## 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 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 .

^{7.11}Moreover, for any with even, we also have since ; that is, and index the same point when is even.

**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 :

**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 ,

*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:

**Next Section:**

Fourier Theorems

**Previous Section:**

Signal Operators