Digital Signal Processing Reference
In-Depth Information
g
where is the full-precision Hilbert filter, is the peak value attained by the
filter, and
Q
is the percentage overshoot of the phase step response of the digital
demodulator. The peak value
g
peak
k
associated with cut-off frequency
F
c
of the
g
peak
Hilbert filter is given by
g
=
0
.
725
F
(7.44)
peak
c
x
(
k
p
)
This was found by iterating , the position of the
p
th peak of the impulse
response function after
k
iterations with starting conditions at
(
p
0
)
, where
x
2
p
−
1
0
x
=
(7.45a)
p
F
c
and
2
p
−
1
2
1
(
k
p
)
−
1
x
=
−
tan
(7.45b)
(
p
−
1
F
π
F
π
F
x
c
c
c
p
The peaks attained by the impulse response function are found by substituting the
x-
values into
2
(
k
)
π
2
sin
(
F
x
)
c
p
(
k
p
)
2
g
(
x
)
=
(7.45c)
(
k
p
)
π
x
The recursion converges somewhat slowly for the first peak at
p
= 1, but is faster
for other peaks. As such we have least squares fitted the peak values and this leads
to the linear response given in (7.44).
7.11.1 Filter Gain
Following on from the dilation process in (7.43), in order to achieve unity gain we
must reduce the filtered output by the same factor. The filter gain
G
is given by
B
−
1
2
−
1
(7.46)
G
=
512
(
+
Q
/
100
)
0
725
F
c
Thus dividing the output of the Hilbert filter by
G
produces a properly scaled
imaginary component of the input signal.
