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Mitigation of Radio Frequency Interference from the Computer Platform to Improve Wireless 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.

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slide1

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

Preliminary Results

Last Updated May 31, 2007

outline
Outline
  • Problem Definition
  • Noise Modeling
  • Estimation of Noise Model Parameters
  • Filtering and Detection
  • Conclusion
  • Future Work
slide3

I. Problem Definition

  • 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

slide5

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
slide6

[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)

slide7

Middleton Class A Model

Probability densityfunction (pdf)

Envelope statistics

Envelope for Gaussian signal has Rayleigh distribution

slide8

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

slide9

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

slide10

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
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
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).
results of em estimator for class a parameters
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 Parameters

Iterations for Parameter A to Converge

estimation of symmetric alpha stable parameters
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
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 estimator1
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
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 data1
Results on Measured RFI Data
  • Modeling PDF as Symmetric Alpha Stable process

fX(x) - PDF

x – noise amplitude

results on measured rfi data2
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
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

wiener filtering linear filter

^

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 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

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

wiener filtering communication performance
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
Coherent Detection
  • Hard decision
  • Bayesian formulation [Spaulding and Middleton, 1977]

corrupted signal

Decision RuleΛ(X)

H1 or H2

coherent detection1
Coherent Detection
  • Equally probable source
  • Optimal detection rule

N: number of samples in vector X

coherent detection in class a noise with 10 4
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
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 approximation1

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]

v conclusion
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
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
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
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
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
slide36

Middleton Class B Model

Envelope Statistics

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

accuracy of middleton noise models
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 moments

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

slide48

Incoherent Detection

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

Small signal approximation

slide49

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

[Widrow et al., 1975]

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