信号与系统习题集

信号与系统练习题集

第一部分：信号与系统的时域分析

一、填空题 1. e?(t?2)u(t)?(t?3)? ( ).

2. The unit step response g(t) 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 x(t?1) and h(t?1) is ( ). 4. The convolution x(t?t1)*?(t?t2)?( ).

5. The unit impulse response h(t) 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 ( ).

?sin 2t8. ? 2? (t)dt? ( ).

??t9. Given two sequences x[n]?{1,2,2,1} and h[n]?{3,6,5}, their convolution x[n]*h[n]? ( ).

10. Given three LTI systems S1, S2 and S3, their unit impulse responses are h1(t), h2(t) and

h3(t) 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 y(t)??x(?)d?,

??tthen 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 e?2tu(t), then the unit impulse response h(t)

is ( ).

14. x(t)??sint(?(t?1)??(t?1))dt? ( ).

0215. 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 x(t)?u(t)?u(t?1) is

s(t)?s(t?1), where s(t) is the unit step response of the system, then the unit impulse response

??is h(t) = ( ).

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

( ).

x(t)+ + ?-?2 3 ?y(t)

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

( ), or h(t) = ( ).

二、选择题

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

A、(t?1)?(t)??(t) B、(1?t)?(1?t)?2?(t) C、?(1?t)?(t)dt??(t) D、?(1?t)?(1?t)dt?1

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

A. y(t) B. y(t?1) C. y(t?2) D. y(t?1)

3. The unit impulse response of an LTI system is h(t) = e?t, this system is ( ).

A. causal and stable B. causal and unstable C. noncausal and unstable D. noncausal and stable

14. x(t)?(2t2?1)?(t?2)dt = ( ).

?1? A. 1 B. 3 C. 9 D. 0

5. For an LTI system, if the input signal is x1(t), the corresponding output response is y1(t), if the input signal is x2(t), the corresponding output response is y2(t). And if the input signal is

ax1(t)?bx2(t), the corresponding output response is ay1(t)?by2(t) ( a and b are arbitrary real

numbers ). Then the system is a ( ) system.

A. linear B. causal C. nonlinear D. time – invariant

6. x(t?t1)*?(t?t2) = ( ).

A. x(t?t1?t2) B. x(t?t1?t2) C. x(t?t1?t2) D. ?(t?t1?t2)

????x(t)?(t?sint)??t??dt? ( ). 7. ???6?? A.

?1?1?? B.?1 C. ? D. ?

6262668. Given two sequences x1[n] and x2[n], their lengths are M and N respectively. The length of the

convolution of x1[n] and x2[n] is ( ).

A．M B．N C．M?N D．M?N?1

9. The unit impulse response of a continuous-time LTI system is h(t)?2?(t)?d?(t), the differential dtequation describing the input-output relation of this system is ( ).

dy(t)dy(t)A.2y(t)??x(t) B. y(t)?2?x(t)

dtdtdx(t)dy(t)dx(t)C. y(t)?2x(t)? D. ?x(t)?2dtdtdt10. The input-output relation of a continuous-time LTI system is described by the differential

d2y(t)dy(t)dx(t)?2?3y(t)?2x(t)?equation: . The unit impulse response of the system h(t) 2dtdtdt( ).

A . does not include ?(t) B. includes ?(t) C. includes

d?(t) D. is uncertain dt11. Signals x1(t) and x2(t) are shown in the following figures. The expression of the convolution x(t)?x1(t)*x2(t) is ( ).

x1(t) 1 (1)

1 -1 0 x2(t)

(1)

1 -1 0

A. u(t?1)?u(t?1) B. u(t?2)?u(t?2) C. u(t?1)?u(t?1) D. u(t?2)?u(t?2)

12. The following block diagram represents a continuous-time LTI system. The unit impulse

response h(t) satisfies ( ).

dy(t)A. ?y(t)?x(t)

dtB. h(t)?x(t)?y(t) C.

dh(t)?h(t)??(t) dtx(t)+ ?-?y(t)

D. h(t)??(t)?y(t)

13. The input-output relationship of a causal continuous-time system is described by the differential equation:

dy(t)dx(t), then the unit step response s(t)? ( ). ?3y(t)?2dtdt11A. 2e?3tu(t) B. e?3tu(t) C. 2e3tu(t) D. e3tu(t)

22三、综合题（分析、计算题）

1. The input-output relationship of a continuous-time LTI system is described by the equation:

y(t)??e?(t??)x(??2)d?，

??ta. Determine the unit impulse response h(t) of the system.

