Question Details

The Laplace Transform of the signal x(t) = u(t − 2)(tu(t)) is given by which of the following expressions? [”” represents convolution operator]

Options

A

e 2 s s 2 ( s 2 )

B

e 2 (s2) s 3


C

s e 2s (s2) 2

D

e 2 s s 3

Show Answer

Correct Answer :

Option D

e 2 s s 3

Solution :

The correct option is:

e - 2 s s 3

Step-by-Step Derivation and Explanation:

The given time-domain signal is a convolution of two separate functions, let's call them x1(t) and x2(t):
x(t)=x1(t)*x2(t)
where:
x1(t)=u(t-2)
x2(t)=tu(t)

1. Using the Convolution Property of the Laplace Transform:
The convolution property states that the convolution of two signals in the time domain corresponds to the multiplication of their respective Laplace transforms in the s-domain:
L{x1(t)*x2(t)}=X1(s)·X2(s)

2. Finding the Laplace Transform of x1(t)=u(t-2):
The standard Laplace transform of a unit step function is:
L{u(t)}=1s
By applying the time-shifting property of the Laplace Transform, L{f(t-t0)u(t-t0)}=e-t0sF(s), we shift u(t) by t0=2:
X1(s)=L{u(t-2)}=e-2ss

3. Finding the Laplace Transform of x2(t)=tu(t):
Using the standard Laplace transform pair for the ramp function:
L{tu(t)}=1s2
Thus, X2(s)=1s2.

4. Calculating the Combined Laplace Transform:
We multiply X1(s) and X2(s) to obtain the transform of the convolved signal:
X(s)=X1(s)·X2(s)
X(s)=e-2ss·1s2=e-2ss3

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