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Wireless Networking and Communications Group
Mitigating Near-field Interference in Laptop
Embedded Wireless Transceivers
Marcel Nassar(1), Kapil Gulati(1) , Arvind K. Sujeeth(1),
Navid Aghasadeghi(1), Brian L. Evans(1), Keith R. Tinsley(2)
(1)
The University of Texas at Austin, Austin, Texas, USA
(2) System Technology Lab, Intel, Hillsborough, Oregon, USA
2008 IEEE International Conference on
Acoustics, Speech, and Signal Processing
3rd April, 2008
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Problem Definition
• Within computing platforms, wireless
transceivers experience radio frequency
interference (RFI) from clocks/busses
PCI Express busses
LCD clock harmonics
Backup
We’ll be using noise and
interference interchangeably
Approach
• Statistical modelling of RFI
• Filtering/detection based on estimation of model parameters
Past Research
Potential reduction in bit error rates by factor of 10 or more
[Spaulding & Middleton, 1977]
2
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
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
Backup
Alpha-Stable Processes
• Models statistical properties of “impulsive” noise
• Approximation for Middleton Class B (broadband) noise
3
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Proposed Contributions
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: Correlation receiver
Wiener filtering and Myriad filtering
4
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Middleton Class A Model
Backup
0.7
5
4
Power Spectrum Magnitude (dB)
Probability density function
0.6
0.5
0.4
0.3
0.2
3
2
1
0
-1
-2
-3
0.1
-4
0
-10
-5
0
Noise amplitude
5
-5
10
A

0.2
0.4
0.6
Frequency
0.8
1
Power Spectral Density for A = 0.15, = 0.8
Probability Density Function for A = 0.15, = 0.8
Parameter
0
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]
5
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Symmetric Alpha Stable Model
Backup
5
0.07
4
Power Spectrum Magnitude (dB)
Probability density function
0.06
0.05
0.04
0.03
0.02
0.01
0
-50
2
1
0
-1
-2
-3
-4
0
Noise amplitude
-5
50
0
0.2
0.4
0.6
Frequency
0.8
1
Power Spectral Density for = 1.5,  = 0 and  = 10
Probability Density Function for  = 1.5,  = 0 and  = 10
Parameter
3
Description
Range
α
Characteristic Exponent. Amount of impulsiveness
α[0,2]
δ
Localization. Analogous to mean

