CDS 270-2: Lecture 6-2 Impact of Communication Noise on Estimation

May 3, 2006 Yasamin Mostofi, Caltech 1

CDS 270-2: Lecture 6-2Impact of Communication Noise

on Estimation over Wireless Links

Yasamin MostofiMay 3, 2006

Goals: To understand the impact of noisy wireless

communication links on estimation over wirelessTo evaluate the performance of Kalman filtering over

noisy mobile links


Reading for May 3th and 5th

Can also be found athttp://www.cds.caltech.edu/~yasi/

• Y. Mostofi and R. Murray, “Receiver Design Principles for Estimation over Fading Channels," Proceedings of Conference on Decision andControl (CDC), December 2005

• Y. Mostofi and R. Murray, “On Dropping Noisy Packets in KalmanFiltering over a Wireless Fading Channel”, Proceedings of American Control Conference (ACC), June 2005

• Y. Mostofi and R. Murray, "Effect of Time-Varying Fading Channels on the Control Performance of a Mobile Sensor Node," Proceedings ofIEEE 1st International Conference on Sensor and Ad Hoc Communications and Networks (Secon), October 2004, Santa Clara, CA 05


SensorSensor SensorSensorSensorSensor

Environment/System

Data Fusion/Processing

Decision &Control Actuator

Network

SensorSensorLocal ProcessingSensor

Networked Sensing, Estimation & Control

Today: Modeling and impact of wireless links


System Model

Dynamical System Observer Wireless

Channel Receiver)(kx )(ky )(ˆ kx

Estimator

#1

#2

Trans-mitter

node#1node#2


System Model

)()()1( kwkAxkx +=+

)()()( kvkCxky +=

Linear dynamical system:

Observation:

QR

:)(kw :)(kv

Zero mean noise with variance of

Zero mean noise with variance of

:)(ˆ kx Kalman filter estimate of )(kx

To focus on communication noise, assume scalar quantities

Dynamical System Observer Wireless

Channel Receiver)(kx )(ky )(ˆ kx

EstimatorTrans-mitter

)(ˆ ky


Wireless Transmission

quantiz-ation

D/A &modulation

wirelesschannel

)(ky demodulation

channelcoding/processing

source decoding

error detection

)(ˆ kyA/D

channel decoding/processing

+

receiver noise

packet drop

Yes

No

Past 4 classes: Studied cases where only noise-free samples came through


Error Detection & Packet Drop• Drop criteria changes depending on the

application• Voice applications are delay-sensitive:

– Calls are dropped only if crucial bits are corrupted– There is error detection for crucial bits– The rest of the error is either corrected or tolerated

• Data applications are not delay-sensitive:– Packets are only kept if no error is detected

• What is optimum for control over wireless? (we will get to this later)


Wireless Communication• Impairments:

– Signal attenuation– Multipath, fading &

shadowing– Time-varying links– Limited bandwidth– Collision

• One measure of link quality: – Received Signal to

Noise Ratio = Ratio of received signal power to receiver noise power


Communication Noise• Impairments result in noisy reception:

• .

• is a function of received Signal to Noise Ratio• Communication noise appears as additional observation

noise• This model provides the right abstraction for estimation

and control

)()()()()()(ˆ knkvkCxknkyky ++=+=

)()(

2 kkn

nσ of variance withnoise ioncommunicat is

)(2 knσ


Communication Noise Variance

:

Function of TX/RX technologies like quantization, noise figure,modulation, channel coding, ..

)(2 knσ

Distribution of :

Function of environmentCommon outdoor model:

– exponential distribution

kψ

quantizationnoise

)(2 knσ

kψ

= communication noise variance

ψk: Received Signal to Noise Ratio at kth transmission


Can the Estimator Know the Variance ofCommunication Noise?

• KF relies on using covariance of the observation noise

• In order for the estimator to know the quality of the communication link, a cross-layer information path is needed

• In general such paths can improve the performance considerably and have gotten considerable interest recently

• However, one has to be cautious since careless use of such paths can ruin the robustness of the system

Application Layer:Estimation &

Control

TCP/IP

Physical Layer


Design Choices

• Estimation & control over wireless links are new applications– Need new design paradigms

• Possible design choices:– Receiver can keep all the samples– Receiver can optimize the packet drop– Cross-layer: estimator can use link quality information

• Consider unstable processes. We are interested in: – Analytical expressions to evaluate the performance of

Kalman filtering over wireless noisy links– Optimizing packet drop (Friday lecture)– Stability condition (Friday lecture)


Performance Evaluation Example

• Consider the following example:– Receiver that keeps all the packets– KF that uses knowledge of channel quality (trust

coefficient)– One class of channels:

– Exponential distributed– Channel gets uncorrelated from one sample to next– In order to focus on communication impact, assume

the following for this example:

0)(2 >= βψβσ for

kn k

kψ

100 === CQR & ,


Performance Evaluation

ikkkkik

kk

kk

PP

PPAP

−−++

+ ×+×

=

ψψψ

ψββ

,.....,, 11,1

2

1

over of average :

dtt

ezzez

PAd

PePAP

PPAEP

z

tz

kk

kkkk

kk

kk

k

∫

∫∞ −

∞ Γ−

+

==Π

ΓΠ×Γ=

+Γ=

+=

)),)(

)()|( 2

0

22

0,1

Expint( whereExpint( with

ββψψβ

βψββ ψ

kP finding in interested are We

[ ]01 ,...,|

2)(ˆ)( ψψ −−=

kkxkxPk

kψ1

=Γ


Performance Evaluation (cont.)

