preliminary results

Wireless Networking and Communications Group

Department of Electrical and

Computer Engineering

Mitigation of Radio Frequency Interferencefrom the Computer Platform to Improve

Wireless Data Communication

Prof. Brian L. EvansGraduate Students Kapil Gulati and Marcel Nassar

Undergraduate Students Navid Aghasadeghi and Arvind Sujeeth

Preliminary Results

Last Updated May 31, 2007

2




Outline

I. Problem Definition

II. Noise Modeling

III. Estimation of Noise Model Parameters

IV. Filtering and Detection

V. Conclusion

VI. Future Work

3




I. Problem Definition

Within computing platforms, wireless transceivers experience radio frequency interference (RFI) from computer subsystems, esp. from clocks (harmonics) and busses

Objectives• Develop offline methods to improve communication performance in the presence of computer platform RFI• Develop adaptive online algorithms for these methods

Approach• Statistical modeling of RFI• Filtering/detection based on estimation of model parameters

We’ll be using noise and interference interchangeably

4




Common Spectral Occupancy

StandardCarrier (GHz)

Wireless Networking

Example Interfering Computer Subsystems

Bluetooth 2.4Personal Area

NetworkGigabit Ethernet, PCI Express Bus,

Memory, LCD

IEEE 802. 11 b/g/n

2.4Wireless LAN

(Wi-Fi)Gigabit Ethernet, PCI Express Bus,

Memory, LCD

IEEE 802.16e

2.5–2.69 3.3–3.8

5.725–5.85

Mobile Broadband(Wi-Max)

PCI Express Bus, Memory, LCD

IEEE 802.11a

5.2Wireless LAN

(Wi-Fi)PCI Express Bus, Memory, LCD

5




II. Noise ModelingRFI is a combination of independent radiation events, and predominantly has non-Gaussian statistics

Statistical-Physical Models (Middleton Class A, B, C)• Independent of physical conditions (universal)• Sum of independent Gaussian and Poisson interference• Models nonlinear phenomena governing electromagnetic interference

Alpha-Stable Processes• Models statistical properties of “impulsive” noise• Approximation to Middleton Class B noise

6




Class A Narrowband interference (“coherent” reception) Uniquely represented by two parameters

Class B Broadband interference (“incoherent” reception) Uniquely represented by six parameters

Class C Sum of class A and class B (approx. as class B)

[Middleton, 1999]

Middleton Class A, B, C Models

7




Middleton Class A Model

00

0 !

2)(

2

2

02

z

zezm

Ae

zwm

z

m m

mA

A

Parameters Description Range

Overlap Index. Product of average number of emissions per second and mean duration of typical emission

A [10-2, 1]

Gaussian Factor. Ratio of second-order moment of Gaussian component to that of non-Gaussian component

Γ [10-6, 1]

1

2!)(

2

2

02

2

2

Am

where

em

Aezf

m

z

m m

mA

Zm

Envelope statisticsProbability densityfunction (pdf)

Envelope for Gaussian signal has Rayleigh distribution

8




Middleton Class A Statistics

Probability Density Function

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025Class A Probability Density Function; A = 0.15, = 0.1

x

PD

F f x(x

)

Example for A = 0.15 and = 0.1

As A → , Class A pdf converges to Gaussian

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

-8

-6

-4

-2

0

2

4

6

8

10

Frequency

Pow

er S

pect

rum

Mag

nitu

de (

dB)

Power Spectal Density of Class A noise, A = 0.15, = 0.1

Power Spectral Density

9




Symmetric Alpha Stable Model

Characteristic Function: ||)( je

Parameters )20(: ,α Characteristic exponent indicative of the thickness

of the tail of impulsiveness of the noise

Localization parameter (analogous to mean)

Dispersion parameter (analogous to variance))0(:

)(: δ-

No closed-form expression for pdf except for α = 1 (Cauchy), α = 2 (Gaussian), α = 1/2 (Levy) and α = 0 (not very useful)

Approximate pdf using inverse transform of power series expansion of characteristic function

