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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