Information Technology Reference
In-Depth Information
If
i
f
is the
i
th harmonic of the fundamental frequency 1/
N
, then the least square
estimates of the coefficients
a
and
(,)
i
ab
will be
i
ax
,
0
2
N
a
x C
,
¦
i
t
it
N
t
N
1
2
b
x S
.
¦
i
t
it
N
t
1
The periodogram then consists of
q
= (
N
-1)/2 values
N
N
If
()
(
a
2
b
2
)
i
= 1, 2, …,
q
,
¦
i
i
i
2
t
1
called intensities at frequency values
f
However, for an even value
N
, which should be set equal to 2
q
, the above
equation holds for
i
= 1, 2, 3, ..., (
q
-1), and the last coefficients are
.
i
1
N
t
a
(1) ,
x
¦
q
t
N
t
1
b
0,
q
and correspondingly
If
() (0. )
I
Na
2
.
q
q
Because in a given time interval the highest frequency is 0.5 cycles per interval,
in the definition of sample spectrum - if
f
i
is the
i
th harmonics of the fundamental
frequency (1/
N
) - the definition of the periodogram is modified to
N
2
2
If
()
(
a
b
)
f
f
2
where
0
dd
In this case
I
(
f
) is referred to as the
sample spectrum
.
0.5.
Example
The simplest Fourier model
X
aa
cos
Z
t
b
sin
Z
t
Z
t
0
t
can, in the case of multidimensional analysis, be represented in matrix form as
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