Radio Frequency Interference Sensing and Mitigation in Wireless Receivers Talk at The University of Texas at Austin Wireless Networking and Communications

Radio Frequency Interference Sensing and Mitigation in

Wireless Receivers

Talk at The University of Texas at Austin

Wireless Networking and Communications Group

7 Oct 2009

Prof. Brian L. Evans

Lead Graduate StudentsAditya Chopra, Kapil Gulati, Yousof Mortazavi and Marcel Nassar

In collaboration with Eddie Xintian Lin, Alberto Alcocer Ochoa,

Chaitanya Sreerama and Keith R. Tinsley at Intel Labs

2

Outline

Problem definition Single carrier single antenna systems

Radio frequency interference modeling Estimation of interference model parameters Filtering/detection

Multi-input multi-output (MIMO) single carrier systems Co-channel interference modeling Conclusions Future work

2


3

Radio Frequency Interference

Electromagnetic interference Limits wireless communication performance Applications of RFI modeling

Sense and mitigate strategies for coexistence of wireless networks and services

Sense and avoid strategies for cognitive radio We focus on sense and mitigate strategies for wireless

receivers embedded in notebooks Platform noise from user’s computer subsystems Co-channel interference from other in-band wireless networks

and services

4


Problem Definition4

Objectives Develop offline methods to improve communication

performance in presence of computer platform RFI Develop adaptive online algorithms for these methods

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

Within computing platforms, wireless transceivers experience radio frequency interference from clocks and busses

We will use noise and interference interchangeably

We will use noise and interference interchangeably

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5


Impact of RFI5

Impact of LCD noise on throughput for an IEEE 802.11g embedded wireless receiver [Shi, Bettner, Chinn, Slattery & Dong, 2006]

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6


Statistical Modeling of RFI6

Radio frequency interference Sum of independent radiation events Predominantly non-Gaussian impulsive statistics

Key statistical-physical models Middleton Class A, B, C models

Independent of physical conditions (canonical) Sum of independent Gaussian and Poisson interference

Symmetric Alpha Stable models Approximation of Middleton Class B model

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7


Assumptions for RFI Modeling7

Key assumptions for Middleton and Alpha Stable models[Middleton, 1977][Furutsu & Ishida, 1961] Infinitely many potential interfering sources with same effective

radiation power Power law propagation loss Poisson field of interferers with uniform intensity

Pr(number of interferers = M |area R) ~ Poisson(M; R) Uniformly distributed emission times Temporally independent (at each sample time)

Limitations Alpha Stable models do not include thermal noise Temporal dependence may exist

8


Our Contributions8

Mitigation of computational platform noise in single carrier, single antenna systems [Nassar, Gulati, DeYoung, Evans & Tinsley, ICASSP 2008, JSPS 2009]

9


Middleton Class A model9

Probability Density Function

1

2!)(

2

2

02

2

2

Am

where

em

Aezf

m

z

m m

mA

Zm

-10 -5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Noise amplitude

Pro

bability d

ensity f

unction

PDF for A = 0.15, = 0.8

A

Parameter

Description RangeOverlap 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]

10


Symmetric Alpha Stable Model10

Characteristic Function

Closed-form PDF expression only forα = 1 (Cauchy), α = 2 (Gaussian),α = 1/2 (Levy), α = 0 (not very useful)

Approximate PDF using inverse transform of power series expansion

Second-order moments do not exist for α < 2 Generally, moments of order > α do not exist

||)( je

PDF for = 1.5, = 0, = 10

-50 0 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Noise amplitude

Pro

babili

ty d

ensity f

unction

Parameter Description Range

Characteristic Exponent. Amount of impulsiveness

Localization. Analogous to mean

Dispersion. Analogous to variance

αδ

]2,0[α

),( ),0(

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11

Example Power Spectral Densities

Middleton Class A Symmetric Alpha Stable

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

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

Overlap Index (A) = 0.15

Gaussian Factor ( = 0.1 Characteristic Exponent ( = 1.5

Localization () = 0Dispersion () = 10

Simulated Densities

12


Estimation of Noise Model Parameters12

Middleton Class A model Based on Expectation Maximization [Zabin & Poor, 1991]

Find roots of second and fourth order polynomials at each iteration Advantage: Small sample size is required (~1000 samples) Disadvantage: Iterative algorithm, computationally intensive

