Question

Choose the eigenfunction(s) of stable linear time-invariant continuous-time systems from the following options.

• A. \( e^{\frac{2\pi t}{3}} \)
• B. \( \cos\left(\frac{2\pi t}{3}\right) \)
• C. \( t^2 \)
• D. \( \sin\left(\frac{2\pi t}{3}\right) \)

Hint:

For linear time-invariant (LTI) systems, complex exponentials \( e^{st} \) are the key eigenfunctions. Other functions like trigonometric functions can be transformed into exponentials, but they aren't direct eigenfunctions.

Answer 1

The Correct Option is A, C

Step 1: Understanding Eigenfunctions of LTI Systems
For continuous-time LTI systems, complex exponentials of the form \( e^{st} \) are eigenfunctions. When passed through an LTI system, the output is simply scaled by a constant. \[ {If } x(t) = e^{st}, { then } y(t) = H(s)e^{st} \] Step 2: Evaluating the Options
(A) \( e^{\frac{2\pi t}{3}} \): This is a complex exponential. Hence, it is an eigenfunction of a continuous-time LTI system. Correct Answer
(B) \( \cos\left(\frac{2\pi t}{3}\right) \): This is not strictly an eigenfunction, but it can be represented as a linear combination of exponentials, so not an eigenfunction in strict form.
(C) \( t^2 \): This is a polynomial function. It is not an eigenfunction, but it is commonly mistaken due to appearing in differential equations. However, in strict terms of eigenfunction definition, this is not an eigenfunction.
(D) \( \sin\left(\frac{2\pi t}{3}\right) \): Same reasoning as for cosine. Not an eigenfunction in strict sense.