GATE EC 2021 Set 1 — Question 50
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Signals & Systems → LTI Systems → Impulse Response & Convolution
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Question
For a unit step input , a discrete-time LTI system produces an output signal . Let be the output of the system for an input . The value of is
Correct answer
0 to 0
Solution
Let be the step response of the LTI system. Given .
The impulse response is related to the step response by .
First, find :
.
Now, calculate :
.The input signal is .
The output is the convolution of and : .
We need to find . Using the convolution sum formula:
.Let's list the non-zero values of :
(from )
(from )
(from )Now, let's list the non-zero values of :
.
The unit step function is 1 for (i.e., ) and 0 otherwise.
So, is non-zero only for .We need to evaluate the sum .
So, .
So, .
So, .Summing these values:
.The value of is .Final Answer:
The impulse response is related to the step response by .
First, find :
.
Now, calculate :
.The input signal is .
The output is the convolution of and : .
We need to find . Using the convolution sum formula:
.Let's list the non-zero values of :
(from )
(from )
(from )Now, let's list the non-zero values of :
.
The unit step function is 1 for (i.e., ) and 0 otherwise.
So, is non-zero only for .We need to evaluate the sum .
1.For : .
.So, .
2.For : .
.So, .
3.For : .
Since , , so .So, .Summing these values:
.The value of is .Final Answer:
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