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. =---)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 2δtt( ). 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 ⎰∞-=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 impulse response 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 x is)1()(--t s t s , where s(t) is the unit step response of the system, then the unit impulse response ish(t) = ( ).17. The block diagram of a continuous-time LTI system is illustrated in the following figure. The differentialequationdescribingtheinput-outputrelationshipofthesystemis( ).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 、)()()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 δ2. 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 yB. )1(-t yC. )2(-t yD. )1(+t y3. The unit impulse response of an LTI system is h(t) = te-, this system is ( ).)(tA. causal and stableB. causal and unstableC. noncausal and unstableD. noncausal and stable4. dt t t t x )2()12()(112-+=⎰-δ = ( ).A. 1B. 3C. 9D. 05. 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 . And if the input signal is)()(21t bx t ax +, the corresponding output response is )()(21t by t ay + ( a and b are arbitrary realnumbers ). Then the system is a ( ) system.A. linearB. causalC. nonlinearD. time – invariant6. )(*)(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 ++δ7. =⎪⎭⎫⎝⎛-+=⎰∞∞-dt t t t t x 6)sin ()(πδ ( ). A.6πB.16-πC. 216-πD. 216+π8. Given two sequences ][1n x and ][2n x , their lengths are M and N respectively. The length of theconvolution of ][1n x and ][2n x is ( ).A ．MB ．NC ．N M +D ．1-+N M9. The unit impulse response of a continuous-time LTI system is dtt 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 dtt dy t y =+ C. dt t dx t x t y )()(2)(+= D. dtt 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 dtt y d )()(2)(3)(2)(22+=++. The unit impulse response of the system h(t) ( ).A . does not include )(t δ B. includes )(t δ C. includesdtt 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 uB. )2()2(--+t u t uC. )1()1(+--t u t uD. )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 dtt dy =+ B. )()()(t y t x t h -= C.)()()(t t h dtt dh δ=+ D. )()()(t y t t h -=δ13. The input-output relationship of a causal continuous-time system is described by the differential equation:dtt 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. )(213t u e t -11(1)(1))(2t x⊕)(t x )t+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:τττd x e t y tt )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 2. 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 theoutput 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=-+dtt 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 dtt 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 -=, as illustrated in the Figure6.(a). Compute y(t) = x(t)*h(t).Figure 2)(t x t5)(t h t131(b). Compute g(t) = dx(t)/dt * h(t).(c). How is g(t) related to y(t)?6. Let ∑∞-∞=--=k tk t t u e t y )3(*)()(δShow that t Ae t y -=)( for 0 ≤ t < 3, and determine the value A. 7. A causal LTI system is described by the differential equation:)()()(2)(3)(22t x dtt dx t y dt t dy dt t y d +=++ If the input signal is )()(2t u e t x 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 x 1(t) in Figure 9(a) is the signal y 1(t) illustrated in Figure 9(b). Determine and sketch carefully the response of the system to the input x 2(t) depicted in Figure 9(c).