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GATE 2014 EC (EC3) – Question 28

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

An all-pass system has $H(z)=(z^{-1}-b)/(1-az^{-1})$, with $|H(e^{-j\omega})|=1$ for every $\omega$. If Re(a) and Im(a) are nonzero, find b.

  1. $a$
  2. $a^*$
  3. $1/a^*$
  4. $1/a$

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Show answer and explanation

Correct answer: (B) $a^*$

Explanation

Set **b=a*,B**. On the unit circle, the numerator magnitude |z⁻¹−a*| equals |z*−a*|=|z−a|, while |1−az⁻¹|=|z−a| because |z|=1. Their ratio therefore has magnitude1. The conjugation is essential for complex a.