1. Mitigating Radio Frequency Interference in Embedded Wireless Receivers Wireless Networking and Communications Group 8 January 2016 Prof. Brian L. Evans Lead Graduate Students Aditya Chopra, Kapil Gulati and Marcel Nassar In collaboration with Keith R. Tinsley and Chaitanya Sreerama 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  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

4. Wireless Networking and Communications Group Common Spectral Occupancy 4 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 Impact of RFI 5  Impact of LCD noise on throughput performance for a 802.11g embedded wireless receiver [J. Shi et al., 2006] Backup

6. Wireless Networking and Communications Group Statistical Modeling of RFI 6  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

7. Wireless Networking and Communications Group Assumptions for RFI Modeling 7  Key Assumptions [Middleton, 1977][Furutsu & Ishida, 1961]  Infinitely many potential interfering sources with same effective radiation power  Power law propagation loss  Poisson field of interferers Pr(number of interferers = M |area R) ~ Poisson  Poisson distributed emission times  Temporally independent (at each sample time)  Limitations  [Alpha Stable]: Does not include thermal noise  Temporal dependence may exist

8. Wireless Networking and Communications Group Our Contributions 8 Mitigation of computational platform noise in single carrier, single antenna systems [Nassar et al., ICASSP 2008]

9. Wireless Networking and Communications Group Middleton Class A model 9  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]

10. Wireless Networking and Communications Group Symmetric Alpha Stable Model 10  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 and  = 10 ParameterDescriptionRange Characteristic Exponent. Amount of impulsiveness Localization. Analogous to mean Dispersion. Analogous to variance Backup

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  Broadband RFI data  80,000 samples collected using 20GSPS scope 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 Backup

13. Wireless Networking and Communications Group Filtering and Detection 13  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

14. Filtering and Detection Methods Filtering  Wiener Filtering (Linear) Detection  Correlation Receiver (Linear)  MAP (Maximum a posteriori probability) detector [Spaulding & Middleton, 1977]  Small Signal Approximation to MAP detector [Spaulding & Middleton, 1977] Filtering  Myriad Filtering [Gonzalez & Arce, 2001]  Hole Punching Detection  Correlation Receiver (Linear)  MAP approximation 14 Wireless Networking and Communications Group Middleton Class A noise Symmetric Alpha Stable noise Backup

15. Wireless Networking and Communications Group Results: Class A Detection 15 Pulse shape Raised cosine 10 samples per symbol 10 symbols per pulse Channel A = 0.35  = 0.5 × 10 -3 Memoryless

16. Wireless Networking and Communications Group Filtering for Alpha Stable Noise 16  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

17. Wireless Networking and Communications Group Filtering for Alpha Stable Noise (Cont..) 17  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 )

18. Wireless Networking and Communications Group Results: Alpha Stable Detection 18 Use dispersion parameter  in place of noise variance to generalize SNR Backup

19. Wireless Networking and Communications Group Extensions to MIMO systems 19  RFI Modeling  Middleton Class A Model for two-antenna systems [McDonald & Blum, 1997]  Closed form PDFs for M x N MIMO system not published  Prior Work  Much prior work assumes independent noise at antennas  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]

20. Wireless Networking and Communications Group Our Contributions 20 2 x 2 MIMO receiver design in the presence of RFI [Gulati et al., Globecom 2008] Backup

21. Wireless Networking and Communications Group Results: RFI Mitigation in 2 x 2 MIMO 21 Complexity Analysis Improvement in communication performance over conventional Gaussian ML receiver at symbol error rate of 10 -2 Communication Performance (A = 0.1,  1 = 0.01,  2 = 0.1,  = 0.4)

22. Wireless Networking and Communications Group Results: RFI Mitigation in 2 x 2 MIMO 22 Complexity Analysis Complexity Analysis for decoding M-QAM modulated signal Communication Performance (A = 0.1,  1 = 0.01,  2 = 0.1,  = 0.4)

23. Wireless Networking and Communications Group Conclusions 23  Radio Frequency Interference from computing platform  Affects wireless data communication transceivers  Models include Middleton models and alpha stable models  RFI mitigation can improve communication performance  Single carrier, single antenna systems  Linear and non-linear filtering/detection methods explored  Single carrier, multiple antenna systems  Studied RFI modeling for 2x2 MIMO systems  Optimal and sub-optimal receivers designed  Bounds on communication performance in presence of RFI

