Talk at Texas Instruments in Dallas, Texas

Talk at Texas Instruments in Dallas, Texas
Wireless Networking and Communications Group Statistical Signal Processing for Sensing and Mitigating Impulsive Noise in Communication Receivers Prof. Brian L. Evans Lead Graduate Students Aditya Chopra, Kapil Gulati and Marcel Nassar In collaboration with Eddie Xintian Lin, Alberto Alcocer Ochoa, Chaitanya Sreerama and Keith R. Tinsley at Intel Labs 12 Jan 2010 Talk at Texas Instruments in Dallas, Texas

Outline Introduction Single carrier single antenna systems
2 Introduction Single carrier single antenna systems Radio frequency interference modeling Estimation of interference model parameters Filtering/detection Multi-input multi-output (MIMO) single carrier systems Co-channel interference modeling Conclusions Future work Wireless Networking and Communications Group

Talk focuses on modeling asynchronous non-periodic noise
Impulsive Noise Almost instantaneous (impulse-like) phenomenon Unwanted “clicks” and “pops” in an audio recording Non-Gaussian statistics Models electromagnetic interference through conduction as well as induction (via radiation) Example sources Stopping and starting of electrical devices Clocks and harmonics Communication transmission Talk focuses on modeling asynchronous non-periodic noise

Radio Frequency Interference
Electromagnetic interference from radiation Limits wireless communication performance Applications of RFI modeling Sense and mitigate strategies for coexistence of wireless networks and services Sense and avoid strategies for cognitive radio We focus on sense and mitigate strategies for wireless receivers embedded in notebooks Platform noise from user’s computer subsystems Co-channel interference from other in-band wireless networks and services

Computational Platform Noise
5 Backup Within computing platforms, wireless transceivers experience radio frequency interference from clocks and busses Objectives Develop offline methods to improve communication performance in presence of computer platform RFI Develop adaptive online algorithms for these methods Approach Statistical modeling of RFI Filtering/detection based on estimated model parameters We will use noise and interference interchangeably Wireless Networking and Communications Group

Impact of RFI 6 Impact of LCD noise on throughput for an IEEE g embedded wireless receiver [Shi, Bettner, Chinn, Slattery & Dong, 2006] Backup Backup Wireless Networking and Communications Group

Statistical Modeling of RFI
7 Radio frequency interference Sum of independent radiation events Predominantly non-Gaussian impulsive statistics Key statistical-physical models Middleton Class A, B, C models Independent of physical conditions (canonical) Sum of independent Gaussian and Poisson interference Symmetric Alpha Stable models Approximation of Middleton Class B model Backup Backup Wireless Networking and Communications Group

Assumptions for RFI Modeling
8 Key assumptions for Middleton and Alpha Stable models [Middleton, 1977][Furutsu & Ishida, 1961] Infinitely many potential interfering sources with same effective radiation power Power law propagation loss Poisson field of interferers with uniform intensity l Pr(number of interferers = M |area R) ~ Poisson(M; lR) Uniformly distributed emission times Temporally independent (at each sample time) Limitations Alpha Stable models do not include thermal noise Temporal dependence may exist Wireless Networking and Communications Group

Our Contributions 9 Mitigation of computational platform noise in single carrier, single antenna systems [Nassar, Gulati, DeYoung, Evans & Tinsley, ICASSP 2008, JSPS 2009] Computer Platform Noise Modelling Evaluate fit of measured RFI data to noise models Middleton Class A model Symmetric Alpha Stable Parameter Estimation Evaluate estimation accuracy vs complexity tradeoffs Filtering / Detection Evaluate communication performance vs complexity tradeoffs Middleton Class A: Correlation receiver, Wiener filtering, and Bayesian detector Symmetric Alpha Stable: Myriad filtering, hole punching, and Bayesian detector Wireless Networking and Communications Group

Middleton Class A model
1010 Probability Density Function PDF for A = 0.15, = 0.8 Parameter 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] Wireless Networking and Communications Group

