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 $f(0)=0$ .^7.12 Moreover, for any $x\in\mathbb{C}^N$ 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 (since all indexing in $\mathbb{C}^N$ is modulo $N$ ).

Theorem: Every function $f(n)$ can be uniquely decomposed into a sum of its even part $f_e(n)$ and odd part $f_o(n)$ , where

$$\begin{eqnarray*} f_e(n) &\isdef & \frac{f(n) + f(-n)}{2} \\ f_o(n) &\isdef & \frac{f(n) - f(-n)}{2}. \end{eqnarray*}$$

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

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

To show uniqueness, let $f(n) = f'_e(n) + f'_o(n)$ denote some other even-odd decomposition. Then $f(n)+f(-n) = 2f_e(n) = 2f'_e(n) \,\,\Rightarrow\,\,f_e(n)=f'_e(n)$ , and $f(n)-f(-n) = 2f_o(n) = 2f'_o(n) \,\,\Rightarrow\,\,f_o(n)=f'_o(n)$ .

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 $(-)$ :

$$\begin{eqnarray*} (+)\cdot(+) &=& (+)\\ (-)\cdot(-) &=& (+)\\ (+)\cdot(-) &=& (-)\\ (-)\cdot(+) &=& (-) \end{eqnarray*}$$

Example: $\cos(\omega_k n)$ , $n\in\mathbb{Z}$ , 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 $\mathbb{C}^N$ 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 $\omega_k=2\pi k/N$ ,

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

More generally,

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

for any even signal $x_e$ and odd signal $x_o$ in $\mathbb{C}^N$ . In terms of inner products (§5.9), we may say that the even part of every real signal is orthogonal to its odd part:

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