The University of Texas at Austin, Austin, Texas / Version 1.1
In-Platform Radio Frequency Interference Mitigation for Wireless Communications
Prof. Brian L. EvansGraduate Students / Kapil Gulati and Marcel Nassar
Undergraduate Students / Navid Aghasadeghi and Arvind Sujeeth
Last Updated: May 23, 2007
Version: 1.1
Project website: http://www.ece.utexas.edu/~bevans/projects/rfi/
Table of Contents
I. Introduction 3
II. RFI Noise Modeling 4
Middleton Class A Model 5
Middleton Class B Model 5
Middleton Class C Model 6
Symmetric Alpha Stable (SaS) Model 6
III. Parameter Estimation 7
Middleton Class A Model 7
Symmetric Alpha Stable (SaS) Model 9
IV. Filtering and Detection 13
Wiener Filtering 14
FIR Wiener Formulation 15
IIR Wiener Formulation 16
Simulation Results 16
Coherent Detection 17
Truncation Analysis and Small Signal Approximation of the Bayesian Detection 19
V. Results on Measured Data 21
VI. Conclusion and Future Work 23
VII. REFERENCES 24
I. Introduction
This project seeks to reduce Radio Frequency Interference (RFI) experienced by the wireless data communication transceivers deployed on a computation platform. In particular, we consider RFI generated by the computing platform itself.
Computing platform subsystems have emissions of varying intensities that interfere with wireless data transceivers. Table 1 lists the wireless data communication standards of current interest and the computing platform subsystems that interfere significantly with these standards. The interference from the computing subsystems is dominated by the clocks that drive these systems.With near-field coupling, the interference signals impact the wireless transceivers.
Standard / Carrier (GHz) / Wireless Networking / Example Interfering Computer SubsystemsBluetooth / 2.4 / Personal Area Network / Gigabit Ethernet, PCI Express Bus, Memory, Processor, LCD
IEEE 802.11b/g / 2.4 / Wireless LAN (Wi-Fi) / Gigabit Ethernet, PCI Express Bus, Memory, Processor, LCD
IEEE 802.11n / 2.4 / High-Speed
Wireless LAN / Gigabit Ethernet, PCI Express Bus, Memory, Processor, LCD
IEEE 802.16e / 2.5–2.69
3.3–3.8
5.725–5.85 / Mobile Broadband
(Wi-Max) / PCI Express Bus, Memory, Processor, LCD
IEEE 802.11a / 5.2 / Wireless LAN (Wi-Fi) / PCI Express Bus, Memory, Processor, LCD
Table 1: Example of computer subsystems interfering with wireless standards ([3], [4], [5])
RFI is a combination of independent radiation events, and predominantly has non-Gaussian statistics. The key statistical-physical models used for RFI modeling are the Middleton’s Class A, B and C noise models [6]. Middleton models are well-suited for modeling the predominantly non-Gaussian random processes that arise from the nonlinear phenomena that governs electromagnetic interference. Symmetric Alpha-Stable processes [10] are also considered in this context primarily due to their mathematically tractable form for parameter estimators and communication detectors. In this report, we will use “noise” and “interference” interchangeably as representing unwanted “signals”.
Mitigation of RFI can be achieved by designing an optimum detector or a non-linear filter based on Middleton models for the interference noise. Parameter estimation for the Middleton models [6] can then be used to design these filters or to aid the optimum detector design. Mitigation of RFI would improve communication performance (e.g. bit error rate) and extend the range of the wireless transceivers.
A brief discussion on the Middleton noise models and Symmetric Alpha Stable Processes has been presented in Section II. Section III then discusses the parameter estimation techniques for these models, with emphasis on Middleton’s Class A models and Symmetric Alpha Stable models. Optimum detector design for Middleton Class A interference is then discussed in Section IV. Section V concludes the report and discusses future work and long-term vision for the project.
