<s>
When	O
the	O
energy	B-Algorithm
of	O
the	O
signal	O
is	O
concentrated	O
around	O
a	O
finite	O
time	O
interval	O
,	O
especially	O
if	O
its	O
total	O
energy	B-Algorithm
is	O
finite	O
,	O
one	O
may	O
compute	O
the	O
energy	B-Algorithm
spectral	O
density	O
.	O
</s>
<s>
The	O
power	O
spectral	O
density	O
(	O
PSD	O
)	O
then	O
refers	O
to	O
the	O
spectral	O
energy	B-Algorithm
distribution	O
that	O
would	O
be	O
found	O
per	O
unit	O
time	O
,	O
since	O
the	O
total	O
energy	B-Algorithm
of	O
such	O
a	O
signal	O
over	O
all	O
time	O
would	O
generally	O
be	O
infinite	O
.	O
</s>
<s>
Obtaining	O
a	O
spectrum	O
from	O
time	O
series	O
such	O
as	O
these	O
involves	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
and	O
generalizations	O
based	O
on	O
Fourier	O
analysis	O
.	O
</s>
<s>
Energy	B-Algorithm
spectral	O
density	O
(	O
ESD	O
)	O
would	O
have	O
units	O
would	O
be	O
V2sHz−1	O
,	O
since	O
energy	B-Algorithm
has	O
units	O
of	O
power	O
multiplied	O
by	O
time	O
(	O
e.g.	O
,	O
watt-hour	O
)	O
.	O
</s>
<s>
In	O
the	O
analysis	O
of	O
random	O
vibrations	O
,	O
units	O
of	O
g2Hz−1	O
are	O
frequently	O
used	O
for	O
the	O
PSD	O
of	O
acceleration	O
,	O
where	O
g	O
denotes	O
the	O
g-force	B-Application
.	O
</s>
<s>
Energy	B-Algorithm
spectral	O
density	O
describes	O
how	O
the	O
energy	B-Algorithm
of	O
a	O
signal	O
or	O
a	O
time	O
series	O
is	O
distributed	O
with	O
frequency	O
.	O
</s>
<s>
Here	O
,	O
the	O
term	O
energy	B-Algorithm
is	O
used	O
in	O
the	O
generalized	O
sense	O
of	O
signal	O
processing	O
;	O
that	O
is	O
,	O
the	O
energy	B-Algorithm
of	O
a	O
signal	O
is	O
:	O
</s>
<s>
The	O
energy	B-Algorithm
spectral	O
density	O
is	O
most	O
suitable	O
for	O
transients	O
—	O
that	O
is	O
,	O
pulse-like	O
signals	O
—	O
having	O
a	O
finite	O
total	O
energy	B-Algorithm
.	O
</s>
<s>
Finite	O
or	O
not	O
,	O
Parseval	O
's	O
theorem	O
(	O
or	O
Plancherel	O
's	O
theorem	O
)	O
gives	O
us	O
an	O
alternate	O
expression	O
for	O
the	O
energy	B-Algorithm
of	O
the	O
signal	O
:	O
</s>
<s>
is	O
the	O
value	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
at	O
frequency	O
(	O
in	O
Hz	O
)	O
.	O
</s>
<s>
Since	O
the	O
integral	O
on	O
the	O
left-hand	O
side	O
is	O
the	O
energy	B-Algorithm
of	O
the	O
signal	O
,	O
the	O
value	O
of	O
can	O
be	O
interpreted	O
as	O
a	O
density	O
function	O
multiplied	O
by	O
an	O
infinitesimally	O
small	O
frequency	O
interval	O
,	O
describing	O
the	O
energy	B-Algorithm
contained	O
in	O
the	O
signal	O
at	O
frequency	O
in	O
the	O
frequency	O
interval	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
energy	B-Algorithm
spectral	O
density	O
of	O
is	O
defined	O
as	O
:	O
</s>
<s>
The	O
function	O
and	O
the	O
autocorrelation	O
of	O
form	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
