---
title: "Nyquist rate of 2cos100πt·cos300πt"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:nyquist-rate-interval-2cos"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/nyquist-rate-interval-2cos
site: BCT Engineering
---

# Nyquist rate of 2cos100πt·cos300πt

Signals and Systems · Chapter 4: Sampling

- Asked in: 2080 Chaitra, 4 marks
- Video: [Nyquist Rate (Solved Problem 2)](https://www.youtube.com/watch?v=-AOGX_95yAQ) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/nyquist-rate-interval-2cos
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

Find the Nyquist rate and interval for the following signal: x(t) = 2cos(100πt) cos(300πt)

## Short answer

Convert the product to a sum: 2cos(100πt)cos(300πt) = cos(400πt) + cos(200πt). The maximum frequency is 200 Hz, so the Nyquist rate is 400 Hz and the Nyquist interval is 2.5 ms.

## Answer

**Step 1: convert the product to a sum** using $2\cos A\cos B = \cos(A+B) + \cos(A-B)$:

$$
\begin{aligned}
x(t) &= 2\cos(100\pi t)\cos(300\pi t) \\
&= \cos(400\pi t) + \cos(200\pi t)
\end{aligned}
$$

**Step 2: frequencies** ($f = \omega/2\pi$):

| Component | $\omega$ (rad/s) | $f$ (Hz) |
|---|---|---|
| $\cos(400\pi t)$ | $400\pi$ | 200 |
| $\cos(200\pi t)$ | $200\pi$ | 100 |

Highest frequency $f_m = 200$ Hz.

**Step 3: Nyquist rate and interval.**

$$
\begin{aligned}
f_N &= 2f_m = 2\times200 = 400\ \text{Hz} \\
T_N &= \frac{1}{f_N} = \frac{1}{400} = 2.5\ \text{ms}
\end{aligned}
$$

**Answer:** Nyquist rate $= 400$ samples/s ($\omega_N = 800\pi$ rad/s); Nyquist interval $= 2.5$ ms.

Note: the highest frequency is 200 Hz, not 150 Hz; multiplying two sinusoids creates the sum frequency.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. 2cosA cosB =
   A. cos(A+B) - cos(A-B)
   B. 2cos(A+B)
   C. cos(A+B) + cos(A-B)
   D. cos(A)+cos(B)
   Answer: C. The factor 2 cancels the 1/2. (Key point: Identity)

2. x(t) = 2cos100πt cos300πt equals
   A. cos300πt + cos100πt
   B. 2cos400πt
   C. cos400πt - cos200πt
   D. cos400πt + cos200πt
   Answer: D. A+B = 400π, A-B = -200π. (Key point: Resulting signal)

3. Nyquist rate
   A. 400 Hz
   B. 300 Hz
   C. 200 Hz
   D. 800 Hz
   Answer: A. fm = 200 Hz, so 2fm = 400 Hz. (Key point: Nyquist rate)

4. Nyquist interval
   A. 5 ms
   B. 2.5 ms
   C. 0.4 ms
   D. 4 ms
   Answer: B. 1/400 = 2.5 ms. (Key point: Nyquist interval)

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