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Improving Wireless Data Transmission Speed and Reliability to Mobile Computing Platforms. Prof. Brian L. Evans 1. Preliminary Results. Improving Wireless Data Transmission Speed and Reliability to Mobile Computing Platforms. - PowerPoint PPT Presentation
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Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
Improving Wireless Data Transmission Speed andReliability to Mobile Computing Platforms
in collaboration with Marcel Nassar1, Kapil Gulati1,Arvind K. Sujeeth1, Navid Aghasadeghi1 and Keith R. Tinsley2
1 The University of Texas at Austin, Austin, Texas USA2 System Technology Lab, Intel, Hillsborough, Oregon USA
American University of Beirut 15th July 2008
Improving Wireless Data Transmission Speed and Reliability to Mobile Computing Platforms
Prof. Brian L. Evans1
Preliminary Results
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
2
Outline
Problem definition
Noise modelling
Estimation of noise model parameters
Filtering and detection
Conclusion and future work
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
3
Problem Definition• Within computing platforms, wireless
transceivers experience radio frequencyinterference (RFI) from clocks/bussesPCI Express bussesLCD clock harmonics
Approach• Statistical modelling of RFI• Filtering/detection based on estimation of model parameters
Previous Research• Potential reduction in bit error rates by factor of 10 or more
[Spaulding & Middleton, 1977]
We’ll be using noise and interference interchangeably
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
4
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-
2005
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
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
5
Computer Platform Noise Modelling• RFI is combination of independent radiation events• Has predominantly non-Gaussian statistics
Statistical-Physical Models (Middleton Class A, B, C)• Independent of physical conditions (universal)• Sum of independent Gaussian and Poisson interference• Models electromagnetic interference
Alpha-Stable Processes• Models statistical properties of “impulsive” noise• Approximation for Middleton Class B (broadband) noise
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Proposed Contributions
Computer Platform Noise Modelling
Evaluate fit of measured RFI data to noise modelsNarrowband Interference: Middleton Class A modelBroadband 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
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
7
Outline
Problem definition
Noise modelling
Estimation of noise model parameters
Filtering and detection
Conclusion and future work
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
8
Middleton Class A Model
A
Parameter 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]
-10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Noise amplitude
Pro
bability d
ensity f
unction
Probability Density Function for A = 0.15, = 0.8
0 0.2 0.4 0.6 0.8 1-5
-4
-3
-2
-1
0
1
2
3
4
5
Frequency
Pow
er
Spectr
um
Magnitude (
dB
)
Power Spectral Density for A = 0.15, = 0.8
Wireless Networking and Communications Group
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9
Middleton Class A Model
A
1
2!)(
2
2
02
2
2
Am
where
em
Aezf
m
z
m m
mA
Zm
Probability density function (pdf)
-10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Noise amplitude
Pro
bability d
ensity f
unction
PDF for A = 0.15, = 0.8
Parameter 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]
Wireless Networking and Communications Group
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Parameter Description Range
Characteristic Exponent. Amount of impulsiveness
Localization. Analogous to mean
Dispersion. Analogous to variance
10
Symmetric Alpha Stable Model
α
Probability Density Function for = 1.5, = 0 and = 10 Power Spectral Density for = 1.5, = 0 and = 10
δ
]2,0[α
),( ),0(
-50 0 500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Noise amplitude
Pro
babili
ty d
ensity f
unction
0 0.2 0.4 0.6 0.8 1-5
-4
-3
-2
-1
0
1
2
3
4
5
Frequency
Pow
er
Spectr
um
Magnitude (
dB
)
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
Parameter Description Range
Characteristic Exponent. Amount of impulsiveness
Localization. Analogous to mean
Dispersion. Analogous to variance
11
α
PDF for = 1.5, = 0 and = 10
δ
]2,0[α
),( ),0(
-50 0 500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Noise amplitude
Pro
babili
ty d
ensity f
unction
Symmetric Alpha Stable Model
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
Does not have second-order moment
||)( je
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12
Outline
Problem definition
Noise modelling
Estimation of noise model parameters
Filtering and detection
Conclusion and future work
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
13
Estimation of Noise Model Parameters
For Middleton Class A Model• Expectation maximization (EM) [Zabin & Poor, 1991]
• Finds roots of second and fourth order polynomials at each iteration• Advantage Small sample size required (~1,000 samples)• Disadvantage Iterative algorithm, computationally intensive
For Symmetric Alpha Stable Model• Based on extreme order statistics [Tsihrintzis & Nikias, 1996]
• Parameter estimators require computations similar to mean and standard deviation.
