113 年 國立中山大學通訊工程研究所乙組《工程數學甲》

Use the Fourier series analysis equation to calculate the coefficients $a_k$ (when $k \neq 0$) for the continuous-time periodic signal

with fundamental frequency $\omega_0 = \pi$.

(A) $a_k = \frac{3}{k\pi} e^{-j k\pi/2} \sin(\frac{k\pi}{2})$
(B) $a_k = \frac{3}{k\pi} e^{-j k\pi} \sin(\frac{k\pi}{2})$
(C) $a_k = \frac{3}{k\pi} e^{-j k\pi} \sin(k\pi)$
(D) $a_k = \frac{3}{2k\pi} e^{-j k\pi/2} \sin(\frac{k\pi}{2})$

Let $x_1(t)$ be a continuous-time periodic signal with fundamental frequency $\omega_1$ and Fourier coefficients $a_k$. Given that $x_2(t) = x_1(1-t) + x_1(t-1)$. Find a relationship between the Fourier series coefficients $b_k$ of $x_2(t)$ and the coefficients $a_k$.

(A) $b_k = e^{-j k \omega_1} (a_k + a_{-k})$
(B) $b_k = e^{-j k \omega_1} (a_k - a_{-k})$
(C) $b_k = j k \omega_1 (a_k + a_{-k})$
(D) $b_k = j k \omega_1 (a_k - a_{-k})$

What is the Fourier transform of $e^{-2|t-1|}$?

(A) $\frac{2e^{-j\omega}}{4 + \omega^2}$
(B) $\frac{4e^{-j\omega}}{4 + \omega^2}$
(C) $\frac{4e^{-j\omega}}{2 + j\omega}$
(D) $\frac{2e^{-j\omega}}{2 + j\omega}$

Use the Fourier transform synthesis equation to determine the inverse Fourier transforms of $X(j\omega) = 2\pi\delta(\omega) + \pi\delta(\omega - 4\pi) + \pi\delta(\omega + 4\pi)$.

(A) $1 + \pi\cos(4\pi t)$
(B) $1 + \sin(4\pi t)$
(C) $1 + \cos(4\pi t)$
(D) $1 + \sin(4\pi t)$

Use the Fourier transform synthesis equation to determine the inverse Fourier transforms of

(A) $\frac{4j\cos^2 t}{\pi t}$
(B) $\frac{4\sin^2 t}{\pi t}$
(C) $\frac{4j\sin^2 t}{\pi t}$
(D) $\frac{4\cos^2 t}{\pi t}$

What is the Laplace transform of $e^{-5t}u(t-1)$?

(A) $\frac{e^{-(s+5)}}{(s+5)}$
(B) $\frac{e^{-(s+5)}}{(s+5)}$
(C) $\frac{e^{-(s-5)}}{(s-5)}$
(D) $\frac{e^{(s+5)}}{(s-5)}$

Determine the inverse Laplace transform of
$X(s) = \frac{2(s + 2)}{s^2 + 7s + 12}, \quad \text{Re}\{s\} > -3$

(A) $4e^{-4t}u(-t) - 2e^{-3t}u(-t)$
(B) $4e^{4t}u(t) - 2e^{3t}u(t)$
(C) $4e^{4t}u(-t) - 2e^{3t}u(-t)$
(D) $4e^{-4t}u(t) - 2e^{-3t}u(t)$

One corner of a rectangular parallelepiped is at (1, 1, 1), and three incident sides extend from this point to (-2, 1, 6), (3, 5, 7) and (0, 1, 6). Please identify the volume of this solid.

(A) 20
(B) 40
(C) 18
(D) 42

Consider the wave equation $\frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2}$ for a string tied to the x-axis at $x=0$ and at $x=\pi$. When the string starts to vibrate, the motion takes place in the xu-plane. Let $u(x, t)$ denote the vertical displacement from the x-axis for $t > 0$ and $a^2$ be a real constant. The initial displacement is $f(x)$, $0 < x < \pi$ and the string is released from rest. Given the product solution $u(x, t) = X(x)T(t)$, which of the following is/are correct?

(A) The wave equation of this string is elliptic.
(B) All the boundary conditions belong to Dirichlet conditions.
(C) The initial condition yields $T'(0) = 0$.
(D) This is referred to as a boundary-value problem.
(E) One of the boundary conditions yields $X(0) = \pi$.

Assume the product solution to the wave equation in Question 9 is found as $u(x, t) = A \cos(\beta a t) \sin(Cx)$. If the initial displacement $f(x)$ is a sinusoidal wave with the amplitude of $1/100$ and the frequency of $3/2\pi$, which of the following is/are correct?

(A) There exists a trivial solution.
(B) $B =$ arbitrary integer.
(C) $C = 3$.
(D) There is no motion at $x = 2\pi/3$ and all other $x$ points vibrate vertically over time.
(E) The vertical displacement is reversed ($-f(x)$) when $t = k\pi/\alpha$, $k = 1,2,3...$

Consider the differential equation $P'(t) = aP(t) - bP(t)^2$ with the initial condition of $P(0) = p_0$, where $a$ and $b$ are positive real constants. Which of the following is INCORRECT?

(A) This is a linear differential equation.
(B) This is a Bernoulli's equation.
(C) None of the critical points are attractors.
(D) When $t \to \infty$, $P(t) = a/b$.
(E) The solution interval is $(-\infty, \infty)$.

Given $X' = \begin{pmatrix} 1 & 2 \\ -1 & -3 \end{pmatrix} X$, which of the following is correct?

(A) There are two distinct eigenvalues.
(B) The stationary point is a repeller.
(C) All solutions will converge onto the origin.
(D) One of the eigenvalues is $\lambda_2 = -1$.
(E) One of the solution vectors is $K_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix} e^{-4t}$.

Define the matrix $A$ as $pq^T$, where $p$ and $q$ are nonzero vectors in $\mathbb{R}^{n \times 1}$. Which of the following statements are true?

(A) rank(A) may be 2.
(B) $q$ is an eigenvector of $A$.
(C) $q^T p$ is an eigenvalue of $A$.
(D) The linear equation $Ax = 0$ has infinitely many solutions.
(E) The linear equation $Ax = b$ with $b = q$ has infinitely many solutions.

Define the matrix $M$ as $I - 2uu^T$, where $u$ is a unit vector in $\mathbb{R}^{n \times 1}$, meaning $u^T u = 1$. Which of the following statements are true?

(A) $M = M^2$.
(B) $M^T M = I$.
(C) $M = M^{-1}$.
(D) $\det(M) = 1$.
(E) $M$ has eigenvalues that are complex with non-zero real components.

Let $N(A)$ denote the null space of $A$, and $R(A)$ denote the range space of $A$. Suppose $u \in R(A)$, $v \in N(A^T)$, and both $u$ and $v$ are non-zero. Define $||x|| = \sqrt{\langle x, x \rangle}$. Suppose $x_1 = u + 2v$, $x_2 = 2u - v$, $||x_1||^2 = 29$, $||x_2||^2 = 41$. Which of the following statements are true?

(A) $||u|| = \sqrt{5}$.
(B) $||u|| = 3$.
(C) $||v|| = 7$.
(D) $||v|| = \sqrt{5}$.
(E) $||u|| + ||v|| = 8$.