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GATE 2022 IN – Question 38

Signals and Systems · Discrete-time systems and digital filters · 2 marks · Multiple choice

Bilinear transformation with sampling period 0.1 s is used to obtain $H_d(z)$ from $H_c(s)=1/[(s+1)(s+3)]$. Then $H_d(z)$ is ______.

  1. $\dfrac{(1+z^{-1})^2}{(19-21z^{-1})(23-17z^{-1})}$
  2. $\dfrac{(1-z^{-1})^2}{(21-19z^{-1})(17-23z^{-1})}$
  3. $\dfrac{(1+z^{-1})^2}{(21-19z^{-1})(23-17z^{-1})}$
  4. $\dfrac{(1+z^{-1})^2}{(21-19z^{-1})(17-23z^{-1})}$

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Correct answer: (C) $\dfrac{(1+z^{-1})^2}{(21-19z^{-1})(23-17z^{-1})}$

Explanation

With T=0.1, substitute s=20(1−z⁻¹)/(1+z⁻¹). Then s+1=(21−19z⁻¹)/(1+z⁻¹) and s+3=(23−17z⁻¹)/(1+z⁻¹). Their reciprocal product gives **C**.