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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
May 3, 2006 Yasamin Mostofi, Caltech 2
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
May 3, 2006 Yasamin Mostofi, Caltech 3
SensorSensor SensorSensorSensorSensor
Environment/System
Data Fusion/Processing
Decision &Control Actuator
Network
SensorSensorLocal ProcessingSensor
Networked Sensing, Estimation & Control
Today: Modeling and impact of wireless links
May 3, 2006 Yasamin Mostofi, Caltech 4
System Model
Dynamical System Observer Wireless
Channel Receiver)(kx )(ky )(ˆ kx
Estimator
#1
#2
Trans-mitter
node#1node#2
May 3, 2006 Yasamin Mostofi, Caltech 5
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
May 3, 2006 Yasamin Mostofi, Caltech 6
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
May 3, 2006 Yasamin Mostofi, Caltech 7
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)
May 3, 2006 Yasamin Mostofi, Caltech 8
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
May 3, 2006 Yasamin Mostofi, Caltech 9
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
May 3, 2006 Yasamin Mostofi, Caltech 10
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
May 3, 2006 Yasamin Mostofi, Caltech 11
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
May 3, 2006 Yasamin Mostofi, Caltech 12
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)
May 3, 2006 Yasamin Mostofi, Caltech 13
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 & ,
May 3, 2006 Yasamin Mostofi, Caltech 14
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
May 3, 2006 Yasamin Mostofi, Caltech 15
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
May 3, 2006 Yasamin Mostofi, Caltech 16
Performance Evaluation (cont.)
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 ∑
=+ =
May 3, 2006 Yasamin Mostofi, Caltech 17
Performance Evaluation (cont.)
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
May 3, 2006 Yasamin Mostofi, Caltech 18
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
May 3, 2006 Yasamin Mostofi, Caltech 19
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
May 3, 2006 Yasamin Mostofi, Caltech 21
dBSNRthresh 40=
dBSNRthresh 30=
dBSNRthresh 20=
dBSNRthresh 10=
dBSNRthresh 0=
KeepingAll packets
20P
dB in SNR
Performance Evaluation (cont.)
May 3, 2006 Yasamin Mostofi, Caltech 22
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
May 3, 2006 Yasamin Mostofi, Caltech 23
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
May 3, 2006 Yasamin Mostofi, Caltech 24
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
May 3, 2006 Yasamin Mostofi, Caltech 25
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
May 3, 2006 Yasamin Mostofi, Caltech 26
Abstraction in the Higher Layer
:
Function of TX/RX technologies like quantization, noise figure,modulation, channel coding, ..
1quantization
noise
)(2 knσ )(kpdrop
)()(2 kpk dropn & σ Distribution of :
Function of environment
kψ
kψkψ
May 3, 2006 Yasamin Mostofi, Caltech 27
:
Function of TX/RX technologies like quantization, noise figure,modulation, channel coding, ..
1quantization
noise
)(2 knσ )(kpdrop
)()(2 kpk dropn & σ
Threshψ
Approximate
Abstraction in the Higher Layer
Distribution of :
Function of environment
kψ
kψkψ
May 3, 2006 Yasamin Mostofi, Caltech 28
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:
May 3, 2006 Yasamin Mostofi, Caltech 29
Next Class
• We will complete the table next time for general communication noise variance and Signal to Noise Ratio distribution