stochastic stability of difference equations arising in...
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Stochastic Stability of Difference Equations Arising in Adaptive
EstimationBob Bitmead
Mechanical & Aerospace Engineering Department UCSD
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Adaptive Estimation - where from?Multipath in Wireless Communications
Reflection and path length differences
Staggered arrival times of copies of signal
Channel impulse response
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Mobile Wireless CommsCell phone, Wi-Fi, Wi-bro, MIMO, WiMAX
binary input signal/datareceived datachannel responsechannel noise
Linear regression
uk
yk
vk
hk
yk = h0uk + h1uk!1 + h2uk!2 + . . . + hnuk!n + vk
= (uk uk!1 . . . uk!n )
!
"
"
#
h0
h1
...hn
$
%
%
&
+ vk
"= !T
k " + vk
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Channel Equalization
Channel inserts inter-symbol interference ISIError-free reception requires knowing θ
Need to estimate from received training dataTraining data – binary data sequences known to both receiver and transmitter
Received signal from training data yields θUse θ to deconvolve/remove ISI
yk = !Tk " + vk
GSM phone frame protocol58 data bits 58 data bits
26 training bits 18%
∧∧
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Training in Mobile TelephonyNeed to accommodate channel equalization
Need to track changes in mobile channelRe-training every 5 ms per caller both ways
Convergence speed is an issue
yk = !Tk " + vk
yk|k!1 = !Tk "k!1
"k = "k!1 + µ!k
!
yk ! yk|k!1
"
"k = " ! "k|k!1
= "k!1 ! µ!k
#
!Tk " ! !T
k "k!1 + vk
$
!k =!
I ! µ"k"Tk
"
!k!1 + µ"kvk
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Vector Stochastic Difference Equations
Evolution of the channel impulse response errorRequirement for exponential convergence
Regressors Φk are (pseudo)-random
Need to consider random matrix products
and exponential convergence (a.s., pth-mean, etc)
!k =
!
k"
i=1
#
I ! µ"i"Ti
$
%
!0 + µ
k&
i=1
'
(
)
*
+
k"
j=i+1
#
I ! µ"j"Tj
$
,
-"ivi
.
/
0
!k =!
I ! µ"k"Tk
"
!k!1 + µ"kvk
Xk = AkXk!1, E|X0| < ! Xk =
!
k"
i=1
Ai
#
X0
limk!"
!k|Xk| ! 0, ! > 1, a.s.
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TheoremA necessary & sufficient condition for xk to be a.s.
exponentially convergent to zero is α=E(ln|ak|)<0
P-th Mean ExampleConsider i.i.d. {ak}
Something Simple - Scalarsxk = akxk!1, {ak} ergodic scalar sequencexk = akak!1ak!2 . . . a2a1x0, E|x0| < !
ak =
!
4, with probability 0.50, with probability 0.5
E|xk| = 2kE|x0|
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Almost sure exponential convergence of products does not imply p-th mean exp convergenceA cute result: p-th mean implies a.s.
Suppose we have p-th mean exp convergence
Markov’s inequality
by the Borel-Cantelli Lemma
First Scalar Results
E|!kxk|
p ! 0 as k ! "
P(!kxk > ") <
E!
!!kxk
!
!
p
"p
exp
! 0!!
k=1
P(!kxk > ") < !
!kxk ! 0 a.s.
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Another Cute Scalar ResultFor i.i.d. {ak} almost sure exp convergence implies
p-th mean exp convergence for p small enough
As p→0 we have
So
E|ak|p
=
!!
|ak|pdP
d
dpE|ak|
p=
!!
d
dp|ak|
p dP =
!!
ln|ak| |ak|p dP
E|ak|p ! 1 and
d
dpE|ak|
p !
!!
ln|ak| dP < 0
E|ak|p
< 1 for some p
E|xk|p = [E|ai|
p]kE|x0|p exp
! 0
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We have considered scalar difference equations with ergodic coefficients {ak}
Almost sure exponential convergence of products depends only on distribution function
No dependence on correlation propertiesP-th mean exp convergence for any p implies a.s. exp convergence
A.s. exp convergence implies p-th mean for some p provided we introduce i.i.d. assumption
Do we need to worry about dependence?
