1. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Improving Wireless Data Transmission Speed and Reliability to Mobile Computing Platforms in collaboration with Marcel Nassar 1, Kapil Gulati 1, Arvind K. Sujeeth 1, Navid Aghasadeghi 1 and Keith R. Tinsley 2 1 The University of Texas at Austin, Austin, Texas USA 2 System Technology Lab, Intel, Hillsborough, Oregon USA American University of Beirut 15 th July 2008 Improving Wireless Data Transmission Speed and Reliability to Mobile Computing Platforms Prof. Brian L. Evans 1 Preliminary Results

2. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 2 Outline Problem definition Noise modelling Estimation of noise model parameters Filtering and detection Conclusion and future work

3. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 3 Problem Definition Within computing platforms, wireless transceivers experience radio frequency interference (RFI) from clocks/busses  PCI Express busses  LCD clock harmonics Approach Statistical modelling of RFI Filtering/detection based on estimation of model parameters Previous Research Potential reduction in bit error rates by factor of 10 or more [Spaulding & Middleton, 1977] We’ll be using noise and interference interchangeably

4. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 4 Common Spectral Occupancy Standard Carrier (GHz)‏ Wireless Networking Interfering Clocks and Busses Bluetooth2.4 Personal Area Network Gigabit Ethernet, PCI Express Bus, LCD clock harmonics IEEE 802. 11 b/g/n 2.4 Wireless LAN (Wi-Fi)‏ Gigabit Ethernet, PCI Express Bus, LCD clock harmonics IEEE 802.16e- 2005 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 Wireless LAN (Wi-Fi)‏ PCI Express Bus, LCD clock harmonics

5. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 5 Computer Platform Noise Modelling RFI is combination of independent radiation events Has predominantly non-Gaussian statistics Statistical-Physical Models (Middleton Class A, B, C)‏ Independent of physical conditions (universal)‏ Sum of independent Gaussian and Poisson interference Models electromagnetic interference Alpha-Stable Processes Models statistical properties of “impulsive” noise Approximation for Middleton Class B (broadband) noise Backup

6. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Proposed Contributions Computer Platform Noise Modelling Evaluate fit of measured RFI data to noise models Narrowband Interference: Middleton Class A model Broadband Interference: Symmetric Alpha Stable Parameter EstimationEvaluate estimation accuracy vs complexity tradeoffs Filtering / DetectionEvaluate 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 6

7. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 7 Outline Problem definition Noise modelling Estimation of noise model parameters Filtering and detection Conclusion and future work

8. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 8 Middleton Class A Model ParameterDescriptionRange 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] Probability Density Function for A = 0.15,  = 0.8 Power Spectral Density for A = 0.15,  = 0.8

9. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 9 Middleton Class A Model Probability density function (pdf) PDF for A = 0.15,  = 0.8 ParameterDescriptionRange 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]

10. Wireless Networking and Communications Group Department of Electrical and Computer Engineering ParameterDescriptionRange Characteristic Exponent. Amount of impulsiveness Localization. Analogous to mean Dispersion. Analogous to variance 10 Symmetric Alpha Stable Model Probability Density Function for  = 1.5,  = 0 and  = 10 Power Spectral Density for  = 1.5,  = 0 and  = 10

11. Wireless Networking and Communications Group Department of Electrical and Computer Engineering ParameterDescriptionRange Characteristic Exponent. Amount of impulsiveness Localization. Analogous to mean Dispersion. Analogous to variance 11 PDF for  = 1.5,  = 0 and  = 10 Symmetric Alpha Stable Model 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  Does not have second-order moment Backup

12. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 12 Outline Problem definition Noise modelling Estimation of noise model parameters Filtering and detection Conclusion and future work

13. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 13 Estimation of Noise Model Parameters For Middleton Class A Model Expectation maximization (EM) [Zabin & Poor, 1991] Finds roots of second and fourth order polynomials at each iteration Advantage Small sample size required (~1,000 samples)‏ Disadvantage Iterative algorithm, computationally intensive For Symmetric Alpha Stable Model Based on extreme order statistics [Tsihrintzis & Nikias, 1996] Parameter estimators require computations similar to mean and standard deviation. Advantage Fast / computationally efficient (non-iterative)‏ Disadvantage Requires large set of data samples (~10,000 samples)‏ Backup

14. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 14 Results of Measured RFI Data for Broadband Noise Data set of 80,000 samples collected using 20 GSPS scope Backup Estimated Parameters Symmetric Alpha Stable Model Localization (δ)0.0043 Distance 0.0514 Characteristic exp. (α)1.2105 Dispersion (γ)0.2413 Middleton Class A Model Overlap Index (A)0.1036 Distance 0.0825 Gaussian Factor (Γ)0.7763 Gaussian Model Mean (µ)0 Distance 0.2217 Variance (σ 2 )1 Distance: Kullback-Leibler divergence

15. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 15 Expectation-Maximization Estimator for Class A Noise PDFs with 11 summation terms 50 simulation runs per setting 1000 data samples Convergence criterion: Iterations for Parameter A to Converge Normalized Mean-Squared Error in A ×10 -3 K = A 

16. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Expectation-Maximization Estimator for Class A Noise For convergence for A  [10 -2, 1], worst- case number of iterations for A = 1 Estimation accuracy vs. number of iterations tradeoff

17. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 17 Mean squared error in estimate of characteristic exponent  Data length (N) of 10,000 samples Results averaged over 100 simulation runs Estimate α and “mean”  directly from data Estimate “variance” γ from α and δ estimates Symmetric Alpha Stable Parameter Estimator

18. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 18 Symmetric Alpha Stable Parameter Estimator Mean squared error in estimate of dispersion (“variance”)   = 5 Mean squared error in estimate of localization (“mean”)   = 10

19. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 19 Outline Problem definition Noise modelling Estimation of noise model parameters Filtering and detection Conclusion and future work

20. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 20 Filtering and Detection – System Model Signal Model Multiple samples/copies of the received signal are available: N path diversity [Miller, 1972] Oversampling by N [Middleton, 1977] Using multiple samples increases gains vs. Gaussian case because impulses are isolated events over symbol period s[n] g tx [n] v[n] g rx [n] Λ(.)‏ Pulse Shape Pre-Filtering Matched Filter Decision Rule Impulsive Noise Alternate Adaptive Model Backup N samples per symbol

21. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 21 Filtering and Detection – Methods Class A Noise Correlation receiver (linear)‏ Wiener filtering (linear)‏ Coherent detection using MAP (Maximum A Posteriori Probability) detector [Spaulding & Middleton, 1977] Small signal approximation to MAP Detector [Spaulding & Middleton, 1977] Symmetric Alpha Stable Noise Correlation receiver (linear)‏ Myriad filtering [Gonzalez & Arce, 2001] MAP approximation Hole punching We assume perfect estimation of noise model parameters Backup

22. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Class A Detection – Results 22 Pulse shape Raised cosine 10 samples per symbol 10 symbols per pulse Channel A = 0.35  = 0.5 × 10 -3 Memoryless MethodComp.Detection Perform. Correl.Low WienerMediumLow MAP Approx. MediumHigh MAPHigh

23. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 23 Hole Punching (Blanking) for Pre-Filtering Sets sample to 0 when sample exceeds threshold [Ambike, 1994]  Large values are impulses and true value cannot be recovered  Replacing large values with zero will not bias (correlation) receiver for two-level constellations  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

24. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 24 Myriad Filtering for Pre-Filtering Sliding window algorithm outputs myriad of sample window Myriad of order k for samples x 1, x 2, …, x N [Gonzalez & Arce, 2001]  As k decreases, less impulsive noise passes through myriad filter  As k→0, filter tends to mode filter (output value with highest freq.) Empirical choice of k: [Gonzalez & Arce, 2001] Developed for images corrupted by additive symmetric alpha stable impulsive noise

25. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 25 Myriad Filter Implementation Given a window of samples x 1,…,x N, find β  [x min, x max ] Optimal myriad algorithm 1.Differentiate objective function polynomial p(β) with respect to β 2.Find roots and retain real roots 3.Evaluate p(β) at real roots and extremum 4.Output β that gives smallest value of p(β) Selection myriad (reduced complexity) 1.Use x 1, …, x N as the possible values of β 2.Pick value that minimizes objective function p(β) Backup

26. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Symmetric Alpha Stable Detection – Results 26 MethodComp.Detection Perform. Hole Punching LowMedium Selection Myriad LowMedium MAP Approx. MediumHigh Optimal Myriad HighMedium Use dispersion parameter  in place of noise variance to generalize SNR

27. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Conclusion – Proposed Contributions Computer Platform Noise Modelling Evaluate fit of measured RFI data to noise models Narrowband Interference: Middleton Class A model Broadband Interference: Symmetric Alpha Stable Parameter EstimationEvaluate estimation accuracy vs complexity tradeoffs Filtering / DetectionEvaluate 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 27

28. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 28 Conclusion – Filtering and Detection T Class A Communication Performance Complexity MAP approximationHighMedium MAPHigh Correlation receiverLow Wiener filteringLowMedium Symmetric Alpha Stable Communication Performance Complexity MAP approximationHighMedium Selection myriadMediumLow Hole punchingMediumLow Optimal myriadMediumHigh

29. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Conclusion – Contributions Publications 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. Software Releases RFI Mitigation Toolbox Version 1.1 Beta(Released November 21 st, 2007) Version 1.0 (Released September 22 nd, 2007) http://users.ece.utexas.edu/~bevans/projects/rfi/software.html Project Web Site http://users.ece.utexas.edu/~bevans/projects/rfi/index.html 29

30. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Conclusion – Future Work on Impulsive Noise Communication performance bounds on single-carrier single-antenna detection Multi-input multi-output (MIMO) single-carrier receivers  Performance analysis of standard MIMO receivers using multivariate noise models  Optimal and sub-optimal maximum likelihood (ML) 2  2 receiver  To be presented at 2008 Globecom Conference in December Multicarrier receivers Modelling co-channel interference Backup 30

31. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Thank you, Questions?

32. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 32 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 Trans. on Signal Processing, vol 49, no. 2, Feb 2001

33. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 33 References (cont…)‏ [8] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of gaussian noise and impulsive noise modeled as an alpha-stable process,” IEEE Signal Processing Letters, vol. 1, pp. 55–57, Mar. 1994. [9] J. G. Gonzalez and G. R. Arce, “Optimality of the myriad filter in practical impulsive- noise enviroments,” IEEE Trans. on Signal Proc, vol. 49, no. 2, pp. 438–441, Feb 2001. [10] E. Kuruoglu, “Signal Processing In Alpha Stable Environments: A Least Lp Approach,” Ph.D. dissertation, University of Cambridge, 1998. [11] J. Haring and A.J. Han Vick, “Iterative Decoding of Codes Over Complex Numbers for Impuslive Noise Channels”, IEEE Trans. On Info. Theory, vol 49, no. 5, May 2003 [12] G. Beenker, T. Claasen, and P. van Gerwen, “Design of smearing filters for data transmission systems,” IEEE Trans. on Comm., vol. 33, Sept. 1985. [13] G. R. Lang, “Rotational transformation of signals,” IEEE Trans. Inform. Theory, vol. IT–9, pp. 191–198, July 1963. [14] 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. [15] 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. 1997.

34. Wireless Networking and Communications Group Department of Electrical and Computer Engineering BACKUP SLIDES

35. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 35 Potential Impact Improve communication performance for wireless data communication subsystems embedded in PCs and laptops  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 range from wireless data communication subsystems to wireless access point Extend results to multiple RF sources on single chip

36. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 36 Soviet high power over-the-horizon radar interference [Middleton, 1999] Fluorescent lights in mine shop office interference [Middleton, 1999] P(ε > ε 0 )‏ ε 0 (dB > ε rms )‏ Percentage of Time Ordinate is Exceeded Magnetic Field Strength, H (dB relative to microamp per meter rms)‏ Accuracy of Middleton Noise Models

37. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 37 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)‏ [Middleton, 1999] Middleton Class A, B, C Models

38. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 38 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  = 0

39. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 39 Coherent Detection – Small Signal Approximation Expand noise pdf p Z (z) by Taylor series about S j = 0 (j=1,2)‏ Optimal decision rule & threshold detector for approximation Optimal detector for approximation is logarithmic nonlinearity followed by correlation receiver We use 100 terms of the series expansion for d/dx i ln p Z (x i ) in simulations Backup

40. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 40 Filtering and Detection – Alpha Stable Model MAP detection: remove nonlinear filter Decision rule is given by (p(.) is the SαS distribution)‏ Approximations for SαS distribution: MethodShortcomingsReference Series ExpansionPoor approximation when series length shortened [Samorodnitsky, 1988] Polynomial Approx.Poor approximation for small x [Tsihrintzis, 1993] Inverse FFTRipples in tails when α < 1 Simulation Results

41. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 41 MAP Detector – PDF Approximation SαS random variable Z with parameters ,  can be written Z = X Y ½ [Kuruoglu, 1998] X is zero-mean Gaussian with variance 2  Y is positive stable random variable with parameters depending on  Pdf of Z can be written as a mixture model of N Gaussians [Kuruoglu, 1998] Mean  can be added back in Obtain f Y (.) by taking inverse FFT of characteristic function & normalizing Number of mixtures (N) and values of sampling points (v i ) are tunable parameters

42. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 42 Bit Error Rate (BER) Performance in Alpha Stable Noise

43. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 43 Class A Parameter Estimation Based on APD (Exceedance Probability Density) Plot

44. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 44 Class A Parameter Estimation Based on Moments Moments (as derived from the characteristic equation)‏ Parameter estimates 2 e 2 = e 4 = e 6 = Odd-order moments are zero [Middleton, 1999]

45. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 45 Middleton Class B Model Envelope Statistics Envelope exceedance probability density (APD) which is 1 – cumulative distribution function

46. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 46 Class B Envelope Statistics

47. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 47 Parameters for Middleton Class B Noise ParametersDescriptionTypical Range Impulsive Index A B  [10 -2, 1] Ratio of Gaussian to non-Gaussian intensity Γ B  [10 -6, 1] Scaling Factor N I  [10 -1, 10 2 ] Spatial density parameter α  [0, 4] Effective impulsive index dependent on α A α  [10 -2, 1] Inflection point (empirically determined)‏ ε B > 0

48. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 48 Class B Exceedance Probability Density Plot

49. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 49 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). Backup

50. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 50 Expectation Maximization Overview

51. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 51 Maximum Likelihood for Sum of Densities

52. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 52 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)‏ Backup

53. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 53 Extreme Order Statistics

54. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 54 Estimator for Alpha-Stable 0 < p < α

55. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 55 Minimize Mean-Squared Error E { |e(n)| 2 } d(n)‏d(n)‏ z(n)‏z(n)‏ d(n)‏d(n)‏ ^ w(n)‏w(n)‏ x(n)‏x(n)‏ w(n)‏w(n)‏ x(n)‏x(n)‏d(n)‏d(n)‏ ^ d(n)‏d(n)‏ e(n)‏e(n)‏ d(n): desired signal d(n): filtered signal e(n): error w(n): Wiener filter x(n): corrupted signal z(n): noise d(n): ^ Wiener Filtering – Linear Filter Optimal in mean squared error sense when noise is Gaussian Model Design

56. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 56 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:  (e j  ) correlation of d and x: r dx (n) autocorrelation of x: r x (n) Wiener FIR Filter: w(n) corrupted signal: x(n) noise: z(n)‏

57. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 57 Wiener Filtering – 100-tap FIR Filter Channel A = 0.35  = 0.5 × 10 -3 SNR = -10 dB Memoryless 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 n n

58. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 58 Incoherent Detection Bayes formulation [Spaulding & Middleton, 1997, pt. II] Small signal approximation

59. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 59 Incoherent Detection Optimal Structure: The optimal detector for the small signal approximation is basically the correlation receiver preceded by the logarithmic nonlinearity. Incoherent Correlation Detector

60. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 60 Coherent Detection – Class A Noise Comparison of performance of correlation receiver (Gaussian optimal receiver) and nonlinear detector [Spaulding & Middleton, 1997, pt. II]

61. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 61 Communication performance of approximation vs. upper bound [Spaulding & Middleton, 1977, pt. I] Correlation Receiver Coherent Detection – Small Signal Approximation Near-optimal for small amplitude signals Suboptimal for higher amplitude signals Antipodal A = 0.35  = 0.5×10 -3

62. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 62 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.

63. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 63 [Widrow et al., 1975] s : signal s+n 0 :corrupted signal n0 : noise n1 : reference input z : system output 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

64. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 64 Coherent Detection in Class A Noise with Γ = 10 -4 SNR (dB)‏ Correlation Receiver Performance A = 0.1

65. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 65 Myriad Filtering Myriad Filters exhibit high statistical efficiency in bell-shaped impulsive distributions like the SαS distributions. Have been used as both edge enhancers and smoothers in image processing applications. In the communication domain, they have been used to estimate a sent number over a channel using a known pulse corrupted by additive noise. (Gonzalez 1996)‏ In this work, we used a sliding window version of the myriad filter to mitigate the impulsiveness of the additive noise. (Nassar et. al 2007)‏

66. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 66 Decision Rule Λ(X)‏ H 1 or H 2 corrupted signal MAP Detection Hard decision Bayesian formulation [Spaulding and Middleton, 1977] Equally probable source

67. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 67 Results

68. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 68 MAP Detector – PDF Approximation SαS random variable Z with parameters ,  can be written Z = X Y ½ [Kuruoglu, 1998]  X is zero-mean Gaussian with variance 2   Y is positive stable random variable with parameters depending on  Pdf of Z can be written as a mixture model of N Gaussians [Kuruoglu, 1998]  Mean  can be added back in  Obtain f Y (.) by taking inverse FFT of characteristic function & normalizing  Number of mixtures (N) and values of sampling points (v i ) are tunable parameters

69. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 69 Hole Punching (Blanking) Filter Sets sample to 0 when sample exceeds threshold [Ambike, 1994] Intuition:  Large values are impulses and true value cannot be recovered  Replace large values with zero will not bias (correlation) receiver  If additive noise were purely Gaussian, then the larger the threshold, the lower the detrimental effect on bit error rate

70. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 70 Complexity Analysis MethodComplexity per symbol Analysis Hole Puncher + Correlation Receiver O(N+S)A decision needs to be made about each sample. Optimal Myriad + Correlation Receiver O(NW 3 +S)Due to polynomial rooting which is equivalent to Eigen-value decomposition. Selection Myriad + Correlation Receiver O(NW 2 +S)Evaluation of the myriad function and comparing it. MAP ApproximationO(MNS)Evaluating approximate pdf (M is number of Gaussians in mixture) N is oversampling factor S is constellation size W is window size

71. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 71 4. Performance Bounds in presence of impulsive noise Channel Capacity Case IShannon 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] ) System Model

72. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 72 Capacity in Presence of Impulsive Noise System Model Capacity

73. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 73 Probability of Error for Uncoded Transmission BPSK uncoded transmission One sample per symbol A = 0.1, Γ = 10 -3 [Haring & Vinck, 2002] Backup

74. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 74 Chernoff Factors for Coded Transmission PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols

75. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Part II Single Carrier, Multiple Antenna Communication Systems

76. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 76 Multiple Input Multiple Output (MIMO) Receivers in Impulsive Noise Statistical Physical Models of Noise Middleton Class A model for two-antenna systems [MacDonald & Blum,1997] Extension to larger than 2  2 case is difficult Statistical Models of Noise Multivariate Alpha Stable Process Mixture of weighted multivariate complex Gaussians as approximation to multivariate Middleton Class A noise [Blum et al., 1997]

77. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 77 MIMO Receivers in Impulsive Noise Key Prior Work Performance analysis of standard MIMO receivers in impulsive noise [Li, Wang & Zhou, 2004] Space-time block coding over MIMO channels with impulsive noise [Gao & Tepedelenlioglu,2007] Assumes uncorrelated noise at antennas Our Contributions Performance analysis of standard MIMO receivers using multivariate noise models Optimal and sub-optimal maximum likelihood (ML) receiver design for 2  2 case

78. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 78 Communication Performance 2 x 2 MIMO system A = 0.1, Γ 1 = Γ 2 = 10 -3 Correlation Coeff. = 0.1 Spatial Multiplexing Mode

79. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Part III Multiple Carriers, Single Antenna Communication Systems

80. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 80 Motivation Impulse noise with impulse event followed by “flat” region Coding and interleaving may improve communication performance In multicarrier modulation, impulsive event in time domain spreads out over all subsymbols thereby reducing effect of impulse Complex number (CN) codes [Lang, 1963] Transmitter forms s = GS, where S contains transmitted symbols, G is a unitary matrix and s contains coded symbols Receiver multiplies received symbols by G -1 Gaussian noise unaffected (unitary transformation is rotation) Orthogonal frequency division multiplexing (OFDM) is special case of CN codes when G is inverse discrete Fourier transform matrix

81. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 81 Noise Smearing Smearing effect Impulsive noise energy distributes over longer symbol time Smearing filters maximize impulse attenuation and minimize intersymbol interference for impulsive noise [Beenker, 1985] Maximum smearing efficiency is where N is number of symbols used in unitary transformation As N  , distribution of impulsive noise becomes Gaussian Simulations [Haring, 2003] When using a transformation involving N = 1024 symbols, impulsive noise case approaches case where only Gaussian noise is present Backup

82. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 82 Haring’s Receiver Simulation Results