Hilbert Transformers - Digital Filter Design Solutions - page 396

If the reference is available as sine and cosine functions, then there is no need to
use the Hilbert transform and low-pass filter along the reference path. In this case,
c t is replaced by sin

H
t

c

t . This latter approach leads to a loss of
absolute phase information but in many demodulation applications only the
temporal change in phase is important. Absolute phase measurement is important
in some metrology applications; for example, those involving optical
interferometry where the optical beam is split beforehand into signal and reference
beams. In fact, phase error relations similar to (7.11) and (7.12) have been derived
for optical metrology applications [5,6], and could be considered generic to phase
measurement techniques.
In practice, the difference between the pre- and post-filtering amplitudes of A
(and B ) is typically of the order of 10 -5 , the latter being the peak-to-peak ripple of
the filter, but depending on application, may be important. As such, the peak-to-
peak specification of both Hilbert and companion filters must be taken into
consideration. In the following sections, we will show typical effects using the
transformers presented in this chapter. In particular, we will look at the
demodulated phase error

ω t and by cos ω

t is constant (i.e., dc phase errors), its
dependence on filter length, carrier frequency, carrier frequency mismatch, and
similar effects when

δ ϕ when ϕ

ϕ t is sinusoidal (i.e., the ac phase errors).

7.4 DC PHASE ERRORS

7.4.1 Evolution of Carrier Frequency

The dc phase error is the error δ ϕ arising from the demodulation process when the

input phase

t is constant. In a noise-free simulation we sampled an 800 Hz carrier
signal at 4 kHz, the former corresponding to normalized carrier frequency of F =
0.4. Figure 7.4 (a) shows the demodulated phase output

ϕ

t,m using the scheme
given in Figure 7.2, when the input phase was 90 o and the filter lengths were all
155. The frequency cut-off of both filters was 0.7. The output exhibits a periodic
phase error with frequency at the second harmonic ( F = 0.8) and a peak-to-peak
error of

ϕ

10 -4 o ( rad). This is consistent with the analysis in (7.11)

and (7.13) for a small mismatch in amplitude where, although the simulated input
amplitudes were identical (i.e., A = B = 1), the pass-band ripple errors of both
filters were of the order of 10 -5 , thereby accounting for the error amplitude. In
general, the error scales with magnitude of the input phase. Figure 7.4 (b) shows
the peak- to-peak phase error plotted against input phase. Because of the linear
relationship at low input phase values, in terms of the % error, this is highest for
lowest values falling off gradually to lower % error at higher phase values. We
also find that the largest peak-to-peak dc phase error occurs when the input phase
is 90 o . As such, in characterizing demodulation performance, we opted to look at
the evolution of the peak-to-peak dc phase error at 90 o , denoted by

± 1.35 × ± 2.36 µ

δ ϕ,max against

carrier frequency F .