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Signal and System

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). dt1.15.Consider 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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12Signal and System

(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)? 1.16.Consider 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?

1.17.Consider 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

2

Signal and System

label carefully each of the following signals:

(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] 221.25.Determine 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):

3x(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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Signal and System

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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Signal and System

(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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