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
.^{7.12} Moreover, for any
with
even, we also have
since
; that is,
and
index
the same point when
is even (since all indexing in
is modulo
).

**Theorem: **Every function
can be
uniquely
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

To show uniqueness, let denote some other even-odd decomposition. Then , and .

**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
,

More generally,

for

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University