Next |
Prev |
Up |
Top
|
Index |
JOS Index |
JOS Pubs |
JOS Home |
Search
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 any even signal
and odd signal
in
. In
terms of inner products (§5.9), we may say that the even part
of every real signal is orthogonal to its odd part:
Next |
Prev |
Up |
Top
|
Index |
JOS Index |
JOS Pubs |
JOS Home |
Search
[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]