16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Last time: Compute the spectrum and integrate to get the mean squared value y 2 =
1
2 π j j ∞
∫ ( ) ( ) xx
( )
∫ > ( ) = 2 π j
∑
(residue at enclosed poles)
Note that in the case of repeated roots of the denominator, a pole of multiple order contributes only a single residue.
To evaluate j ∞
∫ by integrating around a closed contour enclosing the entire left half plane, note that if ( ) → 0 faster than
1 s
for large s , the integral along the curved part of the contour is zero.
If ( ) ~ s k n
as s → ∞ , ∫ > semi-circle
≤ k
R n
π π − − 1) → 0 as R → ∞ if n > 1
Applicable to rational functions; no predictor or smoother. Must factor the spectrum of the input into the following form.
I n
=
2
1
π j
∫ j ∞ −
− ds
Refer to the handout “Tabulated Values of the Integral Form”.
Roots of ( ) and ( ) in left half plane only. Should check the stability of the solution.
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16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Record ( ) and ( ) and process that data.
= ∫
∞
0
− ) d
R xy
= [ w x
τ
1 x t − ) d
1
+ ∫
∞ w d
τ d t
0
( ) ( + τ )
] = x t y t + τ )
1
= ∫
∞
0 w x
τ x t y t ) d
1
+
0
∫
∞ w d
τ x t d t ) d
1
If ( ), ( ) are independent and at least ( ) is zero mean, then R xd
τ = .
R xy
= ∫
∞
0 w x
τ R xx
( ) d
1
If ( ) is wide band relative to the system, approximate it as white.
R xx
= S x
R xy
= ∫
∞
0 w x
= S w x
τ S x
δ τ τ τ
1
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