16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 8
Lecture 13
Last time:
()
()
22
2
1 ,
()
,
a
xx
aT
TR
τ
στ
τ
τ
?
??
+? ≤
? ??
=
? ??
?
>
?
()
22
22
0
2
2
2
2
22
() 1
2()21cos
2
2() 1cos
sin
2
2()
2
T
jj
xx a
T
T
a
a
a
S aed ed
T
ad
T
aT
T
T
aT
T
ωτ ωτ
τ
ω τσ τ
τ
π δω σ ωττ
σ
πδω ω
ω
ω
πδω σ
ω
∞
??
?∞ ?
??
=+?
??
??
??
=+?
??
??
=+?
??
??
????
??
=+
??
??
??
??
∫∫
∫
Amplitude of
xx
S falls off, but not very rapidly.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 2 of 8
Use error between early and late indicator to lock onto signal. Error is a linear
function of shift, within the range (,)TT? .
Return to the 1
st
example process and take the case where the change points are
Poisson distributed.
2
22
2
()
a
xx
S
λσ
ω
ω λ
=
+
Take the limit of this as
2
a
σ and λ become large in a particular relation: to
establish the desired relation, replace
22
2
22
2
()
()
aa
a
xx
k
k
kk
S
k
σσ
λλ
λ σ
ω
ω λ
→
→
=
+
and take the limit as k →∞.
()
22
2
2
2
lim ( ) lim
2
a
xx
kk
a
k
S
k
λσ
ω
λ
σ
λ
→∞ →∞
=
=
Note this is independent of frequency.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 3 of 8
This is defined to be a “white noise” by analogy with white light, which is
supposed to have equal participation by all wavelengths.
Can shape ()x t to the correct spectrum so that it can be analyzed in this manner,
by adding a shaping filter in the state-space formulation.
Definition of a white noise process
White means constant spectral density.
0
( ) , constant
xx
SSω =
0
0
1
()
2
()
j
xx
R Se d
S
τω
τ ω
π
δτ
∞
?∞
=
=
∫
White noise processes only have a defined power density. The variance of a
white noise process is not defined.
If you start with almost any process ()x t ,
lim ( )
a
ax at
→∞
? is a white noise
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 4 of 8
So far we have looked only at the
xx
R and
xx
S of processes. We shall find that if
we wish to determine only the
yy
R or
yy
S (thus the
2
y ) of outputs of linear
systems, all we need to know about the inputs are their
xx
R or
xx
S .
But what if we wanted to know the probability that the error in a dynamic
system would exceed some bound? For this we need the first probability density
function of the system error – an output. Very difficult in general.
The pdf of the output ()yt satisfies the Fokker-Planck partial differential
equation – also called the Kolmagorov forward equation. Applies to a
continuous dynamic system driven by a white noise process.
One case is easy: Gaussian process into a linear system, output is Gaussian.
Gaussian processes are defined by the property that probability density functions
of all order are normal functions.
11 2 2
( , ; , ;... , ) -dimensional normal
nnn
fxtxt xt n=
1
1
2
2
1
()
(2 )
T
xM x
n n
fx e
Mπ
?
?
=
M is the covariance matrix for
1
2
()
()
()
n
x t
x t
x
x t
??
??
??
=
??
??
??
M
Thus, ()
n
f x for all n is determined by M - the covariance matrix for x .
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 5 of 8
()( )
(, )
ij i j
xx i j
M xt xt
R tt
=
=
Thus for a Gaussian process, the autocorrelation function completely defines all
the statistical properties of the process since it defines the probability density
functions of all order. This means:
If
( )
(, )
yx i j xx j i
R tt R t t=?, the process is stationary.
If two processes (), ()x tyt are jointly Gaussian, and are uncorrelated
()
(, ) 0
xy i j
Rtt= , they are independent processes.
Most important: Gaussian input ? linear system ? Gaussian output. In this
case all the statistical properties of the output are determined by the correlation
function of the output – for which we shall require only the correlation functions
for the inputs.
Upcoming lectures will not cover several sections that deal with:
Narrow band Gaussian processes
Fast Fourier Transform
Pseudorandom binary coded signals
These are important topics for your general knowledge.
Characteristics of Linear Systems
Definition of linear system:
If
11
22
() ()
() ()
ut yt
ut yt
→
→
Then
12 12
() () () ()au t bu t ay t by t+→+
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 6 of 8
By characterizing the response to a standard input, the response to any input can
be constructed by superposing responses using the system’s linearity.
(, )wtτ is the weighting function.
0
() lim (, ) ( ) (, ) ( )
i
t
iii
i
yt wt u wt u d
τ
τ ττ τττ
?→
?∞
=?→
∑
∫
The central limit theorem says ()yt→normal if ()ut is white noise.
Stable if every bounded input gives rise to a bounded output.
( , const. bounded for all wt d tττ
∞
?∞
=<∞?
∫
Realizable if causal.
()(, ) 0, wt tτ τ=<
State Space – An alternate characterization for a linear differential system
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 7 of 8
If the input and output are related by an nth order linear differential equation,
once can also relate input to output by a set of n linear first order differential
equations.
() () () () ()
() () ()
x tAtxtBtut
yt Ctxt
=+
=
&
The solution form is:
0
00
() () ()
() (, )( ) (,) ()()
t
t
yt Ctxt
x tttxt tBudτ τττ
=
=Φ + Φ
∫
where (, )t τΦ satisfies:
( , ) ( ) ( , ), ( , )
d
tAtt I
dt
τττΦ=Φ Φ=
Note that any system which can be cast in this form is not only mathematically
realizable but practically realizable as well. Must add a gain times u to y to get
as many zeroes as poles.
For comparison with the weighting function description, take u and y to be
scalars, and take
0
t =?∞. For stable systems, the transition from ?∞ to any finite
time is zero.
Specialize the state space model to single-input, single-output (SISO) and
0
t →?∞:
(, ) 0
() () ()
() (, ) ( ) ( )
() () (, ) ( ) ( )
(, ) () (, ) ( )
T
t
t
T
T
t
yt Ct xt
xt tbud
yt Ct t b u d
wt Ct t b
ττττ
τ τττ
τττ
?∞
?∞
Φ?∞=
=
=Φ
=Φ
=Φ
∫
∫
which we recognize by comparison with the earlier expression for ()yt.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 8 of 8
For an invariant system, the shape of (, )wtτ is independent of the time the input
was applied; the output depends only on the elapsed time since the application
of the input.
( , ) ( ) ( ), 0 by conventionwt wt wtτ ττ→?= =
Stability:
() () const.wt d wt dtττ
∞∞
?∞ ?∞
?= =<∞
∫∫
Note this implies ()wt is Fourier transformable.
Realizability:
( ) 0, ( )
( ) 0, ( 0)
wt t
wt t
τ τ?= <
=<