Biology Reference
In-Depth Information
to identify the prevalent frequencies in the composition of the
function G(t).
Once the power spectrum function f (
) is calculated, it provides
information about the periodic components of the original function G(t).
Suppose, for example, a function G(t) is exactly periodic with period L.
More precisely, suppose we can write G(t) as:
n
f
(
ν
)
G
ð
t
Þ¼
a cos
ð
2
p
t
=
L
Þþ
b sin
ð
2
p
t
=
L
Þ;
where a and b are constants. The power spectrum function f (
n
) will then
have a peak at a frequency
1/L. This, of course, is only true if we
have collected data over a period of time that allows periodic behaviors
to have expressed themselves (i.e., the data should be collected
over a sufficiently long time period T).
n ¼
0
1
1
L
1
ν
A
T
f
(
L
)
A sample graph for f (
n
) is shown in Figure 9-8. Because the connection
between the frequency
n
and the period L is immediate and given
by
1/L, the graph of the power spectrum could be expressed as a
function of
n ¼
or as a function of L (panels A and B of Figure 9-8,
respectively). In endocrinology, the period is more commonly used than
the frequency.
n
0
L
1
T
L
B
Two things are important here:
FIGURE 9-8.
The power spectrum as a function of the
frequency (panel A) and the period (panel B).
1. The width of the peak depends on 1/T, where T is the time interval
over which the data are collected. Typically, collecting data over
longer periods of time will make the peak clearer and better
expressed.
f
(
ν
)
2. For a fixed value T, the height of the peak is proportional to a
2
b
2
.
Thus, increasing the magnitude of the coefficients a and b will make
the peak higher.
þ
0
1
1
L
1
1
L
2
1
...
ν
Assume now that G(t) is a sum of N sine and N cosine waves with
periods L
1
,L
2
,
T
L
N
A
, L
N
. That is:
...
0
1
0
1
0
1
0
1
f
(
L
)
2
p
t
2
p
t
2
t
L
N
p
2
t
L
N
p
@
A
þ
@
A
þ þ
@
A
þ
@
A
G
ð
t
Þ¼
a
1
cos
b
1
sin
a
N
cos
b
N
sin
L
1
L
1
0
1
0
1
X
N
2
p
t
2
p
t
@
A
þ
@
A
:
¼
a
i
cos
b
i
sin
L
i
L
i
0
L
N
L
N
−
1
L
1
T
L
i
¼
1
B
(9-3)
FIGURE 9-9.
The peaks in the power spectrum identify the
prevalent frequencies (panel A) or periods
(panel B).
In this case, the graphs of the power spectrum function are given in
Figure 9-9. As before, the peaks are at the inverse periods
1/L
i
in
panel A and at the periods L
i
in panel B. The heights of the peaks
are proportional to a
i
þ
n
i
¼
b
i
:
