Digital Signal Processing Reference
In-Depth Information
0.25
Scheme 1
F
c
= 0.9
0.2
0.15
Scheme 2
F
c
(LPF) = 0.4
F
c
(LPF1) = 0.9
Scheme 1
F
c
= 0.7
0.1
0.05
Scheme 2
F
c
(LPF) = 0.3
F
c
(LPF1) = 0.7
0
0.1
1
10
100
1000
Modulation Amplitude,
φ
o
(degree)
Figure 7.18
Comparison between demodulation Schemes 1 and 2.
the new real and imaginary parts
x
' and
H
{
x
'}, respectively, are given by
x
′
x
cos
θ
+
H
{
x
}
sin
θ
=
(7.24)
H
{
x
′
}
H
{
x
}
cos
θ
−
x
sin
θ
θ
′
The phase
of the resulting signal in (7.24) is given by
H
{
x
}
−
1
θ
′
=
tan
−
θ
(7.25)
x
Thus, an anticlockwise rotation of the axes introduces a phase lag into the
argument of the analytical signal, whereas a clockwise rotation gives a phase lead.
Note that (7.24) could be represented as
′
′
x
cos
θ
=
A
(7.26)
′
′
H
{
x
}
sin
θ
where
A
turns out to be the same magnitude as the unshifted analytical signal.
Moreover, since both real and imaginary parts of (7.24) involve the unshifted
signal and its Hilbert transform (i.e., both
x
and
H
{
x
}), we can modify the Hilbert
filter coefficients
g
k
directly, so that the coefficients not only imparts 90
o
to the
signal, but also imparts the additional phase delay
θ
. This means that we need two