Symmetric Alpha Stable Model
1111 Characteristic Function Closed-form PDF expression only for α = 1 (Cauchy), α = 2 (Gaussian), α = 1/2 (Levy), α = 0 (not very useful) Approximate PDF using inverse transform of power series expansion Second-order moments do not exist for α < 2 Generally, moments of order > α do not exist Backup PDF for  = 1.5,  = 0,  = 10 Backup Parameter Description Range Characteristic Exponent. Amount of impulsiveness Localization. Analogous to mean Dispersion. Analogous to variance Wireless Networking and Communications Group

Example Power Spectral Densities
Middleton Class A Symmetric Alpha Stable Overlap Index (A) = 0.15 Gaussian Factor (G) = 0.1 Characteristic Exponent (a) = 1.5 Localization (d) = 0 Dispersion (g) = 10 Simulated Densities

Estimation of Noise Model Parameters
13 Middleton Class A model Based on Expectation Maximization [Zabin & Poor, 1991] Find roots of second and fourth order polynomials at each iteration Advantage: Small sample size is required (~1000 samples) Disadvantage: Iterative algorithm, computationally intensive Symmetric Alpha Stable Model Based on Extreme Order Statistics [Tsihrintzis & Nikias, 1996] Parameter estimators require computations similar to mean and standard deviation computations Advantage: Fast / computationally efficient (non-iterative) Disadvantage: Requires large set of data samples (~10000 samples) Backup Backup Wireless Networking and Communications Group

Results on Measured RFI Data
14 14 25 radiated computer platform RFI data sets from Intel 50,000 samples taken at 100 MSPS Estimated Parameters for Data Set #18 Symmetric Alpha Stable Model Localization (δ) 0.0065 KL Divergence 0.0308 Characteristic exp. (α) 1.4329 Dispersion (γ) 0.2701 Middleton Class A Model Overlap Index (A) 0.0854 0.0494 Gaussian Factor (Γ) 0.6231 Gaussian Model Mean (µ) 0.1577 Variance (σ2) 1 KL Divergence: Kullback-Leibler divergence Wireless Networking and Communications Group

15 Best fit for 25 data sets under different platform RFI conditions KL divergence plotted for three candidate distributions vs. data set number Smaller KL value means closer fit Gaussian Class A Alpha Stable

Video over Impulsive Channels
16 Video demonstration for MPEG II video stream 10.2 MB compressed stream from camera (142 MB uncompressed) Compressed file sent over additive impulsive noise channel Binary phase shift keying Raised cosine pulse 10 samples/symbol 10 symbols/pulse length Composite of transmitted and received MPEG II video streams 9dB_correlation.wmv Shows degradation of video quality over impulsive channels with standard receivers (based on Gaussian noise assumption) Additive Class A Noise Value Overlap index (A) 0.35 Gaussian factor (G) 0.001 SNR 19 dB Wireless Networking and Communications Group

Filtering and Detection
Assumption Multiple samples of the received signal are available N Path Diversity [Miller, 1972] Oversampling by N [Middleton, 1977] 17 Impulsive Noise Pulse Shaping Pre-Filtering Matched Filter Detection Rule Middleton Class A noise Symmetric Alpha Stable noise Filtering Wiener Filtering (Linear) Detection Correlation Receiver (Linear) Bayesian Detector [Spaulding & Middleton, 1977] Small Signal Approximation to Bayesian detector [Spaulding & Middleton, 1977] Filtering Myriad Filtering Optimal Myriad [Gonzalez & Arce, 2001] Selection Myriad Hole Punching [Ambike et al., 1994] Detection Correlation Receiver (Linear) MAP approximation [Kuruoglu, 1998] Backup Backup Backup Backup Backup Backup Wireless Networking and Communications Group

Results: Class A Detection
18 Communication Performance Binary Phase Shift Keying Pulse shape Raised cosine 10 samples per symbol 10 symbols per pulse Channel A = 0.35  = 0.5 × 10-3 Memoryless Method Comp. Complexity Detection Perform. Correl. Low Wiener Medium Bayesian S.S. Approx. High Bayesian Backup Backup Backup Wireless Networking and Communications Group