Symmetric Alpha Stable Model Based on Extreme Order Statistics [Tsihrintzis & Nikias, 1996]

Parameter estimators require computations similar to mean and standard deviation computations

Advantage: Fast / computationally efficient (non-iterative) Disadvantage: Requires large set of data samples (~10000 samples)

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1313


Results on Measured RFI Data13

25 radiated computer platform RFI data sets from Intel 50,000 samples taken at 100 MSPS

Estimated Parameters for Data Set #18Symmetric Alpha Stable Model

Localization (δ) 0.0065KL Divergence

0.0308Characteristic exp. (α) 1.4329

Dispersion (γ) 0.2701

Middleton Class A Model

Overlap Index (A) 0.0854 KL Divergence0.0494Gaussian Factor (Γ) 0.6231

Gaussian Model

Mean (µ) 0 KL Divergence0.1577Variance (σ2) 1

KL Divergence: Kullback-Leibler divergence-6 -4 -2 0 2 4 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Noise amplitude

Pro

babi

lity

Den

sity

Fun

ctio

n

Measured PDF

Est. -Stable PDF

Est. Class A PDFEst. Gaussian PDF

Measured PDF

Gaussian PDF

Middleton Class A PDF

Alpha Stable PDF

14

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

Measurement Set Number

Kul

lbac

k-Le

ible

r (K

L) D

iver

genc

e

Estimated Alpha Stable modelEstimated Class A modelEstimated Gaussian model

14

Results on Measured RFI Data

Best fit for 25 data sets under different platform RFI conditions

KL divergence plotted for three candidate distributions vs. data set number

Smaller KL value means closer fit

Gaussian

Class A

Alpha Stable

15

Video over Impulsive Channels

Video demonstration for MPEG II video stream 10.2 MB compressed stream from camera (142 MB uncompressed) Compressed file sent over additive impulsive noise channel Binary phase shift keying

Raised cosine pulse10 samples/symbol10 symbols/pulse length

Composite of transmitted and received MPEG II video streamshttp://www.ece.utexas.edu/~bevans/projects/rfi/talks/

video_demo19dB_correlation.wmv Shows degradation of video quality over impulsive channels with

standard receivers (based on Gaussian noise assumption)Wireless Networking and Communications Group

15

Additive Class A Noise

Value

Overlap index (A) 0.35

Gaussian factor () 0.001

SNR 19 dB

16


Filtering and Detection16

Pulse Shaping Pre-Filtering Matched

FilterDetection

Rule

Impulsive Noise

Middleton Class A noise Symmetric Alpha Stable noise

Filtering Wiener Filtering (Linear)

Detection Correlation Receiver (Linear) Bayesian Detector

[Spaulding & Middleton, 1977] Small Signal Approximation to

Bayesian detector[Spaulding & Middleton, 1977]

Filtering Myriad Filtering

Optimal Myriad [Gonzalez & Arce, 2001]

Selection Myriad Hole Punching

[Ambike et al., 1994]

Detection Correlation Receiver (Linear) MAP approximation

[Kuruoglu, 1998]

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AssumptionMultiple samples of the received signal are available• N Path Diversity [Miller, 1972]• Oversampling by N [Middleton, 1977]

AssumptionMultiple samples of the received signal are available• N Path Diversity [Miller, 1972]• Oversampling by N [Middleton, 1977]

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17


Results: Class A Detection17

Pulse shapeRaised cosine

10 samples per symbol10 symbols per pulse

ChannelA = 0.35

= 0.5 × 10-3

Memoryless

-35 -30 -25 -20 -15 -10 -5 0 5 10 1510

-5

10-4

10-3

10-2

10-1

100

SNR

Bit

Err

or R

ate

(BE

R)

Correlation ReceiverWiener FilteringBayesian DetectionSmall Signal Approximation

Communication Performance Binary Phase Shift Keying

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18


Results: Alpha Stable Detection18

Use dispersion parameter in place of noise variance to generalize SNRUse dispersion parameter in place of noise variance to generalize SNR

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Backup-10 -5 0 5 10 15 20

10-2

10-1

100

Generalized SNR (in dB)

Bit

Err

or R

ate

(BE

R)