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

2. Given an LTI system depicted in Figure 2. Assume that the impulse response of the LTI system is h(t) = e-tu(t), the input signal x(t) = u(t) - u(t-2). Determine and sketch the output response y(t) of the system by evaluating the convolution y(t) = x(t)*h(t).

3. Remember the following identities:

x(t)h(t)Figure 2

y(t) x(t)?x(t)*?(t) x(t?t0)?x(t)*?(t?t0)

?(t?t0)*?(t?t0)??(t)

dy(t)dx(t)dh(t) ?*h(t)?x(t)*dtdtdt4. Consider an LTI system S and a signal x(t)?2e?3tu(t?1). If

x(t)?y(t)

and

dx(t)??3y(t)?e?2tu(t), dtdetermine the impulse response h(t) of S.

5. Let x(t)?u(t?3)?u(t?5) and h(t)?e?3tu(t), as illustrated in the Figure 6.

(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 y(t)?eu(t)*?t?1x(t)t35h(t)1tk?????(t?3k)

Figure 6

Show that y(t)?Ae?t for 0 ≤ t < 3, and determine the value A. 7. A causal LTI system is described by the differential equation:

d2y(t)dy(t)dx(t)?3?2y(t)??x(t) dt2dtdtIf the input signal is x(t)?e?2tu(t), determine the zero-state response y(t) of the system.

8. In this problem, we illustrate one of the most important consequences of the properties of linearity and time invariance. Specifically, once we know the response of a linear system or a linear time-invariant system to a single input or responses to several inputs, we can directly compute the responses to many other input signals.

(a). Consider an LTI system whose response to the signal x1(t) in Figure 9(a) is the signal y1(t) illustrated in Figure 9(b). Determine and sketch carefully the response of the system to the input x2(t) depicted in Figure 9(c).

(b). Determine and sketch the response of the system considered in part (a) to the input x3(t) shown in Figure 9(d).

x3(t)(a) x1(t)2112y1(t)1tx2(t)12(b) 4t21?1(c) 112t1(d) 2?14tFigure 9

第二部分：信号与系统的频域分析

一、填空题

??2,1. The frequency response of an ideal filter is given by H(j?)????0,??100???100?， if the input

signal is x(t)?10cos(80?t)?5cos(120?t), the corresponding output response y(t) = ( ).

2. The Fourier transform of signal x(t)?cos(?0t) is ( ). 3. The Fourier transform of signal x(t)?sin(?0t??6) is ( ).

4. Assume that the Fourier transform of x(t) is denoted as X(j?), then the Fourier transform of

y(t)?ej?0tx(t) is Y(j?) = ( ).

5. The Fourier transform of a continuous – time periodic signal x(t)?( ).

6. It is known that the Fourier transform of x(t) is X(j?)?tx(t) is ( ).

k????aek?jk?0t is X(j?)=

1, then the Fourier transform of j??17. The Fourier transform of signal x(t) is denoted as X(j?), the Fourier transform of (t?1)x(t) is ( ).

8. A time shifting leads to a ( ).

9. The frequency responses of two LTI systems are assumed to be H1(j?) and H2(j?), the frequency response of the interconnection of H1(j?) cascaded by H2(j?) is H(j?) = ( ).

10. A time-domain compression corresponds to a frequency-domain ( ). 11. For a signal x(t), if the condition

????x(t)dt?? is satisfied, then the Fourier transform of x(t)

exists, this condition is ( ) but not ( ).

12. Figure 12 shows a continuous-time signal x(t), its Fourier transform is denoted as X(j?), then X(0)? ( ). (Without evaluatingX(j?)).

13. For a continuous-time LTI system, if the zero-state response of the system to

the input signal

x(t)?e?tu(t) is

1 x(t) y(t)?e?tu(t)?e?2tu(t), then the frequency response of the

system is H(j?)? ( ). 14. The Fourier transform of signal x(t)?( ).

