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、 =---)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 three systems: 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 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 responseis h(t) = ( )、17、 The block diagram of a continuous-time LTI system is illustrated in the following figure 、 The differentialequationdescribingthe input-output relationshipof the systemis( )、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 ( )、A 、 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 = ( )、)(tA 、 )(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 differentialequation: 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 δB 、 includes )(t δC 、 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-11(1)(1))(2t x12、 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 the 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=-+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 、Figure 2⊕)(t x )t+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)、(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) illustratedinFigure9(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)、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 inputsignal is )120cos(5)80cos(10)(t t t x ππ+=, the corresponding output response y(t) = ( )、)(t x t5)(t h tFigure 6131tt2、 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 systemtothe input signal)()(t u e t x t -= is)()()(2t u e t u e t y t t ---=, then the frequency response of thesystem is =)(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 a k , 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 t jIf 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 X 5、 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 、30KHz 8、 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 -)(110、 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 t δ-= 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 themain lobe 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 ofsignal )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 H If 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 withfrequency response ⎩⎨⎧≥=otherwisej H ,015,1)(πωω,••••kka ∠Figure 2determine 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)、 (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, determineFigure 3⊗⊗)t ω)cos(2t ω)(1t x )(2t x(a)⊗)cos(3t ωFigure 4the output response y(t) of the filter 、。