Graphics Programs Reference
In-Depth Information
3.2. The Analytic Signal
The sinusoidal signal
defined in Eq. (3.1) can be written as the real part
x
()
ψ ()
of the complex signal
. More precisely,
Re r
()
e
j
φ
x
()
e
j
2π
f
0
t
x
()
Re
ψ ()
=
{
}
=
{
}
(3.6)
Define the Ðanalytic signalÑ as
ψ ()
v
()
e
j
2π
f
0
t
=
(3.7)
where
v
()
r
()
e
j
φ
x
()
=
(3.8)
and
2
X
()
0
ω 0
≥
ω 0
Ψ()
=
(3.9)
<
Ψ()
is the Fourier transform of
ψ ()
and
X
()
is the Fourier transform of
x
()
. Eq. (3.9) can be written as
Ψ() 2
U
()
X
()
=
(3.10)
where
U
()
ψ ()
is the step function in the frequency domain. Thus, it can be
shown that
is
ψ ()
x
()
j
ô
()
=
+
(3.11)
ô
()
is the Hilbert transform of
x
()
.
Using Eqs. (3.6) and (3.11), one can then write (shown here without proof)
x
()
u
0
I
=
() ω
0
t
cos
u
0
Q
() ω
0
t
sin
(3.12)
which is similar to Eq. (3.4) with
.
ω
0
=
2π
f
0
Using ParsevalÓs theorem it can be shown that the energy associated with the
signal
is
x
()
∞
∫
∞
∫
1
---
1
---
1
---
E
ψ
x
2
u
2
E
x
=
()
d
=
()
d
=
(3.13)
∞
∞
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