Digital Signal Processing Reference
In-Depth Information
7.3.2
Error Propagation in Hilbert Demodulation
This section focuses on the errors associated with the Hilbert demodulation
technique (Scheme 1) and discusses how to minimize them. The inputs
s
t
and c
t
can be written as
c
=
A
sin
ω
t
t
(7.9)
s
=
B
sin(
ω
t
+
ϕ
)
t
t
After transformation and filtering, their amplitudes could be represented
respectively, for the Hilbert transform (H) and low-pass filter (LP), by
A
H
,
A
LP
,
B
H
, and
B
LP
. Thus for example,
H
t
. When these filtered signals are
substituted into (7.6) the measured time-dependent demodulated phase
c
=
A
cos
ω
t
H
ϕ
is
t
,
m
given by
(
A
B
−
A
B
)
cos
ω
t
sin(
ω
t
+
ϕ
)
+
A
B
sin
ϕ
−
1
H
LP
LP
H
t
H
LP
t
ϕ
=
tan
(7.10)
t
,
m
(
A
B
−
A
B
)
sin
ω
t
sin(
ω
t
+
ϕ
)
+
A
B
cos
ϕ
BP
LP
H
H
t
H
H
t
We are really interested in the phase error
δ
ϕ
between the ideal phase
ϕ
t
and the
measured phase
=
,
, and
recognizing that the sign of the error is determined by this convention, we have
ϕ
. Choosing to write the phase error as
δ
ϕ
−
ϕ
t
,
m
ϕ
t
m
t
a
−
(
−
b
+
c
)
tan
ϕ
−
1
t
δ
ϕ
=
tan
(7.11)
2
1
+
c
+
a
tan
ϕ
+
b
tan
ϕ
t
t
where
(
A
B
−
A
B
)
H
LP
LP
H
a
=
cos
ω
t
sin(
ω
t
+
ϕ
)
sec
ϕ
t
t
A
B
H
H
B
LP
b
=
(7.12)
B
H
(
A
B
−
A
B
)
LP
LP
H
H
c
=
sin
ω
t
sin(
ω
t
+
ϕ
)
sec
ϕ
t
t
A
B
H
H
From (7.11), and considering only the amplitudes of the time-dependent signals in
(7.12), we observe that if the phase error
δ
ϕ
is to be zero, then the numerator in
(7.11) must necessarily be zero. This means that
a
=
c
= 0, and
b
= 1. By further
implication,
δ
ϕ
is zero only if
