Radio Frequency Interference Sensing and Mitigation in Wireless Receivers Talk at Intel Labs in Hillsboro, Oregon Wireless Networking and Communications Group 16 March 2009 Prof. Brian L. Evans Lead Graduate Students Aditya Chopra, Kapil Gulati and Marcel Nassar In collaboration with Eddie Xintian Lin, Chaitanya Sreerama, Keith R. Tinsley (technical lead) and Jorge Aguilar Torrentera at Intel Labs

2 Outline  Problem definition  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 Filtering/detection  Multi-input multi-output (MIMO) single carrier systems  Co-channel interference modeling (unpublished)  Conclusions  Future work 2 Wireless Networking and Communications Group

3 Problem Definition 3  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 Within computing platforms, wireless transceivers experience radio frequency interference (RFI) from clocks and busses We will use noise and interference interchangeably Backup

4 Wireless Networking and Communications Group Impact of RFI 4  Impact of LCD noise on throughput for an IEEE g embedded wireless receiver [Shi, Bettner, Chinn, Slattery & Dong, 2006] Backup

5 Wireless Networking and Communications Group Statistical Modeling of RFI 5  Radio Frequency Interference (RFI)  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 Model non-linear phenomenon governing RFI  Symmetric Alpha Stable models Approximation of Middleton Class B model Backup

6 Wireless Networking and Communications Group Assumptions for RFI Modeling 6  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 Pr(number of interferers = M |area R) ~ Poisson(M; R)  Uniformly distributed emission times  Temporally independent (at each sample time)  Limitations  Alpha Stable models do not include thermal noise  Temporal dependence may exist

7 Wireless Networking and Communications Group Our Contributions 7 Mitigation of computational platform noise in single carrier, single antenna systems [Nassar, Gulati, DeYoung, Evans & Tinsley, ICASSP 2008, JSPS 2009]

8 Wireless Networking and Communications Group Middleton Class A model 8  Probability Density Function 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]

9 Wireless Networking and Communications Group Symmetric Alpha Stable Model 9  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 PDF for  = 1.5,  = 0,  = 10 ParameterDescriptionRange Characteristic Exponent. Amount of impulsiveness Localization. Analogous to mean Dispersion. Analogous to variance Backup

10 Example Power Spectral Densities  Middleton Class A  Symmetric Alpha Stable Overlap Index (A) = 0.15 Gaussian Factor (  = 0.1 Characteristic Exponent (  = 1.5 Localization (  ) = 0 Dispersion (  ) = 10

11 Wireless Networking and Communications Group Estimation of Noise Model Parameters 11  Middleton Class A model  Expectation Maximization (EM) [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

12 Wireless Networking and Communications Group Results on Measured RFI Data 12  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) KL Divergence Characteristic exp. (α) Dispersion (γ) Middleton Class A Model Overlap Index (A) KL Divergence Gaussian Factor (Γ) Gaussian Model Mean (µ)0 KL Divergence Variance (σ 2 )1 KL Divergence: Kullback-Leibler divergence

13 Results on Measured RFI Data  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

14 Video over Impulsive Channels  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) Wireless Networking and Communications Group 14 Additive Class A NoiseValue Overlap index (A)0.35 Gaussian factor (  ) SNR19 dB

15 Wireless Networking and Communications Group Filtering and Detection 15 Pulse Shaping Pre-Filtering Matched Filter Detection Rule Impulsive Noise 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 Assumption Multiple samples of the received signal are available N Path Diversity [Miller, 1972] Oversampling by N [Middleton, 1977] Assumption Multiple samples of the received signal are available N Path Diversity [Miller, 1972] Oversampling by N [Middleton, 1977] Backup

16 Wireless Networking and Communications Group Results: Class A Detection 16 Pulse shape Raised cosine 10 samples per symbol 10 symbols per pulse Channel A = 0.35  = 0.5 × Memoryless Communication Performance Binary Phase Shift Keying Backup

