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 $f(0)=0$.^7.8 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

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

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

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

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: $\cos(\omega_k n)$, $n\in{\bf Z}$, 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 odd (even times odd).

Example: $\sin(\omega_k n)\cdot\sin(\omega_l n)$ is even (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 $\omega_k=2\pi k/N$,

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

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 ${\bf C}^N$.