Preliminary Results

1. Mitigation of Radio Frequency Interferencefrom the Computer Platform to ImproveWireless Data Communication Preliminary Results Last Updated October 1, 2007

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

3. 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

5. 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, 1999] Middleton Class A, B, C Models Class ANarrowband interference (“coherent” reception) Uniquely represented by two parameters Class BBroadband interference (“incoherent” reception) Uniquely represented by six parameters Class CSum of class A and class B (approx. as class B)

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

8. Probability Density Function Middleton Class A Statistics As A → , Class A pdf converges to Gaussian Example for A = 0.15 and G = 0.1 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 = 0and “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 = AΓ). Root 2nd-order polynomial. • Given K, maximize A. Root4th-order poly. (after approximation).

13. PDFs with 11 summation terms 50 simulation runs per setting Convergence criterion: Example learning curve Normalized Mean-Squared Error in A ×10-3 EM Estimator for Class A Parameters Using 1000 Samples 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 • AdvantageFast / 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 αand “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. d = 10 g = 5 Mean squared error in estimate of dispersion (“variance”) g Mean squared error in estimate of localization (“mean”) d Results for Symmetric Alpha Stable Parameter Estimator

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 fX(x) - PDF Normalized MSE = 0.0055 x – noise amplitude

19. IV. Filtering and Detection • 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 Corrupted signal Filtered signal Hypothesis Alternate Adaptive Model Filter Decision Rule We assume perfect estimation of noise model parameters Incoherent case

20. ^ d(n) ^ d(n): desired signald(n): filtered signale(n): error w(n): Wiener filter x(n): corrupted signalz(n): noise d(n): ^ d(n) z(n) d(n) x(n) w(n) d(n) x(n) e(n) w(n) Wiener Filtering – Linear Filter • Optimal in mean squared error sense when noise is Gaussian • Model • Design 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. Raised Cosine Pulse Shape n Transmitted waveform corrupted by Class A interference n Received waveform filtered by Wiener filter n Wiener Filtering – 100-tap FIR Filter Pulse shape10 samples per symbol10 symbols per pulse ChannelA = 0.35G = 0.5 × 10-3SNR = -10 dBMemoryless

23. Wiener Filtering – Communication Performance Pulse shapeRaised cosine10 samples per symbol10 symbols per pulse ChannelA = 0.35G = 0.5 × 10-3Memoryless Bit Error Rate (BER) Optimal Detection RuleDescribed next -10 10 SNR (dB) -40 -20 0 -30

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

25. 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 A = 0.1 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 ford/dxi ln pZ(xi) in simulations

28. Correlation Receiver Coherent Detection –Small Signal Approximation AntipodalA = 0.35G = 0.5×10-3 • Near-optimal for small amplitude signals • Suboptimal for higher amplitude signals 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. 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 • 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. 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 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

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. e2 = e4 = e6 = Class A Parameter Estimation Based on Moments • Moments (as derived from the characteristic equation) • Parameter estimates Odd-order momentsare 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 0 < p < α

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.