10




Probability Density Function Power Spectral Density

Symmetric Alpha Stable Statistics

Example: exponent = 1.5, “mean” = 0 and “variance” = 10

||)( je

-50 -40 -30 -20 -10 0 10 20 30 40 500

1

2

3

4

5

6

7

8x 10

-4 PDF for SS noise, = 1.5, =10, = 0

x

Pro

babi

lity

dens

ity f

unct

ion

f X(x

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

-8

-6

-4

-2

0

2

4

6

8

10

Frequency

Pow

er S

pect

rum

Mag

nitu

de (

dB)

Power Spectal Density of S S noise, = 1.5, = 10, = 0×10-4

11




III. Estimation of Noise Model Parameters

For the Middleton Class A Model• Expectation maximization (EM) [Zabin & Poor, 1991]

• Based on envelope statistics (Middleton) • Based on moments (Middleton)

For the Symmetric Alpha Stable Model• Based on extreme order statistics [Tsihrintzis & Nikias, 1996]

For the Middleton Class B Model• No closed-form estimator exists• Approximate methods based on envelope statistics or moments

12




00

0 !

2)(

2

2

02

z

zezm

Ae

zwm

z

m m

mA

2

0

2

2

2),|(;!

),|()(

j

z

j

Aj

j

jj

j

jezAzp

j

eA

Azpzw

Estimation of Middleton Class A Model Parameters

Expectation maximization• E: Calculate log-likelihood function w/ current parameter values• M: Find parameter set that maximizes log-likelihood function

EM estimator for Class A parameters [Zabin & Poor, 1991]

• Expresses envelope statistics as sum of weighted pdfs

Maximization step is iterative• Given A, maximize K (with K = A Γ). Root 2nd-order polynomial.• Given K, maximize A. Root 4th-order poly. (after approximation).

13




Results of EM Estimator for Class A Parameters

PDFs with 11 summation terms50 simulation runs per setting

Convergence criterion:Example learning curve

7

1

1 10ˆ

ˆˆ

n

nn

A

AA

1e-006 1e-005 0.0001 0.001 0.01

10

15

20

25

30

K

Num

ber

of I

tera

tions

Number of Iterations taken by the EM Estimator for A

A = 0.01

A = 0.1

A = 1

Iterations for Parameter A to Converge

1e-006 1e-005 0.0001 0.001 0.01

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

x 10-3

K

Fra

ctio

nal M

SE

= |

(A -

Aes

t) /

A |2

Fractional MSE of Estimator for A

A = 0.01

A = 0.1

A = 1

Normalized Mean-Squared Error in A×10-3

2

)(A

AAANMSE est

est

14




Estimation of Symmetric Alpha Stable Parameters

Based on extreme order statistics [Tsihrintzis & Nikias, 1996]

PDFs of max and min of sequence of independently and identically distributed (IID) data samples follow

• PDF of maximum:

• PDF of minimum:

Extreme order statistics of Symmetric Alpha Stable pdf approach Frechet’s distribution as N goes to infinity

Parameter estimators then based on simple order statistics• Advantage Fast / computationally efficient (non-iterative)• Disadvantage Requires large set of data samples (N ~ 20,000)

)( )](1[ )(

)( )( )(1

:

1:

xfxFNxf

xfxFNxf

XN

Nm

XN

NM

15




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09MSE in estimates of the Characteristic Exponent ()

Characteristic Exponent:

Mea

n S

quar

ed E

rror

(M

SE

)

Mean squared error in estimate of characteristic exponent α

Data length (N) was 10,000 samples

Results averaged over 100 simulation runs

Example on this slide (which is continued on next slide) uses = 5 and = 10

Results for Symmetric Alpha Stable Parameter Estimator

16




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9x 10

-3 MSE in estimates of the Localization Parameter ()

Characteristic Exponent: M

ean

Squ

ared

Err

or (

MS

E)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7MSE in estimates of the Dispersion Parameter ()

Characteristic Exponent:

Mea

n S

quar

ed E

rror

(M

SE

)

Results for Symmetric Alpha Stable Parameter Estimator

Mean squared error in estimate of dispersion (“variance”)

Mean squared error in estimate of localization (“mean”)

= 5 = 10

17




Results on Measured RFI Data

Data set of 80,000 samples collected using 20 GSPS scope

Measured data represents "broadband" noise

Symmetric Alpha Stable Process expected to work well since PDF of measured data is symmetric

Middleton Class A will model PDF beyond a certain point• Middleton Class B envelope PDF has same form as Middleton

Class A envelope PDF beyond an envelope value (inflection point)• we expect the envelope PDF to match closely to Middleton Class

A envelope PDF beyond the inflection point.

18




0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

-3 PDF of real data vs alpha stable

True PDF

Esimtated PDF usingalpha stable modelling

x – noise amplitude

f X(x

) -

PD

F

Estimated Parameters

Localization (δ) -0.0393

Dispersion (γ) 0.5833

Characteristic Exponent (α)

1.5525


Modeling PDF as Symmetric Alpha Stable process

19




0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3 Envelope PDF of real data vs Class A model

True PDF

Esimtated PDF Middleton Class A

Expected: Envelope PDF’s match beyond a certain envelope

Envelope computed via non-linear lowpass filtering obtained via Teager operator, z[n] = (x[n])2 – x[n-1]x[n+1]

z – noise envelope

f Z(z

) –

En

velo

pe

PD

F


Estimated Parameters

Overlap Index (AA) 0.5403

Gaussian Factor (Γ) 0.0096

Modeling envelope PDF using Middleton Class A model

20




Filter Decision RuleFiltered

signal

HypothesisCorrupted

signal

IV. Filtering and Detection

Wiener filtering (linear)• Requires knowledge of signal and noise statistics• Provides benchmark for non-linear methods

Other filtering• Adaptive noise cancellation• Nonlinear filtering

Detection in Middleton Class A and B noise• Coherent detection [Spaulding & Middleton, 1977]

• Incoherent detection [Spaulding & Middleton, 1977]

We assume perfect

estimation of noise model parameters

21




Minimize Mean-Squared Error E { |e(n)|2 }

d(n)

z(n)

d(n)^w(n)

x(n)

w(n)x(n) d(n)^

d(n)

e(n)

d(n): desired signald(n): filtered signale(n): error w(n): Wiener filter x(n): corrupted signalz(n): noise

d(n):^

Wiener Filtering – Linear Filter

Optimal in mean squared error sense when noise is Gaussian

Model

Design

22




Wiener Filtering – Finite Impulse Response (FIR) Case

Wiener-Hopf equations for FIR Wiener filter of order p-1

General solution in frequency domain

)1(

)1(

)0(

)1(

)1(

)0(

0...21

1

1...10 **

pr

r

r

pw

w

w

rprpr

r

prrr

dx

dx

dx

xxx

x

xxx

)()(

)(

)(

)(2

j

zj

d

jd

jx

jdx

ee

e

e

eje

MMSEH

desired signal: d(n)power spectrum: (e j )

correlation of d and x: rdx(n)autocorrelation of x: rx(n)Wiener FIR Filter: w(n)

corrupted signal: x(n)noise: z(n)

1 1 0 )()()(1

0

p-...,,,kkrlkrlwp

ldxx

23




Wiener Filtering – 100-tap FIR Filter

ChannelA = 0.35

= 0.5 × 10-3

SNR = -10 dBMemoryless

Pulse shape10 samples per symbol10 symbols per pulse

Raised Cosine Pulse Shape

Transmitted waveform corrupted by Class A interference

Received waveform filtered by Wiener filter

n

n

n

24




Wiener Filtering – Communication Performance

ChannelA = 0.35

= 0.5 × 10-3

Memoryless

Pulse shapeRaised cosine

10 samples per symbol10 symbols per pulse

SNR (dB)100-10-20-30-40

Bit

Err

or R

ate

(BE

R)

Optimal Detection RuleDescribed next

25




Decision Rule Λ(X) H1 or H2

corrupted signal

Coherent Detection

Hard decision

Bayesian formulation [Spaulding and Middleton, 1977]

1)|()(

)|()()(

2

1

11

22

H

H

HXpHp

HXpHpX

ZSXH

ZSXH

22

11

:

:

26




Coherent Detection

Equally probable source

Optimal detection rule

1)(

)()(

2

1

1

2

H

H

Z

Z

SXp

SXpX

1

,

2!)(

'

'

2

2/

021

22

Am

where

em

Aezp

m

z

m m

mA

N

nZ

mn

N: number of samples in vector X

1

2!

