A control system transfer function is H(s) = 1/s3. Express its impulse response in terms of unit step signala)u(t)&otimes; u(t) &otimes; u(t);&otimes; denotes convolutionb)u(t)&otimes; u(t) &otimes; u(t) &otimes; u(t);&otimes; denotes convolutionc)u(t)× u(t)× u(t);× denotes multiplicationd)u(t)&otimes; u(t);&otimes; denotes convolutionCorrect answer is option 'A'. Can you explain this answer?

Question

A control system transfer function is H(s) = 1/s3. Express its impulse response in terms of unit step signala)u(t)&otimes; u(t) &otimes; u(t);&otimes; denotes convolutionb)u(t)&otimes; u(t) &otimes; u(t) &otimes; u(t);&otimes; denotes convolutionc)u(t)&times; u(t)&times; u(t);&times; denotes multiplicationd)u(t)&otimes; u(t);&otimes; denotes convolutionCorrect answer is option 'A'. Can you explain this answer?

Answer

Convolution in the time domain implies multiplication in S(or frequency) domain

The Laplace transform of any signal h(t) is given by

So we observe that the H(s) = 1/s³, corresponds to a unit step signal convoluted with itself thrice.

Therefore the correct answer is option 1

Answer

Understanding the Transfer Function
The transfer function given is H(s) = 1/s^3. This represents a system with a third-order pole at the origin, indicating a specific response to inputs.
Impulse Response Calculation
To find the impulse response h(t), we need to take the inverse Laplace transform of H(s). The inverse transform of H(s) = 1/s^n is known to be:
- h(t) = (1/n!)*u(t) for n = 1, 2, 3, ...
For n = 3:
- h(t) = (1/3!)*u(t) = (1/6)*u(t).
This indicates that the impulse response h(t) is a scaled unit step function.
Connection with Convolution
The impulse response can also be expressed in terms of convolutions of unit step functions.
- Convolution of u(t) with itself n times yields the function that corresponds to the nth integral of the unit step function.
Thus, the impulse response can be expressed as:
- h(t) = u(t) ⊗ u(t) ⊗ u(t).
This is the convolution of three unit step functions, which is mathematically equivalent to the impulse response we derived.
Options Analysis
Now, analyzing the options given:
- Option A: u(t) ⊗ u(t) ⊗ u(t) - This is indeed the correct representation of the impulse response, as it corresponds to the convolution of three unit step functions.
- Option B: u(t) ⊗ u(t) ⊗ u(t) ⊗ u(t) - This would represent a fourth convolution and is not applicable here.
- Option C: u(t) × u(t) × u(t) - This denotes multiplication, which does not apply to our impulse response.
- Option D: u(t) ⊗ u(t) - This is only a second convolution and does not match the required order.
Thus, the correct answer is option A, as it accurately represents the impulse response derived from the transfer function.

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