(b). Determine and sketch the response of the system considered in part (a) to the input x 3(t) shown in Figure 9(d).tt第二部分：信号与系统的频域分析一、填空题1. The frequency response of an ideal filter is given by ⎪⎩⎪⎨⎧<≥=πωπωω100,0100,2)(j H ， if the input signalis )120cos(5)80cos(10)(t t t x ππ+=, the corresponding outputresponsey(t)=( ).2. The Fourier transform of signal )cos()(0t t x ω= is ( ).3. The Fourier transform of signal )6sin()(0πω+=t t x is ( ).4. Assume that the Fourier transform of )(t x is denoted as )(ωj X , then the Fourier transform of)()(0t x e t y t j ω= is )(ωj Y = ( ).5. The Fourier transform of a continuous – time periodic signal ∑∞-∞==k tjk kea t x 0)(ω is )(ωj X =( ).6. It is known that the Fourier transform of )(t x is 11)(+=ωωj j X , then the Fourier transform of )(t tx is ( ).7. The Fourier transform of signal )(t x is denoted as )(ωj X , the Fourier transform of )()1(t x t - is ( ).8. A time shifting leads to a ( ).9. The frequency responses of two LTI systems are assumed to be 1()H j ω and 2()H j ω, the frequency response of the interconnection of 1()H j ω cascaded by 2()H j ω is ()H j ω =( ).10. A time-domain compression corresponds to a frequency-domain ( ). 11. For a signal x(t), if the condition⎰∞∞-∞<dt t x )( is satisfied, then the Fourier transform of x(t)exists, this condition is ( ) but not ( ).12. Figure 12 shows a continuous-time signal )(t x , its Fourier transform is denoted as )(ωj X , then=)0(X ( ). (Without evaluating )(ωj X ).13. For a continuous-time LTI system, if the zero-state response of the systemtotheinputsignal)()(t u e t x t-= is)()()(2t u e t u e t y t t ---=, then the frequency response of the systemis =)(ωj H ( ).14. The Fourier transform of signal ttt x 4sin )(= is =)(ωj X ( ).15. The inverse Fourier transform of )(ωδ is =)(t x ( ).16. The frequency spectrum includes two parts, one is ( ), the other is ( ). 17. Let )(ωj X denote the Fourier transform of signal )(t x , then the Fourier transform of signal)4cos(*)32()(t tx t y += is =)(ωj Y ( ). (Expressed using )(ωj X ).18. Let )(ωj X denote the Fourier transform of signal )(t x , then the Fourier transform of signal)cos()()(t t x t y π= is =)(ωj Y ( ). (Expressed using )(ωj X ).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⎪⎩⎪⎨⎧>≤=1001001)(ωωωj HWhen 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)()()(t x t y t x S =−→−For k ( ) it is guaranteed that a k = 0.21. Consider a continuous-time LTI system whose frequency response isωωωω)4sin()()(==⎰+∞∞--dt e t h j H tjIf the input to this system is a periodic signal ⎩⎨⎧<≤-<≤=841401)(t t t x with period T = 8, thecorresponding system output is y(t) = ( ).二、选择题1. The frequency response of an ideal lowpass filter is⎪⎩⎪⎨⎧>≤=πωπωω120,0120,2)(j H .If the input signal is )200cos(5)100cos(10)(t t t x ππ+=, the output response is )(t y = ( ).A. )100cos(10t πB. )200cos(10t πC. )100cos(20t πD. )200cos(5t π2. The Fourier transform of the rectangular pulse )1()1()(--+=t u t u t x is ( ).A. )(4ωSaB. )(2ωSaC. )2(2ωSaD. )2(4ωSa3. Let )(ωj X denote the Fourier transform of a signal )(t x , the Fourier transform of jt e t x )( is ( ).A. )(ωωj X e j -B. )(ωωj X e jC. ))1((-ωj XD. ))1((+ωj X4. Let )(ωj X denote the Fourier transform of signal )(t x , the Fourier transform of )1(-t x is ( ).A. )(ωωj X e j -B. )(ωωj X e jC. ))1((-ωj XD.))1((+ωj X5. The Fourier transform of the rectangular pulse )1()()(--=t u t u t x is ( ).A. 2)2(ωωjesa - B. 2)2(ωωjesa C. ωωj e sa -)( D. ωωj e sa )(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 )(t x is 20KHz, the bandwidth of signal )2(t x is ( ).A.20KHzB.40KHzC.10KHzD.30KHz8. Let )(ωj X denote the Fourier transform of signal )(t x , the Fourier transform of dtt dx t )( is ( ).A.