GATE EE 2017 Set 1 — Question 51
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Signals & Systems → Laplace & Z Transforms → Inverse Z Transform
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Question
Consider a causal and stable LTI system with rational transfer function , whose corresponding impulse response begins at . Furthermore, . The poles of are for . The zeros of are all at . Let . The value of equals _______. (Give the answer up to three decimal places.)
Correct answer
0.09 to 0.1
Solution
Step 1: Determine the denominator of from its poles.
The poles are for . These are the roots of .
Proof: for .Step 2: Formulate .
Since the system is causal and the impulse response begins at , , which implies the degree of the numerator equals the degree of the denominator. Given all zeros are at , the numerator is .
Step 3: Find the constant using .
Step 4: Find the impulse response .
is the coefficient of . For , :
Step 5: Calculate .
Rounding to three decimal places, .
The poles are for . These are the roots of .
Proof: for .Step 2: Formulate .
Since the system is causal and the impulse response begins at , , which implies the degree of the numerator equals the degree of the denominator. Given all zeros are at , the numerator is .
Step 3: Find the constant using .
Step 4: Find the impulse response .
is the coefficient of . For , :
Step 5: Calculate .
Rounding to three decimal places, .
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