Stochastic Stability of Difference Equations Arising in Adaptive

EstimationBob Bitmead

Mechanical & Aerospace Engineering Department UCSD

Adaptive Estimation - where from?Multipath in Wireless Communications

Reflection and path length differences

Staggered arrival times of copies of signal

Channel impulse response

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

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%

∧∧

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

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.

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|

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.

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

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

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

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

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

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#

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

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 )

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

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

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

The End