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16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 6
Lecture 19
Last time:
03
1
3
3
3
1
3
3
1
(
)
()
(
)
()0
f
o
r
0
ii
D
is
wR
d
w
R
d
ττ
τ
∞∞
−∞
−∞
−−
−=
≥
∫∫
Solution in the Free Configuration, NonRealTime Filter Case
The applicable condition in this important case is:
1
3
3
3
1
3
3
0
ii
D
is
d
w
R
d
−∞
−∞
for all
1
.
[]
1
1
0
s
ed
∞
−
−∞
=
∫
Since this function of
1
must be zero for all
1
, its transform must also be zero.
13
3
3
0
3
1
3
133
1
3
0
ss
ii
D
is
d
d
w
R
ee
d
d
w
R
τττ
−∞
−∞
−∞
−∞
0
0
() ()
() () 0
ii
is
is
ii
HsSs D
sSs
DsS s
Hs
Ss
=
with
Ds
=
desired signal transfer
is
ss
ns
ii
ss
sn
ns
nn
Ss Ss S s
Ss S s S s S s S s
=+
=+++
This is the optimum filter “transfer function.”
Since it is nonrealtime, it is
usually implemented as a weighting sequence in a digital computer.
The
continuous weighting function is
00
1
( )
2
j
st
j
wt
Hsed
s
j
π
∞
−∞
=
∫
I will give an integral form which is useful in evaluating this integral.
0
typically has poles in both left and right hand planes.
This does not imply
instability in the case of a twosided transform.
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Prof. Vander Velde
Page 2 of 6
()
0
( )
ot
i
w t
d
τ
ττ
∞
−∞
=−
∫
In terms of
s
, the integrals are written:
1.
If the real part of
a
is positive,
1
1
0
1
1!
2
00
na
t
j
st
n
j
te
t
e
n
ds
j
sa
t
π
−−
∞
−∞
⎧
>
⎪
−
=
⎨
+
⎪
<
⎩
∫
2.
If the real part of
a
is negative,
1
1
0
1
2
t
j
st
n
j
t
e
n
ds
j
t
∞
−
⎧
<
⎪
−
=
⎨
+
⎪
>
⎩
∫
In writing power density spectra, which are rational functions of
2
ω
 if rational
at all – we note that
2
is replaced by
2
s
−
.
Note that the inverse transform of
0
Hs
can be done by expanding
0
in a
partial fraction expansion
and integrating each term separately.
This expended problem is more general than the case treated in the text for two
reasons:
1.
A more general expression for the desired output is permitted
2.
Non minimum phase fixed parts of the system,
Fs
, may
be
handled
without stability problems due to cancellation of unstable modes.
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This note was uploaded on 11/07/2011 for the course AERO 16.322 taught by Professor Wallacevandervelde during the Fall '04 term at MIT.
 Fall '04
 WallaceVanderVelde

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