Radio Frequency Interference Sensing and Mitigation in Wireless Receivers

1
Radio Frequency Interference Sensing and
Mitigation in Wireless Receivers
Prof. Brian L. Evans Lead Graduate
Students Aditya 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

• Talk at The University of Texas at Austin

2
Outline
2

• 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

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 users computer subsystems
• Co-channel interference from other in-band
  wireless networks and services

4
Problem Definition
4
Backup
Within computing platforms, wireless transceivers
experience radio frequency interference from
clocks and 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 estimated model
  parameters

We will use noise and interference interchangeably
5
Impact of RFI
5

• 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 RFI
6

• 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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Assumptions for RFI Modeling
7

• Key assumptions for Middleton and Alpha Stable
  modelsMiddleton, 1977Furutsu Ishida, 1961
• Infinitely many potential interfering sources
  with same effective radiation power
• Power law propagation loss
• Poisson field of interferers with uniform
  intensity l
• Pr(number of interferers M area R)
  Poisson(M lR)
• 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 Contributions
8
Mitigation of computational platform noise in
single carrier, single antenna systems Nassar,
Gulati, DeYoung, Evans Tinsley, ICASSP 2008,
JSPS 2009
Computer Platform Noise Modelling Evaluate fit of measured RFI data to noise models Middleton Class A model Symmetric Alpha Stable
Parameter Estimation Evaluate estimation accuracy vs complexity tradeoffs
Filtering / Detection Evaluate communication performance vs complexity tradeoffs Middleton Class A Correlation receiver, Wiener filtering, and Bayesian detector Symmetric Alpha Stable Myriad filtering, hole punching, and Bayesian detector
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Middleton Class A model
9

• Probability Density Function

PDF for A 0.15,?? 0.8
10
Symmetric Alpha Stable Model
10

• Characteristic Function
• Closed-form PDF expression only fora 1
  (Cauchy), a 2 (Gaussian),a 1/2 (Levy), a 0
  (not very useful)
• Approximate PDF using inverse transform of power
  series expansion
• Second-order moments do not exist for a lt 2
• Generally, moments of order gt a do not exist

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PDF for ? 1.5, ? 0, ? 10
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Parameter Description Range
Characteristic Exponent. Amount of impulsiveness
Localization. Analogous to mean
Dispersion. Analogous to variance
11
Example Power Spectral Densities

• Middleton Class A

• Symmetric Alpha Stable

Characteristic Exponent (a) 1.5 Localization
(d) 0 Dispersion (g) 10
Overlap Index (A) 0.15 Gaussian Factor (G)
0.1
Simulated Densities
12
Estimation of Noise Model Parameters
12

• 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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Results on Measured RFI Data
13
13

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

Estimated Parameters for Data Set 18 Estimated Parameters for Data Set 18 Estimated Parameters for Data Set 18
Symmetric Alpha Stable Model Symmetric Alpha Stable Model Symmetric Alpha Stable Model
Localization (d) 0.0065 KL Divergence 0.0308
Characteristic exp. (a) 1.4329 KL Divergence 0.0308
Dispersion (?) 0.2701 KL Divergence 0.0308
Middleton Class A Model Middleton Class A Model Middleton Class A Model
Overlap Index (A) 0.0854 KL Divergence 0.0494
Gaussian Factor (G) 0.6231 KL Divergence 0.0494
Gaussian Model Gaussian Model Gaussian Model
Mean (µ) 0 KL Divergence 0.1577
Variance (s2) 1 KL Divergence 0.1577
KL Divergence Kullback-Leibler divergence
Wireless Networking and Communications Group
14
Results on Measured RFI Data
14

• 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
15

• 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 keyingRaised cosine pulse10
  samples/symbol10 symbols/pulse length
• Composite of transmitted and received MPEG II
  video streams
• http//www.ece.utexas.edu/bevans/projects/rfi/tal
  ks/video_demo19dB_correlation.wmv
• Shows degradation of video quality over impulsive
  channels with standard receivers (based on
  Gaussian noise assumption)

