GATE 2024 EC – Question 44

Networks, Signals and Systems · Continuous-time Signals · 2 marks · Multiple choice

Consider two continuous time signals $x(t)$ and $y(t)$ as shown below.

If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is ______.

$-4X(4f)e^{-j\pi f}$
$-4X(4f)e^{-j4\pi f}$
$-\dfrac14X(f/4)e^{-j\pi f}$
$-\dfrac14X(f/4)e^{-j4\pi f}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $-4X(4f)e^{-j4\pi f}$

Explanation

$x(t)$ is a triangle of height $-1$ centred at $t=-0.5$ with half-width 0.5. $y(t)$ is a triangle of height $+1$ centred at $t=0$ with half-width 2. So $y(t)$ is $x$ stretched by a factor 4 (half-width 0.5 to 2), inverted, and shifted: $y(t)=-x\!\left(\dfrac{t}{4}-0.5\right)=-x\!\left(\dfrac{t-2}{4}\right)$. Time scaling gives $x(t/4)\leftrightarrow4X(4f)$, and the delay by 2 gives $e^{-j2\pi f\cdot2}=e^{-j4\pi f}$. Hence $Y(f)=-4X(4f)e^{-j4\pi f}$.