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GATE 2019 EC – Question 13

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

Let $H(z)$ be the z-transform of a real-valued discrete-time signal $h[n]$. If $P(z)=H(z)H\left(\frac1z\right)$ has a zero at $z=\frac12+\frac12j$, and $P(z)$ has a total of four zeros, which one of the following plots represents all the zeros correctly?

Diagram for GATE 2019 EC question 13
  1. Plot (A)
  2. Plot (B)
  3. Plot (C)
  4. Plot (D)

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Show answer and explanation

Correct answer: (D) Plot (D)

Explanation

Because $h[n]$ is real, zeros occur in conjugate pairs, and $P(z)$ is unchanged under $z\to1/z$, so zeros also occur in reciprocal pairs. Starting from $\frac12+\frac12j$ we get its conjugate $\frac12-\frac12j$ and the reciprocals $1-j$ and $1+j$. Plot (D) shows these four zeros.