Signals and SystemChap11.6 Determine whether or not each of the following signals is periodic: (a): (/4)1()2()j t x t e u t π+= (b): 2[][][]x n u n u n =+- (c): 3[]{[4][14]}k x n n k n k δδ∞=-∞=----∑1.9 Determine whether or not each of the following signals is periodic, If a signal is periodic , specify its fundamental period:(a): 101()j t x t je = (b): (1)2()j t x t e -+= (c):73[]j n x n e π=(d): 3(1/2)/54[]3j n x n e π+= (e): 3/5(1/2)5[]3j n x n e += 1.14 considera periodic signal 1,01()2,12t x t t ≤≤⎧=⎨-<<⎩with period T=2. Thederivative of this signal is related to the “impulse train”()(2)k g t t k δ∞=-∞=-∑,with period T=2. It can be shown that 1122()()()dx t A g t t A g t t dt=-+-. Determine the values of 1A , 1t , 2A , 2t .1.15.Consider a system S with input x[n] and output y[n].This system is obtained through a series interconnection of a system S 1 followed by a system S2. The input-output relationships for S 1 and S 2 are S 1: ],1[4][2][111-+=n x n x n y S 2: ]3[21]2[][222-+-=n x n x n yWhere ][1n x and ][2n x denote input signals.(a) Determine the input-output relationship for system S.(b)Does the input-output relationship of system S change if the order in which S 1 and S 2 are connected in series is reversed(i e ,if S 2 follows S 1)? 1.16.Consider a discrete-time system with input x[n] and output y[n].The input-output relationship for this system is]2[][][-=n x n x n y(a) Is the system memoryless?(b) Determine the output of the system when the input is ][n A δ, where A is any real or complex number. (c) Is the system invertible?1.17.Consider a continuous-time system with input x(t) and output y(t) related by))(sin()(t x t y =(a) Is this system causal? (b) Is this system linear?1.21.A continous-time signal ()x t is shown in Figure P1.21. Sketch and label carefully each of the following signals:(a): (1)x t - (b): (2)x t - (c): (21)x t + (d): (4/2)x t - (e): [()()]()x t x t u t +- (f):()[(3/2)(3/2)]x t t t δδ+--1.22. A discrete-time signal ()x t is shown in Figure P1.22. Sketch andlabel carefully each of the following signals:(a): [4]x n - (b): [3]x n - (c): [3]x n (d): [31]x n + (e): [][3]x n u n - (f): [2][2]x n n δ-- (g):11[](1)[]22n x n x n +- (h): 2[(1)]x n - 1.25.Determine whether or not each of the following continuous-time signals is periodic. If the signal is periodic, determine its fundamental period.(a): ()3cos(4)3x t t π=+ (b): (1)()j t x t e π-= (c):2()[cos(2)]3x t t π=-(d): (){cos(4)()}x t t u t ενπ= (e): (){sin(4)()}x t t u t ενπ= (f): (2)()t n n x t e∞--=-∞=∑1.26. Determine whether or not each of the following discrete-time signals is periodic. If the signal is periodic, determine its fundamental period. (a):6[]sin(1)7x n n π=+ (b): []cos()8nx n π=- (c): 2[]cos()8x n n π= (d):[]cos()cos()24x n n n ππ=(e):[]2cos()sin()2cos()4826x n n n n ππππ=+-+Chap 22.1 Let]3[]1[2][][---+=n n n n x δδδ and ]1[2]1[2][-++=n n n h δδCompute and plot each of the following convolutions: (a)][*][][1n h n x n y = (b)][*]2[][2n h n x n y += (c)]2[*][][3+=n h n x n y2.3 Consider an input x[n] and a unit impulse response h[n] given by],2[)21(][2-=-n u n x n].2[][+=n u n hDetermine and plot the output ].[*][][n h n x n y = 2.7 A linear system S has the relationship[][][2]k y n x k g n k ∞=-∞=-∑Between its input x[n] and its output y[n], where g[n]=u[n]-u[n-4]. (a) Determine y[n] where ]1[][-=n n x δ (b) Determine y[n] where ]2[][-=n n x δ (c) Is S LTI?(d) Determine y[n] when x[n]=u[n] 2.10 Suppose that⎩⎨⎧≤≤=elsewhere t t x ,010,1)( And )/()(αt x t h =,where 10≤<α.