2信号与系统_奥本海默_华科_电信系_英文_试卷

TEST OF HUAZHONG UNIVERSITY OF SCIENCE & TECHNOLOGY (A)
Course: SIGNALS & SYSTEMS (Closed Book) (2008/05/24)
SPECIALTY_________CLASS_________NAME__________No.____________
1. (20 points)Consider the following problems, then fill in the blanks. (2 points for each blank)
(a)
()()=-⎰-dt t t 3sin 2π
ππδ__________________;
(b) The fundamental period of sequence ⎪⎭
⎫ ⎝⎛+=376cos ][n n x π is________________; (c) If a continuous-time system is defined by ()()t x e t y t -=1, then we can determine that it ’s a (linear / nonlinear) _____________, (time invariant / time variant) _________________, (causal / noncausal) _____________ system;
(d) Consider a discrete-time system with the input and output relationship being
[][][2]y n x n x n =-, if the input [][]n A n x δ=, here A is an arbitrary real or complex number,
the output []y n =___________;
(e) If an LTI system with impulse response ()t h 1 is invertible, and its inverse system has an impulse response ()t h 2, then we have 12()()h t h t *=______________;
(f) The constant component of the continuous-time periodic signal ()sin()x t t ω= is________; (g) A signal ()x t with Fourier transform ()ωj X undergoes impulse-train sampling. If
()0=ωj X for s rad /105>ω, then the Nyquist sampling period is___________ second ; (h) Consider a signal ()t x 1 with FT ()ωj X 1. If ()01=ωj X for m ωω>, then for
signal ()⎪⎭
⎫ ⎝⎛=2312t x t x with FT ()ωj X 2, there must be ()02=ωj X for >ω_________.
2．(10 points) ()t x is shown in Figure 1, sketch and label carefully the signals ()ττd x t
⎰∞
--2
and ⎥⎦
⎤⎢⎣⎡⎪⎭⎫ ⎝
⎛+12
t x dt d .
Figure 1
3. (8 points) Verify that if the input signal []x n to an LTI system is periodic with period N , then the output []y n is also a periodic signal with the same period.
4. (10 points)
(a) Evaluate the Laplace transform of signal 2()t
x t t e -= and specify its ROC.
(b) Evaluate the inverse z-transform of ()a z z a z X >⎪⎭
⎫ ⎝⎛+=1ln .
5. (10 points) Consider the discrete-time periodic sequence [][4]k x n n k δ∞
=-∞
=
-∑, determine its
Fourier series coefficients k a and its Fourier transform, and plot []x n and ()j X e ω.
6. (10 points) Consider an ideal low-pass filter with frequency response 1,()0,c
c
H j ωωωωω⎧<⎪=⎨>⎪⎩.
Suppose that the input signal sin()
()at x t t
π=
, please answer the following questions: (a) Determine the output ()y t in the case of c a ω<; (b) Determine the output ()y t in the case of c a ω>; (c) In which case, the input signal is not distorted? And evaluate 2
()y t dt ∞-∞
⎰in this
case.
7. (16 points) Suppose we are given the following information about a causal LTI system described by a linear constant-coefficient second-order differential equation with impulse response ()h t and system function ()H s : 1. If ()1x t =, then ()1y t =-; 2. ()H s has a pole at
1s =-and a zero at 1s =; 3. ()h t doesn’t contain any impulse s and ()
20=+h . Try to answer the following questions: (a) Determine the differential equation and draw a direct-form block diagram for the system; (b) Is the system stable? (c) Determine the output of the system when the input ()()x t u t =and the initial conditions (0)4,(0)3y y --'=-=.
8. (16 points) A discrete-time causal system with system function 1()H z depicted in Figure 2(a), (a) If the system 1()H z is cascaded with another causal system with system function 2()H z to construct a system with system function ()a H z , just as Figure 2(b) shown, to make
21
()10.25j a j H e e
ωω
-=
-, determine 2()H z ; (b) Determine the unit sample response []a h n of the system ()a H z ; (c) Let ()b H z be the system function of a causal system shown in Figure 2(c), to make ()()
ωωj a j b e H e H =, determine the values of 1b 、2b 、3b in Figure 2(c); (d) Write
]n
[]x n
[]x n
[]n a
[]x n []y n
(c)
Figure 2。