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Transcript of Mitigating Radio Frequency Interference in Embedded Wireless Receivers Wireless Networking and...
Mitigating Radio Frequency Interference in
Embedded Wireless Receivers
Wireless Networking and Communications Group
April 21, 2023
Prof. Brian L. Evans
Lead Graduate StudentsAditya Chopra, Kapil Gulati and Marcel Nassar
In collaboration with Keith R. Tinsley and Chaitanya Sreerama at Intel Labs
Outline
Problem definition Single carrier single antenna systems
Radio frequency interference modeling Estimation of interference model parameters Filtering/detection
Multi-input multi-output (MIMO) single carrier systems Conclusions Future work
2
Wireless Networking and Communications Group
Wireless Networking and Communications Group
Problem Definition3
Objectives Develop offline methods to improve communication
performance in presence of computer platform RFI Develop adaptive online algorithms for these methods
Approach Statistical Modeling of RFI Filtering/Detection based on estimated model parameters
Within computing platforms, wireless transceivers experience radio frequency interference (RFI) from clocks and busses
We will use noise and interference interchangeably
We will use noise and interference interchangeably
Wireless Networking and Communications Group
Common Spectral Occupancy4
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
Wireless Networking and Communications Group
Impact of RFI5
Impact of LCD noise on throughput performance for a 802.11g embedded wireless receiver [J. Shi et al., 2006]
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Statistical Modeling of RFI6
Radio Frequency Interference (RFI) Sum of independent radiation events Predominantly non-Gaussian impulsive statistics
Key Statistical-Physical Models Middleton Class A, B, C models
Independent of physical conditions (Canonical) Sum of independent Gaussian and Poisson interference Model non-linear phenomenon governing RFI
Symmetric Alpha Stable models Approximation of Middleton Class B model
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Assumptions for RFI Modeling7
Key Assumptions [Middleton, 1977][Furutsu & Ishida, 1961] Infinitely many potential interfering sources with same
effective radiation power Power law propagation loss Poisson field of interferers
Pr(number of interferers = M |area R) ~ Poisson Poisson distributed emission times Temporally independent (at each sample time)
Limitations [Alpha Stable]: Does not include thermal noise Temporal dependence may exist
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Our Contributions8
Mitigation of computational platform noise in single carrier, single antenna systems [Nassar et al., ICASSP 2008]
Wireless Networking and Communications Group
Middleton Class A model9
Probability Density Function
1
2!)(
2
2
02
2
2
Am
where
em
Aezf
m
z
m m
mA
Zm
-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
A
Parameter
Description RangeOverlap 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
Symmetric Alpha Stable Model10
Characteristic Function
Closed-form PDF expression only forα = 1 (Cauchy), α = 2 (Gaussian),α = 1/2 (Levy), α = 0 (not very useful)
Approximate PDF using inverse transform of power series expansion
Second-order moments do not exist for α < 2 Generally, moments of order > α do not exist
||)( je
PDF for = 1.5, = 0 and = 10
-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
Parameter Description Range
Characteristic Exponent. Amount of impulsiveness
Localization. Analogous to mean
Dispersion. Analogous to variance
αδ
]2,0[α
),( ),0(
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Wireless Networking and Communications Group
Estimation of Noise Model Parameters11
Middleton Class A model Expectation Maximization (EM) [Zabin & Poor, 1991]
Find roots of second and fourth order polynomials at each iteration Advantage: Small sample size is required (~1000 samples) Disadvantage: Iterative algorithm, computationally intensive
Symmetric Alpha Stable Model Based on Extreme Order Statistics [Tsihrintzis & Nikias, 1996]
Parameter estimators require computations similar to mean and standard deviation computations
Advantage: Fast / computationally efficient (non-iterative) Disadvantage: Requires large set of data samples (~10000 samples)
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Results on Measured RFI Data12
Broadband RFI data 80,000 samples collected using 20GSPS scope
-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
Estimated ParametersSymmetric Alpha Stable Model
Localization (δ) 0.0043Distance 0.0514Characteristic 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
Backup
Wireless Networking and Communications Group
