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 also symmetric, but the term symmetric applies also to functions symmetric about a point other than 0 .
Definition: A function is said to be odd if .
An odd function is also called antisymmetric.
Note that every finite odd function must satisfy .7.12 Moreover, for any with even, we also have since ; that is, and index the same point when is even (since all indexing in is modulo ).
Theorem: Every function can be uniquely 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
To show uniqueness, let denote some other even-odd decomposition. Then , and .
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 :
Example: , , is an even signal since .
Example: is an odd signal since .
Example: is an odd signal (even times odd).
Example: is an even signal (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 ,
for any even signal and odd signal in . In terms of inner products (§5.9), we may say that the even part of every real signal is orthogonal to its odd part: