The GATE Grind

GATE 2025 EC – Question 18

Networks, Signals and Systems · Continuous-time Signals · 1 mark · Multiple choice

Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega)=\int_{-\infty}^{\infty}f(t)\exp(-j\omega t)\,dt$ exists.

Which one of the following statements is always TRUE?

  1. $|F(\omega)|\le\int_{-\infty}^{\infty}|f(t)|\,dt$
  2. $|F(\omega)|>\int_{-\infty}^{\infty}|f(t)|\,dt$
  3. $|F(\omega)|\le\int_{-\infty}^{\infty}f(t)\,dt$
  4. $|F(\omega)|\ge\int_{-\infty}^{\infty}f(t)\,dt$

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Show answer and explanation

Correct answer: (A) $|F(\omega)|\le\int_{-\infty}^{\infty}|f(t)|\,dt$

Explanation

$|F(\omega)|=\left|\int f(t)e^{-j\omega t}dt\right|\le\int|f(t)|\,|e^{-j\omega t}|\,dt=\int|f(t)|\,dt$, since $|e^{-j\omega t}|=1$. This holds for every $\omega$. Options C and D fail because $\int f(t)dt$ can be negative or smaller than $|F(\omega)|$.