Time Out
xk = akxk!1 = akak!1ak!2 . . . a2a1x0
E[ln |ak|] < 0
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Another scalar example
P-th mean exponential convergence depends on ε i.i.d. case is exp convergent in 0.25-mean
Destabilized for ε<0.29
ak =
!
4, with probability 0.50, with probability 0.5
Extend previous i.i.d. exampleSame distribution
xk =
!
4k, with probability
(1!!)k!1
2
0, with probability 1 !
(1!!)k!1
2
ak =
!
ak!1, with probability 1 ! !
not(ak!1), with probability !
Now an ergodic Markov process
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Dependence matters
What about the vector/matrix case?
Immediately much harder to get resultsTheorem (i.i.d. case)
The difference equation is a.s. exp convergent to 0 if
Even in the scalar case, the time-dependence enters the picture; not for a.s. exponential convergence but for p-th mean exponential convergence
Xk = AkXk!1 = AkAk!1Ak!2 . . . A2A1X0
!maxE(ATk Ak) < 1
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Matrix results
Dependent ergodic matrix casesProposition:
The a.s. exp convergence in the matrix case is determined by the ergodic distribution and the time dependence
Proof by revealing examples in ℜ2
!maxE(ATk Ak) < 1i.i.d. case if but not only if
Proved using Martingale Convergence Theorem
E!
XTk Xk|Fk!1
"
= E!
Xk!1ATk AkXk!1|Fk!1
"
! !maxE(ATk Ak)XT
k!1Xk!1 < XTk!1Xk!1
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Dependent ergodic matrices
Multiplication of Xk by AkAmplification of component along ψk by a
Amplification orthogonal to ψk by factor b
Dependence of Ak managed by scalar ψk dependence
ψk uniformly distributed on [-π,π]
Dependence introduced via Markovian ψk
Ak = a
!
cos !k
sin!k
"
( cos !k sin!k ) + b
!
! sin !k
cos !k
"
(! sin !k cos !k )
!k ! ["", "]
!k ! ["
2" #,
"
2+ #]
!k = !k!1 + "k mod 2#
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Dependence and a.s. convergence
Exponential a.s. convergence conditionsslowfasti.i.d.
Ak = a
!
cos !k
sin!k
"
( cos !k sin!k ) + b
!
! sin !k
cos !k
"
(! sin !k cos !k )
!k ! U ["", "]
!k ! U ["
2" #,
"
2+ #]
!k ! U ["", "]
i.i.d. case
slow dependence
fast dependence
!k = !k!1 + "k mod 2#, "k i.i.d.
max(a2, b2) < 1
a2b2
< 1
a2
< !
b2
2+
!
b2
2+ 2
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Back to Mobile PhonesMobile channel equalization requires fast convergence of channel estimator
Care is needed in choosing training signalsConstraints are introduced by delay structure
Never i.i.d.
Decision between size of μ and correlation of {uk}
usually PRBN sequence for{uk}
Ak = I ! µ!k!Tk
!Tk = (uk uk!1 uk!2 . . . uk!n )
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Conclusion
Adaptive estimation forms a central part of mobile communications
Some care is needed to understand stochastic stability and exponential convergence
This is affected seriously by dependenceScalar examples mask this effect
Averaging methods prove of value in designRely on separation of time scales
The math is important for understanding limitations
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Multi-Input Multi-Output (MIMO) channelsc.f. the 802.11g/n standardsn transmit antennæ and m receive antennæ
An nxm dynamic channel modelWhat is the best model class;
Finite Impulse Response (FIR)?State-space (IIR)? Parsimony?
What is the best training signal?Experiment design (1920-)
How much training data needs to be sent?nxmx26 bits per 142-bit frame?
Current problems
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Finite-sample performanceAll of the results presented are asymptotic
Convergence, rates, a.s., p-th mean26-bit messages in 142-bit frames are very finite
How does life change?Mixed determinism and probabilityClever methods of using priorsNeed to admit biased estimatorsWillingness to permit failure
Dropped calls ... big dealThey’ll call back
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The End