1. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Mitigating Computer Platform Radio Frequency Interference in Embedded Wireless Transceivers Prof. Brian L. Evans Lead Graduate StudentsKapil Gulati and Marcel Nassar Other Graduate StudentsAditya Chopra and Marcus DeYoung Undergraduate StudentsNavid Aghasadeghi and Arvind K. Sujeeth Preliminary Results February 25, 2008

2. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 2 Outline Problem Definition I: Single Carrier, Single Antenna Communication Systems Noise Modeling Estimation of Noise Model Parameters Filtering and Detection Bounds on Communication Performance II: Single Carrier, Multiple Antenna Communication Systems III: Multiple Carrier, Single Antenna Communication Systems 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 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 estimation of model parameters We’ll be using noise and interference interchangeably Backup

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 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 PART I Single Carrier, Single Antenna Communication Systems

6. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 6 1. Noise Modeling RFI is 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 Backup

7. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 7 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

8. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 8 Middleton Class A Model ParametersDescriptionRange 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 (pdf) Backup Probability Density Function for A = 0.15,  = 0.1

9. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 9 Symmetric Alpha Stable Model Characteristic function: Parameters Characteristic exponent indicative of thickness of tail of impulsiveness Localization (analogous to mean) Dispersion (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) Could approximate pdf using inverse transform of power series expansion of characteristic function Backup

10. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 10 2. Estimation of Noise Model Parameters For the Middleton Class A Model Expectation maximization (EM) [Zabin & Poor, 1991] Based on envelope statistics [Middleton, 1979] Based on moments [Middleton, 1979] 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 Backup Complexity Iterative algorithm At each iteration: Rooting a second order polynomial (Given A, maximize K (= AΓ) ) Rooting a fourth order polynomial (Given K, maximize A) Advantage Small sample size required (~1000 samples) Disadvantage Iterative algorithm, computationally intensive Complexity Parameter estimators are based on simple order statistics Advantage Fast / computationally efficient (non-iterative) Disadvantage Requires large set of data samples (N ~ 10,000) Backup

11. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 11 Results on Measured RFI Data Data set of 80,000 samples collected using 20 GSPS scope Measured data is "broadband" noise Middleton Class B model would match PDF is symmetric Symmetric Alpha Stable Process expected to work well Approximation to Class B model

12. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 12 Results on Measured RFI Data Modeling PDF as Symmetric Alpha Stable process Estimated Parameters Localization (δ)-0.0393 Dispersion ( γ ) 0.5833 Characteristic Exponent (α) 1.5525 Normalized MSE = 0.0055

13. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 13 3. 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 Nonlinear Filter Matched Filter Decision Rule Impulsive Noise Alternate Adaptive Model Backup

14. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 14 Filtering and Detection 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] Alpha Stable Noise Correlation Receiver (linear) MAP Approximation Myriad Filtering [Gonzalez & Arce, 2001] Hole Punching [Ambike et al., 1994] We assume perfect estimation of noise model parameters Backup

15. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 15 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

16. Wireless Networking and Communications Group Department of Electrical and Computer Engineering Class A Detection - Results 16 Pulse shape Raised cosine 10 samples per symbol 10 symbols per pulse Channel A = 0.35  = 0.5 × 10 -3 Memoryless MethodComp.Perf. MAPO(NMK)High Correl.O(N+K)Low WienerO(NW+K)Low Approx.O(MN+K)High K: Constellation Size N: number of samples per symbol M: number of retained terms of the series expansion W: Window Size

17. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 17 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

18. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 18 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

19. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 19 Myriad 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 gets through myriad filter As k→0, filter tends to mode filter (output value with highest freq.) Empirical choice of k: [Gonzalez & Arce, 2001]

20. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 20 Myriad Filtering – 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

21. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 21 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

22. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 22 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

23. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 23 Bit Error Rate (BER) Performance in Alpha Stable Noise

24. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 24 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

25. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 25 Capacity in Presence of Impulsive Noise System Model Capacity

26. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 26 Probability of Error for Uncoded Transmission BPSK uncoded transmission One sample per symbol A = 0.1, Γ = 10 -3 [Haring & Vinck, 2002] Backup

27. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 27 Chernoff Factors for Coded Transmission PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols

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

29. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 29 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]

30. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 30 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

31. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 31 Communication Performance 2 x 2 MIMO system A = 0.1, Γ 1 = Γ 2 = 10 -3 Correlation Coeff. = 0.1 Spatial Multiplexing Mode

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

33. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 33 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

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

35. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 35 Conclusion Radio frequency interference from computing platform Affects wireless data communication transceivers Models include Middleton noise models and alpha stable processes Cancellation can improve communication performance Initial RFI cancellation methods explored Linear (Wiener) and Non-linear filtering (Myriad, Hole Punching) Optimal detection rules (significant gains at low bit rates) Preliminary work Performance bounds in presence of RFI RFI mitigation in multicarrier, MIMO communication systems

36. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 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, accepted for publication. 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 36

37. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 37 Future Work Single carrier, single antenna communication systems Fixed-point (embedded) methods for parameter estimation and detection methods Estimation and detection for Middleton Class B model Single carrier, multiple antenna communication systems MIMO receiver design in presence of RFI Performance bounds for MIMO receivers in presence of RFI Multicarrier Modulation and Coding Explore unitary coding schemes resilient to RFI Investigate multi-layered coding

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

39. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 39 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.

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

41. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 41 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

42. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 42 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

43. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 43 Power Spectral Density Middleton Class A Statistics Envelope statistics Envelope for Gaussian signal has Rayleigh distribution

44. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 44 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

45. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 45 Probability Density FunctionPower Spectral Density Example: exponent  = 1.5, “mean”  = 0 and “variance”  = 10 Symmetric Alpha Stable Statistics ×10 -4

46. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 46 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

47. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 47 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

48. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 48 Class A Parameter Estimation Based on APD (Exceedance Probability Density) Plot

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

50. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 50 Middleton Class B Model Envelope Statistics Envelope exceedance probability density (APD) which is 1 – cumulative distribution function

51. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 51 Class B Envelope Statistics

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

53. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 53 Class B Exceedance Probability Density Plot

54. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 54 Expectation Maximization Overview

55. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 55 Maximum Likelihood for Sum of Densities

56. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 56 EM Estimator for Class A Parameters Using 1000 Samples PDFs with 11 summation terms 50 simulation runs per setting Convergence criterion: Example learning curve Iterations for Parameter A to Converge Normalized Mean-Squared Error in A ×10 -3

57. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 57 Results of EM Estimator for Class A Parameters

58. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 58 Extreme Order Statistics

59. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 59 Estimator for Alpha-Stable 0 < p < α

60. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 60 Mean squared error in estimate of characteristic exponent α Data length (N) was 10,000 samples Results averaged over 100 simulation runs Estimate α and “mean”  directly from data Estimate “variance” γ from α and δ estimates Continued next slide Results for Symmetric Alpha Stable Parameter Estimator

61. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 61 Results for Symmetric Alpha Stable Parameter Estimator Mean squared error in estimate of dispersion (“variance”)   = 5 Mean squared error in estimate of localization (“mean”)   = 10

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

63. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 63 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)

64. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 64 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

65. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 65 Incoherent Detection Bayes formulation [Spaulding & Middleton, 1997, pt. II] Small signal approximation

66. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 66 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

67. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 67 Coherent Detection – Class A Noise Comparison of performance of correlation receiver (Gaussian optimal receiver) and nonlinear detector [Spaulding & Middleton, 1997, pt. II]

68. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 68 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

69. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 69 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.

70. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 70 [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

71. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 71 Region 2 Region 1 Region 3 Gaussian Class A (with same power)

72. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 72 Haring’s Receiver Simulation Results

73. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 73 Coherent Detection in Class A Noise with Γ = 10 -4 SNR (dB) Correlation Receiver Performance A = 0.1

74. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 74 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)

75. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 75 Decision Rule Λ(X) H 1 or H 2 corrupted signal MAP Detection Hard decision Bayesian formulation [Spaulding and Middleton, 1977] Equally probable source

76. Wireless Networking and Communications Group Department of Electrical and Computer Engineering 76 Results