Results: Alpha Stable Detection
19 Backup Communication Performance Same transmitter settings as previous slide Method Comp. Complexity Detection Perform. Hole Punching Low Medium Selection Myriad MAP Approx. High Optimal Myriad Backup Backup Backupc Backup Backup Use dispersion parameter g in place of noise variance to generalize SNR Wireless Networking and Communications Group

Video over Impulsive Channels #2
20 Video demonstration for MPEG II video stream revisited 5.9 MB compressed stream from camera (124 MB uncompressed) Compressed file sent over additive impulsive noise channel Binary phase shift keying Raised cosine pulse 10 samples/symbol 10 symbols/pulse length Composite of transmitted video stream, video stream from a correlation receiver based on Gaussian noise assumption, and video stream for a Bayesian receiver tuned to impulsive noise 9dB.wmv Additive Class A Noise Value Overlap index (A) 0.35 Gaussian factor (G) 0.001 SNR 19 dB Wireless Networking and Communications Group

Video over Impulsive Channels #2
Structural similarity measure [Wang, Bovik, Sheikh & Simoncelli, 2004] Score is [0,1] where higher means better video quality Bit error rates for ~50 million bits sent: 6 x 10-6 for correlation receiver 0 for RFI mitigating receiver (Bayesian) Frame number

Extensions to MIMO systems
22 Radio Frequency Interference Modeling and Receiver Design for MIMO systems RFI Model Spatial Corr. Physical Model Comments Middleton Class A No Yes Uni-variate model Assume independent or uncorrelated noise for multiple antennas Receiver design: [Gao & Tepedelenlioglu, 2007] Space-Time Coding [Li, Wang & Zhou, 2004] Performance degradation in receivers Weighted Mixture of Gaussian Densities Not derived based on physical principles Receiver design: [Blum et al., 1997] Adaptive Receiver Design Bivariate Middleton Class A [McDonald & Blum, 1997] Extensions of Class A model to two-antenna systems Backup Wireless Networking and Communications Group

Our Contributions 23 2 x 2 MIMO receiver design in the presence of RFI [Gulati, Chopra, Heath, Evans, Tinsley & Lin, Globecom 2008] RFI Modeling Evaluated fit of measured RFI data to the bivariate Middleton Class A model [McDonald & Blum, 1997] Includes noise correlation between two antennas Parameter Estimation Derived parameter estimation algorithm based on the method of moments (sixth order moments) Performance Analysis Demonstrated communication performance degradation of conventional receivers in presence of RFI Bounds on communication performance [Chopra , Gulati, Evans, Tinsley, and Sreerama, ICASSP 2009] Receiver Design Derived Maximum Likelihood (ML) receiver Derived two sub-optimal ML receivers with reduced complexity Backup Backup Backup Wireless Networking and Communications Group

Results: RFI Mitigation in 2 x 2 MIMO
24 Improvement in communication performance over conventional Gaussian ML receiver at symbol error rate of 10-2 A Noise Characteristic Improve-ment 0.01 Highly Impulsive ~15 dB 0.1 Moderately Impulsive ~8 dB 1 Nearly Gaussian ~0.5 dB Communication Performance (A = 0.1, 1= 0.01, 2= 0.1, k = 0.4) Wireless Networking and Communications Group

Results: RFI Mitigation in 2 x 2 MIMO
25 Complexity Analysis for decoding M-level QAM modulated signal Receiver Quadratic Forms Exponential Comparisons Gaussian ML M2 Optimal ML 2M2 Sub-optimal ML (Four-Piece) Sub-optimal ML (Two-Piece) Complexity Analysis Communication Performance (A = 0.1, 1= 0.01, 2= 0.1, k = 0.4) Wireless Networking and Communications Group

Co-Channel Interference Modeling
26 26 Region of interferer locations determines interference model [Gulati, Chopra, Evans & Tinsley, Globecom 2009] Symmetric Alpha Stable Middleton Class A Wireless Networking and Communications Group

