Mathematics of the Discrete Fourier Transform (DFT)

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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 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 $$x\$ 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 $${\$ is zero.
Proof: This is readily shown by writing the sum as $$x_o(0) + [x_o(1) + x_o(-1)] + \$ , 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 $$\$ ,

More generally,

for any even signal and odd signal in $${\$ .
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