Filtering and Detection13
System Model
Assumptions: Multiple samples of the received signal are available
N Path Diversity [Miller, 1972]
Oversampling by N [Middleton, 1977]
Multiple samples increase gains vs. Gaussian case Impulses are isolated events over symbol period
Pulse Shapin
g
Pre-Filtering
Matched Filter
Detection Rule
Impulsive Noise
N samples per symbolN samples per symbol
Filtering and Detection Methods
Filtering Wiener Filtering (Linear)
Detection Correlation Receiver (Linear) MAP (Maximum a posteriori
probability) detector [Spaulding & Middleton, 1977]
Small Signal Approximation to MAP detector[Spaulding & Middleton, 1977]
Filtering Myriad Filtering
[Gonzalez & Arce, 2001] Hole Punching
Detection Correlation Receiver (Linear) MAP approximation
14
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Middleton Class A noise Symmetric Alpha Stable noise
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Results: Class A Detection15
Pulse shapeRaised cosine
10 samples per symbol10 symbols per pulse
ChannelA = 0.35
= 0.5 × 10-3
Memoryless
Wireless Networking and Communications Group
Filtering for Alpha Stable Noise16
Myriad Filtering Sliding window algorithm outputs myriad of a sample window Myriad of order k for samples x1,x2,…,xN [Gonzalez & Arce, 2001]
As k decreases, less impulsive noise passes through the myriad filter As k→0, filter tends to mode filter (output value with highest frequency)
Empirical Choice of k [Gonzalez & Arce, 2001]
Developed for images corrupted by symmetric alpha stable impulsive noise
22
11 minargˆ,,
i
N
ikNM xkxxg
1
2),(
k
Wireless Networking and Communications Group
Filtering for Alpha Stable Noise (Cont..)17
Myriad Filter Implementation Given a window of samples, x1,…,xN, find β [xmin, xmax] Optimal Myriad algorithm
1. Differentiate objective function polynomial p(β) with respect to β
2. Find roots and retain real roots3. Evaluate p(β) at real roots and extreme points4. 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)
22
1)(
i
N
ixkp
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Results: Alpha Stable Detection18
-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 SNRUse dispersion parameter in place of noise variance to generalize SNR
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Extensions to MIMO systems19
RFI Modeling Middleton Class A Model for two-antenna systems
[McDonald & Blum, 1997]
Closed form PDFs for M x N MIMO system not published Prior Work
Much prior work assumes independent noise at antennas Performance analysis of standard MIMO receivers in impulsive
noise [Li, Wang & Zhou, 2004]
Space-time block coding over MIMO channels with impulsive noise [Gao & Tepedelenlioglu,2007]
Wireless Networking and Communications Group
Our Contributions20
2 x 2 MIMO receiver design in the presence of RFI[Gulati et al., Globecom 2008]
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Results: RFI Mitigation in 2 x 2 MIMO 21
Complexity Analysis
Improvement in communication performance over conventional Gaussian ML receiver at symbol
error rate of 10-2
Communication Performance (A = 0.1, 1= 0.01, 2= 0.1, = 0.4)
-10 -5 0 5 10 15 20
10-3
10-2
10-1
SNR [in dB]
Vec
tor
Sym
bol E
rror
Rat
e
Optimal ML Receiver (for Gaussian noise)Optimal ML Receiver (for Middleton Class A)Sub-Optimal ML Receiver (Four-Piece)Sub-Optimal ML Receiver (Two-Piece)
Wireless Networking and Communications Group
Results: RFI Mitigation in 2 x 2 MIMO 22
Complexity Analysis
Complexity Analysis for decoding M-QAM modulated signal
Communication Performance (A = 0.1, 1= 0.01, 2= 0.1, = 0.4)
-10 -5 0 5 10 15 20
10-3
10-2
10-1
SNR [in dB]
Vec
tor
Sym
bol E
rror
Rat
e
Optimal ML Receiver (for Gaussian noise)Optimal ML Receiver (for Middleton Class A)Sub-Optimal ML Receiver (Four-Piece)Sub-Optimal ML Receiver (Two-Piece)
Wireless Networking and Communications Group
Conclusions23
Radio Frequency Interference from computing platform Affects wireless data communication transceivers Models include Middleton models and alpha stable models
RFI mitigation can improve communication performance Single carrier, single antenna systems
Linear and non-linear filtering/detection methods explored Single carrier, multiple antenna systems
Studied RFI modeling for 2x2 MIMO systems Optimal and sub-optimal receivers designed Bounds on communication performance in presence of RFI
Wireless Networking and Communications Group
Contributions24
PublicationsM. Nassar, K. Gulati, A. K. Sujeeth, N. Aghasadeghi, B. L. Evans and K. R. Tinsley, “Mitigating Near-field
Interference in Laptop Embedded Wireless Transceivers”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Mar. 30-Apr. 4, 2008, Las Vegas, NV USA.
