The Laplace Transform of the signal x(t) = u(t − 2)(tu(t)) is given by which of the following expressions? [”” represents convolution operator]
Correct Answer :
Solution :
The correct option is:
Step-by-Step Derivation and Explanation:
The given time-domain signal is a convolution of two separate functions, let's call them and :
where:
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:
2. Finding the Laplace Transform of :
The standard Laplace transform of a unit step function is:
By applying the time-shifting property of the Laplace Transform, , we shift by :
3. Finding the Laplace Transform of :
Using the standard Laplace transform pair for the ramp function:
Thus, .
4. Calculating the Combined Laplace Transform:
We multiply and to obtain the transform of the convolved signal:
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