Co-Channel Interference Modeling
27 27 Propose unified framework to derive narrowband interference models for ad-hoc and cellular network environments Key result: tail probabilities (one minus cumulative distribution function) Case 1: Ad-hoc network Case 3-a: Cellular network (mobile user) Wireless Networking and Communications Group

Conclusions Radio Frequency Interference from computing platform
28 Radio Frequency Interference from computing platform Affects wireless data communication transceivers Models include Middleton and alpha stable distributions RFI mitigation can improve communication performance Single carrier, single antenna systems Linear and non-linear filtering/detection methods explored Single carrier, multiple antenna systems Optimal and sub-optimal receivers designed Bounds on communication performance in presence of RFI Results extend to co-channel interference modeling Wireless Networking and Communications Group

RFI Mitigation Toolbox in MATLAB
29 Provides a simulation environment for RFI generation Parameter estimation algorithms Filtering and detection methods Demos for communication performance analysis Latest Toolbox Release Version 1.4 beta, Dec. 14th 2009 Snapshot of a demo Wireless Networking and Communications Group

Publications and Presentations
30 Journal and conference papers M. Nassar, K. Gulati, M. R. DeYoung, B. L. Evans and K. R. Tinsley, “Mitigating Near-Field Interference in Laptop Embedded Wireless Transceivers”, J. of Signal Proc. Systems, Mar 2009, invited paper. M. Nassar, K. Gulati, A. K. Sujeeth, N. Aghasadeghi, B. L. Evans and K. R. Tinsley, “Mitigating Near-field Interference in Laptop Embedded Wireless Transceivers”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Mar. 30-Apr. 4, 2008, Las Vegas, NV USA. K. Gulati, A. Chopra, R. W. Heath Jr., B. L. Evans, K. R. Tinsley, and X. E. Lin, “MIMO Receiver Design in the Presence of Radio Frequency Interference”, Proc. IEEE Int. Global Communications Conf., Nov. 30-Dec. 4th, 2008, New Orleans, LA USA. A. Chopra, K. Gulati, B. L. Evans, K. R. Tinsley, and C. Sreerama, “Performance Bounds of MIMO Receivers in the Presence of Radio Frequency Interference”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Apr , 2009, Taipei, Taiwan, accepted. K. Gulati, A. Chopra, B. L. Evans and K. R. Tinsley, “Statistical Modeling of Co-Channel Interference”, Proc. IEEE Int. Global Communications Conf., Nov. 30-Dec. 4, 2009, Honolulu, HI USA, accepted. K. Gulati, A. Chopra, B. L. Evans and K. R. Tinsley, “Statistical Modeling of Co-Channel Interference in a Field of Poisson and Poisson Distributed Interferers”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Mar , 2010, Dallas, TX USA. Project Web site Wireless Networking and Communications Group

Future Work Extend RFI modeling for
31 Extend RFI modeling for Adjacent channel interference Multi-antenna systems Temporally correlated interference Multi-input multi-output (MIMO) single carrier systems RFI modeling and receiver design Multicarrier communication systems Coding schemes resilient to RFI System level techniques to reduce computational platform generated RFI Backup Wireless Networking and Communications Group

32 Thank You. Questions ? Wireless Networking and Communications Group

References RFI Modeling Parameter Estimation
33 RFI Modeling [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 , May 1999. [2] K.F. McDonald and R.S. Blum. “A physically-based impulsive noise model for array observations”, Proc. IEEE Asilomar Conference on Signals, Systems& Computers, vol 1, 2-5 Nov [3] K. Furutsu and T. Ishida, “On the theory of amplitude distributions of impulsive random noise,” J. Appl. Phys., vol. 32, no. 7, pp. 1206–1221, 1961. [4] J. Ilow and D . Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers”, IEEE transactions on signal processing, vol. 46, no. 6, pp , 1998. Parameter Estimation [5] 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 , Jan [6] 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 , Jun. 1996 RFI Measurements and Impact [7] J. Shi, A. Bettner, G. Chinn, K. Slattery and X. Dong, "A study of platform EMI from LCD panels - impact on wireless, root causes and mitigation methods,“ IEEE International Symposium on Electromagnetic Compatibility, vol.3, no., pp , Aug. 2006 Wireless Networking and Communications Group