Dispersion. Analogous to variance
(
,)
 (0, )
6
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Estimation of Noise Model Parameters
For Middleton Class A Model
• Expectation maximization (EM) [Zabin & Poor, 1991]
Backup
• Finds roots of second and fourth order polynomials at each iteration
• Advantage
Small sample size required (~1000 samples)
• Disadvantage Iterative algorithm, computationally intensive
For Symmetric Alpha Stable Model
• Based on extreme order statistics [Tsihrintzis & Nikias, 1996]
Backup
• 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)
7
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Results of Measured RFI Data for Broadband Noise
Backup
Data set of 80,000 samples collected using 20 GSPS scope
Measured Data Fitting
1
Probability Density Function
Estimated Parameters
Measured PDF
Estimated Alpha
Stable PDF
Estimated Middleton
Class A PDF
Estimated Equi-power
Gaussian PDF
0.8
Symmetric Alpha Stable Model
0.6
Localization (δ)
0.0043
Characteristic exp. (α)
1.2105
Dispersion (γ)
0.2413
Middleton Class A Model
Overlap Index (A)
0.1036
Gaussian Factor (Γ)
0.7763
0.4
Gaussian Model
0.2
0
-5
-4
-3
-2
-1
0
1
Noise amplitude
2
3
4
5
8
Mean (µ)
0
Variance (σ2)
1
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Filtering and Detection – System Model
Signal Model
Pulse Shape
s[n]
Alternate
Adaptive Model
Impulsive
Noise
v[n]
Pre-Filtering
gtx[n]
Backup
Matched
Filter
Decision Rule
grx[n]
Λ(.)
Multiple samples/copies of the received signal are available:
• N path diversity [Miller, 1972]
• Oversampling by N [Middleton, 1977]
N samples per
symbol
Using multiple samples increases gains vs. Gaussian case
because impulses are isolated events over symbol period
9
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
We assume perfect
estimation of noise
model parameters
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]
Backup
Backup
Backup
Alpha Stable Noise
• Correlation Receiver (linear)
• Myriad Filtering [Gonzalez & Arce, 2001]
• MAP Approximation
• Hole Puncher
Backup
Backup
Backup
10
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Class A Detection - Results
Pulse shape
Raised cosine
10 samples per symbol
10 symbols per pulse
Method
11
Comp.
Channel
A = 0.35
 = 0.5 × 10-3
Memoryless
Detection
Perform.
Correl.
Low
Low
Wiener
Medium Low
Approx.
Medium High
MAP
High
High
Department of Electrical and
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Wireless Networking and Communications Group
Alpha Stable Results
10
-1
BER
10
0
Method
Communication Performance ( =0.9, =0, M=12)
10
-2
-10
Matched Filter
Hole Punching
MAP
Myriad
-5
0
5
10
15
20
Comp.
Detection
Perform.
Hole
Punching
Low
Medium
Selection
Myriad
Low
Medium
MAP
Approx.
Medium High
Optimal
Myriad
High
Medium
Generalized SNR
12
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Conclusion
Class
A Noise
T
MAP
High Performance
High Complexity
MAP approximation
High Performance
Medium Complexity
Correlation Receiver
Low Performance
Low Complexity
Wiener Filtering
Low Performance
Medium Complexity
MAP Approximation
High Performance
Medium Complexity
Optimal Myriad
Medium Performance
High Complexity
Selection Myriad
Medium Performance
Low Complexity
Hole Puncher
Medium Performance
Low Complexity
Alpha Stable Noise
13
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Thank you,
Questions?
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
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. 6072, 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 ImpulsiveNoise Environments”, IEEE Trans. on Signal Processing, vol 49, no. 2, Feb 2001
15
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
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 impulsivenoise 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.
16
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
BACKUP SLIDES
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Common Spectral Occupancy
Standard
Carrier
(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
2.5–2.69
3.3–3.8
5.725–5.85
Mobile
Broadband
(Wi-Max)
PCI Express Bus,
LCD clock harmonics
IEEE
802.11a
5.2
Wireless LAN
(Wi-Fi)
PCI Express Bus,
LCD clock harmonics
18
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
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 multiple
RF sources on single chip
19
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
ε0 (dB > εrms)
Magnetic Field Strength, H (dB relative to
microamp per meter rms)
Accuracy of Middleton Noise Models
Percentage of Time Ordinate is Exceeded
P(ε > ε0)
Soviet high power over-the-horizon radar
interference [Middleton, 1999]
Fluorescent lights in mine shop office
interference [Middleton, 1999]
20
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Middleton Class A, B, C Models
[Middleton, 1999]
Class A
Class B
Class C
Narrowband interference (“coherent” reception)
Uniquely represented by two parameters
Broadband interference (“incoherent” reception)
Uniquely represented by six parameters
Sum of class A and class B (approx. as class B)
21
Department of Electrical and
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Wireless Networking and Communications Group
Middleton Class A Model
Class A Probability Density Function; A = 0.15,  = 0.1
0.025
Probability density function (pdf)
0.02
where  m2
 m!
Am
m 0
2 m2
e
0.015
x

PDF f (x)
f Z ( z)  e
A
z2

2
2 m
0.01
m

 A
1 
0.005
0
-10
-8
-6
-4
-2
0
x

4
6
8
10
Probability Density Function for A = 0.15,  = 0.1
Parameters Description
A
2
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]
22
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Symmetric Alpha Stable Model
Characteristic function:  ( )  e
j  ||
PDF for S  S noise,  = 1.5,  =10,  = 0
-4
8
x 10
of thickness of tail of impulsiveness
-  δ   Localization (analogous to mean)
 0
Dispersion (analogous to variance)
6
X
Parameters
Characteristic exponent indicative
α  0, 2
Probability density function f (x)
7
5
4
3
2
1
0
-50
-40
-30
-20
-10
0
x
10
20
30
40
50
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
23
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Results of Measured RFI Data for Broadband Noise
Data set of 80,000 samples collected using 20 GSPS scope
Measured Data Fitting
1
Probability Density Function
Estimated Parameters
Measured PDF
Estimated Alpha
Stable PDF
Estimated Middleton
Class A PDF
Estimated Equi-power
Gaussian PDF
0.8
Symmetric Alpha Stable Model
Localization (δ)
0.0043
Characteristic exp. (α)
1.2105
Dispersion (γ)
0.2413
0.6
Middleton Class A Model
0.4
Overlap Index (A)
0.1036
Gaussian Factor (Γ)
0.7763
KL Divergence
0.0825
Gaussian Model
0.2
0
-5
KL Divergence
0.0514
-4
-3
-2
-1
0
1
Noise amplitude
2
3
4
5
Mean (µ)
0
Variance (σ2)
1
24
KL Divergence
0.2217
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Coherent Detection – Small Signal Approximation
Expand noise pdf pZ(z) by Taylor series about Sj = 0 (j=1,2)

p
(
X
)
p
(
X

S
)

p
(
X
)


p
(
X
)