Matched FilterHole PunchingMAPMyriad

Communication Performance Same transmitter settings as previous slide

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19

Video over Impulsive Channels #2

Video demonstration for MPEG II video stream revisited 5.9 MB compressed stream from camera (124 MB uncompressed) Compressed file sent over additive impulsive noise channel Binary phase shift keying

Raised cosine pulse10 samples/symbol10 symbols/pulse length

Composite of transmitted video stream, video stream from a correlation receiver based on Gaussian noise assumption, and video stream for a Bayesian receiver tuned to impulsive noise

http://www.ece.utexas.edu/~bevans/projects/rfi/talks/video_demo19dB.wmv


19

Additive Class A Noise

Value

Overlap index (A) 0.35

Gaussian factor () 0.001

SNR 19 dB

20

Video over Impulsive Channels #2

Structural similarity measure [Wang, Bovik, Sheikh & Simoncelli, 2004]

Score is [0,1] where higher means better video quality

Frame number

Bit error rates for ~50 million bits sent:

6 x 10-6 for correlation receiver

0 for RFI mitigating receiver (Bayesian)

21


Extensions to MIMO systems21

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22


Our Contributions22

2 x 2 MIMO receiver design in the presence of RFI[Gulati, Chopra, Heath, Evans, Tinsley & Lin, Globecom 2008]

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23

-10 -5 0 5 10 15 20

10-3

10-2

10-1

SNR [in dB]

Vec

tor

Sym

bol E

rror

Rat

e

Optimal ML Receiver (for Gaussian noise)Optimal ML Receiver (for Middleton Class A)Sub-Optimal ML Receiver (Four-Piece)Sub-Optimal ML Receiver (Two-Piece)


Results: RFI Mitigation in 2 x 2 MIMO 23

Improvement in communication performance over conventional Gaussian ML receiver at symbol

error rate of 10-2

Communication Performance (A = 0.1, 1= 0.01, 2= 0.1, = 0.4)

24


Results: RFI Mitigation in 2 x 2 MIMO 24

Complexity Analysis

Complexity Analysis for decoding M-level QAM modulated signal

Communication Performance (A = 0.1, 1= 0.01, 2= 0.1, = 0.4)

-10 -5 0 5 10 15 20

10-3

10-2

10-1

SNR [in dB]

Vec

tor

Sym

bol E

rror

Rat

e

Optimal ML Receiver (for Gaussian noise)Optimal ML Receiver (for Middleton Class A)Sub-Optimal ML Receiver (Four-Piece)Sub-Optimal ML Receiver (Two-Piece)

2525

Co-Channel Interference Modeling


25

Region of interferer locations determines interference model [Gulati, Chopra, Evans & Tinsley, Globecom 2009]

Symmetric Alpha Stable Middleton Class A

2626

Co-Channel Interference Modeling

Propose unified framework to derive narrowband interference models for ad-hoc and cellular network environments Key result: tail probabilities (one minus cumulative distribution function)


26

Case 1: Ad-hoc network Case 3-a: Cellular network (mobile user)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-4

10-3

10-2

10-1

100

Interference threshold (a)

P (

Inte

rfe

rnc

e a

mp

litu

de

> a

)

Simulated Symmetric Alpha Stable

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10-10

10-5

100

Interference threshold (a)

P (

Inte

rfe

ren

ce

am

plit

ud

e >

a)

SimulatedSymmetric Alpha StableGaussianMiddleton Class A

27


Conclusions27

Radio Frequency Interference from computing platform Affects wireless data communication transceivers Models include Middleton and alpha stable distributions

RFI mitigation can improve communication performance Single carrier, single antenna systems

Linear and non-linear filtering/detection methods explored Single carrier, multiple antenna systems

Optimal and sub-optimal receivers designed Bounds on communication performance in presence of RFI

Results extend to co-channel interference modeling

28

RFI Mitigation Toolbox

Provides a simulation environment for RFI generation Parameter estimation

algorithms Filtering and detection methods Demos for communication

performance analysis

Latest Toolbox ReleaseVersion 1.3, Aug 26th 2009


28

Snapshot of a demo

http://users.ece.utexas.edu/~bevans/projects/rfi/software/index.html

29


Other Contributions29

Publications[Journal Articles]M. Nassar, K. Gulati, M. R. DeYoung, B. L. Evans and K. R. Tinsley, “Mitigating Near-Field Interference in

Laptop Embedded Wireless Transceivers”, J. of Signal Proc. Systems, Mar 2009, invited paper. [Conference Papers]M. Nassar, K. Gulati, A. K. Sujeeth, N. Aghasadeghi, B. L. Evans and K. R. Tinsley, “Mitigating Near-field

Interference in Laptop Embedded Wireless Transceivers”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Mar. 30-Apr. 4, 2008, Las Vegas, NV USA.

