mitigating computer platform radio frequency interference in embedded wireless transceivers
DESCRIPTION
Preliminary Results. Mitigating Computer Platform Radio Frequency Interference in Embedded Wireless Transceivers. February 25, 2008. Outline. Problem Definition I: Single Carrier, Single Antenna Communication Systems Noise Modeling Estimation of Noise Model Parameters - PowerPoint PPT PresentationTRANSCRIPT
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 Students Kapil Gulati and Marcel Nassar
Other Graduate Students Aditya Chopra and Marcus DeYoung
Undergraduate Students Navid Aghasadeghi and Arvind K. Sujeeth
Preliminary Results
February 25, 2008
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering2
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
Wireless Networking and Communications Group
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Problem Definition
Within computing platforms, wirelesstransceivers experience radio frequencyinterference (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
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Common Spectral Occupancy
StandardCarrier (GHz)
Wireless Networking
Interfering Clocks and Busses
Bluetooth 2.4Personal Area
NetworkGigabit Ethernet, PCI Express
Bus, LCD clock harmonics
IEEE 802. 11 b/g/n
2.4Wireless 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
5.2Wireless LAN
(Wi-Fi)PCI Express Bus,
LCD clock harmonics
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PART I
Single Carrier, Single Antenna Communication Systems
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1. Noise ModelingRFI 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
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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
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Middleton Class A Model
A
Parameters Description Range
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]
1
2!)(
2
2
02
2
2
Am
where
em
Aezf
m
z
m m
mA
Zm
Probability density function (pdf)
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Probability Density Function for A = 0.15, = 0.1
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.005
0.01
0.015
0.02
0.025Class A Probability Density Function; A = 0.15, = 0.1
x
PD
F f x(x
)
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Symmetric Alpha Stable Model
Characteristic function: ||)( je
Parameters 20,α Characteristic exponent indicative
of thickness of tail of impulsiveness
Localization (analogous to mean)
Dispersion (analogous to variance)0 δ-
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
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-50 -40 -30 -20 -10 0 10 20 30 40 500
1
2
3
4
5
6
7
8x 10
-4 PDF for SS noise, = 1.5, =10, = 0
x
Pro
babi
lity
dens
ity f
unct
ion
f X(x
)
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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
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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 intensiveComplexityParameter estimators are based on simple order statisticsAdvantage Fast / computationally efficient (non-iterative)Disadvantage Requires large set of data samples (N ~ 10,000)
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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
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45Measured PDF
x: noise amplitude
Me
asu
red
PD
F f
X(x
)
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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
2
2ˆ
measuredf
measuredf
estimatedf
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45PDF of measured data vs alpha stable
x: noise amplitude
PD
F f
X(x
)
Measured PDF
Estimated PDF usingalpha stable modeling
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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]gtx[n]
v[n]
grx[n] Λ(.)
Pulse ShapeNonlinear
FilterMatched
Filter Decision Rule
Impulsive Noise
Alternate Adaptive Model
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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
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Coherent Detection – Small Signal Approximation
Expand noise pdf pZ(z) by Taylor series about Sj = 0 (j=1,2)
Optimal decision rule & threshold detector for approximation
Optimal detector for approximation is logarithmic nonlinearity followed by correlation receiver
ji
N
i i
Z
ZjZZjZ sx
XpXpSXpXpSXp
1
)()()()()(
1)(ln1
)(ln1
)(2
1
11
12
H
H
N
iiZ
ii
N
iiZ
ii
xpdxd
s
xpdxd
s
X
We use 100 terms of the
series expansion ford/dxi ln pZ(xi) in simulations
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Class A Detection - Results
16
Pulse shapeRaised cosine
10 samples per symbol10 symbols per pulse
ChannelA = 0.35
= 0.5 × 10-3
Memoryless
Method Comp. Perf.
MAP O(NMK) High
Correl. O(N+K) Low
Wiener O(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
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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:
1)|()(
)|()()(
2
1
11
22
H
H
HXpHp
HXpHpX
Method Shortcomings Reference
Series Expansion Poor approximation when series length shortened
[Samorodnitsky, 1988]
Polynomial Approx. Poor approximation for small x [Tsihrintzis, 1993]
Inverse FFT Ripples in tails when α < 1 Simulation Results
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MAP Detector – PDF Approximation
SαS random variable Z with parameters , can be written Z = X Y½ [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 amixture model of N Gaussians[Kuruoglu, 1998]
• Mean 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
N
iiY
iY
N
i
v
z
vf
vfezp
i
1
2
2
1
2
,0,
2
2
2
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Myriad Filtering
Sliding window algorithm
Outputs myriad of sample window
Myriad of order k for samples x1, x2, … , xN [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]
1
2),(
k
22
11 minargˆ,,
i
N
ikNM xkxxg
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Myriad Filtering – Implementation
Given a window of samples x1,…,xN, find β [xmin, xmax]
Optimal myriad algorithm1. 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 x1,…,xN as the possible values of β
2. Pick value that minimizes objective function p(β)
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22
1)(
i
N
ixkp
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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
hp
hphp Tnx
Tnxnxh
][0
][][
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Complexity Analysis
Method Complexity per symbol
Analysis
Hole Puncher + Correlation Receiver
O(N+S) A decision needs to be made about each sample.