II. RFI Noise Modeling
Two general approaches for modeling electromagnetic interference (EMI) are through physical modeling and through statistical-physical modeling. In physical modeling, each source of EMI would require a different circuit model. Statistical-physical models, on the other hand, provide universal models for accurately modeling EMI from natural and human-made sources. The key statistical-physical models are the Middleton’s Class A, B and C noise models [6]. Middleton model are the most widely accepted model for RFI primarily since these models are canonical, i.e. their mathematical form is independent of the physical environment. Middleton models are classified with respect to the receiver bandwidth:
Class A: Narrowband Noise: Interference Spectrum is narrower than the receiver bandwidth
Class B: Broadband Noise: Interference Spectrum is wider than the receiver bandwidth
Class C: Mixed Case: Sum of Class A and Class B
Figure 1: Classification of Middleton Noise Models
The envelope statistics for the Middleton models have been derived in [6] and are summarized as follows:
Middleton Class A Model
Hence the Class A model [6] is uniquely determined by the following two parameters:
· is “overlap index”. It is the product of the average number of emissions events impinging on the receiver per second and mean duration of a typical interfering source emission, and Î [10-2, 1] in general.
· is the ratio , where is the intensity of the independent Gaussian component, is intensity of the impulsive non-Gaussian component, and Î [10-6, 1] in general.
Middleton Class B Model
The Class B interference model [6] is analytically more complex since two characteristic functions are now needed to approximate the exact characteristic function. Hence we have two expressions for the envelope density, one for small and intermediate envelope values, the other for the larger values, given as follows
Hence the five parameters that uniquely identify the Class B interference are:
· AB is impulsive index, where AB Î [10-2, 1] in general.
· is the ratio , where is the intensity of the independent Gaussian component, is intensity of the impulsive non-Gaussian component, and where Î [10-6, 1] in general.
· a is spatial density propagation parameter equal to (n - m) / g, such that n is the dimension of space of distribution of noise sources (1, 2, or 3), m is the non-negative power law exponent for the emitting sources, and g is the non-negative power law exponent of their propagation, where aÎ [0, 2] in general.
· Aais an effective index that depends on a, where AaÎ [10-1, 1] in general.
· NI is the scaling factor used in normalizing the process to the measured intensity of the process, where NI Î [10-1, 102] in general.
Middleton Class C Model
Since Class C interference is a sum of the Class A and Class B interference models, no specific derivations for Class C are required. Furthermore, Middleton proved in [6] that in Class C can be approximated to Class B models in most cases. Hence, we limit our discussion to Class A and Class B Middleton Models for the remainder of the report.
Symmetric Alpha Stable (SaS) Model
While Middleton Class A and Class B models [6] are known to accurately model RFI sources, their practical applications are limited due to the intractable form of their distributions. In particular, Class B interference model is difficult to use due to the existence of five parameters and also an empirically determined inflection point () [7]. Hence many authors have considered Symmetric Alpha Stable (SaS) model [10] as an approximation to Middleton Class B model. This approximation is particularly accurate for the case of narrowband reception without a Gaussian component, as well as the case of a symmetric pdf without a Gaussian component.
Symmetric Alpha Stable (SaS) models [10] are used to model the statistical properties of an “impulsive” signal. A random variable is said to have (SaS) distribution if its characteristic function is of the form,
Hence the following three parameters uniquely identify a (SaS) distribution,
· is the characteristic exponent and is a measure of the ”thickness” of the tail of the distribution, where in general
· is the localization parameter. It is the mean when and the median when, , where in general
· is scale parameter or the dispersion and is similar to the variance of the Gaussian distribution, where in general
III. Parameter Estimation
Middleton Class A Model
Middleton proposes two estimators for the Class A model [6]. The first is based on empirical observations derived from the plot of envelope threshold (E0) and the probability distribution, P(E > E0), where E is the instantaneous envelope [7]. Although this method yields surprisingly accurate results, it is not practical since it is based on empirical observations. Also, it is affected by finite data record since we have an estimate of distributions based on the observed sample set.
The second method is based on the moments derived from the observed sample set [7]. Since the analytical expression for the characteristic equation for Class A interference model is known, we can derive the expression for all even-order moments (odd-order moments are zero) and relate the parameter to the moments generated. This yields the following result for the parameter estimates,
where, are the second, fourth and sixth order moments of the envelope data respectively. The accuracy of the estimates is affected by finite data record and usually requires a large number of data samples (~10000). Furthermore, we observed that this estimator for is highly dependent on the observation set and yielded poor estimates in many cases.