24. Wireless Networking and Communications Group Contributions 24  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. 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, accepted for publication. 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. 19-24, 2009, Taipei, Taiwan, submitted.  Software Releases RFI Mitigation Toolbox Version 1.1 Beta (Released November 21 st, 2007) Version 1.0(Released September 22 nd, 2007)  Project Website http://users.ece.utexas.edu/~bevans/projects/rfi/index.html

25. Wireless Networking and Communications Group Future Work 25  Modeling RFI to include  Computational platform noise  Co-channel interference  Adjacent channel interference  Multi-input multi-output (MIMO) single carrier systems  RFI modeling and receiver design  Multicarrier communication systems  Coding schemes resilient to RFI  Circuit design guidelines to reduce computational platform generated RFI Backup

26. Wireless Networking and Communications Group 26 Thank You, Questions ?

27. Wireless Networking and Communications Group References 27 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. 1129-1149, May 1999. [2] K.F. McDonald and R.S. Blum. “A physically-based impulsive noise model for array observations”, Proc. IEEE Asilomar Conference on Signals, Systems& Computers, vol 1, 2-5 Nov. 1997. [3] K. Furutsu and T. Ishida, “On the theory of amplitude distributions of impulsive random noise,” J. Appl. Phys., vol. 32, no. 7, pp. 1206–1221, 1961. [4] J. Ilow and D. Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers”, IEEE transactions on signal processing, vol. 46, no. 6, pp. 1601-1611, 1998. Parameter Estimation [5] S. M. Zabin and H. V. Poor, “Efficient estimation of Class A noise parameters via the EM [Expectation-Maximization] algorithms”, IEEE Trans. Info. Theory, vol. 37, no. 1, pp. 60-72, Jan. 1991 [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. 1492-1503, Jun. 1996 RFI Measurements and Impact [7] J. Shi, A. Bettner, G. Chinn, K. Slattery and X. Dong, "A study of platform EMI from LCD panels - impact on wireless, root causes and mitigation methods,“ IEEE International Symposium on Electromagnetic Compatibility, vol.3, no., pp. 626-631, 14-18 Aug. 2006

28. Wireless Networking and Communications Group References (cont…) 28 Filtering and Detection [8] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment- Part I: Coherent Detection”, IEEE Trans. Comm., vol. 25, no. 9, Sep. 1977 [9] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment Part II: Incoherent Detection”, IEEE Trans. Comm., vol. 25, no. 9, Sep. 1977 [10] J.G. Gonzalez and G.R. Arce, “Optimality of the Myriad Filter in Practical Impulsive-Noise Environments”, IEEE Trans. on Signal Processing, vol 49, no. 2, Feb 2001 [11] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of Gaussian noise and impulsive noise modelled as an alpha-stable process,” IEEE Signal Processing Letters, vol. 1, pp. 55–57, Mar. 1994. [12] J. G. Gonzalez and G. R. Arce, “Optimality of the myriad filter in practical impulsive-noise environments,” IEEE Trans. on Signal Proc, vol. 49, no. 2, pp. 438–441, Feb 2001. [13] E. Kuruoglu, “Signal Processing In Alpha Stable Environments: A Least Lp Approach,” Ph.D. dissertation, University of Cambridge, 1998. [14] J. Haring and A.J. Han Vick, “Iterative Decoding of Codes Over Complex Numbers for Impulsive Noise Channels”, IEEE Trans. On Info. Theory, vol 49, no. 5, May 2003 [15] Ping Gao and C. Tepedelenlioglu. “Space-time coding over mimo channels with impulsive noise”, IEEE Trans. on Wireless Comm., 6(1):220–229, January 2007.

29. Wireless Networking and Communications Group Backup Slides 29  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

30. Wireless Networking and Communications Group Impact of RFI 30  Calculated in terms of desensitization (“desense”)  Interference raises noise floor  Receiver sensitivity will degrade to maintain SNR  Desensitization levels can exceed 10 dB for 802.11a/b/g due to computational platform noise [J. Shi et al., 2006] Case Sudy: 802.11b, 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

31. Wireless Networking and Communications Group Middleton Class A, B and C Models 31  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

32. Wireless Networking and Communications Group Middleton Class B Model 32  Envelope Statistics  Envelope exceedence probability density (APD), which is 1 – cumulative distribution function (CDF) Return