pair	O
,	O
a	O
result	O
also	O
known	O
as	O
the	O
Wiener	O
–	O
Khinchin	O
theorem	O
(	O
see	O
also	O
Periodogram	O
)	O
.	O
</s>
<s>
As	O
a	O
physical	O
example	O
of	O
how	O
one	O
might	O
measure	O
the	O
energy	B-Algorithm
spectral	O
density	O
of	O
a	O
signal	O
,	O
suppose	O
represents	O
the	O
potential	O
(	O
in	O
volts	O
)	O
of	O
an	O
electrical	O
pulse	O
propagating	O
along	O
a	O
transmission	O
line	O
of	O
impedance	O
,	O
and	O
suppose	O
the	O
line	O
is	O
terminated	O
with	O
a	O
matched	O
resistor	O
(	O
so	O
that	O
all	O
of	O
the	O
pulse	O
energy	B-Algorithm
is	O
delivered	O
to	O
the	O
resistor	O
and	O
none	O
is	O
reflected	O
back	O
)	O
.	O
</s>
<s>
By	O
Ohm	O
's	O
law	O
,	O
the	O
power	O
delivered	O
to	O
the	O
resistor	O
at	O
time	O
is	O
equal	O
to	O
,	O
so	O
the	O
total	O
energy	B-Algorithm
is	O
found	O
by	O
integrating	O
with	O
respect	O
to	O
time	O
over	O
the	O
duration	O
of	O
the	O
pulse	O
.	O
</s>
<s>
To	O
find	O
the	O
value	O
of	O
the	O
energy	B-Algorithm
spectral	O
density	O
at	O
frequency	O
,	O
one	O
could	O
insert	O
between	O
the	O
transmission	O
line	O
and	O
the	O
resistor	O
a	O
bandpass	O
filter	O
which	O
passes	O
only	O
a	O
narrow	O
range	O
of	O
frequencies	O
(	O
,	O
say	O
)	O
near	O
the	O
frequency	O
of	O
interest	O
and	O
then	O
measure	O
the	O
total	O
energy	B-Algorithm
dissipated	O
across	O
the	O
resistor	O
.	O
</s>
<s>
The	O
value	O
of	O
the	O
energy	B-Algorithm
spectral	O
density	O
at	O
is	O
then	O
estimated	O
to	O
be	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
since	O
the	O
power	O
has	O
units	O
of	O
V2	O
Ω−1	O
,	O
the	O
energy	B-Algorithm
has	O
units	O
of	O
V2sΩ−1	O
=	O
J	O
,	O
and	O
hence	O
the	O
estimate	O
of	O
the	O
energy	B-Algorithm
spectral	O
density	O
has	O
units	O
of	O
JHz−1	O
,	O
as	O
required	O
.	O
</s>
<s>
In	O
many	O
situations	O
,	O
it	O
is	O
common	O
to	O
forget	O
the	O
step	O
of	O
dividing	O
by	O
so	O
that	O
the	O
energy	B-Algorithm
spectral	O
density	O
instead	O
has	O
units	O
of	O
V2Hz−2	O
.	O
</s>
<s>
where	O
is	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
The	O
sampling	O
interval	O
is	O
needed	O
to	O
keep	O
the	O
correct	O
physical	O
units	O
and	O
to	O
ensure	O
that	O
we	O
recover	O
the	O
continuous	O
case	O
in	O
the	O
limit	O
But	O
in	O
the	O
mathematical	O
sciences	O
the	O
interval	O
is	O
often	O
set	O
to	O
1	O
,	O
which	O
simplifies	O
the	O
results	O
at	O
the	O
expense	O
of	O
generality	O
.	O
</s>
<s>
The	O
above	O
definition	O
of	O
energy	B-Algorithm
spectral	O
density	O
is	O
suitable	O
for	O
transients	O
(	O
pulse-like	O
signals	O
)	O
whose	O
energy	B-Algorithm
is	O
concentrated	O
around	O
one	O
time	O
window	O
;	O
then	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
of	O
the	O
signals	O
generally	O
