1. Preliminary Results
Mitigation of Radio Frequency Interference from the Computer Platform to Improve Wireless Data Communication
Prof. Brian L. Evans
Graduate Students
Kapil Gulati and Marcel Nassar
Undergraduate Students
Navid Aghasadeghi and Arvind Sujeeth
Last Updated October 1, 2007

2. Outline
Problem Definition
Noise Modeling
Estimation of Noise Model Parameters
Filtering and Detection
Conclusion
Future Work

3. We’ll be using noise and interference interchangeably
I. Problem Definition
Within computing platforms, wireless transceivers experience radio frequency interference (RFI) from computer subsystems, esp. from clocks (and their harmonics) and busses
Objectives
Develop offline methods to improve communication performance in the presence of computer platform RFI
Develop adaptive online algorithms for these methods
Approach
Statistical modeling of RFI
Filtering/detection based on estimation of model parameters
We’ll be using noise and interference interchangeably

4. Common Spectral Occupancy
Standard
Carrier (GHz)
Wireless Networking
Example Interfering Computer Subsystems
Bluetooth
2.4
Personal Area Network
Gigabit Ethernet, PCI Express Bus, LCD clock harmonics
IEEE b/g/n
Wireless LAN (Wi-Fi)
IEEE e
2.5– –3.8
5.725–5.85
Mobile Broadband (Wi-Max)
PCI Express Bus, LCD clock harmonics
IEEE a
5.2

5. Statistical-Physical Models (Middleton Class A, B, C)
II. Noise Modeling
RFI is a combination of independent radiation events, and predominantly has non-Gaussian statistics
Statistical-Physical Models (Middleton Class A, B, C)
Independent of physical conditions (universal)
Sum of independent Gaussian and Poisson interference
Models nonlinear phenomena governing electromagnetic interference
Alpha-Stable Processes
Models statistical properties of “impulsive” noise
Approximation to Middleton Class B noise

6. Middleton Class A, B, C Models
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)

7. Envelope for Gaussian signal has Rayleigh distribution
Middleton Class A Model
Probability density function (pdf)
Envelope statistics
Envelope for Gaussian signal has Rayleigh distribution
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]

8. Middleton Class A Statistics
As A → , Class A pdf converges to Gaussian
Example for A = 0.15 and G = 0.1
Probability Density Function
Power Spectral Density

9. Symmetric Alpha Stable Model Characteristic Function:
Parameters
Characteristic exponent indicative of the thickness of the tail of impulsiveness of the noise
Localization parameter (analogous to mean)
Dispersion parameter (analogous to variance)
No closed-form expression for pdf except for α = 1 (Cauchy), α = 2 (Gaussian), α = 1/2 (Levy) and α = 0 (not very useful)
Approximate pdf using inverse transform of power series expansion of characteristic function

10. Symmetric Alpha Stable Statistics
Example: exponent a = 1.5, “mean” d = 0 and “variance” g = 10
×10-4
Probability Density Function
Power Spectral Density

11. III. Estimation of Noise Model Parameters
For the Middleton Class A Model
Expectation maximization (EM) [Zabin & Poor, 1991]
Based on envelope statistics (Middleton)
Based on moments (Middleton)
For the Symmetric Alpha Stable Model
Based on extreme order statistics [Tsihrintzis & Nikias, 1996]
For the Middleton Class B Model
No closed-form estimator exists
Approximate methods based on envelope statistics or moments

12. Estimation of Middleton Class A Model Parameters
Expectation maximization
E: Calculate log-likelihood function w/ current parameter values
M: Find parameter set that maximizes log-likelihood function
EM estimator for Class A parameters [Zabin & Poor, 1991]
Expresses envelope statistics as sum of weighted pdfs
Maximization step is iterative
Given A, maximize K (with K = A Γ). Root 2nd-order polynomial.
Given K, maximize A. Root 4th-order poly. (after approximation).