K. Gulati, A. Chopra, R. W. Heath Jr., B. L. Evans, K. R. Tinsley, and X. E. Lin, “MIMO Receiver Design in the Presence of Radio Frequency Interference”, Proc. IEEE Int. Global Communications Conf., Nov. 30-Dec. 4th, 2008, New Orleans, LA USA.

A. Chopra, K. Gulati, B. L. Evans, K. R. Tinsley, and C. Sreerama, “Performance Bounds of MIMO Receivers in the Presence of Radio Frequency Interference”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Apr. 19-24, 2009, Taipei, Taiwan, accepted.

K. Gulati, A. Chopra, B. L. Evans and K. R. Tinsley, “Statistical Modeling of Co-Channel Interference”, Proc. IEEE Int. Global Communications Conf., Nov. 30-Dec. 4, 2009, Honolulu, HI USA, accepted.

Project Websitehttp://users.ece.utexas.edu/~bevans/projects/rfi/index.html

30


Future Work30

Extend RFI modeling for Adjacent channel interference Multi-antenna systems Temporally correlated interference

Multi-input multi-output (MIMO) single carrier systems RFI modeling and receiver design

Multicarrier communication systems Coding schemes resilient to RFI System level techniques to reduce computational platform

generated RFI

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31


31

Thank You.Questions ?

32


References32

RFI Modeling[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] K.F. McDonald and R.S. Blum. “A physically-based impulsive noise model for array observations”, Proc. IEEE Asilomar Conference on Signals, Systems& Computers, vol 1, 2-5 Nov. 1997.

[3] K. Furutsu and T. Ishida, “On the theory of amplitude distributions of impulsive random noise,” J. Appl. Phys., vol. 32, no. 7, pp. 1206–1221, 1961.

[4] J. Ilow and D . Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers”, IEEE transactions on signal processing, vol. 46, no. 6, pp. 1601-1611, 1998.

Parameter Estimation[5] 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

[6] 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

RFI Measurements and Impact[7] J. Shi, A. Bettner, G. Chinn, K. Slattery and X. Dong, "A study of platform EMI from LCD panels -

impact on wireless, root causes and mitigation methods,“ IEEE International Symposium on Electromagnetic Compatibility, vol.3, no., pp. 626-631, 14-18 Aug. 2006

33


References (cont…)33

Filtering and Detection[8] 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[9] 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[10] J.G. Gonzalez and G.R. Arce, “Optimality of the Myriad Filter in Practical Impulsive-Noise

Environments”, IEEE Trans. on Signal Processing, vol 49, no. 2, Feb 2001[11] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of Gaussian

noise and impulsive noise modelled as an alpha-stable process,” IEEE Signal Processing Letters, vol. 1, pp. 55–57, Mar. 1994.

[12] J. G. Gonzalez and G. R. Arce, “Optimality of the myriad filter in practical impulsive-noise environments,” IEEE Trans. on Signal Proc, vol. 49, no. 2, pp. 438–441, Feb 2001.

[13] E. Kuruoglu, “Signal Processing In Alpha Stable Environments: A Least Lp Approach,” Ph.D. dissertation, University of Cambridge, 1998.

[14] J. Haring and A.J. Han Vick, “Iterative Decoding of Codes Over Complex Numbers for Impulsive Noise Channels”, IEEE Trans. On Info. Theory, vol 49, no. 5, May 2003

[15] Ping Gao and C. Tepedelenlioglu. “Space-time coding over mimo channels with impulsive noise”, IEEE Trans. on Wireless Comm., 6(1):220–229, January 2007.

