Digital Signal Processing Reference
In-Depth Information
s
t
LPF
H
-
1
−
s
t
c
t
s
t
Q
H
t
s
H
t
c
HTF
s
t
H
H
t
s
c
−
s
c
−
1
t
t
t
ϕ
=
tan
t
H
H
t
s
c
+
s
c
t
t
t
c
t
s
t
c
LPF
t
I
c
t
H
t
c
HTF
H
t
c
H
s
t
Figure 7.2 S
cheme for the demodulation process using the digital Hilbert transform. LPF = compan-
ion low-pass filter and HTF = Hilbert transforming filter.
The complex phase
ϕ
t
in (7.5) is the angle traced out in time between the in-phase
component
I
and its quadrature
Q
. Clearly, the time-dependent demodulated phase
is
H
t
H
t
s
c
−
s
c
−
1
t
t
ϕ
=
tan
(7.7)
t
H
t
H
t
s
c
+
s
c
t
t
It is worth pointing out that the use of c
t
and its transform is usually not necessary,
since sine and cosine functions could be used instead. However, to avoid the
introduction of an arbitrary phase lag between filtered and unfiltered signal
components the above procedure is advisable. The drawback with this approach is
that the processing may become intensive, but if our aim is to track absolute phase
changes, then the extra processing may be justified. This aspect will be taken up
further in the next section. Moreover, the reasons for using the low-pass filter will
become evident as we look at the implementation and error propagation model.
7.3.1
Implementation
The implementation of the Hilbert transformer is almost identical to that used for
first-order differentiators. Given an input data sequence
x
j
sampled at interval
T
,
the Hilbert transforming filter
g
will produce an output
y
given by
L
1
2
∑
=
y
=
g
(
x
−
x
)
(7.8)
L
j
j
+
k
−
1
L
−
j
+
k
+
1
+
k
K
2
j
1