33. Wireless Networking and Communications Group Middleton Class B Model (cont…) 33  Middleton Class B Envelope Statistics Return

34. Wireless Networking and Communications Group Middleton Class B Model (cont…) 34  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

35. Wireless Networking and Communications Group Accuracy of Middleton Noise Models 35 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

36. Wireless Networking and Communications Group Symmetric Alpha Stable PDF 36  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

37. Wireless Networking and Communications Group Symmetric Alpha Stable Model 37  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

38. Wireless Networking and Communications Group Parameter Estimation: Middleton Class A 38  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

39. Wireless Networking and Communications Group Expectation Maximization Overview 39 Return

40. Wireless Networking and Communications Group Results: EM Estimator for Class A 40 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

41. Wireless Networking and Communications Group Results: EM Estimator for Class A 41 For convergence for A  [10 -2, 1], worst- case number of iterations for A = 1 Estimation accuracy vs. number of iterations tradeoff Return

42. Wireless Networking and Communications Group Parameter Estimation: Symmetric Alpha Stable 42  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

43. Wireless Networking and Communications Group Results: Symmetric Alpha Stable Parameter Estimator 43 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

44. Wireless Networking and Communications Group Results: Symmetric Alpha Stable Parameter Estimator (Cont…) 44 Mean squared error in estimate of dispersion (“variance”)  Mean squared error in estimate of localization (“mean”)  Return

45. Wireless Networking and Communications Group Extreme Order Statistics 45 Return

46. Wireless Networking and Communications Group Parameter Estimators for Alpha Stable 46 0 < p < α Return

47. Results on Measured RFI Data  Best fit for 25 data sets taken under different conditions Return

48. Wireless Networking and Communications Group Wiener Filtering 48  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

49. Wireless Networking and Communications Group Wiener Filter Design 49  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

50. Wireless Networking and Communications Group Results: Wiener Filtering 50  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 × 10 -3 SNR = -10 dB Memoryless Pulse shape 10 samples per symbol 10 symbols per pulse Return

51. Wireless Networking and Communications Group MAP Detection for Class A 51  Hard decision  Bayesian formulation [Spaulding & Middleton, 1977]  Equally probable source Return

52. Wireless Networking and Communications Group MAP Detection for Class A: Small Signal Approx. 52  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

53. Wireless Networking and Communications Group Incoherent Detection 53  Baye’s formulation [Spaulding & Middleton, 1997, pt. II]  Small Signal Approximation Correlation receiver Return

54. Wireless Networking and Communications Group Filtering for Alpha Stable Noise (Cont..) 54  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

55. Wireless Networking and Communications Group MAP Detection for Alpha Stable: PDF Approx. 55  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

56. Wireless Networking and Communications Group Results: Alpha Stable Detection 56 Return

57. Wireless Networking and Communications Group Complexity Analysis for Alpha Stable Detection 57 Return

58. Wireless Networking and Communications Group Performance Bounds (Single Antenna) 58  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

59. Wireless Networking and Communications Group Performance Bounds (Single Antenna) 59  Channel Capacity in presence of RFI System Model Parameters A = 0.1, Γ = 10 -3 Capacity Return

60. Wireless Networking and Communications Group Performance Bounds (Single Antenna) 60  Probability of error for uncoded transmissions BPSK uncoded transmission One sample per symbol A = 0.1, Γ = 10 -3 [Haring & Vinck, 2002] Return

61. Wireless Networking and Communications Group Performance Bounds (Single Antenna) 61  Chernoff factors for coded transmissions PEP: Pairwise error probability N: Size of the codeword Chernoff factor: Equally likely transmission for symbols Return

62. Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 62  Channel Capacity [Chopra et al., submitted to ICASSP 2009] 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

63. Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 63  Channel Capacity in presence of RFI for 2x2 MIMO [Chopra et al., submitted to ICASSP 2009] System Model Capacity Parameters : A = 0.1,  1 = 0.01  2 = 0.1,  = 0.4 Return

64. Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 64  Probability of symbol error for uncoded transmissions [Chopra et al., submitted to ICASSP 2009] 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

65. Wireless Networking and Communications Group Performance Bounds (2x2 MIMO) 65  Chernoff factors for coded transmissions [Chopra et al., submitted to ICASSP 2009] 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

66. Wireless Networking and Communications Group Extensions to Multicarrier Systems 66  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