K. Gulati, A. Chopra, R. W. Heath Jr., B. L. Evans, K. R. Tinsley, and X. E. Lin, ”MIMO Receiver Design in the Presence of Radio Frequency Interference”, Proc. IEEE Int. Global Communications Conf., Nov. 30-Dec. 4th, 2008, New Orleans, LA USA, accepted for publication.
A. Chopra, K. Gulati, B. L. Evans, K. R. Tinsley, and C. Sreerama, ``Performance Bounds of MIMO Receivers in the Presence of Radio Frequency Interference'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Apr. 19-24, 2009, Taipei, Taiwan, submitted.
Software ReleasesRFI Mitigation Toolbox
Version 1.1 Beta (Released November 21st, 2007)Version 1.0 (Released September 22nd, 2007)
Project Websitehttp://users.ece.utexas.edu/~bevans/projects/rfi/index.html
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Future Work25
Modeling RFI to include Computational platform noise Co-channel interference Adjacent channel interference
Multi-input multi-output (MIMO) single carrier systems RFI modeling and receiver design
Multicarrier communication systems Coding schemes resilient to RFI Circuit design guidelines to reduce computational platform
generated RFI
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26
Thank You,Questions ?
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References27
RFI Modeling[1] D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: New
methods and results for Class A and Class B noise models”, IEEE Trans. Info. Theory, vol. 45, no. 4, pp. 1129-1149, May 1999.
[2] K.F. McDonald and R.S. Blum. “A physically-based impulsive noise model for array observations”, Proc. IEEE Asilomar Conference on Signals, Systems& Computers, vol 1, 2-5 Nov. 1997.
[3] K. Furutsu and T. Ishida, “On the theory of amplitude distributions of impulsive random noise,” J. Appl. Phys., vol. 32, no. 7, pp. 1206–1221, 1961.
[4] J. Ilow and D . Hatzinakos, “Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers”, IEEE transactions on signal processing, vol. 46, no. 6, pp. 1601-1611, 1998.
Parameter Estimation[5] S. M. Zabin and H. V. Poor, “Efficient estimation of Class A noise parameters via the EM
[Expectation-Maximization] algorithms”, IEEE Trans. Info. Theory, vol. 37, no. 1, pp. 60-72, Jan. 1991
[6] G. A. Tsihrintzis and C. L. Nikias, "Fast estimation of the parameters of alpha-stable impulsive interference", IEEE Trans. Signal Proc., vol. 44, Issue 6, pp. 1492-1503, Jun. 1996
RFI Measurements and Impact[7] J. Shi, A. Bettner, G. Chinn, K. Slattery and X. Dong, "A study of platform EMI from LCD panels -
impact on wireless, root causes and mitigation methods,“ IEEE International Symposium on Electromagnetic Compatibility, vol.3, no., pp. 626-631, 14-18 Aug. 2006
Wireless Networking and Communications Group
References (cont…)28
Filtering and Detection[8] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment-
Part I: Coherent Detection”, IEEE Trans. Comm., vol. 25, no. 9, Sep. 1977[9] A. Spaulding and D. Middleton, “Optimum Reception in an Impulsive Interference Environment
Part II: Incoherent Detection”, IEEE Trans. Comm., vol. 25, no. 9, Sep. 1977[10] J.G. Gonzalez and G.R. Arce, “Optimality of the Myriad Filter in Practical Impulsive-Noise
Environments”, IEEE Trans. on Signal Processing, vol 49, no. 2, Feb 2001[11] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of Gaussian
noise and impulsive noise modelled as an alpha-stable process,” IEEE Signal Processing Letters, vol. 1, pp. 55–57, Mar. 1994.
