GATE EE 2020 Set 1 — Question 49
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Signals & Systems → Laplace & Z Transforms → Difference Equations
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Question
Which of the following options is true for a linear time-invariant discrete time system that obeys the difference equation:
Correct answer
(C) The system impulse response is non-zero at infinitely many instants.
Solution
The given difference equation is:Taking the Z-transform on both sides:The transfer function is:Analysis of options:
(A) False. Because of the feedback term , depends on all previous values of through the recursion. It is an IIR system.
(B) False. A difference equation alone does not define causality. Causality depends on the Region of Convergence (ROC) of the transfer function or the initial conditions.
(C) True. This is an Infinite Impulse Response (IIR) system due to the pole at . The impulse response will contain a term proportional to (assuming causality), which is non-zero for infinitely many values of (unless ).
(D) False. For , depends on both the input and the initial condition .
(A) False. Because of the feedback term , depends on all previous values of through the recursion. It is an IIR system.
(B) False. A difference equation alone does not define causality. Causality depends on the Region of Convergence (ROC) of the transfer function or the initial conditions.
(C) True. This is an Infinite Impulse Response (IIR) system due to the pole at . The impulse response will contain a term proportional to (assuming causality), which is non-zero for infinitely many values of (unless ).
(D) False. For , depends on both the input and the initial condition .
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