ωωωωd j dX j X )()(- B. ωωωωd j dX j X )()(+- C. ωωωωd j dX j X )()(-- D. ωωωωd j dX j X )()(+ 9. Let )(ωj X denote the Fourier transform of signal )(t x , the Fourier transform of)()(b atx t y += is ( ).A. ωωjab ej X a )( B. ωωjab eja X a -)( C. ωωa b j e a j X a )(1 D. ωωa bj e aj X a -)(1 10. Let )(ωj X denote the Fourier transform of signal )1()1()(--+=t u t u t x , then =)0(X ( ).A. 2B. πC.π21D. 4 11. Let )(ωj X denote the Fourier transform of signal )(t x , the Fourier transform of )1(t x - is( ).A ．ωωj e j X )(--B ．ωωj ej X -)( C ．ωωj ej X --)( D ．ωωj e j X )(-12. Let )(ωj X denote the Fourier transform of signal )(t x , the Fourier transform of)()()(a t t x t y -=δ is ( ).A. ωωja e j X -)(B. ωja e a x -)(C. ωωja e j X )(D. ωja e a x )(13. The Fourier transform of signal )()()(τδτδ-++=t t t x is =)(ωj X ( ).A. ωτcos 21B. ωτcos 2C. ωτsin 21D. ωτsin 214. Let ),()(t e t x tδ-= and ττd x t y t⎰∞-=)()(. The Fourier transform of y(t) is =)(ωj Y ( ).A.ωj 1 B. ωj C.)(1ωπδω+j D. )(1ωπδω+-j 15. Consider the square wave )21()21()(ττ--+=t u t u t x , as τ decreases, the width of the mainlobe of )(ωj X ( ).A. increasesB. decreasesC. does not changeD. can not be determined16. It is known that the bandwidth of x(t) is ω∆, the bandwidth of )12(-t x is ( ).A. ω∆2B. 1-ω∆C. ω∆21D. ）（1-21ω∆ 17. The inverse Fourier transform of 01)(t j e aj j X ωωω+=is =)(t x ( ). A. )()()(0t u e t x t t a +-= B. )()(0)(0t t u e t x t t a +=+- C. )()(0)(0t t u e t x t t a -=-- D. )()()(0t u e t x t t a --=18. The Fourier transform of signal )(t x τ is )2()(τωτωτSa j X =, then the Fourier transform of signal)1()(-=t x t y τ is =)(ωj Y ( ).A. ωωωj e Sa j Y )()(=B. ωωωj e Sa j Y -=)()(C. ωωωj e Sa j Y )(2)(=D. ωωωj e Sa j Y -=)(2)(19. Given an LTI system with its frequency response 21)(+=ωωj j H , it is known that the Fourier transform of the output response y(t) is)3)(2(1)(++=ωωωj j j Y , the input signal )(t x =( ).A. )()(2t u e t x t -=B. )()(3t u e t x t --=-C. )()(3t u e t x t -=D. )()(3t u e t x t =20. The frequency response of an ideal lowpass filter is ⎩⎨⎧≥<=-22)(ωωωωj e j H , its unit impulse response is h(t) = ( ).A.)1(2sin -t t π B. )1()1(2sin --t t π C. )1(sin -t t π D. )1()1sin(--t t π三、综合题(分析、计算)1. Consider a continuous-time LTI system whose frequency response isωωω)4sin()(=j HIf the input to this system is a periodic signal ⎩⎨⎧<≤-<≤=841401)(t t t x with period T = 8, determinethe 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 frequencyresponse ⎩⎨⎧≥=otherwisej H ,015,1)(πωω,determine the output signal y(t).3. Figure 3.a illustrates a communication system. Let X 1(j ω) and X 2(j ω) denote the Fourier transforms of x 1(t) and x 2(t), respectively. It is known that ω1 = 4π, ω2 = 8π, and the frequency response of the ideal bandpass filter is H 1(j ω), the overall output response is y(t).••••kka ∠Figure 2(1). Plot the magnitude of the Fourier transform W(j ω) of w(t).(2). Choose an appropriate frequency ω3, so that the output response is y(t) = x 1(t); (3). Plot the magnitude responses of H 1(j ω) and H 2(j ω).4. Figure 4 shows the Fourier transform )(ωj X of a periodic continuous-time signal )(t x . (1). Write out the expression of )(t x .(2). Let ⎩⎨⎧≤=Otherwisej H ,012,1)(πωω be thefrequency response of an ideal lowpass filter, and )(t x is applied to the filter, determine the output response y(t) of the filter.。
