Improving Wireless Data Transmission Speed and Reliability to Mobile Computing Platforms

1
Improving Wireless Data Transmission Speed and
Reliability to Mobile Computing Platforms
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

• Texas Wireless Summit, Austin, Texas

2
Problem Definition
Within computing platforms, wireless transceivers
experience radio frequency interference (RFI)
from clocks and busses

• Objectives
• Develop offline methods to improve communication
  performance in presence of computer platform RFI
• Develop adaptive online algorithms for these
  methods
• Approach
• Statistical modeling of RFI
• Filtering/detection based on estimated model
  parameters

We will use noise and interference interchangeably
3
Common Spectral Occupancy
Standard Band (GHz) Wireless Networking Interfering Clocks and Busses
Bluetooth 2.4 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 2.52.69 3.33.8 5.7255.85 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
4
Impact of RFI

• Impact of LCD noise on throughput performance for
  a 802.11g embedded wireless receiver Shi et al.,
  2006

Backup
5
Our Contributions
Mitigation of computational platform noise in
single carrier,single antenna systems Nassar et
al., ICASSP 2008
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 Estimation Evaluate estimation accuracy vs complexity tradeoffs
Filtering / Detection Evaluate communication performance vs complexity tradeoffs Middleton Class A Correlation receiver, Wiener filtering and Bayesian detector Symmetric Alpha Stable Myriad filtering, hole punching, and Bayesian detector
6
Power Spectral Densities

• Middleton Class A

• Symmetric Alpha Stable

Parameter valuesa 1.5, d 0 and g 10
Parameter valuesA 0.15 and G 0.1
7
Fitting Measured RFI Data

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

Backup
Estimated Parameters Estimated Parameters Estimated Parameters
Symmetric Alpha Stable Model Symmetric Alpha Stable Model Symmetric Alpha Stable Model
Localization (d) 0.0043 Distance 0.0514
Characteristic exp. (a) 1.2105 Distance 0.0514
Dispersion (?) 0.2413 Distance 0.0514
Middleton Class A Model Middleton Class A Model Middleton Class A Model
Overlap Index (A) 0.1036 Distance 0.0825
Gaussian Factor (G) 0.7763 Distance 0.0825
Gaussian Model Gaussian Model Gaussian Model
Mean (µ) 0 Distance 0.2217
Variance (s2) 1 Distance 0.2217
Distance Kullback-Leibler divergence
8
Fitting Measured RFI Data

• Best fit for other 25 data sets under different
  conditions

Return
9
Filtering and Detection Methods

• Middleton Class A noise

• Symmetric Alpha Stable noise

• Filtering
• Wiener Filtering (Linear)
• Detection
• Correlation Receiver (Linear)
• MAP (Maximum a posteriori probability)
  detectorSpaulding Middleton, 1977
• Small Signal Approximation to MAP
  detectorSpaulding Middleton, 1977

• Filtering
• Myriad FilteringGonzalez Arce, 2001
• Hole Punching
• Detection
• Correlation Receiver (Linear)
• MAP approximation

Backup
Backup
Backup
Backup
Backup
10
Results Class A Detection
Pulse shapeRaised cosine10 samples per symbol10 symbols per pulse ChannelA 0.35? 0.5 10-3Memoryless
Method Comp. Complexity Detection Perform.
Correl. Low Low
Wiener Medium Low
MAP Approx. Medium High
MAP High High
11
Results Alpha Stable Detection
Backup
Method Comp. Complexity Detection Perform.
Hole Punching Low Medium
Selection Myriad Low Medium
MAP Approx. Medium High
Optimal Myriad High Medium
Backup
Use dispersion parameter g in place of noise
variance to generalize SNR
12
Results Class A for 2 ? 2 MIMO
Improvement in communication performance over
conventional Gaussian ML receiver at symbol error
rate of 10-2
Complexity Analysis
A Noise Characteristic Improve-ment
0.01 Highly Impulsive 15 dB
0.1 Moderately Impulsive 8 dB
1 Nearly Gaussian 0.5 dB
Communication Performance (A 0.1, ?1 0.01,
?2 0.1, k 0.4)
13
Conclusions