2!)(

2

1

221

222

2/

021

2/

021

H

H

sx

m m

mA

N

n

sx

m m

mA

N

n

mnn

mnn

em

Ae

em

Ae

X

27




Coherent Detection in Class A Noise with Γ = 10-4

SNR (dB) SNR (dB)

Correlation Receiver Performance

A = 0.1

28




Coherent Detection – Small Signal Approximation

Expand pdf pZ(z) by Taylor series about Sj = 0 (for j=1,2)

Optimal decision rule & threshold detector for approximation

Optimal detector for approximation is logarithmic nonlinearity followed by correlation receiver (see next slide)

ji

N

i i

Z

ZjZZjZ sx

XpXpSXpXpSXp

1

)()()()()(

1)(ln1

)(ln1

)(2

1

11

12

H

H

N

iiZ

ii

N

iiZ

ii

xpdxd

s

xpdxd

s

X

We use 100 terms of the

series expansion ford/dxi ln pZ(xi) in simulations

29




Communication performance of approximation vs. upper bound[Spaulding & Middleton, 1977, pt. I]

Correlation Receiver

Coherent Detection –Small Signal Approximation

Near-optimal for small amplitude signals

Suboptimal for higher amplitude signals

AntipodalA = 0.35 = 0.5×10-3

30




V. Conclusion

Radio frequency interference from computing platform• Affects wireless data communication subsystems• Models include Middleton noise models and alpha stable

processes

RFI cancellation• Extends range of communication systems• Reduces bit error rates

Initial RFI interference cancellation methods explored• Linear optimal filtering (Wiener)• Optimal detection rules (26 dB gain for coherent detection)

31




VI. Future Work

Offline methods• Estimator for single symmetric alpha-stable process plus Gaussian• Estimator for mixture of alpha stable processes plus Gaussian

(requires blind source separation for 1-D time series)• Estimator for Middleton Class B parameters• Quantify communication performance vs. complexity tradeoffs for

Middleton Class A detection

Online methods• Develop fixed-point (embedded) methods for parameter estimation

• Middleton noise models• Mixtures of alpha-stable processes

• Develop embedded implementations of detection methods

32




References

[1] D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: New methods and results for Class A and Class B noise models”, IEEE Trans. Info. Theory, vol. 45, no. 4, pp. 1129-1149, May 1999

[2] S. M. Zabin and H. V. Poor, “Efficient estimation of Class A noise parameters via the EM [Expectation-Maximization] algorithms”, IEEE Trans. Info. Theory, vol. 37, no. 1, pp. 60-72, Jan. 1991

[3] G. A. Tsihrintzis and C. L. Nikias, "Fast estimation of the parameters of alpha-stable impulsive interference", IEEE Trans. Signal Proc., vol. 44, Issue 6, pp. 1492-1503, Jun. 1996

[4] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment-Part I: Coherent Detection”, IEEE Trans. Comm., vol. 25, no. 9, Sep. 1977

[5] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment-Part II: Incoherent Detection”, IEEE Trans. Comm., vol. 25, no. 9, Sep. 1977

[6] B. Widrow et al., “Principles and Applications”, Proc. of the IEEE, vol. 63, no.12, Sep. 1975.




