<s>
A	O
given	O
function	O
or	O
signal	O
can	O
be	O
converted	O
between	O
the	O
time	O
and	O
frequency	O
domains	O
with	O
a	O
pair	O
of	O
mathematical	O
operators	O
called	O
transforms	B-Algorithm
.	O
</s>
<s>
An	O
example	O
is	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
which	O
converts	O
a	O
time	O
function	O
into	O
a	O
complex	O
valued	O
sum	O
or	O
integral	O
of	O
sine	O
waves	O
of	O
different	O
frequencies	O
,	O
with	O
amplitudes	B-Application
and	O
phases	O
,	O
each	O
of	O
which	O
represents	O
a	O
frequency	O
component	O
.	O
</s>
<s>
The	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
converts	O
the	O
frequency-domain	O
function	O
back	O
to	O
the	O
time-domain	O
function	O
.	O
</s>
<s>
Some	O
specialized	O
signal	O
processing	O
techniques	O
use	O
transforms	B-Algorithm
that	O
result	O
in	O
a	O
joint	O
time	O
–	O
frequency	O
domain	O
,	O
with	O
the	O
instantaneous	O
frequency	O
being	O
a	O
key	O
link	O
between	O
the	O
time	O
domain	O
and	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
In	O
addition	O
,	O
looking	O
at	O
a	O
system	O
from	O
the	O
point	O
of	O
view	O
of	O
frequency	O
can	O
often	O
give	O
an	O
intuitive	O
understanding	O
of	O
the	O
qualitative	O
behavior	O
of	O
the	O
system	O
,	O
and	O
a	O
revealing	O
scientific	O
nomenclature	O
has	O
grown	O
up	O
to	O
describe	O
it	O
,	O
characterizing	O
the	O
behavior	O
of	O
physical	O
systems	O
to	O
time	O
varying	O
inputs	O
using	O
terms	O
such	O
as	O
bandwidth	B-Algorithm
,	O
frequency	O
response	O
,	O
gain	O
,	O
phase	O
shift	O
,	O
resonant	O
frequencies	O
,	O
time	O
constant	O
,	O
resonance	O
width	O
,	O
damping	O
factor	O
,	O
Q	O
factor	O
,	O
harmonics	O
,	O
spectrum	O
,	O
power	O
spectral	B-Algorithm
density	I-Algorithm
,	O
eigenvalues	O
,	O
poles	O
,	O
and	O
zeros	O
.	O
</s>
<s>
In	O
using	O
the	O
Laplace	O
,	O
Z-	O
,	O
or	O
Fourier	B-Algorithm
transforms	I-Algorithm
,	O
a	O
signal	O
is	O
described	O
by	O
a	O
complex	O
function	O
of	O
frequency	O
:	O
the	O
component	O
of	O
the	O
signal	O
at	O
any	O
given	O
frequency	O
is	O
given	O
by	O
a	O
complex	O
number	O
.	O
</s>
<s>
The	O
modulus	O
of	O
the	O
number	O
is	O
the	O
amplitude	B-Application
of	O
that	O
component	O
,	O
and	O
the	O
argument	O
is	O
the	O
relative	O
phase	O
of	O
the	O
wave	O
.	O
</s>
<s>
For	O
example	O
,	O
using	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
a	O
sound	O
wave	O
,	O
such	O
as	O
human	O
speech	O
,	O
can	O
be	O
broken	O
down	O
into	O
its	O
component	O
tones	O
of	O
different	O
frequencies	O
,	O
each	O
represented	O
by	O
a	O
sine	O
wave	O
of	O
a	O
different	O
amplitude	B-Application
and	O
phase	O
.	O
</s>
<s>
By	O
discarding	O
the	O
phase	O
information	O
,	O
it	O
is	O
possible	O
to	O
simplify	O
the	O
information	O
in	O
a	O
frequency-domain	O
representation	O
to	O
generate	O
a	O
frequency	O
spectrum	O
or	O
spectral	B-Algorithm
density	I-Algorithm
.	O
</s>
<s>
Although	O
"	O
the	O
"	O
frequency	O
domain	O
is	O
spoken	O
of	O
in	O
the	O
singular	O
,	O
there	O
are	O
a	O
number	O
of	O
different	O
mathematical	O
transforms	B-Algorithm
which	O
are	O
used	O
to	O
analyze	O
time-domain	O
functions	O
and	O
are	O
referred	O
to	O
as	O
"	O
frequency	O
domain	O
"	O
methods	O
.	O
</s>
<s>
These	O
are	O
the	O
most	O
common	O
transforms	B-Algorithm
,	O
and	O
the	O
fields	O
in	O
which	O
they	O
are	O
used	O
:	O
</s>
<s>
Fourier	O
series	O
–	O
periodic	O
signals	O
,	O
oscillating	B-Algorithm
systems	O
.	O
</s>
<s>
Fourier	B-Algorithm
transform	I-Algorithm
–	O
aperiodic	O
signals	O
,	O
transients	O
.	O
</s>
<s>
Z	B-Algorithm
transform	I-Algorithm
–	O
discrete-time	O
signals	O
,	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
.	O
</s>
<s>
Wavelet	B-Algorithm
transform	I-Algorithm
—	O
image	O
analysis	O
,	O
data	B-General_Concept
compression	I-General_Concept
.	O
</s>
<s>
The	O
above	O
transforms	B-Algorithm
can	O
be	O
interpreted	O
as	O
capturing	O
some	O
form	O
of	O
frequency	O
,	O
and	O
hence	O
the	O
transform	B-Algorithm
domain	I-Algorithm
is	O
referred	O
to	O
as	O
a	O
frequency	O
domain	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
maps	O
a	O
function	O
having	O
a	O
discrete	O
time	O
domain	O
into	O
one	O
having	O
a	O
discrete	O
frequency	O
domain	O
.	O
</s>
<s>
The	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
on	O
the	O
other	O
hand	O
,	O
maps	O
functions	O
with	O
discrete	O
time	O
(	O
discrete-time	O
signals	O
)	O
to	O
functions	O
that	O
have	O
a	O
continuous	O
frequency	O
domain	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
periodic	O
signal	O
has	O
energy	O
only	O
at	O
a	O
base	O
frequency	O
and	O
its	O
harmonics	O
.	O
</s>
<s>
This	O
is	O
the	O
usual	O
context	O
for	O
a	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