Optimal Myriad + Correlation Receiver
O(NW3+S) Due to polynomial rooting which is equivalent to Eigen-value decomposition.
Selection Myriad + Correlation Receiver
O(NW2+S) Evaluation of the myriad function and comparing it.
MAP Approximation O(MNS) Evaluating approximate pdf(M is number of Gaussians in mixture)
N is oversampling factor S is constellation size W is window size
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Bit Error Rate (BER) Performance in Alpha Stable Noise
-10 -5 0 5 10 15 20
10-2
10-1
100
Generalized SNR
BE
R
Communication Performance (=0.9, =0, M=12)
Matched FilterHole PunchingMAPMyriad
-10 -5 0 5 10 15 2010
-5
10-4
10-3
10-2
10-1
100
Generalized SNR
BE
R
Communication Performance (=1.5,=0,M=12)
Matched FilterHole PunchingMLMyriad
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4. Performance Bounds in presence of impulsive noise
Channel Capacity
Case I Shannon Capacity in presence of additive white Gaussian noise
Case II (Upper Bound) Capacity in the presence of Class A noiseAssumes 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 noiseAssumes input has Gaussian distribution (e.g. bit interleaved coded modulation (BICM) or OFDM modulation [Haring, 2003])
NXY System Model
)()(
)|()(
);(max}}{),({ 2
NhYh
XYhYh
YXICsX EXExf
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Capacity in Presence of Impulsive Noise
)()(
)|()(
);(max}}{),({ 2
NhYh
XYhYh
YXICsX EXExf
NXY
-40 -30 -20 -10 0 10 200
5
10
15
SNR [in dB]
Cap
acity
(bi
ts/s
ec/H
z)
Channel Capacity
X: Gaussian, N: Gaussian
Y:Gaussian, N:ClassA (A = 0.1, = 10-3)
X:Gaussian, N:ClassA (A = 0.1, = 10-3) System Model
Capacity
)()(
)|()(
);(max}}{),({ 2
NhYh
XYhYh
YXICsX EXExf
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Probability of Error for Uncoded Transmission
)(!
2
0m
AWGNe
m
mA
e Pm
AeP
-40 -30 -20 -10 0 10 2010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
dmin
/ [in dB]
Pro
babi
lity
of e
rror
Probability of error (Uncoded Transmission)
AWGN
Class A: A = 0.1, = 10-3
12 A
m
m
BPSK uncoded transmission
One sample per symbol
A = 0.1, Γ = 10-3
[Haring & Vinck, 2002]
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Chernoff Factors for Coded Transmission
N
kkk ccC
PPEP
1
'
'
),,(min
)(
cc
-20 -15 -10 -5 0 5 10 1510
-3
10-2
10-1
100
dmin
/ [in dB]
Che
rnof
f F
acto
r
Chernoff factors for real channel with various parameters of A and MAP decoding
Gaussian
Class A: A = 0.1, = 10-3
Class A: A = 0.3, = 10-3
Class A: A = 10, = 10-3
PEP: Pairwise error probability
N: Size of the codeword
Chernoff factor:
Equally likely transmission for symbols
),,(min ' kk ccC
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Part IISingle Carrier, Multiple Antenna Communication
Systems
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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]
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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
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Communication Performance
0 5 10 15 20 2510
-5
10-4
10-3
10-2
10-1
100
Performance of MIMO Receivers in Implusive Noise (A = 0.1, 1 =
2 = 10-3; = 0.1)
Vec
tor
Sym
bol E
rror
Rat
e (V
SE
R)
SNR [in dB]
ML (Guassian)
ML (Impulsive)Sub-Optimal ML (Impulsive)
2 x 2 MIMO systemA = 0.1, Γ1 = Γ2 = 10-3 Correlation Coeff. = 0.1
Spatial Multiplexing Mode
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Part III
Multiple Carriers, Single Antenna Communication Systems
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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
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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
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N
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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
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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 ReleasesRFI Mitigation Toolbox
Version 1.1 Beta (Released November 21st, 2007)Version 1.0 (Released September 22nd, 2007)
http://users.ece.utexas.edu/~bevans/projects/rfi/software.html
Project Web Sitehttp://users.ece.utexas.edu/~bevans/projects/rfi/index.html
36
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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
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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
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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.
