**Search Mathematics of the DFT**

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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
,

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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