16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 2
Lecture 15
Last time: Compute the spectrum and integrate to get the mean squared value
2
1
() ( ) ()
2
j
xx
j
yFsFsSsd
jπ
∞
?∞
=?
∫
Cauchy-Residue Theorem
( ) 2 (residue at enclosed poles)F s ds jπ=
∑
∫ >
Note that in the case of repeated roots of the denominator, a pole of multiple
order contributes only a single residue.
To evaluate ()
j
j
Fsds
∞
?∞
∫
by integrating around a closed contour enclosing the
entire left half plane, note that if () 0Fs→ faster than
1
s
for large s , the integral
along the curved part of the contour is zero.
If ()~
n
k
Fs
s
as s →∞,
(1)
semi-circle
() 0 as if 1
n
n
k
Fsds R k R R n
R
ππ
??
≤ =→→∞>
∫ >
Integral tables
Applicable to rational functions; no predictor or smoother. Must factor the
spectrum of the input into the following form.
1()()
2()()
j
n
j
csc s
I ds
jdsdsπ
∞
?∞
?
=
?
∫
Refer to the handout “Tabulated Values of the Integral Form”.
Roots of ()cs and ()ds in left half plane only. Should check the stability of the
solution.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 2 of 2
Application to the problem of System Identification
Record ()x t and ()yt and process that data.
[]
111 1 11
00
11111
00
() ()() ()()
() ()()()()
()()( ) ()()( )
xd
xy
xd
ytwxtd wdtd
R Extyt xtyt
wxtytdwxtdtd
τττ τττ
τττ
τ τττ τ τττ
∞∞
∞∞
=?+?
=+=+
=+?++?
∫∫
∫∫
If (), ()x tdtare independent and at least ()x t is zero mean, then () 0
xd
R τ = .
111
0
() ( ) ( )
xy x xx
R wR dτ ττττ
∞
=?
∫
If ()x t is wide band relative to the system, approximate it as white.
() ()
xx x
RSτ δτ=
111
0
() ( ) ( )
()
xy x x
xx
R wS d
Sw
τ τδτττ
τ
∞
=?
=
∫