P20.1 Consider the signal x(t) = 3e 2’u(t) + 4e 3 ‘u(t). (a) Does the Fourier transform of this… 1 answer below »

P20.1

Consider the signal x(t) = 3e 2’u(t) + 4e 3 ‘u(t).

(a) Does the Fourier transform of this signal converge?

(b) For which of the following values of a does the Fourier transform of x(t)e -”

converge?

(i)

(ii) a = 1

a = 2.5

(iii) a = 3.5

(c) Determine the Laplace transform X(s) of x(t). Sketch the location of the poles

and zeros of X(s) and the ROC.

P20.2

Determine the Laplace transform, pole and zero locations, and associated ROC for

each of the following time functions.

(a) e -“u(t),

a > 0

t

(b) e ~a u(t),

a

t

(c) -e -a u(- t),

a

P20.3

Shown in Figures P20.3-1 to P20.3-4 are four pole-zero plots. For each statement in

Table P20.3 about the associated time function x(t), fill in the table with the cor

responding constraint on the ROC.

(a)

(b)

Im

s plane

O

-2

– Re

2

Figure P20.3-2

P20-1

Signals and Systems

P20-2

(c)

(d)

Im

s plane

X O

-2 2

Re

Figure P20.3-3

Constraint on ROC for Pole-Zero Pattern

x(t)

(a)

(b)

(d)

(i) Fourier

transform

of x(t)e -‘

converges

(ii) x(t) = 0,

t > 10

(iii)x(t) = 0,

t

Table P20.3

P20.4

Determine x(t) for the following conditions if X(s) is given by

X(s) =

1

(s + 1)(s + 2)

_____

(a) x(t) is right-sided

(b) x(t) is left-sided

(c) x(t) is two-sided

P20.5

An LTI system has an impulse response h(t) for which the Laplace transform H(s)

is

H(s) =

h(t)e -dt

= s

1′

Re{s} > -1

Determine the system output y(t) for all t if the input x(t) is given by

x(t) = e t ‘ 2 + 2e – t

3

for all t.The Laplace Transform / Problems

P20-3

P20.6

(a) From the expression for the Laplace transform of x(t), derive the fact that the

Laplace transform of x(t) is the Fourier transform of x(t) weighted by an

exponential.

(b) Derive the expression for the inverse Laplace transform using t

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