**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 beevenif . 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 beoddif . 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.

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 anevensignal since .

Example:is anoddsignal since .

Example:isodd(even times odd).

Example:iseven(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,

foranyeven signal and odd signal in .