Even and Odd Functions

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Even and Odd Functions

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 >>

Even and Odd Functions

Some of the Fourier theorems can be succinctly expressed in terms of even and odd symmetries.

Definition: A function $f(n)$ is said to be even if $f(-n)=f(n)$. An even function is alsosymmetric, but the term symmetric applies also to functions symmetric about a point other than $0$.

Definition: A function $f(n)$ is said to be oddif $f(-n)=-f(n)$. An odd function is also calledantisymmetric.

Note that every odd function $f(n)$ must satisfy $f(0)=0$. Moreover, for any $x\ with $N$ even, we also have $x(N/2)=0$ since $x(N/2)=-x(-N/2)=-x(-N/2+N)=-x(N/2)$, i.e., $N/2$ and $-N/2$ index the same point.

Theorem: Every function $f(n)$ can be decomposed into a sum of its even part$f_e(n)$ and odd part $f_o(n)$, where


Proof: In the above definitions, $f_e$ is even and $f_o$ is odd by construction. Summing, we have


Theorem: The product of even functions is even, the product of odd functions is even, and the product of an even times an odd function is odd.

Proof: Readily shown.

Since even times even is even, odd times odd is even, and even times odd is odd, we can think of even as $(+)$ and odd as $(-)$: $(+)\, $(-)\, and $(+)\.

Example: $\ is an even signal since $\.

Example: $\ is an odd signal since $\.

Example: $\ is odd (even times odd).

Example: $\ is even (odd times odd).

Theorem: The sum of all the samples of an odd signal $x_o$ in ${\ is zero.

Proof: This is readily shown by writing the sum as $x_o(0) + [x_o(1) + x_o(-1)] + \, where the last term only occurs when $N$ is even. Each term so written is zero for an odd signal $x_o$.

Example: For all DFT sinusoidal frequencies $\,


More generally,

for any even signal $x_e$ and odd signal $x_o$ in ${\.

<< Previous page  TOC  INDEX  Next page >>


© 1998-2023 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy