Digital Signal Processing Reference
In-Depth Information
applying a gradient descent algorithm on p. Denoting the sensitivity function as
de
k
dp
¼
dx
k
dp
;
s
k
1
¼
ð
3
:
35
Þ
it can easily be shown that s
k
satisfies the recurrence relation
s
k
¼
rps
k
1
r
2
s
k
2
þ
rx
k
:
ð
3
:
36
Þ
In addition, if
v
k
can be computed recursively
using an exponentially weighted fading function using the recurrence relation
v
k
represents the mean of J
k
ð
p
Þ
, then
v
k
¼
v
k
1
þð
1
Þ
e
k
;
ð
3
:
27
Þ
where
;
0
<<
1, is the forgetting factor. Denoting
^
p
k
as the estimate of p at
time index k, we compute
^
p
k
recursively as
2e
k
s
k
1
v
k
^
p
k
¼ ^
p
k
1
;
ð
3
:
38
Þ
is the step size. The instantaneous normalised frequency estimate f
k
is
obtained from
where
^
p
k
by the formula
:
1
2
p
k
2
f
k
¼
cos
1
ð
3
:
39
Þ
The algorithm for obtaining the sequence
f
f
k
g
of the estimates of the instantaneous
normalised frequency from the sampled signal sequence
f
u
k
g
is as follows.
Choose parameters r
;;
and initial values x
0
;
x
1
;
u
0
;
u
1
;
s
0
;
s
1
;
s
2
;
v
0
;
p
0
such that
v
0
>
0.
For k
¼
1
;
2
;
3
; ...
ð
1
r
2
Þ
2
ð
u
k
u
k
2
Þ
p
k
1
x
k
1
r
2
x
k
2
þ
1. x
k
¼
r
^
p
k
1
s
k
1
r
2
s
k
2
þ
rx
k
3. e
k
¼
x
k
u
k
4.
2. s
k
¼
r
^
v
k
¼
v
k
1
þð
1
Þ
e
k
2e
k
s
k
1
v
k
5.
^
p
k
¼ ^
p
k
1
1
2
^
p
k
2
f
k
¼
cos
1
6.
By smoothing the sequence
f
f
k
g
and then performing a hard decision on the
smoothed sequence, we obtain the demodulated signal. We have chosen to pass the
signal
f
^
p
k
g
through a hard limiter to reduce the computational burden, which is a
requirement in real-time embedded system software development.
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