Mitigating Computer Platform Radio Frequency Interference in Embedded Wireless Transceivers

Wireless Networking and Communications Group

Department of Electrical and Computer Engineering

Mitigating Computer Platform Radio Frequency Interference in

Embedded Wireless Transceivers

Prof. Brian L. Evans

Lead Graduate Students Kapil Gulati and Marcel Nassar

Other Graduate Students Aditya Chopra and Marcus DeYoung

Undergraduate Students Navid Aghasadeghi and Arvind K. Sujeeth

Preliminary Results

February 25, 2008


Department of Electrical and Computer Engineering2

Outline

Problem Definition

I: Single Carrier, Single Antenna Communication Systems

• Noise Modeling

• Estimation of Noise Model Parameters

• Filtering and Detection

• Bounds on Communication Performance

II: Single Carrier, Multiple Antenna Communication Systems

III: Multiple Carrier, Single Antenna Communication Systems

Conclusion and Future Work



Problem Definition

Within computing platforms, wirelesstransceivers experience radio frequencyinterference (RFI) from clocks/busses

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 estimation of model parameters

We’ll be using noise and interference interchangeably

Backup



Common Spectral Occupancy

StandardCarrier (GHz)

Wireless Networking

Interfering Clocks and Busses

Bluetooth 2.4Personal Area

NetworkGigabit Ethernet, PCI Express

Bus, LCD clock harmonics

IEEE 802. 11 b/g/n

2.4Wireless 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

5.2Wireless LAN

(Wi-Fi)PCI Express Bus,

LCD clock harmonics



PART I

Single Carrier, Single Antenna Communication Systems



1. Noise ModelingRFI is 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

Backup



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



Middleton Class A Model

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

Probability density function (pdf)

Backup

Probability Density Function for A = 0.15, = 0.1

-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

)



Symmetric Alpha Stable Model

Characteristic function: ||)( je

Parameters 20,α Characteristic exponent indicative

of thickness of tail of impulsiveness

Localization (analogous to mean)

Dispersion (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)

Could approximate pdf using inverse transform of power series expansion of characteristic function Backup

Backup

-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

)



2. Estimation of Noise Model Parameters

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

• Based on envelope statistics [Middleton, 1979] • Based on moments [Middleton, 1979]

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

Backup

Backup

Complexity• Iterative algorithm• At each iteration:

• Rooting a second order polynomial (Given A, maximize K (= AΓ) )• Rooting a fourth order polynomial (Given K, maximize A)

Advantage Small sample size required (~1000 samples)Disadvantage Iterative algorithm, computationally intensiveComplexityParameter estimators are based on simple order statisticsAdvantage Fast / computationally efficient (non-iterative)Disadvantage Requires large set of data samples (N ~ 10,000)

Backup

Backup

Backup



Results on Measured RFI Data

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

• Measured data is "broadband" noise• Middleton Class B model would match

PDF is symmetric• Symmetric Alpha Stable Process

expected to work well• Approximation to Class B model

-20 -15 -10 -5 0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Measured PDF

x: noise amplitude

Me

asu

red

PD

F f

X(x

)



Results on Measured RFI Data

Modeling PDF as Symmetric Alpha Stable process

Estimated Parameters

Localization (δ) -0.0393

Dispersion (γ) 0.5833

Characteristic Exponent (α)

1.5525

Normalized MSE = 0.0055

2

2ˆ

measuredf

measuredf

estimatedf

-20 -15 -10 -5 0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45PDF of measured data vs alpha stable

x: noise amplitude

PD

F f

X(x

)

Measured PDF

Estimated PDF usingalpha stable modeling



3. Filtering and Detection – System Model

Signal Model

Multiple samples/copies of the received signal are available:• N path diversity [Miller, 1972]

• Oversampling by N [Middleton, 1977]

Using multiple samples increases gains vs. Gaussian case because impulses are isolated events over symbol period

s[n]gtx[n]

v[n]

grx[n] Λ(.)

Pulse ShapeNonlinear

FilterMatched

Filter Decision Rule

Impulsive Noise

Alternate Adaptive Model

Backup



Filtering and Detection

Class A Noise• Correlation Receiver (linear)• Wiener Filtering (linear)• Coherent Detection using MAP (Maximum A posteriori

Probability) detector [Spaulding & Middleton, 1977]

• Small Signal Approximation to MAP Detector[Spaulding & Middleton, 1977]

Alpha Stable Noise• Correlation Receiver (linear)• MAP Approximation• Myriad Filtering [Gonzalez & Arce, 2001]

• Hole Punching [Ambike et al., 1994]

We assume perfect estimation of noise model parameters

Backup

Backup

Backup



Coherent Detection – Small Signal Approximation

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

Optimal decision rule & threshold detector for approximation

Optimal detector for approximation is logarithmic nonlinearity followed by correlation receiver

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

Backup



Class A Detection - Results

16

Pulse shapeRaised cosine

10 samples per symbol10 symbols per pulse

ChannelA = 0.35

= 0.5 × 10-3

Memoryless

Method Comp. Perf.

