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GATE 2020 EC – Question 24

Networks, Signals and Systems · Discrete-time Signals · 1 mark · Multiple choice

Which one of the following pole-zero plots corresponds to the transfer function of an LTI system characterized by the input-output difference equation given below?

$$y[n]=\sum_{k=0}^{3}(-1)^kx[n-k]$$

Diagram for GATE 2020 EC question 24
  1. Pole-zero plot (A)
  2. Pole-zero plot (B)
  3. Pole-zero plot (C)
  4. Pole-zero plot (D)

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Show answer and explanation

Correct answer: (A) Pole-zero plot (A)

Explanation

$H(z)=1-z^{-1}+z^{-2}-z^{-3}=\frac{z^3-z^2+z-1}{z^3}=\frac{(z-1)(z^2+1)}{z^3}$. The zeros are at $z=1$ and $z=\pm j$, and there is a third-order pole at the origin. Plot (A) shows exactly these.