15. The inverse Fourier transform of ?(?) is x(t)? ( ).

sin4t is X(j?)? t-1 0 1 Figure 12 16. The frequency spectrum includes two parts, one is ( ), the other is ( ). 17. Let X(j?) denote the Fourier transform of signal x(t), then the Fourier transform of signal

ty(t)?x(?3)*cos(4t) is Y(j?)?( ). (Expressed using X(j?)).

218. Let X(j?) denote the Fourier transform of signal x(t), then the Fourier transform of signal

y(t)?x(t)cos(?t) is Y(j?)?( ). (Expressed using X(j?)).

19. The period of the periodic square wave increases, the space of the spectral lines ( ).

20. Consider a continuous-time ideal lowpass filter S whose frequency response is

??1 H(j?)????0??100??100

When the input to this filter is a signal x(t) with fundamental period T = π/6 and Fourier series coefficients ak, it is found that

S??y(t)?x(t) x(t)?For k ( ) it is guaranteed that ak = 0.

21. Consider a continuous-time LTI system whose frequency response is

?? H(j?)????j?th(t)edt??sin4(?)?

?10?t?4If the input to this system is a periodic signal x(t)?? with period T = 8, the

??14?t?8corresponding system output is y(t) = ( ).

二、选择题

1. The frequency response of an ideal lowpass filter is

??2, H(j?)????0,??120???120?.

If the input signal is x(t)?10cos(100?t)?5cos(200?t), the output response is y(t)= ( ).

A. 10cos(100?t) B. 10cos(200?t) C. 20cos(100?t) D. 5cos(200?t)

2. The Fourier transform of the rectangular pulse x(t)?u(t?1)?u(t?1) is ( ).

A. 4Sa(?) B. 2Sa(?) C. 2Sa(2?) D. 4Sa(2?)

3. Let X(j?) denote the Fourier transform of a signal x(t), the Fourier transform of x(t)ejt is ( ).

A. e?j?X(j?) B. ej?X(j?) C. X(j(??1)) D. X(j(??1))

4. Let X(j?) denote the Fourier transform of signal x(t), the Fourier transform of x(t?1) is ( ).

A. e?j?X(j?) B. ej?X(j?) C. X(j(??1)) D.X(j(??1)) 5. The Fourier transform of the rectangular pulse x(t)?u(t)?u(t?1) is ( ).

A. sa()e2??j?2 B. sa()e2?j?2 C. sa(?)e?j? D. sa(?)ej?

6. The condition for signal transmission with no distortion is that ( ).

A. The magnitude response is a constant in the passband. B. The phase response is a line cross the origin in the passband.

C. The magnitude response is a constant and the phase response is a line cross the origin in the passband.

D. The phase response is a constant and the magnitude response is a line cross the origin. 7. The bandwidth of a signal x(t) is 20KHz, the bandwidth of signal x(2t) is ( ).

A.20KHz B.40KHz C.10KHz

D.30KHz

8. Let X(j?) denote the Fourier transform of signal x(t), the Fourier transform of t( ).

A.X(j?)??dx(t) is dtdX(j?) d?dX(j?) d?B. ?X(j?)??dX(j?) d?C. ?X(j?)??D. X(j?)??dX(j?) d?9. Let X(j?) denote the Fourier transform of signal x(t), the Fourier transform of

ty(t)?x(?b) is ( ).

aA. aX(j?)ejab? B. aX(ja?)e?jab?1?j?1??j? C. X(j)ea D. X(j)ea

aaaabb10. Let X(j?) denote the Fourier transform of signalx(t)?u(t?1)?u(t?1), then X(0)? ( ).

A. 2 B. ? C.

1? D. 4 211. Let X(j?) denote the Fourier transform of signal x(t), the Fourier transform of x(1?t) is ( ).

A．?X(?j?)e B．X(j?)ej??j? C．X(?j?)e?j? D．X(?j?)ej?

12. Let X(j?) denote the Fourier transform of signal x(t), the Fourier transform of

y(t)?x(t)?(t?a) is ( ).

A. X(j?)e?ja? B. x(a)e?ja? C. X(j?)eja? D. x(a)eja?

13. The Fourier transform of signal x(t)??(t??)??(t??) isX(j?)? ( ).

11A. cos?? B. 2cos?? C. sin?? D. 2sin??

