信号与系统习题集英文版

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 e t 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 three systems: S1 parallel interconnected 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 ?∞-=t

d x t y ττ)()(, then th

e unit impulse response h(t) is ( ). 12. The complete response o

f 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 e t -, then the unit impulse response h(t) is ( ). 14. =++-=?∞dt t t t t x ))1()1((2

sin )(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 x

is )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 ( )。

18. The relationship between the unit impulse response h(t) and unit step response s(t) is s(t) = ( ), or h(t) = ( ).

19. If the unit impulse response h(t) of

二、选择题

1. For each of the following equations, ( ) is true.

A 、)()()1(t t t δδ=-

B 、)(2)1()1(t t t δδ=-+

C 、?∞∞-=+)()()1(t dt t t δδ

D 、?∞

∞-=++1)1()1(dt t t δ 2. Given two continuous-time signals )(t x and )(t h , if the convolution of )(t x and )(t h is denoted by )(t y , then the convolution of signals )1(+t x and )2(-t h is

( ).

A 、)(t y

B 、)1(-t y

C 、)2(-t y

D 、)1(+t y 3. The unit impulse response of an LTI system is h(t) = t e

-, this system is ( ).

A. causal and stable

B. causal and unstable

C. noncausal and unstable

D. noncausal and stable

4. dt t t t x )2()12()(112-+=?

-δ = ( ).

A 、1

B 、3

C 、9

D 、0

5. For an LTI system, if the input signal is )(1t x , the corresponding output response is )(1t y , if the input signal is )(2t x , the corresponding output response is )(2t y .

)(t

And if the input signal is )()(21t bx t ax +, the corresponding output response is )()(21t by t ay + ( a and b are arbitrary real numbers ). Then the system is a ( ) system.

A. linear

B. causal

C. nonlinear

D. time – invariant

6. )(*)(21t t t t x --δ = ( ). A. )(21t t t x -- B. )(21t t t x +-

C. )(21t t t x -+

D. )(21t t t ++δ 7. =??

? ??-+=?∞∞-dt t t t t x 6)sin ()(πδ ( ). A.6π B.16

-π C. 216-π D. 2

16+π 8. Given two sequences ][1n x and ][2n x , their lengths are M and N respectively. The length of the convolution of ][1n x and ][2n x is ( ).

A ．M

B ．N

C ．N M +

D ．1-+N M

9. The unit impulse response of a continuous-time LTI system is dt

t d t t h )()(2)(δδ+=, the differential equation describing the input-output relation of this system is ( ). A.)()()(2t x dt t dy t y =+ B. )()(2)(t x dt

t dy t y =+ C. dt t dx t x t y )()(2)(+= D. dt

t dx t x dt t dy )(2)()(+= 10. The input-output relation of a continuous-time LTI system is described by the

differential equation: dt t dx t x t y dt t dy dt t y d )()(2)(3)(2)(2

2+=++. The unit impulse response of the system h(t) ( ).

A . does not include )(t δ B. includes )(t δ C. includes dt

t d )(δ D. is uncertain 11. Signals )(1t x and )(2t x are shown in the following figures. The expression of

the convolution )(*)()(21t x t x t x = is ( ).

A. )1()1(--+t u t u

B. )2()2(--+t u t u

C. )1()1(+--t u t u

D. )2()2(+--t u t u

12. The following block diagram represents a continuous-time LTI system. The unit impulse response h(t) satisfies ( ).

A.

)()()(t x t y dt

t dy =+ B. )()()(t y t x t h -= C. )()()(t t h dt t dh δ=+ D. )()()(t y t t h -=δ 13. The input-output relationship of a continuous-time system is described by the differential equation: dt

t dx t y dt t dy )(2)(3)(=+, then the unit step response =)(t s ( ). A. )(23t u e t - B. )(213t u e t - C. )(23t u e t D. )(2

13t u e t 三、综合题

1. The input-output relationship of a continuous-time LTI system is described by the equation: τττd x e t y t

t )2()()(-=?∞---， a. Determine the unit impulse response h(t) of the system.

b. Determine the system response y(t) to the input signal )2()1()(--+=t u t u t x .

2. Given an LTI system depicted in Figure (1). Assume that

the impulse response of the LTI system is h(t) = e -t u(t), the

input signal x(t) = u(t) - u(t-2). Determine and sketch the

⊕)(t x )t

+ -1 1 (1)

0 (1) )(2t x Figure (1)

output response y(t) of the system by evaluating the convolution y(t) = x(t)*h(t).

3. Remember the following identities:

)(*)()(t t x t x δ=

)(*)()(00t t t x t t x -=-δ

)()(*)(00t t t t t δδδ=-+

dt

t dh t x t h dt t dx dt t dy )(*)()(*)()(== 4. Consider an LTI system S and a signal )1(2)(3-=-t u e t x t . If

)()(t y t x →

and )()(3)(2t u e t y dt

t dx t -+-→, determine the impulse response h(t) of S.

5. Let )5()3()(---=t u t u t x and )()(3t u e t h t -=.

(a). Compute y(t) = x(t)*h(t).

(b). Compute g(t) = dx(t)/dt * h(t).

(c). How is g(t) related to y(t)?

6. Let )5()3()(---=t u t u t x and )()(3t u e t h t -=, as illustrated in the following figure.

(a). Compute y(t) = x(t)*h(t).

(b). Compute g(t) = dx(t)/dt * h(t).

(c). How is g(t) related to y(t)?

7. Let ∑∞-∞=--=k t k t t u e t y )3(*)()(δ

Show that t Ae t y -=)( for 0 ≤ t < 3, and determine

the value A.

8. A causal LTI system is described by the differential equation:

)()()(2)(3)(22t x dt

t dx t y dt t dy dt t y d +=++ )(t x t 5)

(t h t Figure 6 131

If the input signal is )(

x t-

=, determine the zero-state response y(t) of the

t

e

u

)(2t

system.