Digital Signal Processing Reference
In-Depth Information
Table 7.1
Coefficients for Empirical Relation between Demodulation
Bandwidth and Modulation Amplitude for Scheme 2
LPF
Cut-off
F
c
a
b
A
LPF1 and HTF cut-off, 0.7
(Filter Length,
L
+1 = 155)
0.2 -0.033 -0.229 0.1
0.3 -0.027 -0.194 0.14
0.4 -0.045 -0.1 0.13
0.5 -0.027 -0.185 0.08
LPF1 and HTF cut-off, 0.8
(Filter Length,
L
+1 = 155)
0.2
a
o
ln
ϕ
+
b
B
~
A
ϕ
o
*
dem
0.005
-0.344
0.11
0.3
-0.021
-0.258
0.16
0.4
-0.024
-0.203
0.18
0.5
-0.028
-0.174
0.13
LPF1 and HTF cut-off, 0.9
(Filter Length,
L
+1 = 155)
0.2 -0.002 -0.347 0.1
0.3 -0.018 -0.26 0.15
0.4 -0.025 -0.223 0.2
0.5 -0.026 -0.194 0.18
*
(1)
B
dem
, is in normalized frequency and must be multiplied by the
Nyquist to realize the frequency in Hertz. (2)
o
is measured in degrees.
ϕ
frequency was in the center of the premixing filter. This is shown in Figure 7.18.
These results suggest that both schemes show similar performance levels. The
accuracy of the demodulation bandwidths given in this work is
±
0.01 (normalized
frequency).
Note that the largest bandwidths for Scheme 2 occur when the postmixing
filter cut-off
F
out
is about half that of the premixing bandwidth
F
in
. Thus a
comparison between 0.7 for Scheme 1 and 0.7 and 0.3 for Scheme 2 is adequate.
In general, as the modulation amplitude decreases the demodulation bandwidth
increases exponentially.
7.8 INTRODUCING AN ARBITRARY PHASE SHIFT INTO A SIGNAL
In this section we will take a brief look at how to introduce a known phase shift
into a signal using the digital Hilbert transform. This may be useful in situations
where we want to rephase a signal in relation to a known reference. A simple way
of doing this is to cast the analytical signal into the Argand plane, then rotate its
axes through
with the Cartesian rotator
R
θ
3
. Let the real part of the signal be
x
and its imaginary part be
H
{
x
}. After anticlockwise rotation of its axes through
θ
θ
,
3
′
cos
θ
sin
θ
x
x
and
R
=
=
R
H
{
x
′
}
θ
H
{
x
}
θ
−
sin
θ
cos
θ