MAP O(NMK) High

Correl. O(N+K) Low

Wiener O(NW+K) Low

Approx. O(MN+K) High

K: Constellation Size

N: number of samples per symbol

M: number of retained terms of the series expansion

W: Window Size



Filtering and Detection – Alpha Stable Model

MAP detection: remove nonlinear filter

Decision rule is given by (p(.) is the SαS distribution)

Approximations for SαS distribution:

1)|()(

)|()()(

2

1

11

22

H

H

HXpHp

HXpHpX

Method Shortcomings Reference

Series Expansion Poor approximation when series length shortened

[Samorodnitsky, 1988]

Polynomial Approx. Poor approximation for small x [Tsihrintzis, 1993]

Inverse FFT Ripples in tails when α < 1 Simulation Results



MAP Detector – PDF Approximation

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 amixture 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



Myriad Filtering

Sliding window algorithm

Outputs myriad of sample window

Myriad of order k for samples x1, x2, … , xN [Gonzalez & Arce, 2001]

• As k decreases, less impulsive noise gets through myriad filter• As k→0, filter tends to mode filter (output value with highest freq.)

Empirical choice of k: [Gonzalez & Arce, 2001]

1

2),(

k

22

11 minargˆ,,

i

N

ikNM xkxxg



Myriad Filtering – Implementation

Given a window of samples x1,…,xN, find β [xmin, xmax]

Optimal myriad algorithm1. Differentiate objective function

polynomial p(β) with respect to β

2. Find roots and retain real roots

3. Evaluate p(β) at real roots and extremum

4. 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)

Backup

22

1)(

i

N

ixkp



Hole Punching (Blanking) Filter

Sets sample to 0 when sample exceeds threshold [Ambike, 1994]

Intuition:• Large values are impulses and true value cannot be recovered• Replace large values with zero will not bias (correlation) receiver• If additive noise were purely Gaussian, then the larger the threshold,

the lower the detrimental effect on bit error rate

hp

hphp Tnx

Tnxnxh

][0

][][



Complexity Analysis

Method Complexity per symbol

Analysis

Hole Puncher + Correlation Receiver

O(N+S) A decision needs to be made about each sample.

Optimal Myriad + Correlation Receiver

O(NW3+S) Due to polynomial rooting which is equivalent to Eigen-value decomposition.

Selection Myriad + Correlation Receiver

O(NW2+S) Evaluation of the myriad function and comparing it.

MAP Approximation O(MNS) Evaluating approximate pdf(M is number of Gaussians in mixture)

N is oversampling factor S is constellation size W is window size



Bit Error Rate (BER) Performance in Alpha Stable Noise

-10 -5 0 5 10 15 20

10-2

10-1

100

Generalized SNR

BE

R

Communication Performance (=0.9, =0, M=12)

Matched FilterHole PunchingMAPMyriad

-10 -5 0 5 10 15 2010

-5

10-4

10-3

10-2

10-1

100

Generalized SNR

BE

R

Communication Performance (=1.5,=0,M=12)

Matched FilterHole PunchingMLMyriad



4. Performance Bounds in presence of impulsive noise

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



Capacity in Presence of Impulsive Noise

)()(

)|()(

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

NhYh

XYhYh

YXICsX EXExf

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

Capacity

)()(

)|()(

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

NhYh

XYhYh

YXICsX EXExf



Probability of Error for Uncoded Transmission

)(!

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]

Backup



Chernoff Factors for Coded Transmission

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



Part IISingle Carrier, Multiple Antenna Communication

Systems



Multiple Input Multiple Output (MIMO) Receivers in Impulsive Noise

Statistical Physical Models of Noise• Middleton Class A model for two-antenna systems

[MacDonald & Blum,1997]

• Extension to larger than 2 2 case is difficult

Statistical Models of Noise• Multivariate Alpha Stable Process• Mixture of weighted multivariate complex Gaussians as

approximation to multivariate Middleton Class A noise[Blum et al., 1997]



MIMO Receivers in Impulsive Noise

Key Prior Work• Performance analysis of standard MIMO receivers in impulsive

noise [Li, Wang & Zhou, 2004]

• Space-time block coding over MIMO channels with impulsive noise[Gao & Tepedelenlioglu,2007]

• Assumes uncorrelated noise at antennas

Our Contributions• Performance analysis of standard MIMO receivers using

multivariate noise models• Optimal and sub-optimal maximum likelihood (ML) receiver design

for 2 2 case



Communication Performance

0 5 10 15 20 2510

-5

10-4

10-3

10-2

10-1

100

Performance of MIMO Receivers in Implusive Noise (A = 0.1, 1 =

2 = 10-3; = 0.1)

Vec

tor

Sym

bol E

rror

Rat

e (V

SE

R)

SNR [in dB]

ML (Guassian)

ML (Impulsive)Sub-Optimal ML (Impulsive)

2 x 2 MIMO systemA = 0.1, Γ1 = Γ2 = 10-3 Correlation Coeff. = 0.1

Spatial Multiplexing Mode



Part III

Multiple Carriers, Single Antenna Communication Systems



Motivation

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

out over all subsymbols thereby reducing effect of impulse

Complex number (CN) codes [Lang, 1963]

• Transmitter forms s = GS, where S contains transmitted symbols,G is a unitary matrix and s contains coded symbols

• Receiver multiplies received symbols by G-1

• Gaussian noise unaffected (unitary transformation is rotation)• Orthogonal frequency division multiplexing (OFDM) is special case

of CN codes when G is inverse discrete Fourier transform matrix



Noise Smearing

Smearing effect• Impulsive noise energy distributes over longer symbol time• Smearing filters maximize impulse attenuation and minimize

intersymbol interference for impulsive noise [Beenker, 1985]

• Maximum smearing efficiency is where N is number of symbols used in unitary transformation

• As N , distribution of impulsive noise becomes Gaussian

Simulations [Haring, 2003]

• When using a transformation involving N = 1024 symbols, impulsive noise case approaches case where only Gaussian noise is present

Backup

N



Conclusion

Radio frequency interference from computing platform• Affects wireless data communication transceivers• Models include Middleton noise models and alpha stable processes• Cancellation can improve communication performance

Initial RFI cancellation methods explored• Linear (Wiener) and Non-linear filtering (Myriad, Hole Punching)• Optimal detection rules (significant gains at low bit rates)

Preliminary work• Performance bounds in presence of RFI• RFI mitigation in multicarrier, MIMO communication systems



Contributions

Publications 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, accepted for publication.

Software ReleasesRFI Mitigation Toolbox

Version 1.1 Beta (Released November 21st, 2007)Version 1.0 (Released September 22nd, 2007)

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

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

36



Future Work

Single carrier, single antenna communication systems• Fixed-point (embedded) methods for parameter estimation and

detection methods• Estimation and detection for Middleton Class B model

Single carrier, multiple antenna communication systems• MIMO receiver design in presence of RFI• Performance bounds for MIMO receivers in presence of RFI

Multicarrier Modulation and Coding• Explore unitary coding schemes resilient to RFI• Investigate multi-layered coding



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.

[7] 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



References (cont…)[8] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of

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

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

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

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

[12] G. Beenker, T. Claasen, and P. van Gerwen, “Design of smearing filters for data transmission systems,” IEEE Trans. on Comm., vol. 33, Sept. 1985.

[13] G. R. Lang, “Rotational transformation of signals,” IEEE Trans. Inform. Theory, vol. IT–9, pp. 191–198, July 1963.

[14] 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.

[15] 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.



BACKUP SLIDES



Potential Impact

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

• 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 range from wireless data communication subsystems to wireless access point

Extend results to multipleRF sources on single chip



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



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

Middleton Class A Statistics

00

0 !

2)(

2

2

02

z

zezm

Ae

zwm

z

m m

mA

Envelope statistics

Envelope for Gaussian signal has Rayleigh distribution



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



Probability Density Function Power Spectral Density

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

Symmetric Alpha Stable Statistics ||)( 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



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).

Backup

Backup



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 ~ 10,000)

)( )](1[ )(

)( )( )(1

:

1:

xfxFNxf

xfxFNxf

XN

Nm

XN

NM

Backup

Backup

Backup



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



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]



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)(



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



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



Class B Exceedance Probability Density Plot



Expectation Maximization Overview



Maximum Likelihood for Sum of Densities



EM Estimator for Class A Parameters Using 1000 Samples

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



Results of EM Estimator for Class A Parameters



Extreme Order Statistics



Estimator for Alpha-Stable

0 < p < α



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

Estimate α and “mean” directly from data

Estimate “variance” γ from α and δ estimates

Continued next slide

Results for Symmetric Alpha Stable Parameter Estimator



Results for Symmetric Alpha Stable Parameter Estimator

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: M

ean

Squ

ared

Err

or (

MS

E)

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

= 5

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:

Mea

n S

quar

ed E

rror

(M

SE

)

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

= 10



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



Wiener Filtering – Finite Impulse Response (FIR) CaseWiener-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



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



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



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



Coherent Detection – Class A Noise

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



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



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.



[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



0 500 1000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 500 1000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Region 2

Region 1

Region 3

Gaussian Class A (with same power)



Haring’s Receiver Simulation Results



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

SNR (dB) SNR (dB)

Correlation Receiver Performance

A = 0.1



Myriad Filtering

Myriad Filters exhibit high statistical efficiency in bell-shaped impulsive distributions like the SαS distributions.

Have been used as both edge enhancers and smoothers in image processing applications.

In the communication domain, they have been used to estimate a sent number over a channel using a known pulse corrupted by additive noise. (Gonzalez 1996)

In this work, we used a sliding window version of the myriad filter to mitigate the impulsiveness of the additive noise. (Nassar et. al 2007)



Decision Rule Λ(X) H1 or H2

corrupted signal

MAP Detection

Hard decision

Bayesian formulation [Spaulding and Middleton, 1977]

1)|()(

)|()()(

2

1

11

22

H

H

HXpHp

HXpHpX

ZSXH

ZSXH

22

11

:

:

1)(

)()(

2

1

1

2

H

H

Z

Z

SXp

SXpX

Equally probable source



Results

Mitigating Computer Platform Radio Frequency Interference in Embedded Wireless Transceivers

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