What is the convolution integral c(t) for a system with input x(t) and impulse response h(t), where x(t) = u(t - 1) - u(t - 3) and h(t) = u(t) - u(t - 2) ?a)b)c)d)Correct answer is option 'A'. Can you explain this answer? - EduRev Electrical Engineering (EE) Question

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What is the convolution integral c(t) for a system with input x(t) and impulse response h(t), where x(t) = u(t - 1) - u(t - 3) and h(t) = u(t) - u(t - 2) ?

Correct answer is option 'A'. Can you explain this answer?

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What is the convolution integral c(t) for a system with input x(t) and...

Concept:

By using the impulse response of a system, convolution can be used to calculate a system's zero state response (i.e., its response when it has zero initial conditions) to an arbitrary input. Linearity and superposition are used.

The convolution can be defined as:

u(t)*u(t) = r(t)

Calculation:
Given signals are x(t) = u(t - 1) - u(t - 3) and h(t) = u(t) - u(t - 2)
x(t) and h(t) is graphically represented by:

c(t) = x(t) * h(t)

c(t) = [u(t -1) - u(t - 3)]*[u(t) - u(t -2)]

c(t) = u(t - 1)*u(t) - u(t -1)*u(t - 2) - u(t - 3)*u(t) + u(t - 3)*u(t - 2)

c(t) = r(t - 1) - r(t - 3) - r(t - 3) + r(t + 5)

c(t) = r(t - 1) -2r(t - 3) + r(t + 5)

The output signal is represented as:

NOTES:

If the two analog signals are convolved

The resultant of two signals will have a width equal to the sum of the individual width of two signals being convolved.
The lower extent of the resultant signal will be equal to the sum of individual lower extents l = l₁ + l₂, similarly, the upper extent is the sum of individual upper extents u = u₁ + u₂
The area of resultant convolution is equal to the product of the area of the signals being convolved.

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What is the convolution integral c(t) for a system with input x(t) and impulse response h(t), where x(t) = u(t - 1) - u(t - 3) and h(t) = u(t) - u(t - 2) ?a)b)c)d)Correct answer is option 'A'. Can you explain this answer?

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