---
title: "Aliasing; Nyquist of 2sin40πt + …"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:aliasing-overcome-40pi"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/aliasing-overcome-40pi
site: BCT Engineering
---

# Aliasing; Nyquist of 2sin40πt + …

Signals and Systems · Chapter 4: Sampling

- Asked in: 2073 Magh, 3+3 marks
- Video: [Sampling, Aliasing & Nyquist Theorem](https://www.youtube.com/watch?v=yWqrx08UeUs) — 0612 TV w/ NERDfirst
- 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-overcome-40pi
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

What is aliasing effect and how can we overcome it? Determine Nyquist rate for a continuous time signal x(t) = 2sin40πt + 10cos200πt + 5cos100πt.

## Short answer

Aliasing is distortion when sampling below the Nyquist rate, where high frequencies fold into the baseband as lower frequencies. Overcome it by using an anti-aliasing low-pass filter before sampling and sampling at f_s > 2f_max. For x(t)=2sin40πt+10cos200πt+5cos100πt, frequencies are 20, 100, 50 Hz; f_max=100 Hz. Nyquist rate = 200 Hz.

## Answer

#### Aliasing effect
**Aliasing** is the distortion that occurs when a signal is sampled at a rate lower than its Nyquist rate ($f_s < 2f_M$). The shifted copies of the spectrum in the sampled signal overlap, so a high-frequency component "folds back" and appears as a lower frequency (its *alias*). After overlap the original spectrum cannot be separated by any filter, so the signal cannot be recovered exactly.

Example: a 700 Hz tone sampled at $f_s = 1000$ Hz gives the same samples as a $1000-700 = 300$ Hz tone.

**How to overcome aliasing:**
1. **Sample fast enough:** choose $f_s > 2f_M$ (in practice $f_s$ is kept 2.5 to 5 times $f_M$ to leave a guard band).
2. **Anti-aliasing filter:** pass the signal through an analog low-pass filter with cutoff $f_s/2$ *before* sampling, so that no component above $f_s/2$ reaches the sampler. Real signals and noise are never strictly band-limited, so this filter is always used in practice.
3. **Oversampling:** sample at a much higher rate, then low-pass filter and decimate digitally. This relaxes the sharpness needed in the analog anti-aliasing filter.

#### Nyquist rate of $x(t) = 2\sin 40\pi t + 10\cos 200\pi t + 5\cos 100\pi t$

| Term | $\omega$ (rad/s) | $f$ (Hz) |
|---|---|---|
| $2\sin 40\pi t$ | $40\pi$ | $20$ |
| $10\cos 200\pi t$ | $200\pi$ | $100$ |
| $5\cos 100\pi t$ | $100\pi$ | $50$ |

$f_M = 100$ Hz.
$$
f_N = 2 f_M = 2\times 100 = 200\ \text{Hz}
$$
**Answer: Nyquist rate = 200 Hz (200 samples/s, $\omega_N = 400\pi$ rad/s).**

## Check yourself

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

1. Aliasing occurs when
   A. fs > 2fM
   B. The signal is DC
   C. fs < 2fM
   D. The filter is causal
   Answer: C. Copies overlap and high frequencies fold back. (Key point: Cause)

2. Anti-aliasing filter is placed
   A. After the sampler as an HPF
   B. After reconstruction as an LPF
   C. Before the sampler as an HPF
   D. Before the sampler as an analog LPF with cutoff fs/2
   Answer: D. It removes frequencies above fs/2 first. (Key point: Anti-aliasing filter)

3. Why is the anti-aliasing filter always used?
   A. Real signals and noise are never strictly band-limited
   B. Ideal signals are band-limited
   C. It increases fs
   D. It makes samples quantized
   Answer: A. Out-of-band content would otherwise alias. (Key point: Why needed)

4. Nyquist rate of 2sin40πt + 10cos200πt + 5cos100πt
   A. 100 Hz
   B. 200 Hz
   C. 400 Hz
   D. 50 Hz
   Answer: B. fM = 100 Hz. (Key point: Nyquist rate)

## Close variants set in other exams

- [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×

## Related in Signals and Systems

- [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×
- [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
- [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 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×
- [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×
- [Aliasing; Nyquist of 5cos1500πt − …](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-nyquist-three-tones) — Signals and Systems, chapter 4

## Related in other subjects

- [Nyquist-Kotelnikov sampling theorem; aliasing and aperture](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-kotelnikov-aliasing-aperture) — Communication Systems, chapter 4
- [Nyquist rate and interval for 6cos50πt + 20sin300πt − 10cos100πt](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-rate-6cos50-512) — Communication Systems, chapter 4, asked 2×
- [Sampling theorem, Nyquist criterion and aliasing](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Asampling-nyquist-aliasing) — Instrumentation I, chapter 6
- [PCM bit rate for 4.5 MHz TV signal with 1024 levels](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apcm-tv-4-5mhz-1024) — Communication Systems, chapter 4