11

)(

1

)())((

:10

1

22

2

2 ≥−

ΓΠ

−−

×Γ

Π=

×+Γ

Π

>>

=Γ

−− iAq

AAq

Aqq

Aq

ii

i

i

and arbitrary

an for following the have will We.

with dist.lly exponentia an Consider 1: Lemma

ββψβ

ψ

ψ

))(()(1

21122

0,1−

−−+

+ΓΠ×Γ=

ΓΠ×Γ=

k

kk

kk PA

PAP

AP ψββββ



1: Lemma Using

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

ΓΠ

−−

ΓΠ

Γ= −−

−−

+ 21

21

22

1,1 1

)(

1

)(

AP

APAAP kk

k

ββ

β

..

),(

,2

0,0

20

,,1

kmz

mkz

m

zmzmk

BAB

PABP

=

−=+

Γ=

ΓΠ= ∑

find to is goal The where

:Similarly

β

β

Let ).(),( 2mk

zk PAmzT

−

ΓΠ=

β Then ).,(

0,,1 mzTBP k

m

zmzmk ∑

=+ =



ii

m

ikmi

k

m

z z

mzk

m

i i

mi

k

m

z z

mzk

m

z z

mzmk

mkmk

mkk

m

zmzmk

AmiTB

mTB

miTB

mTB

mzTB

P

kmP

PmzTBP

2

1

01,

0 1

,1

1

,1

0 1

,

0 1

,1,1

11

0,,1

11)1,(

)1,0()1,(

)1,0()1,1(

:11

).,(

−=+=

+⎥⎦

⎤⎢⎣

⎡−+=

+−++=

−≤≤−

=

∑

∑∑

∑∑

∑

+

=+

= +

+

=

−

= += +++

−−−−

−=

+

ξ

ξξ

ξξ

ψ

where

1) Lemma (using for following the in result will over averaging and of

function a as ngSubstituti


Performance Evaluation • Finally

0

0

1,1

,

1

0 1

1,

≠

=

=−

−−

−

= +

−∑

iB

B

iB

i

ki

ki

k

z z

kz

ξ

ξ

)(0

20

,10

2

PAeBP i

k

i

PAkik

i ββ

Γ= ∑

=

Γ

+ Expint

ii A

AB

2

20,0

11

1

−=

Γ=

=Γ

ξ

ψ

ki ≤≤0


Stability Condition

• will be bounded as long as

1)( 002

10

0 =<⇒Γ

> + µµµβ µ Expint where if eAPPP kkk

0≠ψkP

( )( )

)(

,

/2

2

11

1

2

1

k

P

kk

kkPk

kkkk

kk

PeA

PPAEPEEP

PPPPAP

k

kkk

ββ

ψββ

ψββ

β

ψψ

Γ×Γ=

⎟⎟⎠

⎞⎜⎜⎝

⎛

+×

≤=

+×

=

Γ

++

++

Expint

:inequality sJensen' Using

. of function concave a is :Proof

May 3, 2006 Yasamin Mostofi, Caltech 20k

Performance Evaluation

kP

1.4

==

βA


dBSNRthresh 40=

dBSNRthresh 30=

dBSNRthresh 20=

dBSNRthresh 10=

dBSNRthresh 0=

KeepingAll packets

20P

dB in SNR



Summary (so far)

• We looked at a receiver that keeps all the packets and uses a cross-layer information path

• We derived analytical expression to evaluate performance

• We showed that the design is always stable


Possible Extensions

• Derive average estimation error variance for: – Other communication noise variances– General noise variance– General Signal to Noise Ratio distribution– Vector case

• Derive expressions for other moments of estimation error variance


Wireless Transmission

quatiz-ation

D/A &modulation

wirelesschannel

)(ky demodulation

channelcoding/processing

dequan-tization

error detection

)(ˆ kyA/D

decoding/processing

+

receiver noise

packet drop

Yes

No

)()()(ˆ knkyky +=

)()(

2 kkn

nσ of variance withnoise ioncommunicat is


Optimum Design• What is the optimum packet drop for estimation

and control over wireless links?• What are the benefits of using channel

knowledge in the estimator?– Stability & performance

• Consider general cases: general ψ and σn2 and

system parameters• Ideal noise profile: keeping only noise-free

samples– Suitable for non delay-sensitive applications


Abstraction in the Higher Layer

:


1quantization

noise

)(2 knσ )(kpdrop

)()(2 kpk dropn & σ Distribution of :

Function of environment

kψ

kψkψ


:


1quantization

noise

)(2 knσ )(kpdrop

)()(2 kpk dropn & σ

Threshψ

Approximate

Abstraction in the Higher Layer

Distribution of :

Function of environment

kψ

kψkψ


New Design Paradigms

cross-layer no cross-layer

Non-ideal noise profile

Ideal noise-profile

Scenario#3 Scenario#2

?

??

?

drop

keepall

Scenario#1Sinopoli et al. &

Liu et al.

Only keep noise-free samples

Take impact of noisy samples into account

21#,

−< Ap scenariodrop :nodes mobile-non forScenario#1: Sinopoli et al. (TAC 04), to maintain stability:


Next Class

• We will complete the table next time for general communication noise variance and Signal to Noise Ratio distribution

CDS 270-2: Lecture 6-2 Impact of Communication Noise on Estimation

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