GATE 2024 IN Q.27: solution and answer

GATE 2024 · INQ.27 · 1 mark · NAT

The Nyquist sampling frequency for $x (t) = 10 sin^{2} (200 π t)$ is ________ Hz (rounded off to nearest integer).

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The Nyquist sampling frequency is 400 Hz.

Official answer: 400Source: Official answer key, GATE 2024 IN (IISc Bengaluru)

Write the square as a sum of sinusoids using $sin^{2} θ = \frac{1 - cos 2 θ}{2}$ with $θ = 200 π t$ : $x (t) = 10 sin^{2} (200 π t) = 10 \cdot \frac{1 - cos ( 400 π t )}{2} = 5 - 5 cos (400 π t)$
The signal now has only two components: a DC term at $0 Hz$ and a cosine with $ω = 400 π rad/s$ , which is $f = \frac{400 π}{2 π} = 200 Hz$ .
So the signal is band-limited, and its highest frequency is $f_{ma x} = 200 Hz$ . It is not $100 Hz$ , because squaring a sinusoid doubles its frequency.
By the sampling theorem, the Nyquist sampling frequency (Nyquist rate) is $f_{s} = 2 f_{ma x} = 2 \times 200 = 400 Hz$

KEY IDEA

Before you apply $f_{s} = 2 f_{ma x}$ to a non-linear expression, first rewrite it as a sum of pure sinusoids. The identity $sin^{2} θ = \frac{1 - c o s 2 θ}{2}$ turns $10 sin^{2} (200 π t)$ into a DC term plus a cosine at $400 π rad/s$ ( $200 Hz$ ), so $f_{ma x} = 200 Hz$ and the Nyquist rate is $400 Hz$ .

WATCH OUT FOR

Reading $f_{ma x}$ straight off the argument $200 π t$ gives $100 Hz$ and the wrong answer $200 Hz$ , because it misses that squaring doubles the frequency. A second slip is dividing by $2 π$ incorrectly and treating $400 π$ as $400 Hz$ , which gives $800 Hz$ .

From the GATE 2024 IN paper. Every question of this paper · Download the full paper · All 8 Sampling questions, with notes

GATE 2024 IN Q.27: solution and answer

Answer and solution

More Sampling questions