• Radio frequency interference from computing
  platform
• Affects wireless data communication transceivers
• Fit Middleton Class A and symmetric alpha stable
  models
• RFI mitigation can reduce bit error rate by a
  factor of
• 100 for Middleton Class A model, single carrier
  system
• 10 for Middleton Class A model, 2 x 2 MIMO
  system
• 10 for Symmetric Alpha Stable model, single
  carrier system
• Other applications of impulsive noise models
• Co-channel interference
• Adjacent channel interference

14
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.
• 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 21st,
  2007)Version 1.0 (Released September 22nd, 2007)
• Project Websitehttp//users.ece.utexas.edu/bevan
  s/projects/rfi/index.html

15

• Thank You,
• Questions ?

16
References

• 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.
  12061221, 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

17
References (cont)

• 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. 5557, 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. 438441, 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)220229, January 2007.

18
Backup Slides

• 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
19
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

20
Impact of RFI

• 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
21
Statistical Modeling of RFI

• 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
Backup
22
Assumptions for RFI Modeling

• Key Assumptions Middleton, 1977Furutsu
  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

23
Middleton Class A, B and C Models
Return

• Class A Narrowband interference (coherent
  reception) Uniquely represented by 2 parameters
• Class B Broadband interference (incoherent
  reception) Uniquely represented by six
  parameters
• Class C Sum of Class A and Class B (approx. Class
  B)

Backup
24
Middleton Class A model

• Probability Density Function

PDF for A 0.15,?? 0.8
25
Middleton Class B Model

• Envelope Statistics
• Envelope exceedence probability density (APD),
  which is 1 cumulative distribution function
  (CDF)

Return
26
Middleton Class B Model (cont)

• Middleton Class B Envelope Statistics

Return
27
Middleton Class B Model (cont)

• Parameters for Middleton Class B Model

Return
28
Accuracy of Middleton Noise Models
Return
Magnetic Field Strength, H (dB relative to
microamp per meter rms)?
e0 (dB gt erms)?
Percentage of Time Ordinate is Exceeded
P(e gt e0)?
Soviet high power over-the-horizon radar
interference Middleton, 1999
Fluorescent lights in mine shop office
interference Middleton, 1999
29
Symmetric Alpha Stable Model

• Characteristic Function
• Closed-form PDF expression only fora 1
  (Cauchy), a 2 (Gaussian),a 1/2 (Levy), a 0
  (not very useful)
• Approximate PDF using inverse transform of power
  series expansion
• Second-order moments do not exist for a lt 2
• Generally, moments of order gt a do not exist

Backup
PDF for ? 1.5, ? 0 and ? 10
Backup
Parameter Description Range
Characteristic Exponent. Amount of impulsiveness
Localization. Analogous to mean
Dispersion. Analogous to variance
30
Symmetric Alpha Stable 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

Return
31
Symmetric Alpha Stable Model

• Heavy tailed distribution

Return
Density functions for symmetric alpha stable
distributions for different values of
characteristic exponent alpha a) overall density
and b) the tails of densities
32
Estimation of Noise Model Parameters

• 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
Backup
33
Parameter Estimation Middleton Class A

• Expectation Maximization (EM)
• E Step Calculate log-likelihood function \w
  current parameter values
• M Step Find parameter set that maximizes
  log-likelihood function
• EM Estimator for Class A parameters Zabin
  Poor, 1991
• Express envelope statistics as sum of weighted
  PDFs
• Maximization step is iterative
• Given A, maximize K ( AG). Root 2nd order
  polynomial.
• Given K, maximize A. Root 4th order polynomial

Return
Backup
Results
Backup
34
Expectation Maximization Overview
Return
35
Results EM Estimator for Class A
Return
Iterations for Parameter A to Converge
Normalized Mean-Squared Error in A
K A G
PDFs with 11 summation terms 50 simulation runs
per setting
1000 data samples Convergence criterion
36
Results EM Estimator for Class A
Return

• For convergence for A ? 10-2, 1, worst-case
  number of iterations for A 1
• Estimation accuracy vs. number of iterations
  tradeoff

37
Parameter Estimation Symmetric Alpha Stable

• 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 Frechets 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
  (N10,000)

Return
Results
Backup
38
Results Symmetric Alpha Stable Parameter
Estimator
Return

