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GATE 2016 EE – Question 44

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 2 marks · Multiple choice

Suppose $x_1(t)$ and $x_2(t)$ have the Fourier transforms as shown below.

[Figure: $X_1(j\omega)$ is real and made of three triangles: height 0.3 over $\omega$ from $-1$ to 0, height 1 over 0 to 1, and height 0.5 over 1 to 2. $X_2(j\omega)$ is its mirror image: height 0.5 over $-2$ to $-1$, height 1 over $-1$ to 0, and height 0.3 over 0 to 1.]

Which one of the following statements is TRUE?

Diagram for GATE 2016 EE question 44
  1. $x_1(t)$ and $x_2(t)$ are complex and $x_1(t)x_2(t)$ is also complex with nonzero imaginary part
  2. $x_1(t)$ and $x_2(t)$ are real and $x_1(t)x_2(t)$ is also real
  3. $x_1(t)$ and $x_2(t)$ are complex but $x_1(t)x_2(t)$ is real
  4. $x_1(t)$ and $x_2(t)$ are imaginary but $x_1(t)x_2(t)$ is real

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Correct answer: (C) $x_1(t)$ and $x_2(t)$ are complex but $x_1(t)x_2(t)$ is real

Explanation

A real signal needs $X(-\omega) = X^*(\omega)$. Here $X_1$ is real but not even, so $X_1(-\omega) \neq X_1^*(\omega)$, and $x_1(t)$ is complex. The transform of the conjugate signal $x_1^*(t)$ is $X_1^*(-\omega) = X_1(-\omega)$, which is exactly the mirror image $X_2(\omega)$. So $x_2(t) = x_1^*(t)$, and the product $x_1(t)x_2(t) = |x_1(t)|^2$ is real.