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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 $ f(n)$ is said to be even if $ f(-n)=f(n)$.

An even function is also symmetric, but the term symmetric applies also to functions symmetric about a point other than 0.

Definition: A function $ f(n)$ is said to be odd if $ f(-n)=-f(n)$.

An odd function is also called antisymmetric.

Note that every finite odd function $ f(n)$ must satisfy $ f(0)=0$.7.11 Moreover, for any $ x\in{\bf C}^N$ with $ N$ even, we also have $ x(N/2)=0$ since $ x(N/2)=-x(-N/2)=-x(-N/2+N)=-x(N/2)$; that is, $ N/2$ and $ -N/2$ index the same point when $ N$ is even.

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

f_e(n) &\isdef & \frac{f(n) + f(-n)}{2} \\
f_o(n) &\isdef & \frac{f(n) - f(-n)}{2}.

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

$\displaystyle f_e(n) + f_o(n) = \frac{f(n) + f(-n)}{2} + \frac{f(n) - f(-n)}{2} = f(n).

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 $ (-)$:

(+)\cdot(+) &=& (+)\\
(-)\cdot(-) &=& (+)\\
(+)\cdot(-) &=& (-)\\
(-)\cdot(+) &=& (-)

Example: $ \cos(\omega_k n)$, $ n\in{\bf Z}$, is an even signal since $ \cos(-\theta) = \cos(\theta)$.

Example: $ \sin(\omega_k n)$ is an odd signal since $ \sin(-\theta) = -\sin(\theta)$.

Example: $ \cos(\omega_k n)\cdot\sin(\omega_l n)$ is an odd signal (even times odd).

Example: $ \sin(\omega_k n)\cdot\sin(\omega_l n)$ is an even signal (odd times odd).

Theorem: The sum of all the samples of an odd signal $ x_o$ in $ {\bf C}^N$ is zero.

Proof: This is readily shown by writing the sum as $ x_o(0) + [x_o(1) + x_o(-1)]
+ \cdots + x(N/2)$, 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 $ \omega_k=2\pi k/N$,

$\displaystyle \sum_{n=0}^{N-1}\sin(\omega_k n) \cos(\omega_k n) = 0, \; k=0,1,2,\ldots,N-1.

More generally,

$\displaystyle \sum_{n=0}^{N-1}x_e(n) x_o(n) = 0,

for any even signal $ x_e$ and odd signal $ x_o$ in $ {\bf C}^N$. In terms of inner products5.9), we may say that the even part of every real signal is orthogonal to its odd part:

$\displaystyle \left<x_e,x_o\right> = 0

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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