S

p
(
X
)

s

Z
j
Z
Z
ji

x
i

1
i
 

Z
j
N
Z
Optimal decision rule & threshold detector for approximation
N
d
1

s
ln
p
(x

2
i
Z
i) H
1

dx
i
1
i

(X
) N
1
d

H
2
1

s
ln
p
(
x
)

1
i
Z i
dx
i
1
i
We use 100 terms of the
series expansion for
d/dxi ln pZ(xi) in simulations
Optimal detector for approximation is logarithmic nonlinearity
followed by correlation receiver
Backup
25
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Hole Punching (Blanking) Filter
Sets sample to 0 when sample exceeds threshold [Ambike, 1994]
x
[
n
]x
[
n
]
T

hp
h


hp
[
n
]
T
hp
0 x
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
26
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Filtering and Detection – Alpha Stable Model
MAP detection: remove nonlinear filter
Decision rule is given by (p(.) is the SαS distribution)
1

p
(
H
)
p
(
X
|
H
)
2

(
X
)
 2
1

p
(
H
)
p
(
X
|H
)H
1
1
2
H
Approximations for SαS distribution:
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
27
Department of Electrical and
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Wireless Networking and Communications Group
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 
N
z2
 2
2vi
2e
Pdf of Z can be written as a
mixture model of N Gaussians
p,0,zi1
[Kuruoglu, 1998]

fY vi2
f v 
N
Y
2
i
i
1
• 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
28
Department of Electrical and
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Wireless Networking and Communications Group
Bit Error Rate (BER) Performance in Alpha Stable Noise
10
0
Communication Performance ( =0.9, =0, M=12)
0
10
Communication Performance ( =1.5,=0,M=12)
-1
10
-2
-1
10
BER
BER
10
-3
10
10
-2
Matched Filter
Hole Punching
MAP
Myriad
-4
10
Matched Filter
Hole Punching
ML
Myriad
-5
-10
-5
0
5
10
15
20
10
-10
Generalized SNR
29
-5
0
5
10
Generalized SNR
15
20
Department of Electrical and
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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
30
Department of Electrical and
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Wireless Networking and Communications Group
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
Backup
EM estimator for Class A parameters [Zabin & Poor, 1991]
• Expresses envelope statistics as sum of weighted pdfs

 A
m


A
m

2
e
z
e
z

0
2
w
(
z
)

 m
m
!


0
m

0
z

0

w
(z)
j pj(z| A
,
)
2
z
2
j
0
z2
2

j A
A
e
e j
j 
; pj(z| A
,
)2z 2
j!
j
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).
31
Backup
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Estimation of Symmetric Alpha Stable Parameters
Based on extreme order statistics [Tsihrintzis & Nikias, 1996]
Backup
PDFs of max and min of sequence of independently and
identically distributed (IID) data samples follow
N

1
f
(
x
)

N
F
(
x
)f
(
x
)
:
N
X
• PDF of maximum: M
N

1
f
(
x
)

N
[
1

F
(
x
)]
f
(
x
)
m
:
N
X
• 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
Backup
Backup
• Advantage
Fast / computationally efficient (non-iterative)
• Disadvantage Requires large set of data samples (N ~ 10,000)
32
Department of Electrical and
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Wireless Networking and Communications Group
Class A Parameter Estimation
Based on APD (Exceedance
Probability Density) Plot
33
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Wireless Networking and Communications Group
Class A Parameter Estimation Based on Moments
Moments (as derived from the characteristic equation)
e2 =
e4 =
e6 =
Odd-order
moments
are zero
Parameter estimates
[Middleton, 1999]
2
34
Department of Electrical and
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Wireless Networking and Communications Group
Middleton Class B Model
Envelope Statistics
Envelope exceedance probability density (APD) which is 1 – cumulative
distribution function
ˆm  
(
1
)mA
m
m
  

.
ˆ
ˆ
P
(



)

1


1

.
F
1

;
2
;






1
0B

I
0
11
0
m
!
 2  2

m

0
where
, 1F
the
confluent
hypergeome
tric
function
1is

0N
i
ˆ


;
0
2
G
B
1 4


2
'
 ; G
ˆ A
A





B
B
 
'
2G
4
(
1


)2

 
B
B

m
2
2
A
0
/(
2

)
B 
mB
P
(



)

e
e

1
0B

II
!
m

0m

A
B
0
B
0
B
35
Department of Electrical and
Computer Engineering
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Class B Envelope Statistics
Exceedance Probability Density Graph for Class B
Parameters: A = 10-1, A = 1,  = 5, N = 1,  = 1.8
Normalized Envelope Threshold (E0 / Erms )

B
B
I
4.5
4
3.5

3
2.5
PB-II

B
2
1.5

1
PB-I
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P(E > E0)
36
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Parameters for Middleton Class B Noise
Parameters Description
Typical Range
A
B
Impulsive Index
AB  [10-2, 1]

B
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
N
I

A

B
37
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Class B Exceedance
Probability Density Plot
38
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Computer Engineering
Wireless Networking and Communications Group
Expectation Maximization Overview
39
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Computer Engineering
Wireless Networking and Communications Group
Maximum Likelihood for Sum of Densities
40
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
EM Estimator for Class A Parameters Using 1000 Samples
Normalized
Mean-Squared
Error in A
Fractional MSE of Estimator for A
x 10
-3
×10-3
A = 0.01
A = 0.1
A=1
2.6
A

A
est
NMSE
(
A
)
est
2.2
A
2
A = 0.01
A = 0.1
A=1
30
25
Number of Iterations
Fractional MSE = | (A - A
est
) / A |2
2.4
Iterations
forof Iterations
Parameter
AEstimator
to Converge
Number
taken by the EM
for A
2
1.8
1.6
1.4
20
15
1.2
1
10
0.8
1e-006
1e-005
0.0001
K
0.001
PDFs with 11 summation terms
50 simulation runs per setting
0.01
1e-006
1e-005
0.0001
K
0.001
0.01
ˆ A
ˆ
A

7
n
n
1
10
Convergence criterion:
ˆ
A
n
1
Example learning curve
41
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Results of EM Estimator for Class A Parameters
42
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Wireless Networking and Communications Group
Extreme Order Statistics
43
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Estimator for Alpha-Stable
0<p<α
44
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Results for Symmetric Alpha Stable Parameter Estimator
MSE in estimates of the Characteristic Exponent (  )
0.09
Data length (N) was
10,000 samples
0.08
Mean Squared Error (MSE)
0.07
0.06
Results averaged over
100 simulation runs
0.05
0.04
Estimate α and “mean” 
directly from data
0.03
0.02
0.01
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Characteristic Exponent: 
1.6
1.8
Estimate “variance” γ
from α and δ estimates
2
Mean squared error in estimate
of characteristic exponent α
Continued next slide
45
Department of Electrical and
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Results for Symmetric Alpha Stable Parameter Estimator
MSE in estimates of the Dispersion Parameter ()
MSE in estimates of the Localization Parameter ( )
-3
9
x 10
7
8
6
Mean Squared Error (MSE)
Mean Squared Error (MSE)
7
6
5
4
 = 10
3
5
4
3
=5
2
2
1
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Characteristic Exponent: 
1.6
1.8
0
0
2
0.2
0.4
0.6
0.8
1
1.2
1.4
Characteristic Exponent: 
1.6
1.8
2
Mean squared error in estimate Mean squared error in estimate
of localization (“mean”) 
of dispersion (“variance”) 
46
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Wiener Filtering – Linear Filter
Optimal in mean squared error sense when noise is Gaussian
Model
d(n)
^
x(n)
d(n)
w(n)
d(n)
Design
x(n)
d(n):
^
d(n):
e(n):
w(n):
x(n):
z(n):
z(n)
^
w(n)
d(n)
desired signal
filtered signal
error
Wiener filter
corrupted signal
noise
Minimize Mean-Squared
Error E { |e(n)|2 }
e(n)
47
Department of Electrical and
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Wiener Filtering – Finite Impulse Response (FIR) Case
Wiener-Hopf equations for FIR Wiener filter of order p-1
*
*
(
0
)
r

(
0
)r

 r

 ...


0
1
r
p

1
w
dx
x
x
x



w
r
(
1
)
(
1
)


r
1



dx
x








    


 





r
(
p

1
)
w
(
p

1
)






r
p

1
r
p

2
...
r
0




dx

x
x
x


p

1
w
(
l
)
r
(
k

l
)

r
(
k
)k

0
,
1
,
...,
p1

x
dx
l

0
General solution in frequency domain



  
j
j
2

(
e
)

(
e
)
j
dx
d


H
e



j
j
j
MMSE
(
e
)

(
e
)


(
e
 
x
z )
d
48
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)
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Wiener Filtering – 100-tap FIR Filter
Raised Cosine
Pulse Shape
Transmitted waveform corrupted by Class A interference
Pulse shape
10 samples
per symbol
10 symbols
n per pulse
Channel
A = 0.35
 = 0.5 × 10-3
nSNR = -10 dB
Memoryless
Received waveform
filtered by Wiener filter
n
49
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Incoherent Detection
Bayes formulation [Spaulding & Middleton, 1997, pt. II]
p
(
X
|H
)
p
(

)
d


H
1
:X
(
t
)

S
(
t
,

)

Z
(
t
)

1
p
(
X
)


(
X
)


1

p
(
X
)
H
2
:X
(
t
)

S
(
t
,

)

Z
(
t
)
p
(
X
|
H
)
p
(

)
d

2


2
H
1
2
1
a











1
H
2
where
a:amplitud
e
and
φ: phase
Small signal approximation


2


2
N
N



l
(
x
)
cos
t

l
(
x
)
sin
t
H


i
2
i
i
2
i
1




d
i

1
i

1


 1
where
l(x
)

ln
p
(x
)
i
Z
i
2 N
2
N

dx



H
i
2
l
(
x
)
cos
t

l
(
x
)
sin
t


i
1
i
i
1
i



i

1
i

1



50
Department of Electrical and
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Incoherent Detection
Optimal Structure:
Incoherent Correlation Detector
The optimal detector for the small signal approximation is basically
the correlation receiver preceded by the logarithmic nonlinearity.
51
Department of Electrical and
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Coherent Detection – Class A Noise
Comparison of performance
of correlation receiver
(Gaussian optimal receiver)
and nonlinear detector
[Spaulding & Middleton, 1997, pt. II]
52
Department of Electrical and
Computer Engineering
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Coherent Detection –
Small Signal Approximation
Antipodal
A = 0.35
 = 0.5×10-3
Correlation Receiver
Near-optimal for small
amplitude signals
Suboptimal for higher
amplitude signals
Communication performance of
approximation vs. upper bound
[Spaulding & Middleton, 1977, pt. I]
53
Department of Electrical and
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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.
54
Department of Electrical and
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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
s : signal
s+n0 :corrupted signal
n0 : noise
n1 : reference input
z : system output
[Widrow et al., 1975]
55
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Haring’s Receiver Simulation Results
56
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
Coherent Detection in Class A Noise with Γ = 10-4
A = 0.1
Correlation
Receiver
Performance
SNR (dB)
SNR (dB)
57
Department of Electrical and
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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)
58
Department of Electrical and
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MAP Detection
Hard decision
corrupted
signal
Decision Rule
Λ(X)
H1 or H2
Bayesian formulation [Spaulding and Middleton, 1977]
H
S
Z
1:X
1
H
S2Z
2:X
H
1

p
(
H
)
p
(
X
|
H
)
2

(
X
)
 2
1

p
(
H
)
p
(
X
|H
)
H
1
1
2
Equally probable source
H
1
p
(
X

S
)

Z
2

(
X
)

1

p
(
X

S
)
Z
1 H
2
59
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Results
60
Department of Electrical and
Computer Engineering
Wireless Networking and Communications Group
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 
N
Pdf of Z can be written as a
mixture model of N Gaussians
p ,0, z  
[Kuruoglu, 1998]
 2e
i 1

z2
2vi2
fY vi2 
 f v 
N
Y
2
i
i 1
• 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
61
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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]
N
2
g x ,, x   ˆ  arg min  k 2  x   
M
1
N
k


i 1
i

• 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:
k ( ,  ) 

2 
62
1

[Gonzalez & Arce, 2001]
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Myriad Filtering – Implementation
Given a window of samples x1,…,xN, find β  [xmin, xmax]
Optimal myriad algorithm

p(  )   k  xi   
1. Differentiate objective function
i 1
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)
N
2
2

Selection myriad (reduced complexity)
1. Use x1,…,xN as the possible values of β
Backup
2. Pick value that minimizes objective function p(β)
63
Department of Electrical and
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Hole Punching (Blanking) Filter
Sets sample to 0 when sample exceeds threshold [Ambike, 1994]
 x[n] x[n]  Thp
hhp  
x[n]  Thp
 0
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
64
Department of Electrical and
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Complexity Analysis
Method
Complexity
per symbol
Hole Puncher +
Correlation Receiver
O(N+S)
Analysis
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
65
W is window size
Department of Electrical and
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