重大信号与系统作业

Signals and System

Chap1

1.6 Determine whether or not each of the following signals is periodic: (a): x1(t)2ej(t/4)u(t) (b): x2[n]u[n]u[n] (c): x3[n]k{[n4k][n14k]}

1.9 Determine whether or not each of the following signals is periodic, If a signal is periodic , specify its fundamental period:

(a): x1(t)jej10t (b): x2(t)e(1j)t (c):

x3[n]ej7n

(d): x4[n]3ej3(n1/2)/5 (e): x5[n]3ej3/5(n1/2) 1.14 considera periodic signal x(t)1,0t1with period T=2. The

2,1t2derivative of this signal is related to the “impulse train”g(t)with period T=2. It can be shown thatDetermine the values of A1, t1, A2, t2.

k(t2k),

dx(t)A1g(tt1)A2g(tt2). dt a system S with input x[n] and output y[n].This system is obtained through a series interconnection of a system S1 followed by a system S2. The input-output relationships for S1 and S2 are S1: y1[n]2x1[n]4x1[n1], S2: y2[n]x2[n2]x2[n3]

Where x1[n] and x2[n] denote input signals.

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(a) Determine the input-output relationship for system S.

(b)Does the input-output relationship of system S change if the order in which S1 and S2 are connected in series is reversed(i e ,if S2 follows S1)? a discrete-time system with input x[n] and output y[n].The input-output relationship for this system is

y[n]x[n]x[n2]

(a) Is the system memoryless?

(b) Determine the output of the system when the input is A[n], where A is any real or complex number. (c) Is the system invertible?

a continuous-time system with input x(t) and output y(t) related by

y(t)x(sin(t))

(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): x(t1) (b): x(2t) (c): x(2t1) (d): x(4t/2) (e): [x(t)x(t)]u(t) (f):

x(t)[(t3/2)(t3/2)]

1.22. A discrete-time signal x(t)is shown in Figure P1.22. Sketch and label carefully each of the following signals:

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(a): x[n4] (b): x[3n] (c): x[3n] (d): x[3n1] (e): x[n]u[3n] (f): x[n2][n2] (g):

11x[n](1)nx[n] (h): x[(n1)2] 22 whether or not each of the following continuous-time signals is periodic. If the signal is periodic, determine its fundamental period.

(a): x(t)3cos(4t) (b): x(t)ej(t1) (c):

3x(t)[cos(2t)]2

3 (d): x(t){cos(4t)u(t)} (e): x(t){sin(4t)u(t)} (f): x(t)ne(2tn)

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):x[n]sin(6nn1) (b): x[n]cos() (c): 78x[n]cos(n2) 8(d):

x[n]cos(n)cos(n)24

 (e):

x[n]2cos(n)sin(n)2cos(n)

4826

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Chap 2

2.1 Let

x[n][n]2[n1][n3] and h[n]2[n1]2[n1]

Compute and plot each of the following convolutions: (a)y1[n]x[n]*h[n] (b)y2[n]x[n2]*h[n] (c)y3[n]x[n]*h[n2]

2.3 Consider an input x[n] and a unit impulse response h[n] given by

1x[n]()n2u[n2],

2h[n]u[n2].

Determine and plot the output y[n]x[n]*h[n]. 2.7 A linear system S has the relationship

y[n]kx[k]g[n2k]

Between its input x[n] and its output y[n], where g[n]=u[n]-u[n-4]. (a) Determine y[n] where x[n][n1] (b) Determine y[n] where x[n][n2] (c) Is S LTI?

(d) Determine y[n] when x[n]=u[n] 2.10 Suppose that

1,0t1 x(t)0,elsewhereAnd h(t)x(t/),where 01.

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(a) Determine and sketch y(t)x(t)*h(t)

(b) If dy(t)/dt contains only three discontinuities, what is the value of

?

