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Two systems with impulse responses h1(t) and h2(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by
- a)a product of h1(t) and h2(t)
- b)sum of of h1(t) and h2(t)
- c)convolution of h1(t) and h2(t)
- d)subtraction of h2(t) from h1(t)
Correct answer is option 'C'. Can you explain this answer?
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Impulse Response of Cascaded Systems
When two systems are connected in cascade, the overall impulse response of the cascaded system can be found by taking the convolution of the impulse responses of the individual systems.
Explanation
Let's consider two systems with impulse responses h1(t) and h2(t) connected in cascade. When an input impulse is applied to the cascaded system, it will pass through the first system with impulse response h1(t) and then through the second system with impulse response h2(t).
Now, let's denote the input impulse as x(t). The output of the first system can be represented as y1(t) = h1(t) * x(t), where * denotes the convolution operation. This output becomes the input to the second system.
Therefore, the output of the cascaded system can be represented as y(t) = h2(t) * y1(t) = h2(t) * (h1(t) * x(t)).
By using the associative property of convolution, we can rewrite the expression as y(t) = (h2(t) * h1(t)) * x(t).
Thus, the overall impulse response of the cascaded system is given by the convolution of h1(t) and h2(t), i.e., h(t) = h2(t) * h1(t).
Answer
The correct answer is option 'C', which states that the overall impulse response of the cascaded system is the convolution of h1(t) and h2(t).
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Two systems with impulse responsesh1(t) and h2(t) are connected in cascade.Then the overall impulse response of the cascaded system is given bya)a product of h1(t) and h2(t)b)sum of of h1(t) and h2(t)c)convolution ofh1(t) and h2(t)d)subtraction of h2(t) from h1(t)Correct answer is option 'C'. Can you explain this answer?
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