2214. Let x(t)?e?t?(t), and y(t)??x(?)d?. The Fourier transform of y(t) is Y(j?)?( ).

??tA.

111???(?) D. ????(?) B. j? C. j?j?j?1115. Consider the square wave x(t)?u(t??)?u(t??), as ? decreases, the width of the main

22lobe of X(j?) ( ).

A. increases B. decreases C. does not change D. can not be determined

16. It is known that the bandwidth of x(t) is??, the bandwidth of x(2t?1) is ( ).

A. 2?? B. ??-1 C. 17. The inverse Fourier transform of X(j?)?11 ?? D. （??-1）221ej?t0 is x(t)?( ). j??aA. x(t)?e?a(t?t0)u(t) B. x(t)?e?a(t?t0)u(t?t0) C. x(t)?e?a(t?t0)u(t?t0) D. x(t)?e?a(t?t0)u(t)

18. The Fourier transform of signal x?(t) is X?(j?)??Sa(?), then the Fourier transform of

2signal y(t)?x?(t?1) is Y(j?)?( ).

A. Y(j?)?Sa(?)ej? B. Y(j?)?Sa(?)e?j? C. Y(j?)?2Sa(?)ej? D. Y(j?)?2Sa(?)e?j?

19. Given an LTI system with its frequency response H(j?)?1, it is known that the Fourier j??2?transform of the output response y(t) is Y(j?)?x(t)=( ).

1, the input signal

(j??2)(j??3)A. x(t)?e?2tu(t) B. x(t)??e?3tu(?t) C. x(t)?e?3tu(t) D. x(t)?e3tu(t)

?e?j?20. The frequency response of an ideal lowpass filter is H(j?)???0??2, its unit impulse ??2response is h(t) = ( ).

A.

三、综合题(分析、计算)

1. Consider a continuous-time LTI system whose frequency response is

sin(4?)H(j?)?

sin2tsin2(t?1)sintsin(t?1) B. C. D.

?(t?1)?(t?1)?(t?1)?(t?1)??10?t?4If the input to this system is a periodic signal x(t)?? with period T = 8, determine

??14?t?8the corresponding system output y(t).

2. The fundamental frequency of a continuous-time periodic signal is ω0 = π, Figure 2 shows the spectral coefficients of x(t).

(a) Write out the expression of x(t).

(b) If x(t) is applied to an ideal highpass filter with

ak??21???1,frequency response H(j?)???0,determine the output signal y(t).

??15?otherwise?20?10,

??010?ak?20k?k3. Figure 3.a illustrates a communication system. Let X1(jω) and X2(jω) denote the Fourier transforms of x1(t) and

Figure 2

x2(t), respectively. It is known that ω1 = 4π, ω2 = 8π, and the frequency response of the ideal bandpass filter is H1(jω), the overall output response is y(t). (1). Plot the magnitude of the Fourier transform W(jω) of w(t).

(2). Choose an appropriate frequency ω3, so that the output response is y(t) = x1(t); (3). Plot the magnitude responses of H1(jω) and H2(jω).

x1(t)?cos(?1t)x2(t)?cos(?2t)?w(t)H1(j?)z(t)?cos(?3t)y(t)H2(j?) (a)

Figure 3

??1X1(j?)?1??X2(j?)??(b) ?

10.5?8??4???|W(j?)|??4?|H1(j?)|8???8??4????|H2(j?)|4?8???8??4????4?8?

4. Figure 4 shows the Fourier transform X(j?) of a periodic continuous-time signal x(t). (1). Write out the expression of x(t).

?1,(2). Let H(j?)???0,X(j?) be the

(2)??12?Otherwise(1)12010?20??frequency response of an ideal lowpass filter, and x(t) is applied to the filter, determine the output response y(t) of the filter.。

?20??10?Figure 4

5. For a causal LTI system, the input and output signals are x(t)?e?tu(t)?e?3tu(t),

y(t)?(2e?t?2e?4t)u(t), respectively.

(1). Determine the frequency response H(j?). (2). Determine the unit impulse response h(t).

