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### Mitigation of Radio Frequency Interferencefrom the Computer Platform to ImproveWireless Data Communication

Preliminary Results

Last Updated May 31, 2007

Outline

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

- Within computing platforms, wireless transceivers experience radio frequency interference (RFI) from computer subsystems, esp. from clocks (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

- 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 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)

Probability densityfunction (pdf)

Envelope statistics

Envelope for Gaussian signal has Rayleigh distribution

Middleton Class A Statistics

As A → , Class A pdf converges to Gaussian

Example for A = 0.15 and G = 0.1

Power Spectral Density

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 = 0and “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 = AΓ). Root 2nd-order polynomial.
- Given K, maximize A. Root4th-order poly. (after approximation).

PDFs with 11 summation terms

50 simulation runs per setting

Convergence criterion:

Example learning curve

Normalized Mean-Squared Error in A

×10-3

Results of EM Estimator for Class A ParametersIterations 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
- AdvantageFast / computationally efficient (non-iterative)
- Disadvantage Requires large set of data samples (N ~ 20,000)

Results for Symmetric Alpha Stable Parameter Estimator

Data length (N) was 10,000 samples

Results averaged over 100 simulation runs

Example on this slide (which is continued on next slide) uses g = 5 and d = 10

Mean squared error in estimate of characteristic exponent α

Results for Symmetric Alpha Stable Parameter Estimator

g = 5

d = 10

Mean squared error in estimate of dispersion (“variance”) g

Mean squared error in estimate of localization (“mean”) d

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
- Middleton Class A will model PDF beyond a certain point
- Middleton Class B envelope PDF has same form as Middleton Class A envelope PDF beyond an envelope value (inflection point)
- we expect the envelope PDF to match closely to Middleton Class A envelope PDF beyond the inflection point.

Results on Measured RFI Data

- Modeling PDF as Symmetric Alpha Stable process

fX(x) - PDF

x – noise amplitude

Results on Measured RFI Data

- Modeling envelope PDF using Middleton Class A model

Expected: Envelope PDF’s match beyond a certain envelope

Envelope computed via non-linear lowpass filtering obtained via Teager operator, z[n] = (x[n])2 – x[n-1]x[n+1]

fZ(z) – Envelope PDF

z – noise envelope

IV. Filtering and Detection

- Wiener filtering (linear)
- Requires knowledge of signal and noise statistics
- Provides benchmark for non-linear methods
- Other filtering
- Adaptive noise cancellation
- Nonlinear filtering
- Detection in Middleton Class A and B noise
- Coherent detection [Spaulding & Middleton, 1977]
- Incoherent detection[Spaulding & Middleton, 1977]

Hypothesis

Filtered

signal

Corrupted

signal

Filter

Decision Rule

We assume perfect estimation of noise model parameters

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 }

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)

n

Transmitted waveform corrupted by Class A interference

n

Received waveform filtered by Wiener filter

n

Wiener Filtering – 100-tap FIR FilterPulse shape10 samples per symbol10 symbols per pulse

ChannelA = 0.35G = 0.5 × 10-3SNR = -10 dBMemoryless

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

Coherent Detection

- Hard decision
- Bayesian formulation [Spaulding and Middleton, 1977]

corrupted signal

Decision RuleΛ(X)

H1 or H2

Coherent Detection in Class A Noise with Γ = 10-4

A = 0.1

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 ford/dxi ln pZ(xi) in simulations

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]

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 (26 dB gain for coherent detection)

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
- 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.

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

Envelope Statistics

Envelope exceedance probability density (APD) which is 1 – cumulative distribution function

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

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

Estimator for Alpha-Stable

0 < p < α

- Bayes formulation[Spaulding & Middleton, 1997, pt. II]

Small signal approximation

- 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

s : signals+n0 :corrupted signaln0 : noisen1 : reference inputz : system output

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