An efficient parameter estimation method for Class A model has been developed by Zabin and Poor [9] based on the Expectation Maximization (EM) algorithm [8]. It is based on identifying that we can express the envelope probability density as a sum of weighted probability densities, under the constraint that the sum of the weights is equal to one. Let , where denote the parameter set that has to be estimated. Envelope statistics can then be expressed as
The two-steps, the expectation step (E-Step) and the maximization step (M-Step), of the expectation maximization algorithm [8] is hence given as follows.
E-Step: Evaluate, expected value of the log-likelihood function
M-Step: Determine to maximize
The log-likelihood function can then be expressed as follows [9],
where, are the set of N observations of the envelope, and
The maximization step is then developed as a two-step iterative procedure [9], where we first maximize over assuming thatis known and then vice-versa. The first minimization can be expressed as a polynomial of order 2 in and the second minimization can be expressed as a polynomial of order 4 in (after the linearizing approximation). The two step Maximization step can therefore be solved efficiently as it reduces to finding roots to polynomial equations of order 2 and 4, respectively.
The performance of the EM Estimator developed by Zabin and Poor [9] for Class A model has been shown in Figure 2 and Figure 3 for the estimation of parameter. The estimates were calculated using N = 1000 envelope data samples which were generated synthetically based on the envelope distribution. Note that the envelope pdf is expressed as an infinite sum and only the first 11 terms were used in simulations. The results were averaged over 50 Monte-Carlo simulation runs.
Figure 2: Fractional MSE in estimates of parameter using EM algorithm [9] for Class A model
Figure 3: Number of iterations taken by the EM Estimator [9] for Class A Model
Symmetric Alpha Stable (SaS) Model
An efficient and computationally fast estimator was developed by Tsihrintzis and Nikias [10]. It is based on the asymptotic behavior of the extreme order statistics. Following is a short summary of this concept.
Let be a collection of independent realizations of a random variable with the probability density (pdf) and cumulative density (cdf) . Let andbe the maximum and the minimum in the sequence. Statistics ofand are refereed to as extreme-order statistics of the collection. For alpha-stable model, it can be shown (using the theorem for Feasible Asymptotic Distribution for Extreme-Order Statistics [10]), that the density of maxima and minima ( and ) approach the Frechet distributions [10] as . Hence the estimators for the three-parameters of the alpha-stable interference models are derived as follows [10]:
· Estimator for the localization parameter ():
· Estimator for characteristic exponent ():
First subtract the localization parameter from the observations and then segment the resulting “centered” data into L non-overlapping segments, each of length K = N/L.
Consider the following standard deviations corresponding to the maximum and the minimum of the data segment:
With these definitions, the estimator for the characteristic exponent is given as
· Estimator for the dispersion ():
where, , and p is an arbitrary choice for the order ( of fractional moment.
The performance of the extreme order statistics method by Tsihrintzis and Nikias [10] was observed for (dispersion parameter), (localization parameter) and by varying (characteristic exponent) over its entire range,. Estimates were calculated using N=10000 data samples generated synthetically based on the characteristic function of symmetric alpha stable model. The data was segmented into L = 1250 sets for the estimator for the characteristic exponent. Fractional lower order moments of were used in the estimator for the dispersion parameter by choosing .The mean square error in the estimates for the characteristic exponent, dispersion parameter and the localization parameter has been shown in Figure 4, Figure 5 and Figure 6, respectively. The results were averaged over 100 Monte-Carlo simulation runs.
Figure 4: MSE in the estimates of the Characteristic Exponent for N = 10000 synthetic data samples, true parameters {δ = 10 (localization), γ = 5 (dispersion)}, L = 1250, p = α/3.
Figure 5: MSE in the estimates for the dispersion parameter for for N = 10000 synthetic data samples, true parameters {δ = 10 (localization), γ = 5 (dispersion)}, L = 1250, p = α/3.