---
title: "Nyquist rate of 20sin500πt + 10cos300πt − 50cos1000πt"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:sampling-theorem-nyquist-three-tones"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/sampling-theorem-nyquist-three-tones
site: BCT Engineering
---

# Nyquist rate of 20sin500πt + 10cos300πt − 50cos1000πt

Signals and Systems · Chapter 4: Sampling

- Asked in: 2081 Chaitra, 3+3 marks; 2075 Baisakh, 6 marks (asked 2 times)
- Video: [Nyquist Rate (Solved Problem 1)](https://www.youtube.com/watch?v=hRH9edUACcI) — 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/sampling-theorem-nyquist-three-tones
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

Define sampling theorem and Nyquist criteria for sampling. Find the Nyquist rate for the signal: x(t) = 20 sin(500πt) + 10 cos(300πt) − 50 cos(1000πt)

## Short answer

Sampling theorem: a band-limited signal with maximum frequency fm is exactly recoverable from samples if fs ≥ 2fm; the minimum rate is the Nyquist rate, and 1/(2fm) is the Nyquist interval. Component frequencies are 250, 150, and 500 Hz, so fm = 500 Hz. Nyquist rate = 1000 samples/s, interval = 1 ms.

## Answer

#### Sampling theorem

A continuous-time signal that is band-limited to $f_m$ Hz (no frequency components above $f_m$) is completely described by its samples taken at a uniform rate $f_s \ge 2f_m$. The signal can then be exactly recovered from the samples by passing them through an ideal low-pass filter with cutoff between $f_m$ and $f_s - f_m$.

#### Nyquist criterion

- The minimum sampling rate $f_N = 2f_m$ is the **Nyquist rate**; its reciprocal $T_N = \frac{1}{2f_m}$ is the **Nyquist interval** (maximum gap between samples).
- If $f_s < 2f_m$, the spectral copies overlap and **aliasing** occurs. In practice $f_s$ is kept above $2f_m$ (e.g. audio: $f_m = 20$ kHz, $f_s = 44.1$ kHz).

#### Nyquist rate of $x(t) = 20\sin(500\pi t) + 10\cos(300\pi t) - 50\cos(1000\pi t)$

| Term | $\omega$ (rad/s) | $f = \omega/2\pi$ (Hz) |
|---|---|---|
| $20\sin(500\pi t)$ | $500\pi$ | 250 |
| $10\cos(300\pi t)$ | $300\pi$ | 150 |
| $-50\cos(1000\pi t)$ | $1000\pi$ | 500 |

Highest frequency $f_m = 500$ Hz (the amplitudes do not matter).

$$
\begin{aligned}
f_N &= 2f_m = 2\times500 = 1000\ \text{Hz} \\
T_N &= \frac{1}{1000} = 1\ \text{ms}
\end{aligned}
$$

**Answer:** Nyquist rate $= 1000$ samples/s (1 kHz, or $2000\pi$ rad/s); Nyquist interval $= 1$ ms.

## Check yourself

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

1. What does the sampling theorem state?
   A. Any signal is recoverable if fs <= 2fm
   B. A signal band-limited to fm is fully described by samples taken at fs >= 2fm
   C. A signal is recoverable only if fs = fm
   D. Sampling must be done at exactly the signal's average frequency
   Answer: B. With fs >= 2fm, an ideal LPF recovers the signal. (Key point: Sampling theorem)

2. What is the Nyquist interval?
   A. T_N = 2fm
   B. T_N = fm
   C. T_N = 1/(2fm), the maximum gap between samples
   D. The minimum gap between samples
   Answer: C. It is the reciprocal of the Nyquist rate. (Key point: Nyquist interval)

3. Do the amplitudes 20, 10, 50 affect the Nyquist rate?
   A. Yes, the largest amplitude is used
   B. Yes, they are summed
   C. Yes, they are multiplied by the frequency
   D. No, only the highest frequency matters
   Answer: D. Nyquist rate depends only on fm. (Key point: Amplitudes)

4. What is the frequency of -50cos(1000πt)?
   A. 500 Hz
   B. 1000 Hz
   C. 250 Hz
   D. 50 Hz
   Answer: A. f = 1000π/2π = 500 Hz. (Key point: Frequency of 1000πt term)

5. Nyquist rate of 20sin(500πt)+10cos(300πt)-50cos(1000πt)?
   A. 500 Hz
   B. 1000 Hz
   C. 600 Hz
   D. 2000 Hz
   Answer: B. fm = 500 Hz, so 2fm = 1000 Hz. (Key point: Result)

## Close variants set in other exams

- [Nyquist criteria, aliasing; 30sin100πt + …](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Anyquist-criteria-aliasing-example) — Signals and Systems, chapter 4, asked 3×

## Related in Signals and Systems

- [Sampling theorem; Nyquist of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-theorem-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 5×
- [Aliasing; Nyquist rate of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 4×
- [Prove sampling theorem; Nyquist of product](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aprove-sampling-theorem-nyquist) — Signals and Systems, chapter 4, asked 2×
- [Sampling theorem; aliasing cause; Nyquist](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-aliasing-cause-nyquist) — Signals and Systems, chapter 4
- [Nyquist rate of (1/2π)cos50πt·cos500πt](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Anyquist-theorem-product-50-500) — Signals and Systems, chapter 4, asked 2×
- [Nyquist rate of (1/2π)cos4000πt·cos1000πt](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-theorem-nyquist-product) — Signals and Systems, chapter 4, asked 2×

## Related in other subjects

- [Ideal sampling and reconstruction; Nyquist rate of ½π cos(400πt)cos(1000πt)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aideal-sampling-reconstruction-nyquist) — Communication Systems, chapter 4
- [Flat top vs natural sampling; Nyquist rate of sin(800πt)/πt + 10cos2000t](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aflat-top-vs-natural-nyquist-rate) — Communication Systems, chapter 4
- [Natural and flat top sampling; Nyquist rate of sin²(400πt)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anatural-flat-top-sin2-400pit) — Communication Systems, chapter 4
- [Aperture and aliasing; sampling 10cos(20πt)cos(200πt) at 250 samples/s](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-aliasing-10cos20pit-250) — Communication Systems, chapter 4