• Advantage Fast / computationally efficient (non-iterative)• Disadvantage Requires large set of data samples (~10,000 samples)
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-5 -4 -3 -2 -1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Measured Data Fitting
Noise amplitude
Pro
babi
lity
Den
sity
Fun
ctio
n
Measured PDF
Estimated AlphaStable PDFEstimated MiddletonClass A PDF
Estimated Equi-powerGaussian PDF
14
Results of Measured RFI Data for Broadband Noise
Data set of 80,000 samples collected using 20 GSPS scope
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Estimated Parameters
Symmetric Alpha Stable Model
Localization (δ) 0.0043Distance 0.0514
Characteristic exp. (α) 1.2105
Dispersion (γ) 0.2413
Middleton Class A Model
Overlap Index (A) 0.1036 Distance0.0825Gaussian Factor (Γ) 0.7763
Gaussian Model
Mean (µ) 0 Distance0.2217Variance (σ2) 1
Distance: Kullback-Leibler divergence
Wireless Networking and Communications Group
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15
Expectation-Maximization Estimator for Class A Noise
PDFs with 11 summation terms50 simulation runs per setting
1000 data samplesConvergence criterion:
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
Frac
tiona
l MS
E =
| (A
- A
est) /
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
7
1
1 10ˆ
ˆˆ
n
nn
A
AA
K = A
Wireless Networking and Communications Group
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Expectation-Maximization Estimator for Class A Noise
• For convergence for A [10-2, 1], worst-case number of iterations for A = 1
• Estimation accuracy vs. number of iterations tradeoff
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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
Data length (N) of 10,000 samples
Results averaged over 100 simulation runs
Estimate α and “mean” directly from data
Estimate “variance” γ from α and δ estimates
Symmetric Alpha Stable Parameter Estimator
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18
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
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19
Outline
Problem definition
Noise modelling
Estimation of noise model parameters
Filtering and detection
Conclusion and future work
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
20
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 Shape Pre-FilteringMatched
Filter Decision Rule
Impulsive Noise
Alternate Adaptive Model
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N samples per symbol
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21
Filtering and Detection – Methods
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]
Symmetric Alpha Stable Noise• Correlation receiver (linear)• Myriad filtering [Gonzalez & Arce, 2001]
• MAP approximation• Hole punching
We assume perfect estimation of noise model parameters
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Class A Detection – Results
22
Pulse shapeRaised cosine
10 samples per symbol10 symbols per pulse
ChannelA = 0.35
= 0.5 × 10-3
Memoryless
Method Comp. Detection Perform.
Correl. Low Low
Wiener Medium Low
MAP Approx.
Medium High
MAP High High
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23
Hole Punching (Blanking) for Pre-Filtering
Sets sample to 0 when sample exceeds threshold [Ambike, 1994]
Large values are impulses and true value cannot be recovered Replacing large values with zero will not bias (correlation) receiver
for two-level constellations 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
hp
hphp Tnx
Tnxnxh
][0
][][
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24
Myriad Filtering for Pre-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 passes through myriad filter As k→0, filter tends to mode filter (output value with highest freq.)
Empirical choice of k: [Gonzalez & Arce, 2001]
Developed for images corrupted by additive symmetric alpha stable impulsive noise
1
2),(
k
22
11 minargˆ,,
i
N
ikNM xkxxg
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Myriad Filter 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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Symmetric Alpha Stable Detection – Results
26
Method Comp. Detection Perform.
Hole Punching
Low Medium
Selection Myriad
Low Medium
MAP Approx.
Medium High
Optimal Myriad
High Medium-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
Use dispersion parameter in place of noise variance to generalize SNR
Wireless Networking and Communications Group
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Conclusion – Proposed Contributions
Computer Platform Noise Modelling
Evaluate fit of measured RFI data to noise modelsNarrowband Interference: Middleton Class A modelBroadband 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
27
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28
Conclusion – Filtering and Detection
TClass A Communication Performance Complexity
MAP approximation High Medium
MAP High High
Correlation receiver Low Low
Wiener filtering Low Medium
Symmetric Alpha Stable Communication Performance Complexity
MAP approximation High Medium
Selection myriad Medium Low
Hole punching Medium Low
Optimal myriad Medium High
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Conclusion – 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.
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
29
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Conclusion – Future Work on Impulsive Noise
• Communication performance bounds on single-carrier single-antenna detection
• Multi-input multi-output (MIMO) single-carrier receivers Performance analysis of standard MIMO receivers using
multivariate noise models Optimal and sub-optimal maximum likelihood (ML) 2 2 receiver
To be presented at 2008 Globecom Conference in December
• Multicarrier receivers
• Modelling co-channel interference
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Thank you,Questions?
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32
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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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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36
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
Wireless Networking and Communications Group
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37
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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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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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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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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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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Class A Parameter Estimation Based on APD (Exceedance Probability Density) Plot
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44
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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45
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)(
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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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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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Class B Exceedance Probability Density Plot
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49
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).
Backup
Backup
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
50
Expectation Maximization Overview
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
51
Maximum Likelihood for Sum of Densities
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
52
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 andComputer Engineering
53
Extreme Order Statistics
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
54
Estimator for Alpha-Stable
0 < p < α
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
55
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 andComputer Engineering
56
Wiener Filtering – Finite Impulse Response (FIR) Case
Wiener-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 andComputer Engineering
57
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 andComputer Engineering
58
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 andComputer Engineering
59
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
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
60
Coherent Detection – Class A Noise
Comparison of performance of correlation receiver (Gaussian optimal receiver) and nonlinear detector [Spaulding & Middleton, 1997, pt. II]
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
61
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 andComputer Engineering
62
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 andComputer Engineering
63
[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 andComputer Engineering
64
Coherent Detection in Class A Noise with Γ = 10-4
SNR (dB) SNR (dB)
Correlation Receiver Performance
A = 0.1
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
65
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)
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
66
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 andComputer Engineering
67
Results
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
68
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
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
69
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
][][
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering70
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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering71
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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering72
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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering73
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]
Backup
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering74
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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering
Part IISingle Carrier, Multiple Antenna Communication
Systems
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering76
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]
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering77
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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering78
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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering
Part III
Multiple Carriers, Single Antenna Communication Systems
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering80
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
Wireless Networking and Communications Group
Department of Electrical and Computer Engineering81
Noise Smearing
Smearing effect• Impulsive noise energy distributes over longer symbol time• Smearing filters maximize impulse attenuation and minimize
intersymbol interference for impulsive noise [Beenker, 1985]
• Maximum smearing efficiency is where N is number of symbols used in unitary transformation
• As N , distribution of impulsive noise becomes Gaussian
Simulations [Haring, 2003]
• When using a transformation involving N = 1024 symbols, impulsive noise case approaches case where only Gaussian noise is present
Backup
N
Wireless Networking and Communications Group
Department of Electrical andComputer Engineering
82
Haring’s Receiver Simulation Results