Environmental Engineering Reference
In-Depth Information
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
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