Additive Class A Noise Value
Overlap index (A) 0.35
Gaussian factor (G) 0.001
SNR 19 dB
Wireless Networking and Communications Group
16
Filtering and Detection

• Assumption
• Multiple samples of the received signal are
  available
• N Path Diversity Miller, 1972
• Oversampling by N Middleton, 1977

16
Impulsive Noise
Pulse Shaping
Pre-Filtering
Matched Filter
Detection Rule
Middleton Class A noise
Symmetric Alpha Stable noise

• Filtering
• Wiener Filtering (Linear)
• Detection
• Correlation Receiver (Linear)
• Bayesian DetectorSpaulding Middleton, 1977
• Small Signal Approximation to Bayesian
  detectorSpaulding Middleton, 1977

• Filtering
• Myriad Filtering
• Optimal Myriad Gonzalez Arce, 2001
• Selection Myriad
• Hole Punching Ambike et al., 1994
• Detection
• Correlation Receiver (Linear)
• MAP approximationKuruoglu, 1998

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Results Class A Detection
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Communication Performance
Binary Phase Shift Keying
Pulse shapeRaised cosine10 samples per symbol10 symbols per pulse ChannelA 0.35? 0.5 10-3Memoryless
Method Comp. Complexity Detection Perform.
Correl. Low Low
Wiener Medium Low
Bayesian S.S. Approx. Medium High
Bayesian High High
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Results Alpha Stable Detection
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Communication Performance
Same transmitter settings as previous slide
Method Comp. Complexity Detection Perform.
Hole Punching Low Medium
Selection Myriad Low Medium
MAP Approx. Medium High
Optimal Myriad High Medium
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Use dispersion parameter g in place of noise
variance to generalize SNR
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Video over Impulsive Channels 2
19

• 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 keyingRaised 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/tal
  ks/video_demo19dB.wmv

Additive Class A Noise Value
Overlap index (A) 0.35
Gaussian factor (G) 0.001
SNR 19 dB
Wireless Networking and Communications Group
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Video over Impulsive Channels 2

• Structural similarity measure Wang, Bovik,
  Sheikh Simoncelli, 2004
• Score is 0,1 where higher means better video
  quality

Bit error rates for 50 million bits sent 6 x
10-6 for correlation receiver 0 for RFI
mitigating receiver (Bayesian)
Frame number
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Extensions to MIMO systems
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Radio Frequency Interference Modeling and Receiver Design for MIMO systems Radio Frequency Interference Modeling and Receiver Design for MIMO systems Radio Frequency Interference Modeling and Receiver Design for MIMO systems Radio Frequency Interference Modeling and Receiver Design for MIMO systems
RFI Model Spatial Corr. Physical Model Comments
Middleton Class A No Yes Uni-variate model Assume independent or uncorrelated noise for multiple antennas
Middleton Class A Receiver design Gao Tepedelenlioglu, 2007 Space-Time CodingLi, Wang Zhou, 2004 Performance degradation in receivers Receiver design Gao Tepedelenlioglu, 2007 Space-Time CodingLi, Wang Zhou, 2004 Performance degradation in receivers Receiver design Gao Tepedelenlioglu, 2007 Space-Time CodingLi, Wang Zhou, 2004 Performance degradation in receivers
Weighted Mixture of Gaussian Densities Yes No Not derived based on physical principles
Weighted Mixture of Gaussian Densities Receiver designBlum et al., 1997 Adaptive Receiver Design Receiver designBlum et al., 1997 Adaptive Receiver Design Receiver designBlum et al., 1997 Adaptive Receiver Design
Bivariate Middleton Class AMcDonald Blum, 1997 Yes Yes Extensions of Class A model to two-antenna systems
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Our Contributions
22
2 x 2 MIMO receiver design in the presence of
RFIGulati, Chopra, Heath, Evans, Tinsley Lin,
Globecom 2008
RFI Modeling Evaluated fit of measured RFI data to the bivariate Middleton Class A model McDonald Blum, 1997 Includes noise correlation between two antennas
Parameter Estimation Derived parameter estimation algorithm based on the method of moments (sixth order moments)
Performance Analysis Demonstrated communication performance degradation of conventional receivers in presence of RFI Bounds on communication performanceChopra , Gulati, Evans, Tinsley, and Sreerama, ICASSP 2009
Receiver Design Derived Maximum Likelihood (ML) receiver Derived two sub-optimal ML receivers with reduced complexity
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Results RFI Mitigation in 2 x 2 MIMO
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Improvement in communication performance over
conventional Gaussian ML receiver at symbol error
rate of 10-2
A Noise Characteristic Improve-ment
0.01 Highly Impulsive 15 dB
0.1 Moderately Impulsive 8 dB
1 Nearly Gaussian 0.5 dB
Communication Performance (A 0.1, ?1 0.01,
?2 0.1, k 0.4)
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Results RFI Mitigation in 2 x 2 MIMO
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Complexity Analysis for decoding M-level QAM
modulated signal
Receiver Quadratic Forms Exponential Comparisons
Gaussian ML M2 0 0
Optimal ML 2M2 2M2 0
Sub-optimal ML (Four-Piece) 2M2 0 2M2
Sub-optimal ML (Two-Piece) 2M2 0 M2
Complexity Analysis
Communication Performance (A 0.1, ?1 0.01,
?2 0.1, k 0.4)
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Co-Channel Interference Modeling
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25

