# Decimation Theorem (Aliasing Theorem)

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Decimation Theorem (Aliasing Theorem)

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website. NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page  TOC  INDEX  Next page >>

## Decimation Theorem (Aliasing Theorem)

Theorem: For all , Proof: Let denote the frequency index in the aliased spectrum, and let . Then is length , where is the decimation factor. We have Since , the sum over becomes using the closed form expression for a geometric series derived earlier. We see that the sum over effectively samples every samples. This can be expressed in the previous formula by defining which ranges only over the nonzero samples: Since the above derivation also works in reverse, the theorem is proved.

Here is an illustration of the Decimation Theorem in Matlab:

>> N=4;
>> x = 1:N;
>> X = fft(x);
>> x2 = x(1:2:N);
>> fft(x2)                         % FFT(Decimate(x,2))

ans =


 4    -2


>> (X(1:N/2) + X(N/2 + 1:N))/2     % (1⁄2) Alias(X,2)

ans =


4.0000   -2.0000</pre></p><p>An illustration of aliasing in the frequency domain is shown in


Fig. 8.10.<< Previous page  TOC  INDEX  Next page >>

© 1998-2019 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy