Mitigating Radio Frequency Interference in Embedded Wireless Receivers

Mitigating Radio Frequency Interference in

Embedded Wireless Receivers

Wireless Networking and Communications Group

April 21, 2023

Prof. Brian L. Evans

Lead Graduate StudentsAditya Chopra, Kapil Gulati and Marcel Nassar

In collaboration with Keith R. Tinsley and Chaitanya Sreerama at Intel Labs

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 Conclusions Future work

2



Problem Definition3

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 (RFI) from clocks and busses

We will use noise and interference interchangeably

We will use noise and interference interchangeably


Common Spectral Occupancy4

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


Impact of RFI5

Impact of LCD noise on throughput performance for a 802.11g embedded wireless receiver [J. Shi et al., 2006]

Backup


Statistical Modeling of RFI6

Radio Frequency Interference (RFI) 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 Model non-linear phenomenon governing RFI

Symmetric Alpha Stable models Approximation of Middleton Class B model

Backup

Backup


Assumptions for RFI Modeling7

Key Assumptions [Middleton, 1977][Furutsu & Ishida, 1961] Infinitely many potential interfering sources with same

effective radiation power Power law propagation loss Poisson field of interferers

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

Limitations [Alpha Stable]: Does not include thermal noise Temporal dependence may exist


Our Contributions8

Mitigation of computational platform noise in single carrier, single antenna systems [Nassar et al., ICASSP 2008]


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]


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 and = 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(

Backup

Backup


Estimation of Noise Model Parameters11

Middleton Class A model Expectation Maximization (EM) [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)

Backup

Backup


Results on Measured RFI Data12

Broadband RFI data 80,000 samples collected using 20GSPS scope

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

0.2

0.4

0.6

0.8

1

Measured Data Fitting

Noise amplitude

Pro

babi

lity

Den

sity

Fun

ctio

n

Measured PDF

Estimated AlphaStable PDFEstimated MiddletonClass A PDF

Estimated Equi-powerGaussian PDF

Estimated ParametersSymmetric Alpha Stable Model

Localization (δ) 0.0043Distance 0.0514Characteristic exp. (α) 1.2105

Dispersion (γ) 0.2413

Middleton Class A Model

Overlap Index (A) 0.1036 Distance0.0825Gaussian Factor (Γ) 0.7763

Gaussian Model

Mean (µ) 0 Distance0.2217Variance (σ2) 1

Distance: Kullback-Leibler divergence

Backup


Filtering and Detection13

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 Shapin

g

Pre-Filtering

Matched Filter

Detection Rule

Impulsive Noise

N samples per symbolN samples per symbol

Filtering and Detection Methods

Filtering Wiener Filtering (Linear)

Detection Correlation Receiver (Linear) MAP (Maximum a posteriori

probability) detector [Spaulding & Middleton, 1977]

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

Filtering Myriad Filtering

[Gonzalez & Arce, 2001] Hole Punching

Detection Correlation Receiver (Linear) MAP approximation

14


Middleton Class A noise Symmetric Alpha Stable noise

BackupBackup

Backup

Backup

Backup


Results: Class A Detection15

Pulse shapeRaised cosine

10 samples per symbol10 symbols per pulse

ChannelA = 0.35

= 0.5 × 10-3

Memoryless


Filtering for Alpha Stable Noise16

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


Filtering for Alpha Stable Noise (Cont..)17

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


Results: Alpha Stable Detection18

-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

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

Backup

Backup


Extensions to MIMO systems19

RFI Modeling Middleton Class A Model for two-antenna systems

[McDonald & Blum, 1997]

Closed form PDFs for M x N MIMO system not published Prior Work

Much prior work assumes independent noise at antennas 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]


Our Contributions20

2 x 2 MIMO receiver design in the presence of RFI[Gulati et al., Globecom 2008]

Backup


Results: RFI Mitigation in 2 x 2 MIMO 21

Complexity Analysis

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)

-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 22

Complexity Analysis

Complexity Analysis for decoding M-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)


Conclusions23

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

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

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

Studied RFI modeling for 2x2 MIMO systems Optimal and sub-optimal receivers designed Bounds on communication performance in presence of RFI


Contributions24

PublicationsM. 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, accepted for publication.

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, submitted.

Software ReleasesRFI Mitigation Toolbox

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

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


Future Work25

Modeling RFI to include Computational platform noise Co-channel interference Adjacent channel interference

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

Multicarrier communication systems Coding schemes resilient to RFI Circuit design guidelines to reduce computational platform

generated RFI

Backup


26

Thank You,Questions ?


References27

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


References (cont…)28

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.


Backup Slides29

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


Impact of RFI30

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

Return


Middleton Class A, B and C Models31

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]

Return

Backup


Middleton Class B Model32

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


Middleton Class B Model (cont…)33

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


Middleton Class B Model (cont…)34

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

Return


Accuracy of Middleton Noise Models35

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


Symmetric Alpha Stable PDF36

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


Symmetric Alpha Stable Model37

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

Return


Parameter Estimation: Middleton Class A38

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

Backup

Results Backup


Expectation Maximization Overview39

Return


Results: EM Estimator for Class A40

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


Results: EM Estimator for Class A41

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

• Estimation accuracy vs. number of iterations tradeoff

Return


Parameter Estimation: Symmetric Alpha Stable42

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


Results: Symmetric Alpha Stable Parameter Estimator43

• 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


Results: Symmetric Alpha Stable Parameter Estimator (Cont…)

44

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


Extreme Order Statistics45

Return


Parameter Estimators for Alpha Stable46

0 < p < α

Return

Results on Measured RFI Data

Best fit for 25 data sets taken under different conditions Return


Wiener Filtering48

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


Wiener Filter Design49

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


Results: Wiener Filtering50

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


MAP Detection for Class A51

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


MAP Detection for Class A: Small Signal Approx.52

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


Incoherent Detection53

Baye’s 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


Filtering for Alpha Stable Noise (Cont..)54

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


MAP Detection for Alpha Stable: PDF Approx.55

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


Results: Alpha Stable Detection56

Return


Complexity Analysis for Alpha Stable Detection57

Return


Performance Bounds (Single Antenna)58

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



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



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



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


Performance Bounds (2x2 MIMO)62

Channel Capacity [Chopra et al., submitted to ICASSP 2009]

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

NXY System Model

)()(

)|()(

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

NhYh

XYhYh

YXICsX EXExf

Return



Channel Capacity in presence of RFI for 2x2 MIMO[Chopra et al., submitted to ICASSP 2009]

NXY System Model

Capacity

)()(

)|()(

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

NhYh

XYhYh

YXICsX EXExf

-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 = 0.012 = 0.1, = 0.4

Return



Probability of symbol error for uncoded transmissions[Chopra et al., submitted to ICASSP 2009]

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



Chernoff factors for coded transmissions[Chopra et al., submitted to ICASSP 2009]

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


Extensions to Multicarrier Systems66

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

spreads out over all subcarriers, reducing the 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 [Haring 2003] As number of subcarriers increase, impulsive

noise case approaches the Gaussian noise case.

Return

Documents

Mitigating Radio Frequency Interference in Embedded Wireless Receivers