---
title: "Aliasing; Nyquist rate of 1 + cos + sin"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:aliasing-nyquist-1-cos-sin"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/aliasing-nyquist-1-cos-sin
site: BCT Engineering
---

# Aliasing; Nyquist rate of 1 + cos + sin

Signals and Systems · Chapter 4: Sampling

- Asked in: 2076 Baisakh, 2+4 marks; 2076 Bhadra, 2+4 marks; 2074 Bhadra, 2+4 marks (asked 3 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/aliasing-nyquist-1-cos-sin
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

What is aliasing in sampling? Determine the Nyquist rate for the continuous time signal x(t) = 1 + cos(2,000πt) + sin(4,000πt)

## Short answer

Aliasing is spectral overlap when fs < 2fm, making high frequencies appear as false low frequencies. Component frequencies are DC, 1000 Hz, and 2000 Hz, so fm = 2000 Hz. Nyquist rate = 4000 samples/s (4 kHz), Nyquist interval = 0.25 ms.

## Answer

#### Aliasing

**Aliasing** is the overlap of the shifted spectra of a sampled signal when the sampling frequency is less than twice the highest frequency ($f_s < 2f_m$). High-frequency components then appear as false low frequencies ("aliases"), and the original signal cannot be recovered. For example, a 7 kHz tone sampled at 10 kHz looks like a 3 kHz tone.

#### Nyquist rate of $x(t) = 1 + \cos(2000\pi t) + \sin(4000\pi t)$

Find the frequency of each term using $\omega = 2\pi f$:

| Term | $\omega$ (rad/s) | $f$ (Hz) |
|---|---|---|
| $1$ | 0 | 0 (DC) |
| $\cos(2000\pi t)$ | $2000\pi$ | 1000 |
| $\sin(4000\pi t)$ | $4000\pi$ | 2000 |

Highest frequency: $f_m = 2000$ Hz.

$$
\begin{aligned}
\text{Nyquist rate } f_N &= 2f_m = 2 \times 2000 = 4000\ \text{Hz} \\
\text{Nyquist interval } T_N &= \frac{1}{f_N} = \frac{1}{4000} = 0.25\ \text{ms}
\end{aligned}
$$

**Answer:** Nyquist rate $= 4000$ samples/s (4 kHz), i.e. $\omega_s = 8000\pi$ rad/s; Nyquist interval $= 0.25$ ms.

## Check yourself

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

1. What is aliasing?
   A. Loss of amplitude due to quantization
   B. Overlap of shifted spectra when fs < 2fm, so high frequencies appear as false low ones
   C. Overlap of spectra when fs > 2fm
   D. Adding noise to the signal during reconstruction
   Answer: B. Spectral copies overlap and the signal cannot be recovered. (Key point: Aliasing definition)

2. What frequency does sin(4000πt) have?
   A. 4000 Hz
   B. 1000 Hz
   C. 2000 Hz
   D. 4000π Hz
   Answer: C. f = ω/2π = 4000π/2π = 2000 Hz. (Key point: Frequency of terms)

3. What does the constant 1 contribute to the highest frequency?
   A. 1 Hz
   B. It doubles the highest frequency
   C. It sets fm = 1000 Hz
   D. Nothing; it is DC at 0 Hz
   Answer: D. A constant has frequency 0 and does not affect fm. (Key point: DC term)

4. What is the Nyquist rate of 1 + cos(2000πt) + sin(4000πt)?
   A. 4000 Hz
   B. 2000 Hz
   C. 8000 Hz
   D. 3000 Hz
   Answer: A. fm = 2000 Hz so fN = 2fm = 4000 Hz. (Key point: Nyquist rate)

## Close variants set in other exams

- [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×

## Related in Signals and Systems

- [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
- [Band-limited signals; aliasing; Nyquist rate](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Abandlimited-aliasing-nyquist) — Signals and Systems, chapter 4
- [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×
- [Aliasing; Nyquist of 2sin40πt + …](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-overcome-40pi) — Signals and Systems, chapter 4, asked 2×
- [Aliasing; Nyquist of 6cos50πt + …](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-overcome-50pi) — 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

- [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
- [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
- [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