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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