exist	O
.	O
</s>
<s>
That	O
is	O
the	O
reason	O
that	O
we	O
cannot	O
use	O
the	O
energy	B-Algorithm
spectral	O
density	O
itself	O
,	O
which	O
is	O
that	O
diverging	O
integral	O
,	O
in	O
such	O
cases	O
.	O
</s>
<s>
In	O
analyzing	O
the	O
frequency	O
content	O
of	O
the	O
signal	O
,	O
one	O
might	O
like	O
to	O
compute	O
the	O
ordinary	O
Fourier	B-Algorithm
transform	I-Algorithm
;	O
however	O
,	O
for	O
many	O
signals	O
of	O
interest	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
does	O
not	O
formally	O
exist	O
.	O
</s>
<s>
Now	O
,	O
if	O
we	O
divide	O
the	O
time	O
convolution	B-Language
above	O
by	O
the	O
period	O
and	O
take	O
the	O
limit	O
as	O
,	O
it	O
becomes	O
the	O
autocorrelation	O
function	O
of	O
the	O
non-windowed	O
signal	O
,	O
which	O
is	O
denoted	O
as	O
,	O
provided	O
that	O
is	O
ergodic	O
,	O
which	O
is	O
true	O
in	O
most	O
,	O
but	O
not	O
all	O
,	O
practical	O
cases	O
..	O
</s>
<s>
From	O
here	O
we	O
see	O
,	O
again	O
assuming	O
the	O
ergodicity	O
of	O
,	O
that	O
the	O
power	O
spectral	O
density	O
can	O
be	O
found	O
as	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
autocorrelation	O
function	O
(	O
Wiener	O
–	O
Khinchin	O
theorem	O
)	O
.	O
</s>
<s>
Just	O
as	O
with	O
the	O
energy	B-Algorithm
spectral	O
density	O
,	O
the	O
definition	O
of	O
the	O
power	O
spectral	O
density	O
can	O
be	O
generalized	O
to	O
discrete	O
time	O
variables	O
.	O
</s>
<s>
Just	O
as	O
before	O
,	O
from	O
here	O
we	O
recast	O
these	O
products	O
as	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
time	O
convolution	B-Language
,	O
which	O
when	O
divided	O
by	O
the	O
period	O
and	O
taken	O
to	O
the	O
limit	O
becomes	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
cross-correlation	O
function	O
.	O
</s>
<s>
voltage	O
or	O
current	O
)	O
,	O
their	O
Fourier	B-Algorithm
transforms	I-Algorithm
and	O
are	O
usually	O
restricted	O
to	O
positive	O
frequencies	O
by	O
convention	O
.	O
</s>
<s>
Depending	O
on	O
what	O
is	O
known	O
about	O
the	O
signal	O
,	O
estimation	O
techniques	O
can	O
involve	O
parametric	B-General_Concept
or	O
non-parametric	B-General_Concept
approaches	O
,	O
and	O
may	O
be	O
based	O
on	O
time-domain	O
or	O
frequency-domain	O
analysis	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
common	O
parametric	B-General_Concept
technique	O
involves	O
fitting	O
the	O
observations	O
to	O
an	O
autoregressive	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
A	O
common	O
non-parametric	B-General_Concept
technique	O
is	O
the	O
periodogram	O
.	O
</s>
<s>
The	O
spectral	O
density	O
is	O
usually	O
estimated	O
using	O
Fourier	B-Algorithm
transform	I-Algorithm
methods	O
(	O
such	O
as	O
the	O
Welch	O
method	O
)	O
,	O
but	O
other	O
techniques	O
such	O
as	O
the	O
maximum	O