13. EM Estimator for Class A Parameters Using 1000 Samples
Normalized Mean-Squared Error in A
×10-3
PDFs with 11 summation terms
50 simulation runs per setting
Convergence criterion:
Example learning curve
Iterations for Parameter A to Converge

14. Estimation of Symmetric Alpha Stable Parameters
Based on extreme order statistics [Tsihrintzis & Nikias, 1996]
PDFs of max and min of sequence of independently and identically distributed (IID) data samples follow
PDF of maximum:
PDF of minimum:
Extreme order statistics of Symmetric Alpha Stable pdf approach Frechet’s distribution as N goes to infinity
Parameter estimators then based on simple order statistics
Advantage Fast / computationally efficient (non-iterative)
Disadvantage Requires large set of data samples (N ~ 10,000)

15. Results for Symmetric Alpha Stable Parameter Estimator
Data length (N) was 10,000 samples
Results averaged over 100 simulation runs
Estimate α and “mean” δ directly from data
Estimate “variance” γ from α and δ estimates
Continued next slide
Mean squared error in estimate of characteristic exponent α

16. Results for Symmetric Alpha Stable Parameter Estimator
Mean squared error in estimate of localization (“mean”) d
d = 10
Mean squared error in estimate of dispersion (“variance”) g
g = 5

17. Results on Measured RFI Data
Data set of 80,000 samples collected using 20 GSPS scope
Measured data represents "broadband" noise
Symmetric Alpha Stable Process expected to work well since PDF of measured data is symmetric (approximates Middleton Class B model better)

18. Results on Measured RFI Data
Modeling PDF as Symmetric Alpha Stable process
Estimated Parameters
Localization (δ)
Dispersion (γ)
0.5833
Characteristic Exponent (α)
1.5525
fX(x) - PDF
Normalized MSE =
x – noise amplitude

19. IV. Filtering and Detection
Corrupted
signal
Filtered
signal
Hypothesis
Alternate Adaptive Model
Filter
Decision Rule
Wiener filtering (linear)
Requires knowledge of signal and noise statistics
Provides benchmark for non-linear methods
Detection in Middleton Class A and B noise
Coherent detection [Spaulding & Middleton, 1977]
Nonlinear filtering
Myriad filtering
Particle Filtering
We assume perfect estimation of noise model parameters
Incoherent case

20. Wiener Filtering – Linear Filter
Optimal in mean squared error sense when noise is Gaussian
Model
Design
d(n): desired signal d(n): filtered signal e(n): error w(n): Wiener filter x(n): corrupted signal z(n): noise
d(n): ^
d(n)
z(n) ^
w(n)
x(n)
w(n)
x(n)
d(n) ^
e(n)
Minimize Mean-Squared Error E { |e(n)|2 }

21. Wiener Filtering – Finite Impulse Response (FIR) Case
Wiener-Hopf equations for FIR Wiener filter of order p-1
General solution in frequency domain
desired signal: d(n) power spectrum: F(e j w) correlation of d and x: rdx(n) autocorrelation of x: rx(n) Wiener FIR Filter: w(n) corrupted signal: x(n) noise: z(n)

22. Wiener Filtering – 100-tap FIR Filter
Pulse shape 10 samples per symbol 10 symbols per pulse
Raised Cosine Pulse Shape
Transmitted waveform corrupted by Class A interference
Received waveform filtered by Wiener filter n
Channel A = 0.35 G = 0.5 × 10-3 SNR = -10 dB Memoryless

23. Wiener Filtering – Communication Performance
Pulse shape Raised cosine 10 samples per symbol 10 symbols per pulse
Channel A = 0.35 G = 0.5 × 10-3 Memoryless
Bit Error Rate (BER)
Optimal Detection Rule Described next
-40
-30
-20
-10 10
SNR (dB)

24. Bayesian formulation [Spaulding and Middleton, 1977]
Coherent Detection
Hard decision
Bayesian formulation [Spaulding and Middleton, 1977]
corrupted signal
Decision Rule Λ(X)
H1 or H2

25. Equally probable source
Coherent Detection
Equally probable source
Optimal detection rule
N: number of samples in vector X

26. Coherent Detection in Class A Noise with Γ = 10-4
Correlation Receiver Performance
SNR (dB)
SNR (dB)

27. Coherent Detection – Small Signal Approximation
Expand pdf pZ(z) by Taylor series about Sj = 0 (for j=1,2)
Optimal decision rule & threshold detector for approximation
Optimal detector for approximation is logarithmic nonlinearity followed by correlation receiver (see next slide)
We use 100 terms of the series expansion for d/dxi ln pZ(xi) in simulations

