# Even and Odd Functions

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

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# Even and Odd Functions

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

Definition: A function is said to be even if . An even function is alsosymmetric, but the term symmetric applies also to functions symmetric about a point other than .

Definition: A function is said to be oddif . An odd function is also calledantisymmetric.

Note that every odd function must satisfy . Moreover, for any with even, we also have since , i.e., and index the same point.

Theorem: Every function can be decomposed into a sum of its even part and odd part , where

Proof: In the above definitions, is even and 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.

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 in is zero.

Proof: This is readily shown by writing the sum as , where the last term only occurs when is even. Each term so written is zero for an odd signal .

Example: For all DFT sinusoidal frequencies ,

More generally,

for any even signal and odd signal in .

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