• Region of interferer locations determines
  interference model Gulati, Chopra, Evans
  Tinsley, Globecom 2009

Symmetric Alpha Stable
Middleton Class A
Wireless Networking and Communications Group
26
Co-Channel Interference Modeling
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26

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

Case 3-a Cellular network (mobile user)
Case 1 Ad-hoc network
Wireless Networking and Communications Group
27
Conclusions
27

• 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

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RFI Mitigation Toolbox
28

• Provides a simulation environment for
• RFI generation
• Parameter estimation algorithms
• Filtering and detection methods
• Demos for communication performance analysis
• Latest Toolbox Release
• Version 1.3, Aug 26th 2009

Snapshot of a demo
http//users.ece.utexas.edu/bevans/projects/rfi/s
oftware/index.html
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Other Contributions
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• 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 Website
• http//users.ece.utexas.edu/bevans/projects/rfi/
  index.html

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Future Work
30

• 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

• Thank You.
• Questions ?

32
References
32

• 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.
  12061221, 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. 5557, 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. 438441, 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)220229, January 2007.

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Backup Slides
34

• 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

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35
Common Spectral Occupancy
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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 2.52.69 3.33.8 5.7255.85 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
36
Impact of RFI
36

• 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)

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37
Impact of LCD clock on 802.11g
37

• Pixel clock 65 MHz
• LCD Interferers and 802.11g center frequencies

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LCD Interferers 802.11g Channel Center Frequency Difference of Interference from Center Frequencies Impact
2.410 GHz Channel 1 2.412 GHz 2 MHz Significant
2.442 GHz Channel 7 2.442 GHz 0 MHz Severe
2.475 GHz Channel 11 2.462 GHz 13 MHz Just outside Ch. 11. Impact minor
38
Middleton Class A, B and C Models
38
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• 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)

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39
Middleton Class B Model
39

• Envelope statistics
• Envelope exceedence probability density (APD),
  which is 1 cumulative distribution function
  (CDF)

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40
Middleton Class B Model (cont)
40

• Middleton Class B envelope statistics

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41
Middleton Class B Model (cont)
41

• Parameters for Middleton Class B model

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42
Accuracy of Middleton Noise Models
42
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Magnetic Field Strength, H (dB relative to
microamp per meter rms)?
e0 (dB gt erms)?
Percentage of Time Ordinate is Exceeded
P(e gt e0)?
Soviet high power over-the-horizon radar
interference Middleton, 1999
Fluorescent lights in mine shop office
interference Middleton, 1999
43
Symmetric Alpha Stable PDF
43

• 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

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44
Symmetric Alpha Stable Model
44

• Heavy tailed distribution

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Density functions for symmetric alpha stable
distributions for different values of
characteristic exponent alpha a) overall density
and b) the tails of densities
45
Parameter Estimation Middleton Class A
45

• 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 ( AG). Root 2nd order
  polynomial.
• Given K, maximize A. Root 4th order polynomial