• Data length (N) of 10,000 samples
• Results averaged over 100 simulation runs
• Estimate a and mean g directly from data
• Estimate variance g from a and d estimates

Mean squared error in estimate of characteristic
exponent a
39
Results Symmetric Alpha Stable Parameter
Estimator (Cont)
Return
Mean squared error in estimate of dispersion
(variance) ?
Mean squared error in estimate of localization
(mean) ?
40
Extreme Order Statistics
Return
41
Parameter Estimators for Alpha Stable
Return
0 lt p lt a
42
Filtering and Detection

• System Model
• Assumptions
• Multiple samples of the received signal are
  available
• N Path Diversity Miller, 1972
• Oversampling by N Middleton, 1977
• Multiple samples increase gains vs. Gaussian case
• Impulses are isolated events over symbol period

Impulsive Noise
Pulse Shaping
Pre-Filtering
Matched Filter
Detection Rule
N samples per symbol
43
Wiener Filtering

• Optimal in mean squared error sense in presence
  of Gaussian noise

Return
Model

d(n) desired signald(n) filtered
signale(n) error w(n) Wiener filter x(n)
corrupted signalz(n) noise
Design
Minimize Mean-Squared Error E e(n)2
44
Wiener Filter Design

• Infinite Impulse Response (IIR)
• Finite Impulse Response (FIR)
• Wiener-Hopf equations for order p-1

Return
desired signal d(n)power spectrum ?(e j
?) correlation of d and x rdx(n)autocorrelat
ion of x rx(n)Wiener FIR Filter w(n)
corrupted signal x(n)noise z(n)?
45
Results Wiener Filtering

• 100-tap FIR Filter

Return
Pulse shape10 samples per symbol10 symbols per
pulse
ChannelA 0.35? 0.5 10-3SNR -10
dBMemoryless
46
Filtering for Alpha Stable Noise

• Myriad Filtering
• Sliding window algorithm outputs myriad of a
  sample window
• Myriad of order k for samples x1,x2,,xN
  Gonzalez Arce, 2001
• As k decreases, less impulsive noise passes
  through the myriad filter
• As k?0, filter tends to mode filter (output value
  with highest frequency)
• Empirical Choice of k Gonzalez Arce, 2001
• Developed for images corrupted by symmetric alpha
  stable impulsive noise

47
Filtering for Alpha Stable Noise (Cont..)

• Myriad Filter Implementation
• Given a window of samples, x1,,xN, find ß ?
  xmin, xmax
• Optimal Myriad algorithm
• Differentiate objective function polynomial p(ß)
  with respect to ß
• Find roots and retain real roots
• Evaluate p(ß) at real roots and extreme points
• Output ß that gives smallest value of p(ß)
• Selection Myriad (reduced complexity)
• Use x1, , xN as the possible values of ß
• Pick value that minimizes objective function p(ß)

48
MAP Detection for Class A

• Hard decision
• Bayesian formulation Spaulding Middleton,
  1977
• Equally probable source

Return
49
MAP Detection for Class A Small Signal Approx.

• Expand noise PDF pZ(z) by Taylor series about Sj
  0 (j1,2)?
• Approximate MAP detection rule
• Logarithmic non-linearity correlation receiver
• Near-optimal for small amplitude signals

Return
We use 100 terms of the series expansion
ford/dxi ln pZ(xi) in simulations
50
Incoherent Detection

• Bayes formulation Spaulding Middleton, 1997,
  pt. II
• Small Signal Approximation

Return
Correlation receiver
51
Filtering for Alpha Stable Noise (Cont..)

• 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
52
MAP Detection for Alpha Stable PDF Approx.

