum_{i=-\infty}^{\infty} P(f - i/T_b)$ based on (1).
Hint: $P_s(f) = \sum_{n=-\infty}^{\infty} P(nT_b) \delta(f - n/T_b)$ is the Fourier transform of $p_s(t) = \sum_{n=-\infty}^{\infty} p(nT_b) \delta(t - nT_b)$.

(c) (6%) Determine the filter $c(t)$ that maximizes the output signal-to-noise ratio (SNR) for given $g(t)$. You may suppose $a_0$ is transmitted, and examine the SNR of $y(iT_b)|_{i=0} = y(T_b)$.

Let $\phi_1(t),...,\phi_N(t), 0 \le t \le T$, be real-valued orthonormal functions, and let $s_i(t) = \sum_{j=1}^{N} s_{ij}\phi_j(t)$ for $i=1,2,...,M$ be real-valued signals. Define $s_i = (s_{i1}, s_{i2}, ..., s_{iN})^T$ where $T$ denotes the transpose of the vector. (Note that $||s_i||^2 = s_i^T s_i$.)

(a) (6%) Show that $\int_0^T s_i(t)s_j(t) dt = s_i^T s_j$ and $\int_0^T (s_i(t) - s_j(t))^2 dt = ||s_i - s_j||^2$.

Let $W(t)$ be a real-valued Gaussian process with zero-mean and the autocorrelation function $R_W(\tau) = E[W(t)W(t+\tau)] = \sigma_W^2 \rho(|\tau|)$. Define, for $j=1, 2, ..., N$, the random variables $W_j = \int_0^T W(t)\phi_j(t) dt$.

(b) (6%) Derive $E\{W_j\}$ and $E\{W_j W_k\}$ for $j \ne k$.

Now, assume $M=2$, and that $s_1(t) = 1, 0 \le t \le T/2$ and $s_1(t) = -1, T/2 < t \le T$.
Let $s_2(t) = \begin{cases} 1, & 0 \le t \le T/2 \\ -1, & T/2 < t \le T \end{cases}$.
Moreover, assume that we observe $x(t) = s_i(t) + w(t), 0 \le t \le T$, with $i=1$ or $2$, and $w(t)$ a sample realization of the Gaussian process $W(t)$ defined above.

(c) (3%) Find a smallest set of orthonormal functions $\phi_1(t),...,\phi_N(t)$ such that $s_i(t) = \sum_{j=1}^{N} s_{ij}\phi_j(t)$ for $i=1,2$. Let $s_1$ and $s_2$ be vectors as defined above. Compute $d = ||s_1 - s_2||$.

(d) (6%) Give a maximum likelihood (ML) decision rule for estimate $s(t)$ of the signal $s_i(t)$ in $x(t)$, and derive the error probability of the ML estimate $\hat{s}(t)$, i.e., compute $P(\hat{s}(t) \ne s_i(t))$.
Please express $P(\hat{s}(t) \ne s_i(t))$ in terms of the function
$\text{erfc}(u) = \frac{2}{\sqrt{\pi}} \int_u^\infty e^{-z^2} dz$
You may need the probability density function (pdf) $f(v)$ of a Gaussian r.v. $\nu$ with zero mean and variance $\sigma_W^2$:
$f(\nu) = \frac{1}{\sqrt{\pi \sigma_W^2}} \exp\left(-\frac{\nu^2}{\sigma_W^2}\right)$

(This problem is a continuation of Problem 3, but can be solved independently.) Now, assume $M = 4$ and consider the passband signals
$s_i(t) = a_i \sqrt{\frac{2E}{T}} \cos(2\pi f_c t), \quad 0 \le t \le T$
for $i = 1, 2, 3, 4$ with $a_1 = 1, a_2 = 3, a_3 = 5, a_4 = 7$, and $f_c$ an integer multiple of $1/T$.

(a) (1%) Define orthonormal function $\phi_1(t), 0 \le t \le T$, and specify with respect to $\phi_1(t)$ the message points $s_i$ for $i = 1, 2, 3, 4$.

(b) (3%) Draw a constellation of the message points $s_i$, and label each $s_i$ with 2 bits using Gray mapping. Give the value of $d_{\min} = \min \{ ||s_i - s_j|| : i \ne j \}$.

(c) (8%) Suppose that $x(t) = s_i(t) + w(t)$ for some $s_i(t)$ with probability $P(s_i(t)) = 1/4$ for each $i$, and $w(t)$ a sample realization of $W(t)$ as defined in Problem 3. Form a ML estimate $\hat{s}(t)$ of the signal $s_i(t)$, given $x(t)$; derive the averaged probability of symbol error $P(\hat{s}(t) \ne s_i(t))$.
Hint: you may need erfc(.) and the pdf $f(v)$ given in Problem 3(g).

Consider binary data modulation at a carrier frequency of $f_c$ in a Rayleigh fading channel, given that the complex envelope of the received signal $y(t)$ be written as $\tilde{y}(t) = A e^{-j\Phi} d(t)$, if the AWGN noise term (with power spectral density $S_N(f) = N_0/2$) is not included, where $d(t)$ is a NRZ waveform with an amplitude of 1 and with bit period $T$, $A$ is a random amplitude with Rayleigh pdf $f_A(a) = (\alpha/\sigma^2) \exp(-a^2/(2\sigma^2))$, and $\Phi$ is random phase uniformly distributed between 0 and $2\pi$.

(a) (4%) Find out the equation of the received signal $y(t)$ and explain the effect of the channel on the received signal.

(b) (4%) Assume DPSK is used and given that the bit error probability of DPSK in AWGN as $P_e = (1/2)e^{-E_b/N_0}$, prove that DPSK over a Rayleigh fading channel has an average bit error rate of $1/[2(1+\overline{\gamma})]$, where $\overline{\gamma}$ is the average $E_b/N_0$.

(c) (4%) If BPSK modulation is used instead, draw the block diagram of an L antenna space diversity receiver structure with a maximal ratio combiner and explain how it works.

(d) (4%) Following (c), prove that the maximal ratio combiner provides the largest SNR for data detection among all linear combining schemes.