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Results
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46
Expectation Maximization Overview
46
Return
47
Results EM Estimator for Class A
47
Return
Iterations for Parameter A to Converge
Normalized Mean-Squared Error in A
K A G
PDFs with 11 summation terms 50 simulation runs
per setting
1000 data samples Convergence criterion
48
Results EM Estimator for Class A
48
Return

• For convergence for A ? 10-2, 1, worst-case
  number of iterations for A 1
• Estimation accuracy vs. number of iterations
  tradeoff

49
Parameter Estimation Symmetric Alpha Stable
49

• 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 Frechets 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
  (N10,000)

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Results
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50
Parameter Est. Symmetric Alpha Stable Results
50
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• Data length (N) of 10,000 samples
• Results averaged over 100 simulation runs
• Estimate a and mean g directly from data
• Estimate variance g from a and d estimates

Mean squared error in estimate of characteristic
exponent a
51
Parameter Est. Symmetric Alpha Stable Results
51
Return
Mean squared error in estimate of dispersion
(variance) ?
Mean squared error in estimate of localization
(mean) ?
52
Extreme Order Statistics
52
Return
53
Parameter Estimators for Alpha Stable
53
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0 lt p lt a
54
Filtering and Detection
54

• 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

Impulsive Noise
Pulse Shaping
Pre-Filtering
Matched Filter
Detection Rule
N samples per symbol
55
Wiener Filtering
55

• Optimal in mean squared error sense in presence
  of Gaussian noise

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Model

d(n) desired signald(n) filtered
signale(n) error w(n) Wiener filter x(n)
corrupted signalz(n) noise
Design
Minimize Mean-Squared Error E e(n)2
56
Wiener Filter Design
56

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

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desired signal d(n)power spectrum ?(e j
?) correlation of d and x rdx(n)autocorrelat
ion of x rx(n)Wiener FIR Filter w(n)
corrupted signal x(n)noise z(n)?
57
Results Wiener Filtering
57

• 100-tap FIR Filter

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Pulse shape10 samples per symbol10 symbols per
pulse
ChannelA 0.35? 0.5 10-3SNR -10
dBMemoryless
58
MAP Detection for Class A
58

• Hard decision
• Bayesian formulation Spaulding Middleton,
  1977
• Equally probable source

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59
MAP Detection for Class A Small Signal Approx.
59

• Expand noise PDF pZ(z) by Taylor series about Sj
  0 (j1,2)?
• Approximate MAP detection rule
• Logarithmic non-linearity correlation receiver
• Near-optimal for small amplitude signals

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We use 100 terms of the series expansion
ford/dxi ln pZ(xi) in simulations
60
Incoherent Detection
60

• Bayesian formulation Spaulding Middleton,
  1997, pt. II
• Small signal approximation

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Correlation receiver
61
Filtering for Alpha Stable Noise
61

• 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

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62
Filtering for Alpha Stable Noise (Cont..)
62

• Myriad filter implementation
• Given a window of samples, x1,,xN, find ß ?
  xmin, xmax
• Optimal Myriad algorithm
• Differentiate objective function polynomial p(ß)
  with respect to ß
• Find roots and retain real roots
• Evaluate p(ß) at real roots and extreme points
• Output ß that gives smallest value of p(ß)
• Selection Myriad (reduced complexity)
• Use x1, , xN as the possible values of ß
• Pick value that minimizes objective function p(ß)

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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

Return
64
MAP Detection for Alpha Stable PDF Approx.
64

• SaS random variable Z with parameters a , d, g
  can be written Z X Y½ Kuruoglu, 1998
• X is zero-mean Gaussian with variance 2 g
• Y is positive stable random variable with
  parameters depending on a
• PDF of Z can be written as a mixture model of N
  GaussiansKuruoglu, 1998
• Mean d 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