• SaS random variable Z with parameters a , d, g
  can be written Z X Y½ Kuruoglu, 1998
• X is zero-mean Gaussian with variance 2 g
• Y is positive stable random variable with
  parameters depending on a
• PDF of Z can be written as a mixture model of N
  GaussiansKuruoglu, 1998
• Mean d can be added back in
• Obtain fY(.) by taking inverse FFT of
  characteristic function normalizing
• Number of mixtures (N) and values of sampling
  points (vi) are tunable parameters

Return
53
Results Alpha Stable Detection
Return
54
Complexity Analysis for Alpha Stable Detection
Return
Method Complexity per symbol Analysis
Hole Puncher Correlation Receiver O(NS) A decision needs to be made about each sample.
Optimal Myriad Correlation Receiver O(NW3S) Due to polynomial rooting which is equivalent to Eigen-value decomposition.
Selection Myriad Correlation Receiver O(NW2S) Evaluation of the myriad function and comparing it.
MAP Approximation O(MNS) Evaluating approximate pdf(M is number of Gaussians in mixture)
55
Performance Bounds (Single Antenna)

• Channel Capacity

Return
System Model
Case I Shannon Capacity in presence of additive white Gaussian noise
Case II (Upper Bound) Capacity in the presence of Class A noise Assumes that there exists an input distribution which makes output distribution Gaussian (good approximation in high SNR regimes)
Case III (Practical Case) Capacity in presence of Class A noise Assumes input has Gaussian distribution (e.g. bit interleaved coded modulation (BICM) or OFDM modulation Haring, 2003)
56
Performance Bounds (Single Antenna)

• Channel Capacity in presence of RFI

Return
System Model
Capacity
ParametersA 0.1, G 10-3
57
Performance Bounds (Single Antenna)

• Probability of error for uncoded transmissions

Return
Haring Vinck, 2002
BPSK uncoded transmission One sample per symbol A
0.1, G 10-3
58
Performance Bounds (Single Antenna)

• Chernoff factors for coded transmissions

Return
PEP Pairwise error probability N Size of the
codeword Chernoff factor Equally likely
transmission for symbols
59
Extensions to MIMO systems

• 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

60
Our Contributions
2 x 2 MIMO receiver design in the presence of
RFIGulati et al., Globecom 2008
RFI Modeling Evaluated fit of measured RFI data to the bivariate Middleton Class A model McDonald Blum, 1997 Includes noise correlation between two antennas
Parameter Estimation Derived parameter estimation algorithm based on the method of moments (sixth order moments)
Performance Analysis Demonstrated communication performance degradation of conventional receivers in presence of RFI Bounds on communication performanceChopra et al., submitted to ICASSP 2009
Receiver Design Derived Maximum Likelihood (ML) receiver Derived two sub-optimal ML receivers with reduced complexity
Backup
61
Performance Bounds (2x2 MIMO)

• Channel Capacity Chopra et al., submitted to
  ICASSP 2009

Return
System Model
Case I Shannon Capacity in presence of additive white Gaussian noise
Case II (Upper Bound) Capacity in presence of bivariate Middleton Class A noise. Assumes that there exists an input distribution which makes output distribution Gaussian for all SNRs.
Case III (Practical Case) Capacity in presence of bivariate Middleton Class A noise Assumes input has Gaussian distribution
62
Performance Bounds (2x2 MIMO)

• Channel Capacity in presence of RFI for 2x2
  MIMOChopra et al., submitted to ICASSP 2009

Return
System Model
Capacity
ParametersA 0.1, G1 0.01, G2 0.1, k 0.4
63
Performance Bounds (2x2 MIMO)

• Probability of symbol error for uncoded
  transmissionsChopra et al., submitted to ICASSP
  2009

Return
Pe Probability of symbol error S Transmitted
code vector D(S) Decision regions for MAP
detector Equally likely transmission for symbols
ParametersA 0.1, G1 0.01, G2 0.1, k 0.4
64
Performance Bounds (2x2 MIMO)

• Chernoff factors for coded transmissionsChopra
  et al., submitted to ICASSP 2009

Return
PEP Pairwise error probabilityN Size of the
codewordChernoff factorEqually likely
transmission for symbols
ParametersG1 0.01, G2 0.1, k 0.4
65
Results RFI Mitigation in 2 x 2 MIMO
Complexity Analysis for decoding M-QAM modulated
signal
Receiver Quadratic Forms Exponential Comparisons
Gaussian ML M2 0 0
Optimal ML 2M2 2M2 0
Sub-optimal ML (Four-Piece) 2M2 0 3M2
Sub-optimal ML (Two-Piece) 2M2 0 M2
Complexity Analysis
Communication Performance (A 0.1, ?1 0.01,
?2 0.1, k 0.4)
66
Extensions to Multicarrier Systems
Return

• 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.

67
Future Work

• 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