Digital Signal Processing Reference
In-Depth Information
90.0005
3
(
b
)
(
a
)
2.5
90.00025
2
90
1.5
1
89.99975
0.5
89.9995
0
0 0 0 0 0 0
Time/(Sampling Period)
0
20
40
60
80
100
Phase,
φ
t
(degrees)
ϕ
t,m
when input phase is 90
o
, for carrier frequency at 0.4 (
f
N
=
2 kHz). (b) Profile of peak-to-peak dc phase error over a range of input phases. Filter order
L
= 154.
Figure 7.4
(a) Demodulated phase output
Figure 7.5 shows how this maximum error evolves with carrier frequency. We
readily observe that when the carrier frequency is well within the pass-band of
the Hilbert transform and away from its transition edges, the errors tend to be
much smaller than when on the edges. Thus, in this case, the working frequency
range that will guarantee an error of 3
10
-4
o
or less using floating-point arithmetic
is 0.1 to 0.6. From this viewpoint, it is advisable to arrange for the carrier
frequency to be as central as possible in this frequency range and away from the
transition edges of the transform.
×
7.4.2
Phase Step Response
The phase step response is the analog of the unit step response propagating
through a low-pass filter as discussed in earlier chapters. However, in this case, a
unit phase step (measured in degrees) is introduced via
ϕ
t
into the Hilbert
demodulation scheme and its response observed; this corresponds to an input
phase step of 0.0175 radian. The phase step is between 90
o
and 91
o
since
demodulation in this region tends to give the worst-case phase errors. The carrier
frequency is chosen to be at the center of the usable range of the demodulation
process, which in this case is 0.4. Figure 7.6 shows a typical phase step response
for the said demodulation scheme.
7.4.3 Frequency Step Response
If there is a simultaneous step change in carrier frequency in both signal and
reference there is evidence of spiking on the demodulated output. Figure 7.7(a)