References (cont…) Filtering and Detection
34 Filtering and Detection [8] 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 [9] 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 [10] J.G. Gonzalez and G.R. Arce, “Optimality of the Myriad Filter in Practical Impulsive-Noise Environments”, IEEE Trans. on Signal Processing, vol 49, no. 2, Feb 2001 [11] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of Gaussian noise and impulsive noise modelled as an alpha-stable process,” IEEE Signal Processing Letters, vol. 1, pp. 55–57, Mar [12] J. G. Gonzalez and G. R. Arce, “Optimality of the myriad filter in practical impulsive-noise environments,” IEEE Trans. on Signal Proc, vol. 49, no. 2, pp. 438–441, Feb 2001. [13] E. Kuruoglu, “Signal Processing In Alpha Stable Environments: A Least Lp Approach,” Ph.D. dissertation, University of Cambridge, 1998. [14] J. Haring and A.J. Han Vick, “Iterative Decoding of Codes Over Complex Numbers for Impulsive Noise Channels”, IEEE Trans. On Info. Theory, vol 49, no. 5, May 2003 [15] Ping Gao and C. Tepedelenlioglu. “Space-time coding over mimo channels with impulsive noise”, IEEE Trans. on Wireless Comm., 6(1):220–229, January 2007. Wireless Networking and Communications Group

Backup Slides Most backup slides are linked to the main slides
35 Most backup slides are linked to the main slides Miscellaneous topics not covered in main slides Performance bounds for single carrier single antenna system in presence of RFI Backup Wireless Networking and Communications Group

Common Spectral Occupancy
36 Return Standard Carrier (GHz) Wireless Networking Interfering Clocks and Busses Bluetooth 2.4 Personal Area Network Gigabit Ethernet, PCI Express Bus, LCD clock harmonics IEEE b/g/n Wireless LAN (Wi-Fi) IEEE e 2.5– –3.8 5.725–5.85 Mobile Broadband (Wi-Max) PCI Express Bus, LCD clock harmonics IEEE a 5.2 Wireless Networking and Communications Group

Impact of RFI Calculated in terms of desensitization (“desense”)
37 Calculated in terms of desensitization (“desense”) Interference raises noise floor Receiver sensitivity will degrade to maintain SNR Desensitization levels can exceed 10 dB for a/b/g due to computational platform noise [J. Shi et al., 2006] Case Sudy: b, Channel 2, desense of 11dB More than 50% loss in range Throughput loss up to ~3.5 Mbps for very low receive signal strengths (~ -80 dbm) Return Wireless Networking and Communications Group

Just outside Ch. 11. Impact minor
Impact of LCD clock on g 38 Pixel clock 65 MHz LCD Interferers and g center frequencies Return LCD Interferers 802.11g Channel Center Frequency Difference of Interference from Center Frequencies Impact 2.410 GHz Channel 1 2.412 GHz ~2 MHz Significant 2.442 GHz Channel 7 ~0 MHz Severe 2.475 GHz Channel 11 2.462 GHz ~13 MHz Just outside Ch. 11. Impact minor Wireless Networking and Communications Group

Middleton Class A, B and C Models
39 [Middleton, 1999] Return Class A Narrowband interference (“coherent” reception) Uniquely represented by 2 parameters Class B Broadband interference (“incoherent” reception) Uniquely represented by six parameters Class C Sum of Class A and Class B (approx. Class B) Backup Wireless Networking and Communications Group

Middleton Class B Model
40 Envelope statistics Envelope exceedence probability density (APD), which is 1 – cumulative distribution function (CDF) Return Wireless Networking and Communications Group

Middleton Class B Model (cont…)
41 Middleton Class B envelope statistics Return Wireless Networking and Communications Group