entropy	O
method	O
can	O
also	O
be	O
used	O
.	O
</s>
<s>
The	O
spectral	B-Algorithm
centroid	I-Algorithm
of	O
a	O
signal	O
is	O
the	O
midpoint	O
of	O
its	O
spectral	O
density	O
function	O
,	O
i.e.	O
</s>
<s>
The	O
spectral	B-Algorithm
edge	I-Algorithm
frequency	I-Algorithm
(	O
SEF	O
)	O
,	O
usually	O
expressed	O
as	O
"	O
SEF	O
x	O
"	O
,	O
represents	O
the	O
frequency	O
below	O
which	O
x	O
percent	O
of	O
the	O
total	O
power	O
of	O
a	O
given	O
signal	O
are	O
located	O
;	O
typically	O
,	O
x	O
is	O
in	O
the	O
range	O
75	O
to	O
95	O
.	O
</s>
<s>
It	O
is	O
more	O
particularly	O
a	O
popular	O
measure	O
used	O
in	O
EEG	B-Application
monitoring	O
,	O
in	O
which	O
case	O
SEF	O
has	O
variously	O
been	O
used	O
to	O
estimate	O
the	O
depth	O
of	O
anesthesia	O
and	O
stages	O
of	O
sleep	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
remote	B-Application
sensing	I-Application
using	O
a	O
spectrometer	O
,	O
the	O
spectral	O
envelope	O
of	O
a	O
feature	O
is	O
the	O
boundary	O
of	O
its	O
spectral	O
properties	O
,	O
as	O
defined	O
by	O
the	O
range	O
of	O
brightness	O
levels	O
in	O
each	O
of	O
the	O
spectral	O
bands	O
of	O
interest	O
.	O
</s>
<s>
This	O
is	O
the	O
basis	O
of	O
a	O
number	O
of	O
spectral	O
analysis	O
techniques	O
such	O
as	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
and	O
wavelets	O
.	O
</s>
<s>
For	O
transfer	O
functions	O
(	O
e.g.	O
,	O
Bode	B-Algorithm
plot	I-Algorithm
,	O
chirp	O
)	O
the	O
complete	O
frequency	O
response	O
may	O
be	O
graphed	O
in	O
two	O
parts	O
:	O
power	O
versus	O
frequency	O
and	O
phase	O
versus	O
frequency	O
—	O
the	O
phase	O
spectral	O
density	O
,	O
phase	O
spectrum	O
,	O
or	O
spectral	O
phase	O
.	O
</s>
<s>
Although	O
these	O
are	O
also	O
Fourier	B-Algorithm
transform	I-Algorithm
pairs	O
,	O
there	O
is	O
no	O
symmetry	O
(	O
as	O
there	O
is	O
for	O
the	O
autocorrelation	O
)	O
forcing	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
be	O
real-valued	O
.	O
</s>
<s>
The	O
concept	O
and	O
use	O
of	O
the	O
power	O
spectrum	O
of	O
a	O
signal	O
is	O
fundamental	O
in	O
electrical	O
engineering	O
,	O
especially	O
in	O
electronic	O
communication	O
systems	O
,	O
including	O
radio	O
communications	O
,	O
radars	B-Application
,	O
and	O
related	O
systems	O
,	O
plus	O
passive	B-Application
remote	I-Application
sensing	I-Application
technology	O
.	O
</s>
<s>
The	O
spectrum	O
analyzer	O
measures	O
the	O
magnitude	O
of	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
STFT	B-Algorithm
)	O
of	O
an	O
input	O
signal	O
.	O
</s>
<s>
If	O
the	O
signal	O
being	O
analyzed	O
can	O
be	O
considered	O
a	O
stationary	B-Algorithm
process	I-Algorithm
,	O
the	O
STFT	B-Algorithm
is	O
a	O
good	O
smoothed	O
estimate	O
of	O
its	O
power	O
spectral	O
density	O
.	O
</s>
