GATE EE 2020 Set 1 — Question 48
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Signals & Systems → Laplace & Z Transforms → Causality & Stability (ROC)
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Question
The causal realization of a system transfer function having poles at and zeroes at will be
Correct answer
(B) unstable, complex, allpass
Solution
1.Stability: A causal system is stable if and only if all its poles lie in the left half of the -plane (LHP). The given poles are at and . Since has a positive real part (lies in the RHP), the system is unstable.
2.Real/Complex Coefficients: For a transfer function to have real coefficients, its poles and zeros must occur in complex conjugate pairs.
- The conjugate of pole is . However, is given as a zero, not a pole.
- The conjugate of pole is . However, is given as a zero, not a pole.
3.Allpass Characteristic: A system is allpass if for every pole at , there is a zero at .
- For pole , the required zero is . This matches one of the given zeros.
- For pole , the required zero is . This matches the other given zero.
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