2.11 Let

x(t)u(t3)u(t5) and h(t)e3tu(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)? 2.20 Evaluate the following integrals: (a

u0(t)cos(t)dt

(b)0sin(2t)(t3)dt (c)

555u1(1)cos(2)d

2.27 We define the area under a continuous-time signal v(t) as

Avv(t)dt

Show that if y(t)x(t)*h(t), then

AyAxAh

2.40 (a) an LTI system with input and output related through the equation

y(t)e(t)x(2)d

tWhat 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.

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Chap 3

3.1 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) are

*a1a12,a3a34j.

Express x(t) in the form

x(t)Akcos(ktk)

k0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] are

*j/3 a01,a2ej/4,a2ej/4,a4a42eExpress x[n] in the form

x[n]A0Aksin(knk)

k13.3 For the continuous-time periodic signal

x(t)2cos(25t)4sin(t) 33Determine the fundamental frequency 0 and the Fourier series coefficients ak such that

x(t)kaekjk0t.

3.5 Let x1(t) be a continuous-time periodic signal with fundamental frequency 1 and Fourier coefficients ak. Given that

x2(t)x1(1t)x1(t1)

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How is the fundamental frequency 2 of x2(t) related to ?Also, find a relationship between the Fourier series coefficients bk of x2(t) and the coefficients ak 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 ak. 3. ak0 for |k|0.

12 4 0|x(t)|2dt1.

2Specify two different signals that satisfy these conditions.

3.13 Consider a continuous-time LTI system whose frequency response is

H(j)h(t)ejtdtsin(4)

If the input to this system is a periodic signal

1,0t4 x(t)1,4t8With period T=8,determine the corresponding system output y(t). 3.15 Consider a continuous-time ideal lowpass filter S whose frequency response is

1,.......100H(j)

0,.......100When the input to this filter is a signal x(t) with fundamental period

T/6and Fourier series coefficientsak, it is found that

x(t)y(t)x(t).

SFor what values of k is it guaranteed that ak0?

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a continuous-time LTI system S whose frequency response is H(j)1,||250

0,otherwiseWhen the input to this system is a signal x(t) with fundamental period

T/7 and Fourier series coefficients ak,it is found that the output y(t)

is identical to x(t).

For what values of k is it guaranteed that ak0?

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Chap 4

4.1 Use the Fourier transform analysis equation(4.9)to calculate the Fourier transforms of;

(a)e2(t1)u(t1) (b)e2|t1|

Sketch 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)(t1)(t1) (b)

d{u(2t)u(t2)} dtSketch 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 X(j)|X(j)|ej|X(j)|2{u(3)u(3)} 3X(j)

2X(j),where

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), express the Fourier transforms of the signals listed below in the terms of X(j).You may find useful the Fourier transform properties listed in Table4.1. (a)x1(t)x(1t)x(1t) (b)x2(t)x(3t6)

d2(c) x3(t)2x(t1)

dt4.11 Given the relationships

y(t)x(t)h(t)

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And

g(t)x(3t)h(3t)

And given that x(t) has Fourier transform X(j) and h(t) has Fourier

transform H(j),use Fourier transform properties to show that g(t) has the form

g(t)Ay(Bt)

Determine the values of A and B.

4.13 Let x(t) be a signal whose Fourier transform is

X(j)()()(5)

And let

h(t)u(t)u(t2)

(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 X(j).Suppose we are given the following facts: 1. x(t) is real and nonnegative.

2. F1{1j)X(j)}Ae2tu(t),where A is independent of t. 3.|X(j)|d2.

Determine a closed-form expression for x(t).

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

Chap 6

6.1 Consider a continuous-time LTI system with frequency response

H(j)|H(j)|eH(j)and real impulse response h(t). Suppose

that we apply an input x(t)cos(0t0) to this system .The resulting output can be shown to be of the form

y(t)Ax(tt0)

Where A is a nonnegative real number representing an amplitude-scaling factor and t0 is a time delay. (a)Express A in terms of |H(j)|. (b)Express t0 in terms of

H(j0)

6.3 Consider the following frequency response for a causal and stable LTI system:

H(j)1j 1j(a) Show that |H(j)|A,and determine the values of A.

