The GATE Grind

GATE 2016 EE – Question 45

Signals and Systems · Linear time invariant and causal systems · 2 marks · Multiple choice

The output of a continuous-time, linear time-invariant system is denoted by $T\{x(t)\}$ where $x(t)$ is the input signal. A signal $z(t)$ is called eigen-signal of the system $T$, when $T\{z(t)\} = \gamma z(t)$, where $\gamma$ is a complex number, in general, and is called an eigenvalue of $T$. Suppose the impulse response of the system $T$ is real and even. Which of the following statements is TRUE?

  1. $\cos(t)$ is an eigen-signal but $\sin(t)$ is not
  2. $\cos(t)$ and $\sin(t)$ are both eigen-signals but with different eigenvalues
  3. $\sin(t)$ is an eigen-signal but $\cos(t)$ is not
  4. $\cos(t)$ and $\sin(t)$ are both eigen-signals with identical eigenvalues

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $\cos(t)$ and $\sin(t)$ are both eigen-signals with identical eigenvalues

Explanation

For a real, even impulse response the frequency response $H(j\omega) = \int h(t)\cos(\omega t)\,dt$ is real, with no imaginary part. Passing $\cos(t)$ through the system gives $H(j1)\cos(t)$, and passing $\sin(t)$ gives $H(j1)\sin(t)$. Both are eigen-signals with the same eigenvalue $H(j1)$.