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

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

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

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 802. 11 b/g/n Wireless LAN (Wi-Fi) 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

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

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)

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]

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

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

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

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

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).

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

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)

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

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

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)

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

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

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 }

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)

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

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)

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

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

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

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

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]

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

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

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)

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

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

BACKUP SLIDES

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

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

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

Class B Envelope Statistics

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

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]

Class B Exceedance Probability Density Plot

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 e2 = e4 = e6 = Odd-order moments are zero [Middleton, 1999] 2

Expectation Maximization Overview

Maximum Likelihood for Sum of Densities

Results of EM Estimator for Class A Parameters

Extreme Order Statistics

Estimator for Alpha-Stable

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

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.

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

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.

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