GATE EC 2016 Set 1 — Question 16
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Question
Which one of the following is an eigen function of the class of all continuous-time, linear, time-invariant systems ( denotes the unit-step function)?
Correct answer
(C) e^(jω₀ t)
Solution
Complex exponentials of the form are eigenfunctions of linear time-invariant (LTI) systems. For a continuous-time LTI system with impulse response , if the input is , the output is given by:where is the frequency response of the system at . Since the output is a scalar multiple of the input, is an eigenfunction. Sinusoids like and are not strictly eigenfunctions because they can undergo a phase shift, which is not a simple scalar multiplication. is not an eigenfunction due to the windowing effect of the step function.
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