# OpenStax-CNX

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# Aliasing

Module by: Justin Romberg, Don Johnson. E-mail the authors

Summary: This module introduces the idea of aliasing and gives examples of it in sampling and reconstruction problems.

## Introduction

When considering the reconstruction of a signal, you should already be familiar with the idea of the Nyquist rate. This concept allows us to find the sampling rate that will provide for perfect reconstruction of our signal. If we sample at too low of a rate (below the Nyquist rate), then problems will arise that will make perfect reconstruction impossible - this problem is known as aliasing. Aliasing occurs when there is an overlap in the shifted, perioidic copies of our original signal's FT, i.e. spectrum.

In the frequency domain, one will notice that part of the signal will overlap with the periodic signals next to it. In this overlap the values of the frequency will be added together and the shape of the signals spectrum will be unwantingly altered. This overlapping, or aliasing, makes it impossible to correctly determine the correct strength of that frequency. Figure 1 provides a visual example of this phenomenon:

## Aliasing and Sampling

If we sample too slowly, i.e., T>π ΩB : Ωs <2 ΩB T ΩB Ωs 2 ΩB We cannot recover the signal from its samples due to aliasing.

### Example 1

Let f1 t f1 t have CTFT.

Let f2 t f2 t have CTFT.

• What does the DTFT of f 1 , s n= f1 nT f 1 , s n f1 n T look like?
• What does the DTFT of f 2 , s n= f2 nT f 2 , s n f2 n T look like?
• Do any other signals have the same DTFT as f 1 , s n f 1 , s n and f 2 , s n f 2 , s n ?

CONCLUSION: If we sample below the Nyquist frequency, there are many signals that could have produced that given sample sequence.

Why the term "aliasing"? Because the same sample sequence can represent different CT signals (as opposed to when we sample above the Nyquist frequency, then the sample sequence represents a unique CT signal).

### Example 2

ft=cos2πt f t 2 t

Case 1: Sample Ωs =(8π)radsec Ωs 8 rad sec T=14sec T 1 4 sec .

#### Note:

Ωs >2 ΩB Ωs 2 ΩB

Case 2: Sample w Ωs =(83π)radsec w Ωs 8 3 rad sec T=34sec T 3 4 sec .

#### Note:

Ωs <2 ΩB Ωs 2 ΩB

When we run the DTFT from Case #2 through the reconstruction steps, we realize that we end up with the following cosine: f ~ t=cosπ2t f ~ t 2 t This is a "stretched" out version of our original. Clearly, our sampling rate was not high enough to ensure correct reconstruction from the samples.

You may have seen some effects of aliasing such as a wagon wheel turning backwards in a western movie. Aliasing in images can result in Moire Patterns. Here is an example of an image that has Moire artifacts as a result of scanning at too low a frequency.

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##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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