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.
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 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 ,
for any even signal and odd signal in .