所屬科目：中山◆電機◆工程數學甲
1.Given a continuous-time periodic signal Let ak and ω0, be the coefficients and fundamental frequency of the Fourier series of f(t) respectively. Then which of the following statements is correct? (A)(B)(C)(D)a0＜5,ω0＜2,a5＞0. (E) None of the above statements are correct.
2.Consider a discrete signal x(n) = cos(2nr / N), where integer N is the fundamental period. Let a be the coefficients of the discrete-time Fourier series of x(n). Then which of the following statements is correct? (A) a1=-1/2,a2=1/2. (B) a1=a2=j/2 (C) (D) (E) None of the above statements are correct.
3.Consider an LTI system whose impulse response is G(jω)=1/(a+ jω), a ＞ 0. Suppose that there is an input signal X(jω)=1/(a+ jω). Assume that the output signal is y(t)= , where u(t) is a unit-step signal. Then which of the following statements is correct? (A)β+y=-a (B) βy=-a (C) β-y=t+a (D) βy=at (E) None of the above statements are correct.
4.Consider the following three systems, where [n] or x(t) is the system input, y[n] or y(t) denotes the system output, and I.Which of the following statements is correct? (A) I is time-invariant, II is linear, Ill is causal. (B) I is memoryless, II is causal, III is nonlincar. (C) I is stable, II is linear, III is memoryless. (D) I is causal, II is memoryless, III is linear. (E) None of the above statements are correct.
5. Let z =1- . Assume that , whereθ denotes the principal argument. Then which of the following statements is correct? (A) r =2,θ =π/5 (B)(C)(D) (E) None of the above statements are correct.
6.Which one of the following functions, where z = x+ jy is a complex variable, is analytic? (A) f(2)=, where =x-jy. (B) f(z) = xy+jx (C)f(z)=x2ーjy2 (D) f(z)=x2+y2+j2xy (E) None of the above statements are correct.
7.Let z be a complex number. Which of the following statements is correct? (A) Log(z1/z2) = Log(z1)+ Log(z2), where Log(z) is the principal value of the complex logarithm. (B) cos(j) is not a real value.(C) , n=0,±1,±2,......(D) z2+2z-ez=,where exp(2)=ez. (E) None of the above statements are correct.
8. Let f(z)=z/(z2+9), and C be a circle|z - j2|=4 in counterclockwise direction. The evaluation of. Then which of the following statements is correct? (A) a＜0,β＞0 (B) a ＞0,β＞0 (C)a=0,1＜β＜4 (D) a＞0, -3＜β＜4 (E) None of the above statements are correct.
9. Let f(z)=, and C be the right-hand half of the circle|z|=2 form z = j2 to z =-j2. Compute the value of S. f(z)dz =a+ jB. Then which of the following statements is correct? (A) a=0,-15＜β＜0 (B) a＞0,0＜β＜15 (C)-12＜a+β＜l2 (D)-2＜aβ＜-12 (E) None of the above statements are correct
10. The Laurent series of is Which of the following statements is correct? (A) aβ=1/6 (B) aβ=1/12 (C) a+β=5/6 (D)a+β=1/3 (E) None of the above statements are correct.
(a) (5%) Suppose that m=n = 3, A = . Find the set of all solutions to the -4 2 equation Ax = b if it is consistent. Otherwise, find vector p to solve and, moreover peR(A) compute the value of .
(b) (5%) When the equation Ax = b is unsolvable, we may consider the so-called least squares problen to find a set of solutions, having the least squares error, fom solving a normal equation. Suppose that rank(A) =k ＜min(m, n) and let A = BC be a full rank decomposition of A . Use the known matrices B, C, and b to describe the unique projection vector p of b onto R(A) with the least || b-pll2.
(a) (4%) Denote the angle between f1 and f2 by θ. Find all possible values of a2such that θ = π/4.
(A) (4%) Find all possible values of β such that, the inverse of L, does not exist.
(b) (6%) Suppose that exists. Find the matrix representation of corresponding to the ordered base s{x+1，x - 1} and for P2 and R2, respectively.
(A) (15%) Let u(t) ≡ 0 and the initial conditions be x1(0)=x2(0)=.Find the solutions of the differential equations.
(B) (5%) Let initial conditions be x1(0) = x2(0) = c1(0) =, and u(t) be the unit step function. Does the solutions of the differential equations converge to constant values as time approaches infinity? Justify your answers.
15. (10%) Evaluate the following integral
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(B)
(C)
(D)a0＜5,ω0＜2,a5＞0. (E) None of the above statements are correct.
(D) 
(E) None of the above statements are correct.
, where u(t) is a unit-step signal. Then which of the following statements is correct? (A)β+y=-a (B) βy=-a (C) β-y=t+a (D) βy=at (E) None of the above statements are correct.
Which of the following statements is correct? (A) I is time-invariant, II is linear, Ill is causal. (B) I is memoryless, II is causal, III is nonlincar. (C) I is stable, II is linear, III is memoryless. (D) I is causal, II is memoryless, III is linear. (E) None of the above statements are correct.
. Assume that 
, whereθ denotes the principal argument. Then which of the following statements is correct? (A) r =2,θ =π/5 (B)
(C)
(D)
(E) None of the above statements are correct.
, where 
=x-jy. (B) f(z) = xy+jx (C)f(z)=x2ーjy2 (D) f(z)=x2+y2+j2xy (E) None of the above statements are correct.
, n=0,±1,±2,......(D) z2+2z-ez=
,where exp(2)=ez. (E) None of the above statements are correct.
. Then which of the following statements is correct? (A) a＜0,β＞0 (B) a ＞0,β＞0 (C)a=0,1＜β＜4 (D) a＞0, -3＜β＜4 (E) None of the above statements are correct.
, and C be the right-hand half of the circle|z|=2 form z = j2 to z =-j2. Compute the value of S. f(z)dz =a+ jB. Then which of the following statements is correct? (A) a=0,-15＜β＜0 (B) a＞0,0＜β＜15 (C)-12＜a+β＜l2 (D)-2＜aβ＜-12 (E) None of the above statements are correct
is
Which of the following statements is correct? (A) aβ=1/6 (B) aβ=1/12 (C) a+β=5/6 (D)a+β=1/3 (E) None of the above statements are correct. 
. Find the set of all solutions to the -4 2 equation Ax = b if it is consistent. Otherwise, find vector p to solve 
and, moreover peR(A) compute the value of 
.
, the inverse of L, does not exist.
exists. Find the matrix representation of 
for P2 and R2, respectively. 
.Find the solutions of the differential equations.
, and u(t) be the unit step function. Does the solutions of the differential equations converge to constant values as time approaches infinity? Justify your answers.