34


Backup Slides34

Most backup slides are linked to the main slides Miscellaneous topics not covered in main slides

Performance bounds for single carrier single antenna system in presence of RFI Backup

35


Common Spectral Occupancy35

Standard Carrier (GHz)

Wireless Networking Interfering Clocks and Busses

Bluetooth 2.4 Personal Area Network

Gigabit Ethernet, PCI Express Bus, LCD clock harmonics

IEEE 802. 11 b/g/n 2.4 Wireless LAN

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

LCD clock harmonics

IEEE 802.16e

2.5–2.69 3.3–3.8

5.725–5.85

Mobile Broadband(Wi-Max)

PCI Express Bus,LCD clock harmonics

IEEE 802.11a 5.2 Wireless LAN

(Wi-Fi)PCI Express Bus,

LCD clock harmonics

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36


Impact of RFI36

Calculated in terms of desensitization (“desense”) Interference raises noise floor Receiver sensitivity will degrade to maintain SNR

Desensitization levels can exceed 10 dB for 802.11a/b/g due to computational platform noise [J. Shi et al., 2006]Case Sudy: 802.11b, Channel 2, desense of 11dB More than 50% loss in range Throughput loss up to ~3.5 Mbps for very low receive signal strengths

(~ -80 dbm)

floor noise RX

ceInterferenfloor noise RXlog10 10desense

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37

Impact of LCD clock on 802.11g

Pixel clock 65 MHz LCD Interferers and 802.11g center frequencies


37

Return

38


Middleton Class A, B and C Models38

Class A Narrowband interference (“coherent” reception)Uniquely represented by 2 parameters

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

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

[Middleton, 1999]

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39


Middleton Class B Model39

Envelope statistics Envelope exceedence probability density (APD), which is 1 – cumulative

distribution function (CDF)

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)(

Return

40


Middleton Class B Model (cont…)40

Middleton 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

Return

41


Middleton Class B Model (cont…)41

Parameters for Middleton Class B model

B

I

B

B

A

N

A

Parameters

Description Typical RangeImpulsive 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

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42


Accuracy of Middleton Noise Models42

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

ag

neti

c Fi

eld

Str

en

gth

, H

(d

B r

ela

tive t

o

mic

roam

p p

er

mete

r rm

s)

Return

43


Symmetric Alpha Stable PDF43

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

Return

44


Symmetric Alpha Stable Model44

Heavy tailed distribution

Density functions for symmetric alpha stable distributions for different values of characteristic exponent alpha: a) overall density

and b) the tails of densities

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45


Parameter Estimation: Middleton Class A45

Expectation Maximization (EM) E Step: Calculate log-likelihood function \w current parameter values M Step: Find parameter set that maximizes log-likelihood function

EM Estimator for Class A parameters [Zabin & Poor, 1991] Express envelope statistics as sum of weighted PDFs

Maximization step is iterative Given A, maximize K (= A). Root 2nd order polynomial. Given K, maximize A. Root 4th order polynomial

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

Return

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Results Backup

46


Expectation Maximization Overview46

Return

47


Results: EM Estimator for Class A47

PDFs with 11 summation terms50 simulation runs per setting

1000 data samplesConvergence criterion:

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

2

)(A

AAANMSE est

est

7

1

1 10ˆ

ˆˆ

n

nn

A

AA

K = A

Return

48


Results: EM Estimator for Class A48

• For convergence for A [10-2, 1], worst-case number of iterations for A = 1

• Estimation accuracy vs. number of iterations tradeoff

Return

49


Parameter Estimation: Symmetric Alpha Stable49

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

PDFs of max and min of sequence of i.i.d. data samples 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~10,000)

)( )](1[ )(

)( )( )(1

:

1:

xfxFNxf

xfxFNxf

XN

Nm

XN

NM

Return

Results Backup

50


Parameter Est.: Symmetric Alpha Stable Results50

• Data length (N) of 10,000 samples

• Results averaged over 100 simulation runs

• Estimate α and “mean” directly from data

• Estimate “variance” from α and δ estimates

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

Return

51


Parameter Est.: Symmetric Alpha Stable Results51

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 ()


Mea

n S

quar

ed E

rror

(M

SE

)

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

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

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 ()


Mea

n S

quar

ed E

rror

(M

SE

)