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BACKUP SLIDES
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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 multipleRF sources on single chip
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Soviet high power over-the-horizon radar interference [Middleton, 1999]
Fluorescent lights in mine shop office interference [Middleton, 1999]
P(ε > ε0)
ε 0 (
dB
> ε
rms)
Percentage of Time Ordinate is ExceededM
agne
tic F
ield
Str
engt
h, H
(dB
rel
ativ
e to
m
icro
amp
per
met
er r
ms)
Accuracy of Middleton Noise Models
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10
-8
-6
-4
-2
0
2
4
6
8
10
Frequency
Pow
er S
pect
rum
Mag
nitu
de (
dB)
Power Spectal Density of Class A noise, A = 0.15, = 0.1
Power Spectral Density
Middleton Class A Statistics
00
0 !
2)(
2
2
02
z
zezm
Ae
zwm
z
m m
mA
Envelope statistics
Envelope for Gaussian signal has Rayleigh distribution
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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
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Probability Density Function Power Spectral Density
Example: exponent = 1.5, “mean” = 0 and “variance” = 10
Symmetric Alpha Stable Statistics ||)( je
-50 -40 -30 -20 -10 0 10 20 30 40 500
1
2
3
4
5
6
7
8x 10
-4 PDF for SS noise, = 1.5, =10, = 0
x
Pro
babi
lity
dens
ity f
unct
ion
f X(x
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10
-8
-6
-4
-2
0
2
4
6
8
10
Frequency
Pow
er S
pect
rum
Mag
nitu
de (
dB)
Power Spectal Density of S S noise, = 1.5, = 10, = 0×10-4
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00
0 !
2)(
2
2
02
z
zezm
Ae
zwm
z
m m
mA
2
0
2
2
2),|(;!
),|()(
j
z
j
Aj
j
jj
j
jezAzp
j
eA
Azpzw
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).
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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)
)( )](1[ )(
)( )( )(1
:
1:
xfxFNxf
xfxFNxf
XN
Nm
XN
NM
Backup
Backup
Backup
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering48
Class A Parameter Estimation Based on APD (Exceedance Probability Density) Plot
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering49
Class A Parameter Estimation Based on Moments
Moments (as derived from the characteristic equation)
Parameter estimates
2
e2 =
e4 =
e6 =
Odd-order momentsare zero
[Middleton, 1999]
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Department of Electrical and Computer Engineering50
Middleton Class B Model
Envelope StatisticsEnvelope exceedance probability density (APD) which is 1 – cumulative distribution function
Bm
mBA
IIB
BB
BBB
i
B
mm
mIB
mBB em
AeP
GG
AA
G
N
Fwhere
mF
m
m
AP
00
)2/(01
''
200
11
00110
001
220
!)(
2
4
)1(4
1;
2ˆ;
2ˆ
function trichypergeomeconfluent theis,
ˆ;2;2
1.2
1.!
ˆ)1(ˆ1)(
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering51
Class B Envelope Statistics
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Exceedance Probability Density Graph for Class B Parameters: A = 10-1, A
B = 1,
B = 5, N
I = 1, = 1.8
No
rma
lize
d E
nve
lop
e T
hre
sho
ld (
E 0 /
Erm
s)
P(E > E0)
PB-I
PB-II
B
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Department of Electrical and Computer Engineering52
Parameters for Middleton Class B Noise
B
I
B
B
A
N
A
Parameters Description Typical Range
Impulsive Index AB [10-2, 1]
Ratio of Gaussian to non-Gaussian intensity ΓB [10-6, 1]
Scaling Factor NI [10-1, 102]
Spatial density parameter α [0, 4]
Effective impulsive index dependent on α A α [10-2, 1]
Inflection point (empirically determined) εB > 0
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Department of Electrical and Computer Engineering53
Class B Exceedance Probability Density Plot
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Department of Electrical and Computer Engineering54
Expectation Maximization Overview
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Maximum Likelihood for Sum of Densities
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EM Estimator for Class A Parameters Using 1000 Samples
PDFs with 11 summation terms50 simulation runs per setting
Convergence criterion:Example learning curve
7
1
1 10ˆ
ˆˆ
n
nn
A
AA
1e-006 1e-005 0.0001 0.001 0.01
10
15
20
25
30
K
Num
ber
of I
tera
tions
Number of Iterations taken by the EM Estimator for A
A = 0.01
A = 0.1
A = 1
Iterations for Parameter A to Converge
1e-006 1e-005 0.0001 0.001 0.01
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
x 10-3
K
Fra
ctio
nal M
SE
= |
(A -
Aes
t) /
A |2
Fractional MSE of Estimator for A
A = 0.01
A = 0.1
A = 1
Normalized Mean-Squared Error in A×10-3
2
)(A
AAANMSE est
est
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Department of Electrical and Computer Engineering57
Results of EM Estimator for Class A Parameters