Middleton Class B Model (cont…)
42 Parameters for Middleton Class B model Return Parameters 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 Wireless Networking and Communications Group

Accuracy of Middleton Noise Models
43 Return 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] Wireless Networking and Communications Group

Symmetric Alpha Stable PDF
44 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  = 0 Return Wireless Networking and Communications Group

Symmetric Alpha Stable Model
45 Heavy tailed distribution Return Density functions for symmetric alpha stable distributions for different values of characteristic exponent alpha: a) overall density and b) the tails of densities Wireless Networking and Communications Group

Parameter Estimation: Middleton Class A
46 Expectation Maximization (EM) E Step: Calculate log-likelihood function \w current parameter values M Step: Find parameter set that maximizes log-likelihood function EM Estimator for Class A parameters [Zabin & Poor, 1991] Express envelope statistics as sum of weighted PDFs Maximization step is iterative Given A, maximize K (= AG). Root 2nd order polynomial. Given K, maximize A. Root 4th order polynomial Return Backup Results Backup Wireless Networking and Communications Group

Expectation Maximization Overview
47 Return Wireless Networking and Communications Group

Results: EM Estimator for Class A
48 Return Normalized Mean-Squared Error in A Iterations for Parameter A to Converge K = A G PDFs with 11 summation terms 50 simulation runs per setting 1000 data samples Convergence criterion: Wireless Networking and Communications Group

Results: EM Estimator for Class A
49 Return For convergence for A  [10-2, 1], worst-case number of iterations for A = 1 Estimation accuracy vs. number of iterations tradeoff Wireless Networking and Communications Group

Parameter Estimation: Symmetric Alpha Stable
50 Based on extreme order statistics [Tsihrintzis & Nikias, 1996] PDFs of max and min of sequence of i.i.d. data samples 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) Return Results Backup Wireless Networking and Communications Group

Parameter Est.: Symmetric Alpha Stable Results
51 Return Data length (N) of 10,000 samples Results averaged over 100 simulation runs Estimate α and “mean” g directly from data Estimate “variance” g from α and δ estimates Mean squared error in estimate of characteristic exponent α Wireless Networking and Communications Group

Parameter Est.: Symmetric Alpha Stable Results
52 Return Mean squared error in estimate of localization (“mean”)  Mean squared error in estimate of dispersion (“variance”)  Wireless Networking and Communications Group

Extreme Order Statistics

Parameter Estimators for Alpha Stable
54 Return 0 < p < α Wireless Networking and Communications Group

Filtering and Detection
55 System model Assumptions Multiple samples of the received signal are available N Path Diversity [Miller, 1972] Oversampling by N [Middleton, 1977] Multiple samples increase gains vs. Gaussian case Impulses are isolated events over symbol period Impulsive Noise Pulse Shaping Pre-Filtering Matched Filter Detection Rule N samples per symbol Wireless Networking and Communications Group

Wiener Filtering 56 Optimal in mean squared error sense in presence of Gaussian noise Return Model d(n)‏ z(n)‏ ^ w(n)‏ x(n)‏ ^ d(n): desired signal d(n): filtered signal e(n): error w(n): Wiener filter x(n): corrupted signal z(n): noise Design w(n)‏ x(n)‏ d(n)‏ ^ e(n)‏ Minimize Mean-Squared Error E { |e(n)|2 } Wireless Networking and Communications Group

Wiener Filter Design Infinite Impulse Response (IIR)
57 Infinite Impulse Response (IIR) Finite Impulse Response (FIR) Weiner-Hopf equations for order p-1 Return desired signal: d(n) power spectrum: (e j ) correlation of d and x: rdx(n) autocorrelation of x: rx(n) Wiener FIR Filter: w(n) corrupted signal: x(n) noise: z(n)‏ Wireless Networking and Communications Group

Results: Wiener Filtering
58 100-tap FIR Filter Return 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 =  = 0.5 × SNR = -10 dB Memoryless Wireless Networking and Communications Group

