GATE EE 2016 Set 1 — Question 45
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Signals & Systems → Fourier Analysis & Sampling → Frequency Response
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Question
The output of a continuous-time, linear time-invariant system is denoted by where is the input signal. A signal is called eigen-signal of the system , when , where is a complex number, in general, and is called an eigenvalue of . Suppose the impulse response of the system is real and even. Which of the following statements is TRUE?
Correct answer
(D) cos(t) and sin(t) are both eigen-signals with identical eigenvalues
Solution
For a linear time-invariant (LTI) system with impulse response , the output for an input is given by the convolution . An eigen-signal satisfies the property , where is the eigenvalue. Complex exponentials are known to be eigen-functions of LTI systems with eigenvalues equal to the frequency response at that frequency:where .Given that the impulse response is real and even, its Fourier transform is also real and even. Let , where is a real-valued even function ().Now, consider the input . Using Euler's formula:The output is:Since is even, . Thus:This shows is an eigen-signal with eigenvalue .Similarly, for :The output is:This shows is also an eigen-signal with the same eigenvalue .Therefore, both and are eigen-signals with identical eigenvalues.
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