(a) Determine and sketch )(*)()(t h t x t y =(b) If dt t dy /)( contains only three discontinuities, what is the value ofα?2.11 Let)5()3()(---=t u t u t x and )()(3t u e t h t -=(a) Compute )(*)()(t h t x t y =. (b) Compute )(*)/)(()(t h dt t dx t g =. (c) How is g(t) related to y(t)? 2.20 Evaluate the following integrals: (adt t t u )cos()(0⎰∞∞-(b)⎰+50)3()2sin(dt t t δπ (c)⎰--551)2cos()1(τπττd u2.27 We define the area under a continuous-time signal )(t v as⎰∞∞-=dt t v A v )(Show that if )(*)()(t h t x t y =, thenh x y A A A =2.40 (a) an LTI system with input and output related through the equationτττd x e t y tt )2()()(-=⎰∞---What is the impulse response h(t) for this system?(b) Determine the response of the system when the input x(t) is as shown in Figure P2.40.Chap 33.1 A continuous-time periodic signal x(t) is real value and has a fundamental period T=8. The nonzero Fourier series coefficients for x(t) arej a a a a 4,2*3311====--.Express x(t) in the form)cos()(0k k k k t A t x φω+=∑∞=3.2 A discrete-time periodic signal x[n] is real valued and has a fundamental period N=5.The nonzero Fourier series coefficients for x[n] are10=a ,4/2πj e a --=,4/2πj e a =,3/*442πj ea a ==- Express x[n] in the form)sin(][10k k k k n A A n x φω++=∑∞=3.3 For the continuous-time periodic signal)35sin(4)32cos(2)(t t t x ππ++= Determine the fundamental frequency 0ω and the Fourier series coefficients k a such thattjk k kea t x 0)(ω∑∞-∞==.3.5 Let 1()x t be a continuous-time periodic signal with fundamental frequency 1ω and Fourier coefficients k a . Given that211()(1)(1)x t x t x t =-+-How is the fundamental frequency 2ω of 2()x t related to ?Also, find a relationship between the Fourier series coefficients k b of 2()x t and the coefficients k a You may use the properties listed in Table 3.1. 3.8 Suppose we are given the following information about a signal x(t): 1. x(t) is real and odd.2. x(t) is periodic with period T=2 and has Fourier coefficients k a .3. 0=k a for 0||>k .4 1|)(|21202=⎰dt t x .Specify two different signals that satisfy these conditions.3.13 Consider a continuous-time LTI system whose frequency response is⎰∞∞--==ωωωω)4sin()()(dt e t h j H t jIf the input to this system is a periodic signal⎩⎨⎧<≤-<≤=84,140,1)(t t t x With period T=8,determine the corresponding system output y(t). 3.15 Consider a continuous-time ideal lowpass filter S whose frequency response is1, (100)()0, (100)H j ωωω⎧≤⎪=⎨>⎪⎩When the input to this filter is a signal x(t) with fundamental period6/π=T and Fourier series coefficients k a , it is found that)()()(t x t y t x S=→.For what values of k is it guaranteed that 0=k a ?3.35.Consider a continuous-time LTI system S whose frequency response is 1,||250()0,H j otherwise ωω≥⎧=⎨⎩When the input to this system is a signal x(t) with fundamental period/7T π= and Fourier series coefficients k a ,it is found that the output y(t)is identical to x(t).For what values of k is it guaranteed that 0k a =?Chap 44.1 Use the Fourier transform analysis equation(4.9)to calculate the Fourier transforms of;(a))1()1(2---t u e t (b)|1|2--t eSketch and label the magnitude of each Fourier transform.4.2 Use the Fourier transform analysis equation(4.9) to calculate the Fourier transforms of: (a))1()1(-++t t δδ (b))}2()2({-+--t u t u dtdSketch and label the magnitude of each Fourier transform.4.5 Use the Fourier transform synthesis equation(4.8) to determine the inverse Fourier transform of ()()|()|j X j X j X j e ωωω= ,where|()|2{(3)(3)}X j u u ωωω=+-- 3()2X j ωωπ=-+Use your answer to determine the values of t for which x(t)=0. 