17 Wireless Networking and Communications Group Results: Alpha Stable Detection 17 Use dispersion parameter  in place of noise variance to generalize SNR Backup Communication Performance Same transmitter settings as previous slide Backupc Backup

18 Wireless Networking and Communications Group Extensions to MIMO systems 18 Backup

19 Wireless Networking and Communications Group Our Contributions 19 2 x 2 MIMO receiver design in the presence of RFI [Gulati, Chopra, Heath, Evans, Tinsley & Lin, Globecom 2008] Backup

20 Wireless Networking and Communications Group Results: RFI Mitigation in 2 x 2 MIMO 20 Improvement in communication performance over conventional Gaussian ML receiver at symbol error rate of Communication Performance (A = 0.1,  1 = 0.01,  2 = 0.1,  = 0.4)

21 Wireless Networking and Communications Group Results: RFI Mitigation in 2 x 2 MIMO 21 Complexity Analysis Complexity Analysis for decoding M-level QAM modulated signal Communication Performance (A = 0.1,  1 = 0.01,  2 = 0.1,  = 0.4)

22 Co-Channel Interference Modeling Wireless Networking and Communications Group 22  Alpha Stable and Middleton Class A distributions applicable  Region of interferer locations determines interference model Symmetric Alpha Stable Middleton Class A

23 Co-Channel Interference Modeling  Propose u nified framework to derive narrowband interference models for ad-hoc and cellular network environments  Key Result: Tail Probabilities Wireless Networking and Communications Group 23 Case 1: Ad-hoc networkCase 2-a: Cellular network (Base Station) Interference threshold (a)

24 Wireless Networking and Communications Group Conclusions 24  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

25 RFI Mitigation Toolbox  Provides a simulation environment for  RFI generation  Parameter estimation algorithms  Filtering and detection methods  Demos for communication performance analysis Latest Toolbox Release Version 1.2, Feb 7 th 2009 Wireless Networking and Communications Group 25 Snapshot of a demo

26 Wireless Networking and Communications Group Other Contributions 26  Publications [Journal Articles] M. Nassar, K. Gulati, M. R. DeYoung, B. L. Evans and K. R. Tinsley, “Mitigating Near-Field Interference in Laptop Embedded Wireless Transceivers”, Journal of Signal Processing Systems, invited paper. [Conference Papers] 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.  Project Website

27 Wireless Networking and Communications Group Future Work 27  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

28 Wireless Networking and Communications Group 28 Thank You. Questions ?

29 Wireless Networking and Communications Group References 29 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 [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, [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 , 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 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

30 Wireless Networking and Communications Group References (cont…) 30 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 [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 [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 [13] E. Kuruoglu, “Signal Processing In Alpha Stable Environments: A Least Lp Approach,” Ph.D. dissertation, University of Cambridge, [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.

31 Wireless Networking and Communications Group Backup Slides 31  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

32 Wireless Networking and Communications Group Common Spectral Occupancy 32 Standard Carrier (GHz) Wireless Networking Interfering Clocks and Busses Bluetooth2.4 Personal Area Network Gigabit Ethernet, PCI Express Bus, LCD clock harmonics IEEE b/g/n 2.4 Wireless LAN (Wi-Fi) Gigabit Ethernet, PCI Express Bus, LCD clock harmonics IEEE e 2.5– – –5.85 Mobile Broadband (Wi-Max) PCI Express Bus, LCD clock harmonics IEEE a 5.2 Wireless LAN (Wi-Fi) PCI Express Bus, LCD clock harmonics Return

33 Wireless Networking and Communications Group Impact of RFI 33  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

34 Impact of LCD clock on g  Pixel clock 65 MHz  LCD Interferers and g center frequencies Wireless Networking and Communications Group 34 Return

35 Wireless Networking and Communications Group Middleton Class A, B and C Models 35  Class A Narrowband interference (“coherent” reception) Uniquely represented by 2 parameters  Class B Broadband interference (“incoherent” reception) Uniquely represented by six parameters  Class CSum of Class A and Class B (approx. Class B) [Middleton, 1999] Return Backup

36 Wireless Networking and Communications Group Middleton Class B Model 36  Envelope Statistics  Envelope exceedence probability density (APD), which is 1 – cumulative distribution function (CDF) Return

37 Wireless Networking and Communications Group Middleton Class B Model (cont…) 37  Middleton Class B Envelope Statistics Return

38 Wireless Networking and Communications Group Middleton Class B Model (cont…) 38  Parameters for Middleton Class B Model 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 Return

39 Wireless Networking and Communications Group Accuracy of Middleton Noise Models 39 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) ‏ Return

40 Wireless Networking and Communications Group Symmetric Alpha Stable PDF 40  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  Return

41 Wireless Networking and Communications Group Symmetric Alpha Stable Model 41  Heavy tailed distribution Density functions for symmetric alpha stable distributions for different values of characteristic exponent alpha: a) overall density and b) the tails of densities Return

42 Wireless Networking and Communications Group Parameter Estimation: Middleton Class A 42  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 (= A  ). Root 2 nd order polynomial. Given K, maximize A. Root 4 th order polynomial Return Backup Results Backup

43 Wireless Networking and Communications Group Expectation Maximization Overview 43 Return

44 Wireless Networking and Communications Group Results: EM Estimator for Class A 44 PDFs with 11 summation terms 50 simulation runs per setting 1000 data samples Convergence criterion: Iterations for Parameter A to ConvergeNormalized Mean-Squared Error in A K = A  Return

45 Wireless Networking and Communications Group Results: EM Estimator for Class A 45 For convergence for A  [10 -2, 1], worst- case number of iterations for A = 1 Estimation accuracy vs. number of iterations tradeoff Return

46 Wireless Networking and Communications Group Parameter Estimation: Symmetric Alpha Stable 46  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

47 Wireless Networking and Communications Group Results: Symmetric Alpha Stable Parameter Estimator 47 Data length (N) of 10,000 samples Results averaged over 100 simulation runs Estimate α and “mean”  directly from data Estimate “variance”  from α and δ estimates Mean squared error in estimate of characteristic exponent α Return

48 Wireless Networking and Communications Group Results: Symmetric Alpha Stable Parameter Estimator (Cont…) 48 Mean squared error in estimate of dispersion (“variance”)  Mean squared error in estimate of localization (“mean”)  Return

49 Wireless Networking and Communications Group Extreme Order Statistics 49 Return

50 Wireless Networking and Communications Group Parameter Estimators for Alpha Stable 50 0 < p < α Return

51 Wireless Networking and Communications Group Filtering and Detection 51  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 Pulse Shaping Pre-Filtering Matched Filter Detection Rule Impulsive Noise N samples per symbol

52 Wireless Networking and Communications Group Wiener Filtering 52  Optimal in mean squared error sense in presence of Gaussian noise 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 ^ Model Design Return

53 Wireless Networking and Communications Group Wiener Filter Design 53  Infinite Impulse Response (IIR)  Finite Impulse Response (FIR)  Weiner-Hopf equations for order p-1 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) ‏ Return

54 Wireless Networking and Communications Group Results: Wiener Filtering 54  100-tap FIR Filter Raised Cosine Pulse Shape Transmitted waveform corrupted by Class A interference Received waveform filtered by Wiener filter n n n Channel A = 0.35  = 0.5 × SNR = -10 dB Memoryless Pulse shape 10 samples per symbol 10 symbols per pulse Return

55 Wireless Networking and Communications Group MAP Detection for Class A 55  Hard decision  Bayesian formulation [Spaulding & Middleton, 1977]  Equally probable source Return

56 Wireless Networking and Communications Group MAP Detection for Class A: Small Signal Approx. 56  Expand noise PDF p Z (z) by Taylor series about S j = 0 (j=1,2)‏  Approximate MAP detection rule  Logarithmic non-linearity + correlation receiver  Near-optimal for small amplitude signals Correlation Receiver We use 100 terms of the series expansion for d/dx i ln p Z (x i ) in simulations Return

