Digital Signal Processing Reference
In-Depth Information
80
1.4
(
a
)
(
b
)
60
1
40
0.6
20
0.2
0
-0.2
-20
-0.6
-40
-1
-60
-80
-1.4
0
50
100
150
200
250
300
0
50
100
150
200
250
300
Time/(Sampling Period)
Time/(Sampling Period)
Figure 7.9
The ac phase output using Hilbert demodulation scheme;
F
m
= 0.025. (a) ϕ
o
= 60
o
and (b)
ϕ
10
−6 o
. Filter length and cut-off frequency for Hilbert and low pass-filters were 155 and 0.7,
respectively. Carrier frequency = 0.4.
o
= 1
×
ac phase demodulation, in particular, the nature of the errors arising from the
process. Treating
ϕ
t
as the time-dependent ac phase term, we can write
ϕ
=
ϕ
sin(
π
F
m
t
)
(7.15)
t
o
where
o
is the
modulation amplitude and
F
m
is the normalized modulation
frequency. Figure 7.9 shows the output of the demodulation scheme of Figure 7.2
for
ϕ
10
-6 o
and
F
m
= 0.025 in (7.15). Certainly, using floating-point
arithmetic, the scheme is able to demodulate amplitudes of the order of one part in
a 10
6
. In general, the minimum detectable phase is more than 100 times smaller
than the modulation amplitude when the former is larger than roundoff noise. In
the next subsection, we will investigate ac demodulation error limits when
floating-point arithmetic is used.
o
= 60
o
and 1
ϕ
×
7.6.1
AC Phase Errors
The ac phase error is the error
δ
ϕ
(see (7.11)) arising from the demodulation
process when the input phase
t
is an ac signal described by (7.15). Figure 7.10
shows typical ac phase errors corresponding to the demodulated signals shown in
Figure 7.9. We may readily observe that the errors exhibit the second harmonic
component at
F
= 0.8, which is similar to dc phase errors, but carries an envelope
at the modulation frequency
F
m
(=0.025). The peak-to-peak error for a modulation
ϕ
