Environmental Engineering Reference
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where the values of the k coefficients give the weight of the k harmonics in the original
signal
( t ).
If the signal is not periodic and it is defined in only a finite interval, for example
φ
[
π, + π
], the harmonic analysis can be applied after the function
φ
( t ) is extended
periodically beyond the initial interval by means of the relation
( t ), by
assuming the mean of the two limiting values at the discontinuity points (i.e., at the
odd multiplies of
φ
( t
+
2
π
)
= φ
π
).
The Fourier analysis takes a more compact aspect if the complex notation is intro-
duced by the Euler formula, e i α =
cos
α +
i sin
α
. In this way, the sines and cosines
functions can be written as
1
1
2 ( e ikt
e ikt )
2 ( e ikt
e ikt )
cos kt
=
+
,
sin kt
=
,
(A.3)
and the Fourier series takes the form
+∞
α k e ikt
φ
( t )
=
,
(A.4)
k
=−∞
where
π
π φ
1
2
( t ) e ikt d t
α
=
.
(A.5)
k
π
If the function
φ
( t ) has a period T
=
2
π
, the angular frequency
ω =
2
π/
T is
t . In this way the Fourier series becomes
introduced and t is replaced with
ω
T / 2
+∞
1
T
α k e ik ω t
( t ) e ik ω t d t .
φ
( t )
=
,
=
2 φ
(A.6)
k
T
/
k
=−∞
Finally, the constraint of periodicity can also be removed by allowing the length
of the interval to tend to infinity (i.e., T
→∞
). In this case the Fourier integral is
replaced with the Fourier transform:
1
2
) e i ω t d
φ
( t )
=
F (
ω
ω,
(A.7)
π
−∞
where F (
ω
) is the Fourier transform of
φ
( t ),
−∞ φ
( t ) e i ω t d t
F (
ω
)
=
.
(A.8)
If the rescaled frequency f
= ω/
(2
π
)
=
1
/
T is used in place of
ω
, the previous
relations become
−∞ φ
F ( f ) e 2 π ift d f
( t ) e 2 π ift d t
φ
( t )
=
,
F ( f )
=
.
(A.9)
−∞
Equations ( A.7 )-( A.9 ) establish a one-to-one link between
φ
( t )and F (
ω
)[or F ( f )],
generally indicated as
) and called a transform pair . Consequently we can
analyze the same physical process from two different viewpoints: in the time domain
φ
( t )
F (
ω  Search WWH ::

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