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering58
Extreme Order Statistics
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Department of Electrical and Computer Engineering59
Estimator for Alpha-Stable
0 < p < α
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Department of Electrical and Computer Engineering60
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09MSE in estimates of the Characteristic Exponent ()
Characteristic Exponent:
Mea
n S
quar
ed E
rror
(M
SE
)
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
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Results for Symmetric Alpha Stable Parameter Estimator
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
7MSE in estimates of the Dispersion Parameter ()
Characteristic Exponent: M
ean
Squ
ared
Err
or (
MS
E)
Mean squared error in estimate of dispersion (“variance”)
= 5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
7
8
9x 10
-3 MSE in estimates of the Localization Parameter ()
Characteristic Exponent:
Mea
n S
quar
ed E
rror
(M
SE
)
Mean squared error in estimate of localization (“mean”)
= 10
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering62
Minimize Mean-Squared Error E { |e(n)|2 }
d(n)
z(n)
d(n)^w(n)
x(n)
w(n)x(n) d(n)^
d(n)
e(n)
d(n): desired signald(n): filtered signale(n): error w(n): Wiener filter x(n): corrupted signalz(n): noise
d(n):^
Wiener Filtering – Linear Filter
Optimal in mean squared error sense when noise is Gaussian
Model
Design
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering63
Wiener Filtering – Finite Impulse Response (FIR) CaseWiener-Hopf equations for FIR Wiener filter of order p-1
General solution in frequency domain
)1(
)1(
)0(
)1(
)1(
)0(
0...21
1
1...10 **
pr
r
r
pw
w
w
rprpr
r
prrr
dx
dx
dx
xxx
x
xxx
)()(
)(
)(
)(2
j
zj
d
jd
jx
jdx
ee
e
e
eje
MMSEH
desired signal: d(n)power spectrum: (e j )
correlation of d and x: rdx(n)autocorrelation of x: rx(n)Wiener FIR Filter: w(n)
corrupted signal: x(n)noise: z(n)
1 1 0 )()()(1
0
p-...,,,kkrlkrlwp
ldxx
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering64
Wiener Filtering – 100-tap FIR Filter
ChannelA = 0.35
= 0.5 × 10-3
SNR = -10 dBMemoryless
Pulse shape10 samples per symbol10 symbols per pulse
Raised Cosine Pulse Shape
Transmitted waveform corrupted by Class A interference
Received waveform filtered by Wiener filter
n
n
n
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering65
Incoherent Detection
Bayes formulation [Spaulding & Middleton, 1997, pt. II]
)(),()(:2
)(),()(:1
2
1
tZtStXH
tZtStXH
1)(
)(
)()|(
)()|(
)(2
1
1
2
1
2
H
H
Xp
Xp
dpHXp
dpHXp
X
φ: phaseea:amplituda
and where
Small signal approximation
)(xpdx
d)l(xwhere
txltxl
txltxl
iZi
iH
H
N
iii
N
iii
N
iii
N
iii
ln 1
sin)(cos)(
sin)(cos)(
2
1
2
11
2
11
2
12
2
12
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering66
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
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Department of Electrical and Computer Engineering67
Coherent Detection – Class A Noise
Comparison of performance of correlation receiver (Gaussian optimal receiver) and nonlinear detector [Spaulding & Middleton, 1997, pt. II]
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Department of Electrical and Computer Engineering68
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
AntipodalA = 0.35 = 0.5×10-3
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering69
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.
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering70
[Widrow et al., 1975]
s : signals+n0 :corrupted signaln0 : noisen1 : reference inputz : 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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering71
0 500 1000-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 500 1000-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Region 2
Region 1
Region 3
Gaussian Class A (with same power)
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering72
Haring’s Receiver Simulation Results
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Department of Electrical and Computer Engineering73
Coherent Detection in Class A Noise with Γ = 10-4
SNR (dB) SNR (dB)
Correlation Receiver Performance
A = 0.1
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Department of Electrical and Computer Engineering74
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)
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Department of Electrical and Computer Engineering75
Decision Rule Λ(X) H1 or H2
corrupted signal
MAP Detection
Hard decision
Bayesian formulation [Spaulding and Middleton, 1977]
1)|()(
)|()()(
2
1
11
22
H
H
HXpHp
HXpHpX
ZSXH
ZSXH
22
11
:
:
1)(
)()(
2
1
1
2
H
H
Z
Z
SXp
SXpX
Equally probable source
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering76
Results