MAP Detection for Class A
59 Hard decision Bayesian formulation [Spaulding & Middleton, 1977] Equally probable source Return Wireless Networking and Communications Group

MAP Detection for Class A: Small Signal Approx.
60 Expand noise PDF pZ(z) by Taylor series about Sj = 0 (j=1,2)‏ Approximate MAP detection rule Logarithmic non-linearity + correlation receiver Near-optimal for small amplitude signals Return We use 100 terms of the series expansion for d/dxi ln pZ(xi) in simulations Correlation Receiver Wireless Networking and Communications Group

Incoherent Detection 61 Bayesian formulation [Spaulding & Middleton, 1997, pt. II] Small signal approximation Return Correlation receiver Wireless Networking and Communications Group

Filtering for Alpha Stable Noise
62 Myriad filtering Sliding window algorithm outputs myriad of a sample window Myriad of order k for samples x1,x2,…,xN [Gonzalez & Arce, 2001] As k decreases, less impulsive noise passes through the myriad filter As k→0, filter tends to mode filter (output value with highest frequency) Empirical Choice of k [Gonzalez & Arce, 2001] Developed for images corrupted by symmetric alpha stable impulsive noise Return Wireless Networking and Communications Group

Filtering for Alpha Stable Noise (Cont..)
63 Myriad filter implementation Given a window of samples, x1,…,xN, find β  [xmin, xmax] Optimal Myriad algorithm Differentiate objective function polynomial p(β) with respect to β Find roots and retain real roots Evaluate p(β) at real roots and extreme points Output β that gives smallest value of p(β) Selection Myriad (reduced complexity) Use x1, …, xN as the possible values of β Pick value that minimizes objective function p(β) Return Wireless Networking and Communications Group

Filtering for Alpha Stable Noise (Cont..)
64 Hole punching (blanking) filters Set sample to 0 when sample exceeds threshold [Ambike, 1994] Large values are impulses and true values can be recovered Replacing large values with zero will not bias (correlation) receiver for two-level constellation If additive noise were purely Gaussian, then the larger the threshold, the lower the detrimental effect on bit error rate Communication performance degrades as constellation size (i.e., number of bits per symbol) increases beyond two Return Wireless Networking and Communications Group

MAP Detection for Alpha Stable: PDF Approx.
65 SαS random variable Z with parameters a , d, g can be written Z = X Y½ [Kuruoglu, 1998] X is zero-mean Gaussian with variance 2 g Y is positive stable random variable with parameters depending on a PDF of Z can be written as a mixture model of N Gaussians [Kuruoglu, 1998] Mean d can be added back in Obtain fY(.) by taking inverse FFT of characteristic function & normalizing Number of mixtures (N) and values of sampling points (vi) are tunable parameters Return Wireless Networking and Communications Group

Results: Alpha Stable Detection

Complexity Analysis for Alpha Stable Detection
67 Return Method Complexity per symbol Analysis Hole Puncher + Correlation Receiver O(N+S) A decision needs to be made about each sample. Optimal Myriad + Correlation Receiver O(NW3+S) Due to polynomial rooting which is equivalent to Eigen-value decomposition. Selection Myriad + Correlation Receiver O(NW2+S) Evaluation of the myriad function and comparing it. MAP Approximation O(MNS) Evaluating approximate pdf (M is number of Gaussians in mixture) Wireless Networking and Communications Group

Bivariate Middleton Class A Model
68 Joint spatial distribution Return Parameter Description Typical Range Overlap Index. Product of average number of emissions per second and mean duration of typical emission Ratio of Gaussian to non-Gaussian component intensity at each of the two antennas Correlation coefficient between antenna observations Wireless Networking and Communications Group

69 Return 50,000 baseband noise samples represent broadband interference Estimated Parameters Bivariate Middleton Class A Overlap Index (A) 0.313 2D- KL Divergence 1.004 Gaussian Factor (G1) 0.105 Gaussian Factor (G2) 0.101 Correlation (k) -0.085 Bivariate Gaussian Mean (µ) 2D- KL Divergence Variance (s1) 1 Variance (s2) Marginal PDFs of measured data compared with estimated model densities Wireless Networking and Communications Group

