The GATE Grind

GATE 2025 IN – Question 51

Signals and Systems · Signals and linear time-invariant systems · 2 marks · Multiple select

Choose the eigenfunctions of stable linear time-invariant continuous-time systems.

  1. $e^{j(2\pi/3)t}$
  2. $\cos((2\pi/3)t)$
  3. $2^t$
  4. $\sin((2\pi/3)t)$

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Show answer and explanation

Correct answer: (A) $e^{j(2\pi/3)t}$; (C) $2^t$

Explanation

For a complex exponential input $e^{st}$, convolution gives $y(t)=e^{st}\int h(\tau)e^{-s\tau}\,d\tau=H(s)e^{st}$ whenever the transform exists.

Thus exponentials retain their shape and acquire a scalar multiplier.

Options A and C are exponentials ($2^t=e^{(\ln2)t}$), giving the official answer **A,C**.

A real sinusoid can gain phase and need not remain a scalar multiple.

For C, convergence at $s=\ln2$ must hold; BIBO stability alone guarantees convergence on the imaginary axis, not at every complex $s$.