(b)Determine which of the following statements is true about (),the group delay of the system.(Note ()d(H(j))/d,where expressed in a form that does not contain any discontinuities.) 1.()0 for 0 2.()0 for 0 3 ()0 for 0

6.5 Consider a continuous-time ideal bandpass filter whose frequency response is

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H(j)is

1,c||3c H(j)0,elsewhere(a) If h(t) is the impulse response of this filter, determine a function g(t) such that

h(t)sinctg(t) t(b) As c is increased, dose the impulse response of the filter get more concentrated or less concentrated about the origin?

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Chap 7

7.1 A real-valued signal x(t) is know to be uniquely determined by its samples when the sampling frequency is s10,000.For what values of

 is X(j) 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 c1,000.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) T0.5103 (b) T2103 (c) T104

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)x(t)1cos(2,000t)sin(4,000t)

sin(4,000t) tsin(4,000t)2) (c) x(t)(t(b)x(t)7.4 Let x(t) be a signal with Nyquist rate0. Determine the Nyquist rate for each of the following signals: (a)x(t)x(t1) (b)

dx(t) dt13

(c)x2(t) (d)x(t)cos0t

7.9 Consider the signal

x(t)(sin50t2) tWhich we wish to sample with a sampling frequency of s150 to obtain a signal g(t) with Fourier transform G(j).Determine the maximum value of 0 for which it is guaranteed that

G(j)75X(j) for ||0

Where X(j) is the Fourier transform of x(t).

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Chap 8

8.1 Let x(t) be a signal for which X(j)0 when ||>M .Another signal y(t)

is

specified

as

having

the

Fourier

transform

Y(j)2X(j(c)).Determine a signal m(t) such that

x(t)y(t)m(t)

8.3 Let x(t) be a real-valued signal for which

X(j)0 when ||2,000.Amplitude modulation is performed to

produce the signal

g(t)x(t)sin(2,000t)

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 2,000 and passband gain of 2. Determine y(t).

FigureP8.3

8.22 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) shown in Figure P8.22(b). Determine and sketch Y(j), the spectrum of y(t).

8.28 In Section 8.4 we discussed the implementation of single-sideband modulation using 900 phase-shift networks, and in Figure8.21 and 8.22 we specifically illustrated the system and associated

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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) illustrated in Figure8.22, sketch

Y1(j),Y2(j),and Y(j) for the system in FigureP8.28(a), and

demonstrate that only the upper sidebands are retained.

(b) For X(j) imaginary, as illustrated in FigureP8.28(b), sketch

Y1(j),Y2(j) and Y(j) for the system in FigureP8.28(a), and

demonstrate that , for this case also, only the upper sidebands are retained.

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Chap 9

9.2 Consider the signal x(t)e5tu(t1) 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 t0 such that the Laplace transform G(s) of g(t)Ae5tu(tt0) 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:

11 s1s3s1(b) 2

s1(a)

s31(c) 2

ss1

9.7 How many signals have a Laplace transform that may be expressed as

(s1) in its region of convergence?

(s2)(s3)(s2s1)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.

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 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 g(t)e2tx(t) and G(j)[ the Fourier transform of g(t)] converges, determine whether x(t) is left sided, right sided, or two sided. 9.9 Given that

eatu(t)1,Re{s}Re{a} sasDetermine the inverse Laplace transform of

X(s)2(s2),Re{s}3

s27s129.10 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): H1(s)(b): H2(s)1,e{s}1

(s2)(s3)s1,e{s}

s2s12s2,e{s}1 (c): H3(s)2s2s19.13 Let g(t)x(t)x(t) ,Where x(t)etu(t). And the Laplace transform of g(t) is G(s)the constants and .

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s,1e{s}1. Determine the values of 2s1