[12] J. G. Gonzalez and G. R. Arce, “Optimality of the myriad filter in practical impulsive-noise environments,” IEEE Trans. on Signal Proc, vol. 49, no. 2, pp. 438–441, Feb 2001.
[13] E. Kuruoglu, “Signal Processing In Alpha Stable Environments: A Least Lp Approach,” Ph.D. dissertation, University of Cambridge, 1998.
[14] J. Haring and A.J. Han Vick, “Iterative Decoding of Codes Over Complex Numbers for Impulsive Noise Channels”, IEEE Trans. On Info. Theory, vol 49, no. 5, May 2003
[15] Ping Gao and C. Tepedelenlioglu. “Space-time coding over mimo channels with impulsive noise”, IEEE Trans. on Wireless Comm., 6(1):220–229, January 2007.
Wireless Networking and Communications Group
Backup Slides29
Most backup slides are linked to the main slides Miscellaneous topics not covered in main slides
Performance bounds for single carrier single antenna system in presence of RFI Backup
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Impact of RFI30
Calculated in terms of desensitization (“desense”) Interference raises noise floor Receiver sensitivity will degrade to maintain SNR
Desensitization levels can exceed 10 dB for 802.11a/b/g due to computational platform noise [J. Shi et al., 2006]Case Sudy: 802.11b, Channel 2, desense of 11dB More than 50% loss in range Throughput loss up to ~3.5 Mbps for very low receive signal strengths
(~ -80 dbm)
floor noise RX
ceInterferenfloor noise RXlog10 10desense
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Middleton Class A, B and C Models31
Class A Narrowband interference (“coherent” reception)Uniquely represented by 2 parameters
Class B Broadband interference (“incoherent” reception)Uniquely represented by six parameters
Class C Sum of Class A and Class B (approx. Class B)
[Middleton, 1999]
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Backup
Wireless Networking and Communications Group
Middleton Class B Model32
Envelope Statistics Envelope exceedence probability density (APD), which is 1 – cumulative
distribution function (CDF)
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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Middleton Class B Model (cont…)33
Middleton 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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Middleton Class B Model (cont…)34
Parameters for Middleton Class B Model
B
I
B
B
A
N
A
Parameters
Description Typical RangeImpulsive 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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Accuracy of Middleton Noise Models35
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
ag
neti
c Fi
eld
Str
en
gth
, H
(d
B r
ela
tive t
o
mic
roam
p p
er
mete
r rm
s)
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Symmetric Alpha Stable PDF36
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
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Symmetric Alpha Stable Model37
Heavy tailed distribution
Density functions for symmetric alpha stable distributions for different values of characteristic exponent alpha: a) overall density
and b) the tails of densities
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Parameter Estimation: Middleton Class A38
Expectation Maximization (EM) E Step: Calculate log-likelihood function \w current parameter values M Step: Find parameter set that maximizes log-likelihood function
EM Estimator for Class A parameters [Zabin & Poor, 1991] Express envelope statistics as sum of weighted PDFs
Maximization step is iterative Given A, maximize K (= A). Root 2nd order polynomial. Given K, maximize A. Root 4th order polynomial
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
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Backup
Results Backup
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Expectation Maximization Overview39
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Results: EM Estimator for Class A40
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
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
2
)(A
AAANMSE est
est
7
1
1 10ˆ
ˆˆ
n
nn
A
AA
K = A
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Results: EM Estimator for Class A41
• 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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Parameter Estimation: Symmetric Alpha Stable42
Based on extreme order statistics [Tsihrintzis & Nikias, 1996]
PDFs of max and min of sequence of i.i.d. data samples PDF of maximum PDF of minimum
Extreme order statistics of Symmetric Alpha Stable PDF approach Frechet’s distribution as N goes to infinity
Parameter Estimators then based on simple order statistics Advantage: Fast/computationally efficient (non-iterative) Disadvantage: Requires large set of data samples (N~10,000)
)( )](1[ )(
)( )( )(1
:
1:
xfxFNxf
xfxFNxf
XN
Nm
XN
NM
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Results Backup
Wireless Networking and Communications Group
Results: Symmetric Alpha Stable Parameter Estimator43