BACKUP SLIDES

34




Potential Impact

Improve communication performance for wireless data communication subsystems embedded in PCs and laptops

Extend range from the wireless data communication subsystems to the wireless access point

Achieve higher bit rates for the same bit error rate and range, and lower bit error rates for the same bit rate and range

Extend the results to multiple RF sources on a single chip

35




Symmetric Alpha Stable Process PDF

Closed-form expression does not exist in general

Power series expansions can be derived in some cases

Standard symmetric alpha stable model for localization parameter = 0

36




Middleton Class B Model

Envelope StatisticsEnvelope exceedance probability density (APD) which is 1 – cumulative distribution function

Bm

mBA

IIB

BB

BBB

i

B

mm

mIB

mBB em

AeP

GG

AA

G

N

Fwhere

mF

m

m

AP

00

)2/(01

''

200

11

00110

001

220

!)(

2

4

)1(4

1;

2ˆ;

2ˆ

function trichypergeomeconfluent theis,

ˆ;2;2

1.2

1.!

ˆ)1(ˆ1)(

37




Class B Envelope Statistics

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Exceedance Probability Density Graph for Class B Parameters: A = 10-1, A

B = 1,

B = 5, N

I = 1, = 1.8

No

rma

lize

d E

nve

lop

e T

hre

sho

ld (

E 0 /

Erm

s)

P(E > E0)

PB-I

PB-II

B

38




Parameters for Middleton Class B Noise

B

I

B

B

A

N

A

Parameters Description Typical Range

Impulsive Index AB [10-2, 1]

Ratio of Gaussian to non-Gaussian intensity ΓB [10-6, 1]

Scaling Factor NI [10-1, 102]

Spatial density parameter α [0, 4]

Effective impulsive index dependent on α A α [10-2, 1]

Inflection point (empirically determined) εB > 0

39




Soviet high power over-the-horizon radar interference [Middleton, 1999]

Fluorescent lights in mine shop office interference [Middleton, 1999]

P(ε > ε0)

ε 0 (

dB

> ε

rms)

Percentage of Time Ordinate is ExceededM

agne

tic F

ield

Str

engt

h, H

(dB

rel

ativ

e to

m

icro

amp

per

met

er r

ms)

Accuracy of Middleton Noise Models

40




Class B Exceedance Probability Density Plot

41




Class A Parameter Estimation Based on APD (Exceedance Probability Density) Plot

42




Class A Parameter Estimation Based on Moments

Moments (as derived from the characteristic equation)

Parameter estimates

2

e2 =

e4 =

e6 =

Odd-order momentsare zero

[Middleton, 1999]

43




Expectation Maximization Overview

44




Maximum Likelihood for Sum of Densities

45




Results of EM Estimator for Class A Parameters

46




Extreme Order Statistics

47




Estimator for Alpha-Stable

0 < p < α

48




Incoherent Detection

Bayes formulation [Spaulding & Middleton, 1997, pt. II]

)(),()(:2

)(),()(:1

2

1

tZtStXH

tZtStXH

1)(

)(

)()|(

)()|(

)(2

1

1

2

1

2

H

H

Xp

Xp

dpHXp

dpHXp

X

φ: phaseea:amplituda

and where

Small signal approximation

)(xpdx

d)l(xwhere

txltxl

txltxl

iZi

iH

H

N

iii

N

iii

N

iii

N

iii

ln 1

sin)(cos)(

sin)(cos)(

2

1

2

11

2

11

2

12

2

12

49




Incoherent Detection

Optimal Structure:

The optimal detector for the small signal approximation is basically the correlation receiver preceded by the logarithmic nonlinearity.

Incoherent Correlation Detector

50




Coherent Detection – Class A Noise

Comparison of performance of correlation receiver (Gaussian optimal receiver) and nonlinear detector [Spaulding & Middleton, 1997, pt. II]

51




Volterra Filters

Non-linear (in the signal) polynomial filter

By Stone-Weierstrass Theorem, Volterra signal expansion can model many non-linear systems, to an arbitrary degree of accuracy. (Similar to Taylor expansion with memory).

Has symmetry structure that simplifies computational complexity Np = (N+p-1) C p instead of Np. Thus for N=8 and p=8; Np=16777216 and (N+p-1) C p = 6435.

52




[Widrow et al., 1975]

s : signals+n0 :corrupted signaln0 : noisen1 : reference inputz : system output

Adaptive Noise Cancellation

Computational platform contains multiple antennas that can provide additional information regarding the noise

Adaptive noise canceling methods use an additional reference signal that is correlated with corrupting noise

preliminary results

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