Frequency Aliasing Calculator (with Practical Examples)

Introduction

If a signal is sampled at a rate that is lower than the Nyquist rate, it will be aliased.

Calculator

Enter the sampling frequency and the input frequency. The calculator provides the frequency at which the input will appear in the output spectrum.

Background

What is Aliasing?

Frequency Aliasing or simply Aliasing is a phenomenon that occurs when a signal is sampled at a rate that is too low to accurately represent its true frequency content.

This happens when the Nyquist-Shannon sampling criterion is not followed, which states that the sampling rate should be at least twice the highest frequency component of the signal being sampled.

When aliasing occurs, higher frequency components of the signal are “folded-back” and therefore erroneously represented as lower frequency components in the sampled signal. This can result in distortion and loss of information in the reconstructed signal. Furthermore, it can result in higher frequency signals overlapping with lower frequency signals that fall at the same frequencies.

Practical Examples

In this post we provided a Nyquist frequency calculator. At input frequencies above this value, the input signal will be aliased.

As an example, if the sampling frequency is 10 kHz and the input frequency is 7 kHz (higher than the Nyquist frequency of 5 kHz) then it will appear at 3 kHz (using the calculator on this page). This is an example of aliasing.

Now let’s say we had two input signals, one at 7 kHz and the other at 3 kHz, then the aliased 7 kHz signal will overlap with the non-aliased 3 kHz signal and it will erroneously appear that there’s only one input signal.

If we know that signals can be as high as 7 kHz, then the Nyquist rate is calculated to be 14 kHz. This is the minimum frequency that the input signal should be sampled at to prevent aliasing.

When designing a digital sampling system (called a digitizer) it is not uncommon for the sampling rate to exceed the Nyquist frequency by more than a factor of two. This is called oversampling.

Anti-aliasing Filter

In a receiver system, an anti-aliasing filter is typically used to eliminate unwanted signals before they reach the digitizer. In the above example, a low pass filter with a cut-off frequency at 5 kHz would eliminate the 7 kHz signal and pass the 3 kHz signal through. This way the former would not make its way to the digitizer and only the 3 kHz signal would show up in the FFT output.

Note: This assumes a brick wall filter that eliminates the 7 kHz signal entirely. However, there’s practically no such filter and therefore there will be a small or attenuated component of the 7 kHz signal that will alias back.

Example of Sampling a Square Wave

Let’s take the example of a square wave with frequency of 1 kHz. This calculator can be used to find the harmonics. They appear at odd multiples of 1 kHz. In other words, 3 kHz, 5 kHz, 7 kHz, etc. with decreasing amplitudes.

Now if the sampling frequency is 2 kHz, then the harmonics all alias with the fundamental frequency of 1 kHz.

All the components will therefore overlap. Reconstructing this with a digital-to-analog converter would result in a 1 kHz sine wave with an amplitude that’s the sum of the overlapping components.

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