4.6 Given that x(t) has the Fourier transform ()X j ω, express the Fourier transforms of the signals listed below in the terms of ()X j ω.You may find useful the Fourier transform properties listed in Table4.1. (a))1()1()(1t x t x t x --+-= (b))63()(2-=t x t x(c) )1()(223-=t x dtd t x4.11 Given the relationships)()()(t h t x t y *=And)3()3()(t h t x t g *=And given that x(t) has Fourier transform )(ωj X and h(t) has Fouriertransform )(ωj H ,use Fourier transform properties to show that g(t) has the form)()(Bt Ay t g =Determine the values of A and B.4.13 Let x(t) be a signal whose Fourier transform is()()()(5)X j ωδωδωπδω=+-+-And let()()(2)h t u t u t =--(a) Is x(t) periodic? (b) Is ()*()x t h t periodic?(c) Can the convolution of two aperiodic signals be periodic?4.14 Consider a signal x(t) with Fourier transform )(ωj X .Suppose we are given the following facts: 1. x(t) is real and nonnegative.2. ),()}()1{21t u Ae j X j F t --=+ωωwhere A is independent of t.3.⎰∞∞-=πωω2|)(|d j X .Determine a closed-form expression for x(t).Chap 66.1 Consider a continuous-time LTI system with frequency response()()|()|H j H j H j e ωωω= and real impulse response h(t). Supposethat we apply an input 00()cos()x t t ωφ=+ to this system .Theresulting output can be shown to be of the form0()()y t Ax t t =-Where A is a nonnegative real number representing an amplitude-scaling factor and 0t is a time delay.(a)Express A in terms of |()|H j ω.(b)Express 0t in terms of 0()H j ω6.3 Consider the following frequency response for a causal and stable LTI system:1()1j H j j ωωω-=+ (a) Show that |()|H j A ω=,and determine the values of A.(b)Determine which of the following statements is true about ()τω,the group delay of the system.(Note ()(())/d H j d τωωω=- ,where ()H j ω is expressed in a form that does not contain any discontinuities.)1.()0 0for τωω=>2.()0 0for τωω>>3 ()0 0for τωω<>6.5 Consider a continuous-time ideal bandpass filter whose frequency response is⎩⎨⎧≤≤=elsewherej H c c ,03||,1)(ωωωω (a) If h(t) is the impulse response of this filter, determine a function g(t)such that)(sin )(t g tt t h c πω= (b) As c ω is increased, dose the impulse response of the filter get moreconcentrated or less concentrated about the origin?Chap 77.1 A real-valued signal x(t) is know to be uniquely determined by its samples when the sampling frequency is 10,000s ωπ=.For what values of ω is ()X j ω guaranteed to be zero?7.2 A continuous-time signal x(t) is obtained at the output of an ideal lowpass filter with cutoff frequency 1,000c ωπ=.If impulse-train sampling is performed on x(t), which of the following sampling periods would guarantee that x(t) can be recovered from its sampled version using an appropriate lowpass filter?(a) 30.510T -=⨯(b) 3210T -=⨯(c) 410T -=7.3 The frequency which, under the sampling theorem, must be exceeded by the sampling frequency is called the Nyquist rate. Determine the Nyquist rate corresponding to each of the following signals:(a)()1cos(2,000)sin(4,000)x t t t ππ=++ (b)sin(4,000)()t x t tππ=(c) 2sin(4,000)()()t x t t ππ= 7.4 Let x(t) be a signal with Nyquist rate 0ω. Determine the Nyquist ratefor each of the following signals:(a)()(1)x t x t +- (b)()dx t dt(c)2()x t(d)0()cos x t t ω 7.9 Consider the signal2sin 50()()t x t tππ= Which we wish to sample with a sampling frequency of 150s ωπ= to obtain a signal g(t) with Fourier transform ()G j ω.Determine the maximum value of 0ω for which it is guaranteed that0()75() ||G