信号与系统signal and system英文版往年考试题1

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 2πππδ__________________;(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 forsignal ()⎪⎭⎫ ⎝⎛=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⎰∞--2and ⎥⎦⎤⎢⎣⎡⎪⎭⎫ ⎝⎛+12t x dt d .Figure 13. (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()tx 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 itsFourier 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,ccH j ωωωωω⎧<⎪=⎨>⎪⎩.Suppose that the input signal sin()()at x t tπ=, 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 thiscase.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 at1s =-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 make21()10.25j a j H e eωω-=-, 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。

信号与系统习题集第一部分：时域分析 一、填空题 1. =---)3()()2(t t u et δ()。

2. The unit step response )(t g is the zero-state response when the input signal is ( ).3. Given two continuous – time signals x(t) and h(t), if their convolution is denoted by y(t), then the convolution of )1(-t x and )1(+t h is ( ).4. The convolution =+-)(*)(21t t t t x δ( ).5. The unit impulse response )(t h is the zero-state response when the input signal is ( ).6. A continuous – time LTI system is stable if its unit impulse response satisfies the condition: ( ).7. A continuous – time LTI system can be completely determined by its ( ).8. =⎰∞∞-(t)dt 2sin 2 δt t ( ).9. Given two sequences }1,2,2,1{][=n x and }5,6,3{][=n h , their convolution=][*][n h n x ( ).10. Given three LTI systems S1, S2 and S3, their unit impulse responses are )(1t h ,)(2t h and )(3t h respectively. Now, construct an LTI system S using these threesystems: S1 parallelinterconnected by S2, then series interconnected by S3. the unit impulse response of the system S is ( ).11. It is known that the zero-stat response of a system to the input signal x(t) is⎰∞-=td x t y ττ)()(, then the unit impulse response h(t) is ( ).12. The complete response of an LTI system can be expressed as a sum of its zero-state response and its ( ) response.13. It is known that the unit step response of an LTI system is )(2t u et-, then the unit impulseresponse h(t) is ( ).14. =++-=⎰∞dt t t t t x ))1()1((2sin )(0δδπ( ).15. We can build a continuous-time LTI system using the following three basic operations: ( ) , ( ), and ( ). 16. The zero-state response of an LTI system to the input signal )1()()(--=t u t u t xis )1()(--t s t s , where s(t) is the unit step response of the system, then the unit impulse response h(t) is ( ).17. The block diagram of a continuous-time LTI system is illustrated in the following figure. The differential equation describing the input-output relationship of the system is ( )。

Test （Chapter 1,2,3,4）参考答案I. Blank filling (Total scores 60, each scores 5. ) 1、)0(1f a（70％） 2、2/1-（96％）3、)()(t t u e t δ+--（92％）4、Piecewise function can be comprised by rectangular window functions(可以用矩形窗函数构造分段函数). Consider a signal )]1()([)(--=t u t u t t f ，the even component of )(t f is )(t f e ＝ ，the odd component of )(t f is)(t f o ＝)]1()1()(2[2+---t u t u t u t（46％） )]1()1([2--+t u t u t（92％） 5、Consider a signal )]1()([)(--=t u t u t t e ，and)]1()([)(--=-t u t u e t h t ，so =)(*)(t h t e 。

(Express your answer by rectangular window function likeI.4) 先做微分⎪⎩⎪⎨⎧<<-<<+-=--其他,021),2(10,1)(1t t e t e t t y t（76％）6、Necessary and sufficient conditions for stability of the systemisitsimpulseresponse)(t h should be【int(f(x),-inf,inf)＝dx x f ⎰∞∞-)(】∞<⎰∞∞-dt t h )(（37％）7、consider a signal )1()()(--=t u t u t e ，则=)(*)(t e t e (Express your answer by rectangular window function like I.4))]2()1()[2()]1()([----+--t u t u t t u t u t （87％）,012,120,t t t t others <<⎧⎪-<<⎨⎪⎩8、Known Fourier transform of f(t) is )(ωj F ，so the Fourier transform of )4(t f - is 。

5. The impulse response of a LTI system is h(t) 1 [u(t) u(t 4)],the step
4
response for the system is (
).
(A) s(t)
1 4
t[u(t)
u(t
4)]
u(t
4)
(B) s(t) 1 [r(t) r(t 4)] , here
2. Consider sampling x(t) Sa(10t) ， determine the maximum of
sampling interval T so that there will be no aliasing, Tmax
(s).
3. Write or Sketch the spectrum of Rectangular pulse x[n]:
2. 所有答案请直接答在试卷上；
3．考试形式：闭 卷；
4. 本试卷共 3 大题，满分 100 分， 考试时间 120 分钟。
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一、 填空题（4 分/每题，共 20 分）
1. A system has input x1(t) and output y1(t) . If the system has properties, then the input and output pairs has the relationship: input is x2 (t) x1(t 2),s3ox1o(tutp3)ut is y2 (t) y1(t 2) 。 3y1(t 3)
)。
(D)
0.64 1 0.8e j2