57 Wireless Networking and Communications Group Incoherent Detection 57  Bayesian formulation [Spaulding & Middleton, 1997, pt. II]  Small Signal Approximation Correlation receiver Return

58 Wireless Networking and Communications Group Filtering for Alpha Stable Noise 58  Myriad Filtering  Sliding window algorithm outputs myriad of a 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 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

59 Wireless Networking and Communications Group Filtering for Alpha Stable Noise (Cont..) 59  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 extreme points 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 ) Return

60 Wireless Networking and Communications Group Filtering for Alpha Stable Noise (Cont..) 60  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

61 Wireless Networking and Communications Group MAP Detection for Alpha Stable: PDF Approx. 61  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 Return

62 Wireless Networking and Communications Group Results: Alpha Stable Detection 62 Return

63 Wireless Networking and Communications Group Complexity Analysis for Alpha Stable Detection 63 Return

64 Wireless Networking and Communications Group Bivariate Middleton Class A Model 64  Joint Spatial Distribution Return

65 Wireless Networking and Communications Group Results on Measured RFI Data 65  50,000 baseband noise samples represent broadband interference Marginal PDFs of measured data compared with estimated model densities Return

66  2 x 2 MIMO System  Maximum Likelihood (ML) Receiver  Log-Likelihood Function Wireless Networking and Communications Group System Model 66 Sub-optimal ML Receivers approximate Return

67 Wireless Networking and Communications Group Sub-Optimal ML Receivers 67  Two-Piece Linear Approximation  Four-Piece Linear Approximation chosen to minimize Approximation of Return

68 Wireless Networking and Communications Group Results: Performance Degradation 68  Performance degradation in receivers designed assuming additive Gaussian noise in the presence of RFI 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,  = 0.4 Severe degradation in communication performance in high-SNR regimes Return

69 Wireless Networking and Communications Group Performance Bounds (Single Antenna) 69  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 Return

70 Wireless Networking and Communications Group Performance Bounds (Single Antenna) 70  Channel Capacity in presence of RFI System Model Parameters A = 0.1, Γ = Capacity Return

71 Wireless Networking and Communications Group Performance Bounds (Single Antenna) 71  Probability of error for uncoded transmissions BPSK uncoded transmission One sample per symbol A = 0.1, Γ = [Haring & Vinck, 2002] Return

72 Wireless Networking and Communications Group Performance Bounds (Single Antenna) 72  Chernoff factors for coded transmissions PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols Return

73 System Model Wireless Networking and Communications Group 73 Return

74 Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 74  Channel Capacity Case IShannon 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 System Model Return

75 Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 75  Channel Capacity in presence of RFI for 2x2 MIMO System Model Capacity Parameters : A = 0.1,  1 = 0.01,  2 = 0.1,  = 0.4 Return

76 Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 76  Probability of symbol error for uncoded transmissions Parameters : A = 0.1,  1 = 0.01  2 = 0.1,  = 0.4 Pe: Probability of symbol error S: Transmitted code vector D(S): Decision regions for MAP detector Equally likely transmission for symbols Return

77 Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 77  Chernoff factors for coded transmissions PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols Parameters :  1 = 0.01  2 = 0.1,  = 0.4 Return

78 Performance Bounds (2x2 MIMO)  Cutoff Rates for coded transmissions  Similar metric to channel capacity  Relates transmission rate (R) to Pe for a length T codes Wireless Networking and Communications Group 78 Return

79 Performance Bounds (2x2 MIMO) Wireless Networking and Communications Group 79  Cutoff Rate Return

80 Wireless Networking and Communications Group Extensions to Multicarrier Systems 80  Impulse noise with impulse event followed by “flat” region  Coding may improve communication performance  In multicarrier modulation, impulsive event in time domain spreads out over all subcarriers, reducing the 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  [Haring 2003] As number of subcarriers increase, impulsive noise case approaches the Gaussian noise case. Return