GATE 2018 EE Q.16: solution and answer

Official answer: ASource: Official answer key, GATE 2018 EE (IIT Guwahati)

1. What an eigenfunction is. For a system operator $T\{\cdot \}$, an input $x(t)$ is an eigenfunction if the output is the same signal multiplied by a constant: $$y(t)=T\{x(t)\}=kx(t)$$ The constant $k$ may be complex but must not depend on $t$. It is called the eigenvalue.

2. Check with the standard LTI eigenfunction. Take $x(t)=e^{j{\omega }_{0}t}$. The convolution integral gives $$y(t)={\int }_{-\infty }^{\infty }h(\tau )e^{j{\omega }_0(t-\tau )}d\tau =e^{j{\omega }_{0}t}{\int }_{-\infty }^{\infty }h(\tau )e^{-j{\omega }_0\tau }d\tau =H({\omega }_0)x(t)$$ The output is $kx(t)$ with $k=H({\omega }_0)$, a constant. This is the form in (A).

3. Option (B): $k{e}^{j\omega t}x(t)$ multiplies the input by $e^{j\omega t}$, which changes with time (for $\omega \ne 0$). The output is therefore not a constant times $x(t)$, so (B) fails the definition.

4. Option (C): $x(t)e^{j\omega t}$ is the input modulated, i.e. shifted in frequency. The multiplying factor again depends on $t$, so (C) fails too.

5. Option (D): $kH(\omega )$ does not contain $x(t)$ at all. It is not a scaled copy of the input, so it cannot be the output for an eigenfunction input.

6. (A) is the only option where the output equals the input times a constant. Answer: (A).