• Data length (N) of 10,000 samples
• Results averaged over 100 simulation runs
• Estimate α and “mean” directly from data
• Estimate “variance” from α and δ estimates
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
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Results: Symmetric Alpha Stable Parameter Estimator (Cont…)
44
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:
Mea
n S
quar
ed E
rror
(M
SE
)
Mean squared error in estimate of dispersion (“variance”)
Mean squared error in estimate of localization (“mean”)
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:
Mea
n S
quar
ed E
rror
(M
SE
)
Return
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Extreme Order Statistics45
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Parameter Estimators for Alpha Stable46
0 < p < α
Return
Results on Measured RFI Data
Best fit for 25 data sets taken under different conditions Return
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Wiener Filtering48
Optimal in mean squared error sense in presence of Gaussian noise
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
^
Model
Design
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Wiener Filter Design49
Infinite Impulse Response (IIR)
Finite Impulse Response (FIR) Weiner-Hopf equations for order p-1
)(eΦ+)(eΦ
)(eΦ=
)(eΦ
)(eΦ=eH
jωz
jωd
jωd
jωx
jωdxjω
MMSE
2
10,1,... 1
0
-p,=k(k)r=l)(kw(l)rp
=ldxx
)(pr
)(r
)(r=
)w(p
)w(
)w(
rprpr
r
prrr
dx
dx
dx
xxx
x
xxx
1
1
0
1
1
0
0...21
1
1...10
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)
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Results: Wiener Filtering50
100-tap FIR FilterRaised Cosine
Pulse Shape
Transmitted waveform corrupted by Class A interference
Received waveform filtered by Wiener filter
n
n
n
ChannelA = 0.35 = 0.5 ×
10-3
SNR = -10 dB
Memoryless
Pulse shape
10 samples per symbol10 symbols per pulse
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MAP Detection for Class A51
Hard decision Bayesian formulation [Spaulding & Middleton, 1977]
Equally probable source
Z+S=X:H
Z+S=X:H
22
11 1
2
1
11
22
H
H
)H|X)p(p(H
)H|X)p(p(H=)XΛ(
1
2
1
1
2
H
H
Z
Z
)SX(p
)SX(p=)XΛ(
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MAP Detection for Class A: Small Signal Approx.52
Expand noise PDF pZ(z) by Taylor series about Sj = 0 (j=1,2)
Approximate MAP detection rule
Logarithmic non-linearity + correlation receiver Near-optimal for small amplitude signals
ji
N
=i i
Z
ZjΤ
ZZjZ sx
)X(p)X(p=S)X(p)X(p)SX(p
1
Correlation Receiver
1 ln1
ln1
2
1
11i
12i
H
H
N
=iiZ
i
N
=iiZ
i
)(xpdxd
s
)(xpdxd
s
)XΛ(
We use 100 terms of the series expansion for
d/dxi ln pZ(xi) in simulations
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Incoherent Detection53
Baye’s formulation [Spaulding & Middleton, 1997, pt. II]
Small Signal Approximation
Z(t)+θ)(t,S=X(t):H
Z(t)+θ)(t,S=X(t):H
22
11
1
2
1
1
2
1
2
H
H
θ
θ
)X(p
)X(p=
)p(θp(θH|Xp(
)p(θp(θH|Xp(
=)XΛ(
phase :φamplitude:a
φ
a=θ and where
ln
1
sincos
sincos
2
1
2
11
2
11
2
12
2
12
)(xpdx
d=)l(xwhere
tω)l(x+tω)l(x
tω)l(x+tω)l(x
iZi
i
H
H
N
=iii
N
=iii
N
=iii
N
=iii
Correlation receiver
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Wireless Networking and Communications Group
Filtering for Alpha Stable Noise (Cont..)54
Hole Punching (Blanking) Filters Set sample to 0 when sample exceeds threshold [Ambike, 1994]
Large values are impulses and true values can be recovered Replacing large values with zero will not bias (correlation) receiver for
two-level constellation If additive noise were purely Gaussian, then the larger the threshold,
the lower the detrimental effect on bit error rate Communication performance degrades as constellation size
(i.e., number of bits per symbol) increases beyond two
hp
hp
T>nx
Tnxnx
][0
][][hhp
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Wireless Networking and Communications Group
MAP Detection for Alpha Stable: PDF Approx.55
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 a mixture 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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Wireless Networking and Communications Group
Results: Alpha Stable Detection56
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Wireless Networking and Communications Group
Complexity Analysis for Alpha Stable Detection57
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Wireless Networking and Communications Group
Performance Bounds (Single Antenna)58
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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Wireless Networking and Communications Group
Performance Bounds (Single Antenna)59
Channel Capacity in presence of RFI
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
ParametersA = 0.1, Γ = 10-3
Capacity
)()(
)|()(
);(max}}{),({ 2
NhYh
XYhYh
YXICsX EXExf
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Wireless Networking and Communications Group
Performance Bounds (Single Antenna)60
Probability of error for uncoded transmissions
)(!