Return

52


Extreme Order Statistics52

Return

53


Parameter Estimators for Alpha Stable53

0 < p < α

Return

54


Filtering and Detection54

System model

Assumptions Multiple samples of the received signal are available

N Path Diversity [Miller, 1972]

Oversampling by N [Middleton, 1977]

Multiple samples increase gains vs. Gaussian case Impulses are isolated events over symbol period

Pulse Shaping Pre-Filtering Matched

FilterDetection

Rule

Impulsive Noise

N samples per symbolN samples per symbol

55


Wiener Filtering55

Optimal in mean squared error sense in presence of Gaussian noise

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

^

Model

Design

Return

56


Wiener Filter Design56

Infinite Impulse Response (IIR)

Finite Impulse Response (FIR) Weiner-Hopf equations for order p-1

)(eΦ+)(eΦ

)(eΦ=

)(eΦ

)(eΦ=eH

jωz

jωd

jωd

jωx

jωdxjω

MMSE

2

10,1,... 1

0

-p,=k(k)r=l)(kw(l)rp

=ldxx

)(pr

)(r

)(r=

)w(p

)w(

)w(

rprpr

r

prrr

dx

dx

dx

xxx

x

xxx

1

1

0

1

1

0

0...21

1

1...10

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)

Return

57


Results: Wiener Filtering57

100-tap FIR FilterRaised Cosine

Pulse Shape

Transmitted waveform corrupted by Class A interference

Received waveform filtered by Wiener filter

n

n

n

ChannelA = 0.35 = 0.5 ×

10-3

SNR = -10 dB

Memoryless

Pulse shape

10 samples per symbol10 symbols per pulse

Return

58


MAP Detection for Class A58

Hard decision Bayesian formulation [Spaulding & Middleton, 1977]

Equally probable source

Z+S=X:H

Z+S=X:H

22

11 1

2

1

11

22

H

H

)H|X)p(p(H

)H|X)p(p(H=)XΛ(

1

2

1

1

2

H

H

Z

Z

)SX(p

)SX(p=)XΛ(

Return

59


MAP Detection for Class A: Small Signal Approx.59

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

Approximate MAP detection rule

Logarithmic non-linearity + correlation receiver Near-optimal for small amplitude signals

ji

N

=i i

Z

ZjΤ

ZZjZ sx

)X(p)X(p=S)X(p)X(p)SX(p

1

Correlation Receiver

1 ln1

ln1

2

1

11i

12i

H

H

N

=iiZ

i

N

=iiZ

i

)(xpdxd

s

)(xpdxd

s

)XΛ(

We use 100 terms of the series expansion for

d/dxi ln pZ(xi) in simulations

Return

60


Incoherent Detection60

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

Small signal approximation

Z(t)+θ)(t,S=X(t):H

Z(t)+θ)(t,S=X(t):H

22

11

1

2

1

1

2

1

2

H

H

θ

θ

)X(p

)X(p=

)p(θp(θH|Xp(

)p(θp(θH|Xp(

=)XΛ(

phase :φamplitude:a

φ

a=θ and where

ln

1

sincos

sincos

2

1

2

11

2

11

2

12

2

12

)(xpdx

d=)l(xwhere

tω)l(x+tω)l(x

tω)l(x+tω)l(x

iZi

i

H

H

N

=iii

N

=iii

N

=iii

N

=iii

Correlation receiver

Return

61


Filtering for Alpha Stable Noise61

Myriad filtering Sliding window algorithm outputs myriad of a sample window Myriad of order k for samples x1,x2,…,xN [Gonzalez & Arce, 2001]

As k decreases, less impulsive noise passes through the myriad filter As k→0, filter tends to mode filter (output value with highest frequency)

Empirical Choice of k [Gonzalez & Arce, 2001]

Developed for images corrupted by symmetric alpha stable impulsive noise

22

11 minargˆ,,

i

N

ikNM xkxxg

1

2),(

k

Return

62


Filtering for Alpha Stable Noise (Cont..)62

Myriad filter implementation Given a window of samples, x1,…,xN, find β [xmin, xmax] Optimal Myriad algorithm

1. Differentiate objective function polynomial p(β) with respect to β

2. Find roots and retain real roots3. Evaluate p(β) at real roots and extreme points4. Output β that gives smallest value of p(β)

Selection Myriad (reduced complexity)1. Use x1, …, xN as the possible values of β

