Digital Signal Processing Reference
In-Depth Information
c
t
¼ð
10
Þ
. Then the noisy measured signal will be
We also define a vector
y
k
¼ c
t
x
k
1
þ
n
k
. Now let the unknown initial condition vector be
x
0
. Then we
have
0
@
1
A
0
@
1
A
;
0
@
1
A
x
0
þ
y
1
y
2
.
y
N
t
c
n
1
n
2
.
n
N
t
2
c
¼
.
ð
3
:
60
Þ
t
N
c
which is of the form
. Now we know that the least-squares solution for
the initial conditions of the oscillator is
y ¼ Ax
0
þ n
AÞ
1
t
t
.
We do not need to do these computational gyrations. Fortunately, we can use a
fine numerically efficient recursive method known as a Kalman filter. The following
set of equations provide a recursive solution and update the desired estimate of the
signal
x
0
¼ðA
A
y
^
y
k
on receipt of new data:
x ¼ x
k
;
e
k
¼ c
t
x
y
k
;
P ¼ P
k
;
t
1
þ c
Pc
ð
3
:
61
Þ
x
k
þ
1
¼ x
t
e
k
;
Pc
c
Pc
t
P
P
k
þ
1
¼
P
t
;
1
þ c
Pc
t
y
k
¼ c
x
k
þ
1
:
is the desired filtered signal
We recommend you to look at the above as a numerical method for implementing
Þ
1
t
k
t
t
y
k
¼ c
ð
A
A
A
y:
ð
3
:
62
Þ
P
0
¼
10
3
The choice of
I
is good and we can start with any arbitrary value of
x
k
.
Programs given in the appendix will throw more light on this.
3.12.3 Sampling in Space
Consider an antenna consisting of the uniform linear array (ULA) shown in
Figure 3.35. We assume a radiating narrowband source transmitting in all direc-
tions. The signal as seen by the ULA at a given instant of time is a sine wave or part
of it, depending on the interelement spacing. For this reason it is called sampling in
space. The set of signals measured by the ULA is noisy and given as y
n
¼
x
n
þ
n
,
where
n
is a zero-mean uncorrelated random sequence.
This example aims at obtaining a set of coefficients for a pair of FIR filters which
satisfy the property that the output of the filter pair acts as a state estimate for a
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