j X j for ωωωω=≤Where ()X j ω is the Fourier transform of x(t).Chap 88.1 Let x(t) be a signal for which M ()0 when ||> X j ωωω=.Another signal y(t) is specified as having the Fourier transform ()2(())c Y j X j ωωω=-.Determine a signal m(t) such that8.3 Let x(t) be a real-valued signal for which()0 ||2,000X j when ωωπ=>.Amplitude modulation is performed toproduce the signal()()sin(2,000)g t x t t π=A proposed demodulation technique is illustrated in Figure P8.3 where g(t) is the input, y(t) is the output, and the ideal lowpass filter has cutoff frequency 2,000πand passband gain of 2. Determine y(t).FigureP8.38.22 In Figure P8.22(a), a system is shown with input signal x(t) and output signal y(t).The input signal has the Fourier transform ()X j ω shown in Figure P8.22(b). Determine and sketch ()Y j ω, the spectrum of y(t).8.28 In Section 8.4 we discussed the implementation of single-sideband modulation using 090 phase-shift networks, and in Figure8.21 and 8.22 we specifically illustrated the system and associated()()()x t y t m t =spectra required to retain the lower sidebands.Figure P8.28(a) shows the corresponding system required to retain the upper sidebands.(a) With the same ()X j ω illustrated in Figure8.22, sketch12(),()Y j Y j ωω,and ()Y j ω for the system in FigureP8.28(a), and demonstrate that only the upper sidebands are retained.(b) F or ()X j ω imaginary, as illustrated in FigureP8.28(b), sketch12(),()Y j Y j ωω and ()Y j ω for the system in FigureP8.28(a), and demonstrate that , for this case also, only the upper sidebands are retained.Chap 99.2 Consider the signal 5()(1)t x t e u t -=- and denote its Laplace transform by X(s).(a)Using eq.(9.3),evaluate X(s) and specify its region of convergence. (b)Determine the values of the finite numbers A and 0t such that theLaplace transform G(s) of 50()()t g t Ae u t t -=-- has the same algebraic form as X(s).what is the region of convergence corresponding to G(s)?9.5 For each of the following algebraic expressions for the Laplace transform of a signal, determine the number of zeros located in the finite s-plane and the number of zeros located at infinity: (a)1113s s +++ (b) 211s s +- (c) 3211s s s -++9.7 How many signals have a Laplace transform that may be expressed as2(1)(2)(3)(1)s s s s s -++++ in its region of convergence? Solution:There are 4 poles in the expression, but only 3 of them have different real part.∴ The s-plane will be divided into 4 strips which parallel to the jw-axis and have no cut-across.∴ There are 4 signals having the same Laplace transform expression.9.8 Let x(t) be a signal that has a rational Laplace transform with exactly two poles located at s=-1 and s=-3. If 2()() ()t g t e x t and G j ω=[ the Fourier transform of g(t)] converges, determine whether x(t) is left sided, right sided, or two sided. 9.9 Given that1(),{}Re{}sat e u t Re s a s a -↔>-+ Determine the inverse Laplace transform of22(2)(),Re{}3712s X s s s s +=>-++ 9.10 Using geometric evaluation of the magnitude of the Fourier transform from the corresponding pole-zero plot ,determine, for each of the following Laplace transforms, whether the magnitude of the corresponding Fourier transform is approximately lowpass, highpass, or bandpass. (a): 11(),{}1(2)(3)H s e s s s =ℜ>-++ (b): 221(),{}12s H s e s s s =ℜ>-++ (c): 232(),{}121s H s e s s s =ℜ>-++ 9.13 Let ()()()g t x t x t α=+- ,Where ()()t x t e u t β-=. And the Laplace transform of g(t) is 2(),1{}11s G s e s s =-<ℜ<-. Determine the values of the constants αand β.。