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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Wireless Networking and Communications Group
Performance Bounds (Single Antenna)61
Chernoff factors for coded transmissions
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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Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)62
Channel Capacity [Chopra et al., submitted to ICASSP 2009]
Case I Shannon Capacity in presence of additive white Gaussian noise
Case II (Upper Bound) Capacity in presence of bivariate Middleton Class A noise. Assumes that there exists an input distribution which makes output distribution Gaussian for all SNRs.
Case III (Practical Case) Capacity in presence of bivariate Middleton Class A noiseAssumes input has Gaussian distribution
NXY System Model
)()(
)|()(
);(max}}{),({ 2
NhYh
XYhYh
YXICsX EXExf
Return
Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)63
Channel Capacity in presence of RFI for 2x2 MIMO[Chopra et al., submitted to ICASSP 2009]
NXY System Model
Capacity
)()(
)|()(
);(max}}{),({ 2
NhYh
XYhYh
YXICsX EXExf
-40 -30 -20 -10 0 10 200
5
10
15
20
25
SNR [in dB]
Mut
ual I
nfor
mat
ion
(bits
/sec
/Hz)
Channel Capacity with Gaussian noiseUpper Bound on Mutual Information with Middleton noiseGaussian transmit codebook with Middleton noise
Parameters:A = 0.1, 1 = 0.012 = 0.1, = 0.4
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Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)64
Probability of symbol error for uncoded transmissions[Chopra et al., submitted to ICASSP 2009]
Parameters:A = 0.1, 1 = 0.012 = 0.1, = 0.4
Pe: Probability of symbol error
S: Transmitted code vector
D(S): Decision regions for MAP detector
Equally likely transmission for symbols
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Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)65
Chernoff factors for coded transmissions[Chopra et al., submitted to ICASSP 2009]
N
ttt ssC
ssPPEP
1
'
'
),,(min
)(
PEP: Pairwise error probabilityN: Size of the codewordChernoff factor:Equally likely transmission for symbols
),,(min ' kk ccC
-30 -20 -10 0 10 20 30 4010
-8
10-6
10-4
10-2
100
dt2 / N
0 [in dB]
Che
rnof
f Fac
tor
Middleton noise (A = 0.5)Middleton noise (A = 0.1)Middleton noise (A = 0.01)Gaussian noise
Parameters:1 = 0.012 = 0.1, = 0.4
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Wireless Networking and Communications Group
Extensions to Multicarrier Systems66
Impulse noise with impulse event followed by “flat” region Coding may improve communication performance In multicarrier modulation, impulsive event in time domain
spreads out over all subcarriers, reducing the effect of impulse Complex number (CN) codes [Lang, 1963]
Unitary transformations Gaussian noise is unaffected (no change in 2-norm Distance) Orthogonal frequency division multiplexing (OFDM) is a
special case: Inverse Fourier Transform [Haring 2003] As number of subcarriers increase, impulsive
noise case approaches the Gaussian noise case.
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