GATE 2025 EE Q.61: solution and answer

GATE 2025 · EEQ.61 · 2 marks · NAT

An ideal low pass filter has frequency response given by
$H (j ω) = {1, 0, ∣ ω ∣ \leq 200 π otherwise$
Let $h (t)$ be its time domain representation. Then $h (0) =$ ______ (round off to the nearest integer).

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

$h (0) = 200$

Official answer: 200Source: Official answer key, GATE 2025 EE (IIT Roorkee)

Start from the inverse Fourier transform: $h (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} H (j ω) e^{j ω t} d ω$
Set $t = 0$ . Then $e^{j ω t} = 1$ , so $h (0)$ is the area under $H (j ω)$ divided by $2 π$ : $h (0) = \frac{1}{2 π} \int_{- \infty}^{\infty} H (j ω) d ω$
$H (j ω) = 1$ only for $- 200 π \leq ω \leq 200 π$ and is $0$ everywhere else. The area is therefore the width of the passband, $400 π$ : $h (0) = \frac{1}{2 π} \int_{- 200 π}^{200 π} 1 d ω = \frac{400 π}{2 π} = 200$
Check against the closed form. $h (t) = \frac{1}{2 π} \int_{- 200 π}^{200 π} e^{j ω t} d ω = \frac{1}{2 π} \cdot \frac{e ^{j 200 π t} - e ^{- j 200 π t}}{j t} = \frac{sin ( 200 π t )}{π t}$ Using $sin x \approx x$ for small $x$ : $h (0) = t \to 0 lim \frac{sin ( 200 π t )}{π t} = t \to 0 lim \frac{200 π t}{π t} = 200$
The result is already a whole number, so no rounding is needed: $h (0) = 200$ .

KEY IDEA

The value of a signal at $t = 0$ is the total area under its Fourier transform divided by $2 π$ : $h (0) = \frac{1}{2 π} \int H (j ω) d ω$ . For an ideal rectangular low-pass response of height 1 and cutoff $ω_{c}$ , that area is $2 ω_{c}$ , so $h (0) = ω_{c} / π$ . Here $ω_{c} = 200 π$ gives $200$ .

WATCH OUT FOR

Leaving out the $\frac{1}{2 π}$ factor gives $400 π \approx 1257$ . Taking $200 π$ as a frequency in hertz, or using only the positive half of the passband (width $200 π$ instead of $400 π$ ), gives $h (0) = 100$ . The passband runs from $- 200 π$ to $+ 200 π$ rad/s, and both halves count.

From the GATE 2025 EE paper. Every question of this paper · Download the full paper · All 22 Fourier transform questions, with notes

GATE 2025 EE Q.61: solution and answer

Answer and solution

More Fourier transform questions