Return
65
Results Alpha Stable Detection
65
Return
66
Complexity Analysis for Alpha Stable Detection
66
Return
Method Complexity per symbol Analysis
Hole Puncher Correlation Receiver O(NS) A decision needs to be made about each sample.
Optimal Myriad Correlation Receiver O(NW3S) Due to polynomial rooting which is equivalent to Eigen-value decomposition.
Selection Myriad Correlation Receiver O(NW2S) Evaluation of the myriad function and comparing it.
MAP Approximation O(MNS) Evaluating approximate pdf(M is number of Gaussians in mixture)
67
Bivariate Middleton Class A Model
67

• Joint spatial distribution

Return
Parameter Description Typical Range
Overlap Index. Product of average number of emissions per second and mean duration of typical emission
Ratio of Gaussian to non-Gaussian component intensity at each of the two antennas
Correlation coefficient between antenna observations
68
Results on Measured RFI Data
68
Return

• 50,000 baseband noise samples represent broadband
  interference

Estimated Parameters Estimated Parameters Estimated Parameters
Bivariate Middleton Class A Bivariate Middleton Class A Bivariate Middleton Class A
Overlap Index (A) 0.313 2D-KL Divergence1.004
Gaussian Factor (G1) 0.105 2D-KL Divergence1.004
Gaussian Factor (G2) 0.101 2D-KL Divergence1.004
Correlation (k) -0.085 2D-KL Divergence1.004
Bivariate Gaussian Bivariate Gaussian Bivariate Gaussian
Mean (µ) 0 2D-KL Divergence1.6682
Variance (s1) 1 2D-KL Divergence1.6682
Variance (s2) 1 2D-KL Divergence1.6682
Correlation (k) -0.085 2D-KL Divergence1.6682
Marginal PDFs of measured data compared with
estimated model densities
69
System Model
69
Return

• 2 x 2 MIMO System
• Maximum Likelihood (ML) receiver
• Log-likelihood function

Sub-optimal ML Receivers approximate
70
Sub-Optimal ML Receivers
70

• Two-piece linear approximation
• Four-piece linear approximation

Return
Approximation of
chosen to minimize
71
Results Performance Degradation
71

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

Return

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

Severe degradation in communication performance
in high-SNR regimes
72
Performance Bounds (Single Antenna)
72

• Channel capacity

Return
System Model
Case I Shannon Capacity in presence of additive white Gaussian noise
Case II (Upper Bound) Capacity in the presence of Class A noise Assumes 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 noise Assumes input has Gaussian distribution (e.g. bit interleaved coded modulation (BICM) or OFDM modulation Haring, 2003)
73
Performance Bounds (Single Antenna)
73

• Channel capacity in presence of RFI

Return
System Model
Capacity
ParametersA 0.1, G 10-3
74
Performance Bounds (Single Antenna)
74

• Probability of error for uncoded transmissions

Return
Haring Vinck, 2002
BPSK uncoded transmission One sample per symbol A
0.1, G 10-3
75
Performance Bounds (Single Antenna)
75

• Chernoff factors for coded transmissions

Return
PEP Pairwise error probability N Size of the
codeword Chernoff factor Equally likely
transmission for symbols
76
System Model
76
Return
77
Performance Bounds (2x2 MIMO)
77

• Channel capacity

Return
System Model
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 noise Assumes input has Gaussian distribution
78
Performance Bounds (2x2 MIMO)
78

• Channel capacity in presence of RFI for 2x2 MIMO

Return
System Model
Capacity
ParametersA 0.1, G1 0.01, G2 0.1, k 0.4
79
Performance Bounds (2x2 MIMO)
79

• Probability of symbol error for uncoded
  transmissions

Return
Pe Probability of symbol error S Transmitted
code vector D(S) Decision regions for MAP
detector Equally likely transmission for symbols
ParametersA 0.1, G1 0.01, G2 0.1, k 0.4
80
Performance Bounds (2x2 MIMO)
80

• Chernoff factors for coded transmissions

Return
PEP Pairwise error probabilityN Size of the
codewordChernoff factorEqually likely
transmission for symbols
ParametersG1 0.01, G2 0.1, k 0.4
81
Performance Bounds (2x2 MIMO)
81

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

Return
82
Performance Bounds (2x2 MIMO)
82

• Cutoff rate

Return
83
Extensions to Multicarrier Systems
83
Return

• 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