Digital Signal Processing Reference
In-Depth Information
0.16
0.3
0.14
0.4
0.12
0.1
0.2
0.08
0.5
0.06
0.04
0.02
F
out
= 0.2
0
0.1
1
10
100
1000
Modulation Amplitude,
φ
o
(degree)
Figure 7.17
Dependence of demodulation bandwidth on cut-off frequency of the postmixing filter,
LPF, and modulation amplitude
ϕ
o
when the cut-off is 0.7 for both premixing filters. This refers to the
scheme shown in Figure 7.14. The carrier frequency is at the center of premixing filter bandwidth.
F
c
= 0.7 and length 155. It is clear that as the modulation amplitude is increased,
the bandwidth decreases. In general, the higher the cut-off of the postmixing filter,
the larger the demodulation bandwidth as would be expected since more Bessel
components become available to represent the modulated signal, and therefore
less distortion in the demodulated signal. In this approach, the carrier frequency is
positioned at the center of the premixing filter bandwidth. Given these conditions,
we may write an empirical relationship between the demodulation bandwidth and
modulation amplitude in the form
a
o
ln
ϕ
+
b
B
~
A
ϕ
(7.23)
o
dem
where
A
,
a
, and
b
are constants determined from least squares fits to the
demodulated phase in a noise-free setting. We have determined these constants for
premixing filters with cut-off frequencies of 0.7, 0.8, and 0.9 and for filter lengths
of 155 each. These are given in Table 7.1. Note that the demodulation bandwidth
B
dem
is normalized and must be multiplied by the Nyquist frequency to realize the
correct bandwidth in Hertz. Moreover, the modulation amplitude
ϕ
o
in (7.23) is
measured in degrees. Perhaps a quick comparison between demodulation Schemes
1 and 2 in terms of the demodulation bandwidth is appropriate.
In order to make a fair comparison between the said schemes, we compare the
modulation bandwidth-amplitude profiles for a cut-off frequency of 0.7 for
Scheme 1, and, 0.7 and 0.3 for the pre- (LPF1/HTF) and postmixing (LPF) filters,
respectively, for Scheme 2. The lengths of all filters were 155, and the carrier