2. Pick value that minimizes objective function p(β)

22

1)(

i

N

ixkp

Return

63


Filtering for Alpha Stable Noise (Cont..)63

Hole punching (blanking) filters Set sample to 0 when sample exceeds threshold [Ambike, 1994]

Large values are impulses and true values can be recovered Replacing large values with zero will not bias (correlation) receiver for

two-level constellation If additive noise were purely Gaussian, then the larger the threshold,

the lower the detrimental effect on bit error rate Communication performance degrades as constellation size

(i.e., number of bits per symbol) increases beyond two

hp

hp

T>nx

Tnxnx

][0

][][hhp

Return

64


MAP Detection for Alpha Stable: PDF Approx.64

SαS random variable Z with parameters , can be written Z = X Y½ [Kuruoglu, 1998] X is zero-mean Gaussian with variance 2 Y is positive stable random variable with parameters depending on

PDF of Z can be written as a mixture model of N Gaussians[Kuruoglu, 1998]

Mean can be added back in Obtain fY(.) by taking inverse FFT of characteristic function & normalizing Number of mixtures (N) and values of sampling points (vi) are tunable

parameters

N

iiY

iY

N

i

v

z

vf

vfezp

i

1

2

2

1

2

,0,

2

2

2

Return

65


Results: Alpha Stable Detection65

Return

66


Complexity Analysis for Alpha Stable Detection66

Return

67


Bivariate Middleton Class A Model67

Joint spatial distribution Return

68


Results on Measured RFI Data68

50,000 baseband noise samples represent broadband interference

-4 -3 -2 -1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Noise amplitude

Pro

ba

bili

ty D

en

sity

Fu

nct

ion

Measured PDFEstimated MiddletonClass A PDFEqui-powerGaussian PDF

Marginal PDFs of measured data compared with estimated model densities

Return

69

2 x 2 MIMO System

Maximum Likelihood (ML) receiver

Log-likelihood function


System Model69

Sub-optimal ML Receiversapproximate

Return

70


Sub-Optimal ML Receivers70

Two-piece linear approximation

Four-piece linear approximation

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

z

Ap

pro

xma

tion

of

(z)

(z)

1(z)

2(z)

chosen to minimizeApproximation of

Return

71


Results: Performance Degradation71

Performance degradation in receivers designed assuming additive Gaussian noise in the presence of RFI

-10 -5 0 5 10 15 2010

-5

10-4

10-3

10-2

10-1

100

SNR [in dB]

Vec

tor

Sym

bol E

rror

Rat

e

SM with ML (Gaussian noise)SM with ZF (Gaussian noise)Alamouti coding (Gaussian noise)SM with ML (Middleton noise)SM with ZF (Middleton noise)Alamouti coding (Middleton noise)

Simulation Parameters•4-QAM for Spatial Multiplexing (SM) transmission mode•16-QAM for Alamouti transmission strategy•Noise Parameters:A = 0.1, 1= 0.01, 2= 0.1, = 0.4

Severe degradation in communication performance in

high-SNR regimes

Return

72


Performance Bounds (Single Antenna)72

Channel capacity

Case I Shannon Capacity in presence of additive white Gaussian noise

Case II (Upper Bound) Capacity in the presence of Class A noiseAssumes that there exists an input distribution which makes output distribution Gaussian (good approximation in high SNR regimes)

Case III (Practical Case) Capacity in presence of Class A noiseAssumes input has Gaussian distribution (e.g. bit interleaved coded modulation (BICM) or OFDM modulation [Haring, 2003])

NXY System Model

)()(

)|()(

);(max}}{),({ 2

NhYh

XYhYh

YXICsX EXExf

Return

73



Channel capacity in presence of RFI

NXY

-40 -30 -20 -10 0 10 200

5

10

15

SNR [in dB]

Cap

acity

(bi

ts/s

ec/H

z)

Channel Capacity

X: Gaussian, N: Gaussian

Y:Gaussian, N:ClassA (A = 0.1, = 10-3)

X:Gaussian, N:ClassA (A = 0.1, = 10-3)

System Model

ParametersA = 0.1, Γ = 10-3

Capacity

)()(

)|()(

);(max}}{),({ 2

NhYh

XYhYh

YXICsX EXExf

Return

74



Probability of error for uncoded transmissions

)(!