Sub-optimal ML Receivers
System Model 70 Return 2 x 2 MIMO System Maximum Likelihood (ML) receiver Log-likelihood function Sub-optimal ML Receivers approximate Wireless Networking and Communications Group

Sub-Optimal ML Receivers
71 Two-piece linear approximation Four-piece linear approximation Return Approximation of chosen to minimize Wireless Networking and Communications Group

Results: Performance Degradation
72 Performance degradation in receivers designed assuming additive Gaussian noise in the presence of RFI Return Simulation Parameters 4-QAM for Spatial Multiplexing (SM) transmission mode 16-QAM for Alamouti transmission strategy Noise Parameters: A = 0.1, 1= 0.01, 2= 0.1, k = 0.4 Severe degradation in communication performance in high-SNR regimes Wireless Networking and Communications Group

Performance Bounds (Single Antenna)
73 Channel capacity Return System Model Case I Shannon Capacity in presence of additive white Gaussian noise Case II (Upper Bound) Capacity in the presence of Class A noise Assumes that there exists an input distribution which makes output distribution Gaussian (good approximation in high SNR regimes) Case III (Practical Case) Capacity in presence of Class A noise Assumes input has Gaussian distribution (e.g. bit interleaved coded modulation (BICM) or OFDM modulation [Haring, 2003]) Wireless Networking and Communications Group

74 Channel capacity in presence of RFI Return System Model Capacity Parameters A = 0.1, Γ = 10-3 Wireless Networking and Communications Group

75 Probability of error for uncoded transmissions Return [Haring & Vinck, 2002] BPSK uncoded transmission One sample per symbol A = 0.1, Γ = 10-3 Wireless Networking and Communications Group

76 Chernoff factors for coded transmissions Return PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols Wireless Networking and Communications Group

System Model 77 Return Wireless Networking and Communications Group

Performance Bounds (2x2 MIMO)
78 Channel capacity Return System Model Case I Shannon Capacity in presence of additive white Gaussian noise Case II (Upper Bound) Capacity in presence of bivariate Middleton Class A noise. Assumes that there exists an input distribution which makes output distribution Gaussian for all SNRs. Case III (Practical Case) Capacity in presence of bivariate Middleton Class A noise Assumes input has Gaussian distribution Wireless Networking and Communications Group

79 Channel capacity in presence of RFI for 2x2 MIMO Return System Model Capacity Parameters: A = 0.1, G1 = 0.01, G2 = 0.1, k = 0.4 Wireless Networking and Communications Group

80 Probability of symbol error for uncoded transmissions Return Pe: Probability of symbol error S: Transmitted code vector D(S): Decision regions for MAP detector Equally likely transmission for symbols Parameters: A = 0.1, G1 = 0.01, G2 = 0.1, k = 0.4 Wireless Networking and Communications Group

81 Chernoff factors for coded transmissions Return PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols Parameters: G1 = 0.01, G2 = 0.1, k = 0.4 Wireless Networking and Communications Group

82 Cutoff rates for coded transmissions Similar measure as channel capacity Relates transmission rate (R) to Pe for a length T codes Return Wireless Networking and Communications Group

83 Cutoff rate Return Wireless Networking and Communications Group

Extensions to Multicarrier Systems
84 Impulse noise with impulse event followed by “flat” region Coding may improve communication performance In multicarrier modulation, impulsive event in time domain spreads over all subcarriers, reducing effect of impulse Complex number (CN) codes [Lang, 1963] Unitary transformations Gaussian noise is unaffected (no change in 2-norm Distance) Orthogonal frequency division multiplexing (OFDM) is a special case: Inverse Fourier Transform As number of subcarriers increase, impulsive noise case approaches the Gaussian noise case [Haring 2003] Return Wireless Networking and Communications Group

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Talk at Texas Instruments in Dallas, Texas

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