(3). Determine the differential equation describing the input-output relationship of the system.

?1,1???36. The frequency response of an ideal bandpass filter is H(j?)??, the unit impulse

?0,Otherwiseresponse is denoted as h(t), we now have that h(t)?sintg(t), determine g(t). ?t7. A continuous-time signal x(t)?cos(?t) is sampled by the impulse train p(t)?getting xp(t), where, T= 0.5s.

(1). Plot the Fourier transform X(j?) of x(t). (2). Plot the Fourier transform Xp(j?) of xp(t).

?1,(3). Let H(j?)???0,4????8?otherwisek?????(t?kT)

? be the frequency response of an ideal bandpass filter. If

xp(t) is applied to the filter, the output response is denoted as y(t), plot the Fourier transform

Y(j?) of y(t).

(4). By observingY(j?), write out the expression of y(t).

1?8??4?H(j?)?4?图4 X(j?)8???8??4????Xp(j?)4?8???8??4????Y(j?)4?8??

8. Figure 8 illustrates a communication system, The Fourier transforms of the input and output signals x(t) and y(t) are denoted as X(j?) and Y(j?), respectively. If x(t) = cos(0.5πt), determine y(t) and plot Y(j?).

x(t) ?8??4????4?8??H1(j?) ?cos(3?t)

H2(j?) y(t) cos(5?t)

H1(j?) Figure 8 H2(j?) 1 ?5? ?3? 3? 5? ?

?3? 1 3? ?

9. Let X(jω) denote the Fourier transform of the signal x(t) depicted in Figure P4.25.

x1(t)(a). Find ?X(j?).

(b). Find X(j0).

?1?2?112t(c). Evaluate

???X(j?)d?.

Figure P4.25.a.

?(d). Evaluate

????X(j?)2sin??2ej2?d?.

(e). Evaluate

???X(j?)d?.

(f). Sketch the inverse Fourier transform of Re{ X(jω)}.

Note: You should perform all these calculations without explicitly evaluating X(jω). 10. Consider an LTI system whose response to the input

x(t)?e?t?e?3tu(t)

??is y(t)??2e?t?2e?4t?u(t) (a). Find the frequency response of this system. (b). Determine the system’s impulse response.

(c). Find the differential equation relating the input and the output of this system.

sint11. Let g(t)?x(t)cos2t*

?tAssuming that x(t) is real and X(jω) = 0 for |ω| ≥ 1, show that there exists an LTI system S such that x(t)?g(t). (Note: find the relationship between x(t) and g(t))

S第三部分：信号与系统的s域分析

一、填空题

1. The ROC of the Laplace transform X(s)?transform is X(s)= ( ).

2. It is known that the LTI system described by H(s)?H(s) is ( ).

3. The system function of a causal LTI system is H(s)?s?2, then the differential equation

s2?4s?31 is stable, then the ROC of

(s?2)(s?3)11 is Re{s}??3, then the inverse ?s?3s?1describing the input-output relationship of the system is ( ). 4. The Laplace transform of signal e?2tu(t) is ( ), the associated ROC is ( ). 5. Assume that the system function of an anticausal LTI system is H(s)?1, then the frequency s?2response H(j?)? ( ), the unit impulse response h(t)? ( ).

6. It is known that a causal continuous-time LTI system H(s) is stable, then all of the poles of H(s) are located in ( ).

7. It is known that the Laplace transform of signal x(t) is X(s)?x(t)*?(t?1)? ( ).

d2y(t)dy(t)dx(t)8. The differential equation ?2?2y(t)??3x(t) describes the input-output 2dtdtdt1,Re{s}??1, then s?1relationship of an LTI system, then the system function is H(s) = ( ).

9. The Laplace transform of a causal signal x(t) is denoted as X(s), then the Laplace transform of signal

t???x(??1)d?is ( ).

10. Given a continuous-time LTI system, it is known that the zero-state response to arbitrary input x(t) is x(t-t0), t0 > 0, then the system function of the system is H(s) = ( ). 11. The Laplace transform of a continuous-time signal x(t) is X(s)?x(t) = ( ).

12. Figure 1.12 illustrates an LTI system, its system function is H(s) = ( ).

1e?s,Re{s}?0, then

s(2s?1)

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