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present (Kottegoda, 1980). Therefore, if one or more persistence are present
in the time series, spectral technique is often used. A non-stationary periodic
time series
x
(
t
) may be expanded into a Fourier series using following
expression:
Ç
A
[
A
sin (2
S
k f t
)
B
cos (2
S
k f t
)]
x
(
t
) =
(88)
0
k
k
k
1
where
f
= frequency and rest of the parameters are the same as expressed in
Eqn. (75). The frequency can be given as:
1
Z
f
=
(89)
P
2
S
where
P
= period of the function (base period) and Z = angular frequency.
With reference to Eqn. (89), two spectral density functions
S
(Z) and
S
(
f
) can
be related as follows:
Sf
S
(90)
The density function
S
(Z) is related to Fourier transformation
F
(Z) as
follows:
S
(Z) =
2()
F
()
Z
S
(Z) =
(91)
Z
Similarly,
S
(
f
) =
Ff
()
(92)
f
Considering an infinitesimal portion of frequency in the range of (
f
,
f
+
df
), the spectrum
S
(
f
)
df
represents the contribution of components with
frequencies in the range (
f
,
f
+
df
) to the total variance.
The spectral density function
S
(
f
) for a discrete process can be written as
(Box and Jenkins, 1976):
Ç
J J
2
cos (2
S
fk
)
S
(
f
) =
- <
f
<
(93)
0
k
k
1
However, the spectral density function for a continuous process can be
written as:
U
Ô
()cos(2
k
S
f k dk
)
S
(
f
) =
-<
f
<
(94)
Negative frequencies obtained with above expressions can be avoided
and integral of the normalized spectral density function can be maintained
over the entire range at one, a one-sided spectral density function
G
(
f
) =
2
S
(
f
) can be written, where
f
varies only over (0, ) and zero elsewhere
(Shahin et al., 1993). The function
G
(
f
) may be expressed as:
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