28. Coherent Detection – Small Signal Approximation
Antipodal A = 0.35 G = 0.5×10-3
Near-optimal for small amplitude signals
Suboptimal for higher amplitude signals
Correlation Receiver
Communication performance of approximation vs. upper bound [Spaulding & Middleton, 1977, pt. I]

29. Myriad Filtering – Introduction [Gonzalez & Arce, 2001]
Outputs “sample myriad” of elements in sliding window
Sample myriad
Given set of samples x1, …, xN and linearity parameter k>0, sample myriad of order k is
Linearity parameter k
As k, sample myriad converges to sample average
As k0, sample myriad converges to a mode-type myriad (good performance in highly impulsive noise
As k decreases, filter becomes more resilient to impulsive noise

30. Myriad Filtering – Design [Gonzalez & Arce, 2001]
Optimal in highly impulsive alpha stable distributions
Proof is not constructive for designing myriad filters
Choose k empirically using different graphs or formulas based on alpha stable process parameters such as
Other characteristics
More efficient in terms of impulse noise mitigation than median filter
Tunable parameters which can be adapted to changing environments
Weighted version can be further optimized to suit given application

31. Radio frequency interference from computing platform
V. Conclusion
Radio frequency interference from computing platform
Affects wireless data communication subsystems
Models include Middleton noise models and alpha stable processes
RFI cancellation
Extends range of communication systems
Reduces bit error rates
Initial RFI interference cancellation methods explored
Linear optimal filtering (Wiener)
Optimal detection rules (significant gains at low bit rates)

32. VI. Future Work Offline methods Online methods
Estimator for single symmetric alpha-stable process plus Gaussian
Estimator for mixture of alpha stable processes plus Gaussian (requires blind source separation for 1-D time series)
Estimator for Middleton Class B parameters
Quantify communication performance vs. complexity tradeoffs for Middleton Class A detection
Online methods
Develop fixed-point (embedded) methods for parameter estimation of Middleton Noise models, mixtures of alpha-stable processes
Develop embedded implementations of detection methods

33. 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 , 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 , 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 , 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
[7] J.G. Gonzalez and G.R. Arce, “Optimality of the Myriad Filter in Practical Impulsive-Noise Environments”, IEEE Transactions on Signal Processing, vol 49, no. 2, Feb 2001

34. BACKUP SLIDES

35. Potential Impact
Improve communication performance for wireless data communication subsystems embedded in PCs and laptops
Extend range from the wireless data communication subsystems to the wireless access point
Achieve higher bit rates for the same bit error rate and range, and lower bit error rates for the same bit rate and range
Extend the results to multiple RF sources on a single chip

36. 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 d = 0

37. Middleton Class B Model Envelope Statistics
Envelope exceedance probability density (APD) which is 1 – cumulative distribution function

38. Class B Envelope Statistics

39. Parameters for Middleton Class B Noise
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

40. Accuracy of Middleton Noise Models
Magnetic Field Strength, H (dB relative to
microamp per meter rms)
ε0 (dB > εrms)
Percentage of Time Ordinate is Exceeded
P(ε > ε0)
Soviet high power over-the-horizon radar interference [Middleton, 1999]
Fluorescent lights in mine shop office interference [Middleton, 1999]

41. Class B Exceedance Probability Density Plot

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

43. Class A Parameter Estimation Based on Moments
Moments (as derived from the characteristic equation)
Parameter estimates
e2 =
e4 =
e6 =
Odd-order moments are zero [Middleton, 1999] 2

44. Expectation Maximization Overview

45. Maximum Likelihood for Sum of Densities

46. Results of EM Estimator for Class A Parameters

47. Extreme Order Statistics

48. Estimator for Alpha-Stable

49. Incoherent Detection
Bayes formulation [Spaulding & Middleton, 1997, pt. II]
Small signal approximation

50. Incoherent Detection Optimal Structure:
Incoherent Correlation Detector
The optimal detector for the small signal approximation is basically the correlation receiver preceded by the logarithmic nonlinearity.

51. Coherent Detection – Class A Noise
Comparison of performance of correlation receiver (Gaussian optimal receiver) and nonlinear detector [Spaulding & Middleton, 1997, pt. II]

52. 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= and (N+p-1) C p = 6435.

53. 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
[Widrow et al., 1975]
s : signal s+n0 :corrupted signal n0 : noise n1 : reference input z : system output