2

0m

AWGNe

m

mA

e Pm

AeP

-40 -30 -20 -10 0 10 2010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

dmin

/ [in dB]

Pro

babi

lity

of e

rror

Probability of error (Uncoded Transmission)

AWGN

Class A: A = 0.1, = 10-3

12 A

m

m

BPSK uncoded transmission

One sample per symbol

A = 0.1, Γ = 10-3

[Haring & Vinck, 2002]

Return

75



Chernoff factors for coded transmissions

N

kkk ccC

PPEP

1

'

'

),,(min

)(

cc

-20 -15 -10 -5 0 5 10 1510

-3

10-2

10-1

100

dmin

/ [in dB]

Che

rnof

f F

acto

r

Chernoff factors for real channel with various parameters of A and MAP decoding

Gaussian

Class A: A = 0.1, = 10-3

Class A: A = 0.3, = 10-3

Class A: A = 10, = 10-3

PEP: Pairwise error probability

N: Size of the codeword

Chernoff factor:

Equally likely transmission for symbols

),,(min ' kk ccC

Return

76

System Model


76

Return

77


Performance Bounds (2x2 MIMO)77

Channel capacity

Case I Shannon Capacity in presence of additive white Gaussian noise

Case II (Upper Bound) Capacity in presence of bivariate Middleton Class A noise. Assumes that there exists an input distribution which makes output distribution Gaussian for all SNRs.

Case III (Practical Case) Capacity in presence of bivariate Middleton Class A noiseAssumes input has Gaussian distribution

System Model

Return

78



Channel capacity in presence of RFI for 2x2 MIMO

System Model

Capacity

-40 -30 -20 -10 0 10 200

5

10

15

20

25

SNR [in dB]

Mut

ual I

nfor

mat

ion

(bits

/sec

/Hz)

Channel Capacity with Gaussian noiseUpper Bound on Mutual Information with Middleton noiseGaussian transmit codebook with Middleton noise

Parameters:A = 0.1, 1 = 0.01, 2 = 0.1, = 0.4

Return

79



Probability of symbol error for uncoded transmissions

Parameters:A = 0.1, 1 = 0.012 = 0.1, = 0.4

Pe: Probability of symbol error

S: Transmitted code vector

D(S): Decision regions for MAP detector

Equally likely transmission for symbols

Return

80



Chernoff factors for coded transmissions

N

ttt ssC

ssPPEP

1

'

'

),,(min

)(

PEP: Pairwise error probabilityN: Size of the codewordChernoff factor:Equally likely transmission for symbols

),,(min ' kk ccC

-30 -20 -10 0 10 20 30 4010

-8

10-6

10-4

10-2

100

dt2 / N

0 [in dB]

Che

rnof

f Fac

tor

Middleton noise (A = 0.5)Middleton noise (A = 0.1)Middleton noise (A = 0.01)Gaussian noise

Parameters:1 = 0.012 = 0.1, = 0.4

Return

81

Performance Bounds (2x2 MIMO)

Cutoff rates for coded transmissions Similar measure as channel capacity Relates transmission rate (R) to Pe for a length T codes


81

Return

82

Performance Bounds (2x2 MIMO)


82

Cutoff rate

-30 -20 -10 0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5

4

SNR [in dB]

Cut

off R

ate

[bits

/tran

smis

sion

]

BPSK, Middleton noiseBPSK, Gaussian noiseQPSK, Middleton noiseQPSK, Gaussian noise16QAM, Middleton noise16QAM, Gaussian noise

Return

83


Extensions to Multicarrier Systems83

Impulse noise with impulse event followed by “flat” region Coding may improve communication performance In multicarrier modulation, impulsive event in time domain

spreads over all subcarriers, reducing effect of impulse Complex number (CN) codes [Lang, 1963]

Unitary transformations Gaussian noise is unaffected (no change in 2-norm Distance) Orthogonal frequency division multiplexing (OFDM) is a

special case: Inverse Fourier Transform As number of subcarriers increase, impulsive noise case

approaches the Gaussian noise case [Haring 2003]

Return

Documents

Radio Frequency Interference Sensing and Mitigation in Wireless Receivers Talk at The University of Texas at Austin Wireless Networking and Communications