radio frequency interference sensing and mitigation in wireless receivers talk at the university of...
TRANSCRIPT
Radio Frequency Interference Sensing and Mitigation in
Wireless Receivers
Talk at The University of Texas at Austin
Wireless Networking and Communications Group
7 Oct 2009
Prof. Brian L. Evans
Lead Graduate StudentsAditya Chopra, Kapil Gulati, Yousof Mortazavi and Marcel Nassar
In collaboration with Eddie Xintian Lin, Alberto Alcocer Ochoa,
Chaitanya Sreerama and Keith R. Tinsley at Intel Labs
2
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 Co-channel interference modeling Conclusions Future work
2
Wireless Networking and Communications Group
3
Radio Frequency Interference
Electromagnetic interference Limits wireless communication performance Applications of RFI modeling
Sense and mitigate strategies for coexistence of wireless networks and services
Sense and avoid strategies for cognitive radio We focus on sense and mitigate strategies for wireless
receivers embedded in notebooks Platform noise from user’s computer subsystems Co-channel interference from other in-band wireless networks
and services
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Wireless Networking and Communications Group
Problem Definition4
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 from clocks and busses
We will use noise and interference interchangeably
We will use noise and interference interchangeably
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Wireless Networking and Communications Group
Impact of RFI5
Impact of LCD noise on throughput for an IEEE 802.11g embedded wireless receiver [Shi, Bettner, Chinn, Slattery & Dong, 2006]
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Wireless Networking and Communications Group
Statistical Modeling of RFI6
Radio frequency interference 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
Symmetric Alpha Stable models Approximation of Middleton Class B model
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Wireless Networking and Communications Group
Assumptions for RFI Modeling7
Key assumptions for Middleton and Alpha Stable models[Middleton, 1977][Furutsu & Ishida, 1961] Infinitely many potential interfering sources with same effective
radiation power Power law propagation loss Poisson field of interferers with uniform intensity
Pr(number of interferers = M |area R) ~ Poisson(M; R) Uniformly distributed emission times Temporally independent (at each sample time)
Limitations Alpha Stable models do not include thermal noise Temporal dependence may exist
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Wireless Networking and Communications Group
Our Contributions8
Mitigation of computational platform noise in single carrier, single antenna systems [Nassar, Gulati, DeYoung, Evans & Tinsley, ICASSP 2008, JSPS 2009]
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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]
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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, = 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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Example Power Spectral Densities
Middleton Class A Symmetric Alpha Stable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10
-8
-6
-4
-2
0
2
4
6
8
10
Frequency
Pow
er S
pect
rum
Mag
nitu
de (
dB)
Power Spectal Density of Class A noise, A = 0.15, = 0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10
-8
-6
-4
-2
0
2
4
6
8
10
Frequency
Pow
er S
pect
rum
Mag
nitu
de (
dB)
Power Spectal Density of S S noise, = 1.5, = 10, = 0
Overlap Index (A) = 0.15
Gaussian Factor ( = 0.1 Characteristic Exponent ( = 1.5
Localization () = 0Dispersion () = 10
Simulated Densities
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Wireless Networking and Communications Group
Estimation of Noise Model Parameters12
Middleton Class A model Based on Expectation Maximization [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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Wireless Networking and Communications Group
Results on Measured RFI Data13
25 radiated computer platform RFI data sets from Intel 50,000 samples taken at 100 MSPS
Estimated Parameters for Data Set #18Symmetric Alpha Stable Model
Localization (δ) 0.0065KL Divergence
0.0308Characteristic exp. (α) 1.4329
Dispersion (γ) 0.2701
Middleton Class A Model
Overlap Index (A) 0.0854 KL Divergence0.0494Gaussian Factor (Γ) 0.6231
Gaussian Model
Mean (µ) 0 KL Divergence0.1577Variance (σ2) 1
KL Divergence: Kullback-Leibler divergence-6 -4 -2 0 2 4 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Noise amplitude
Pro
babi
lity
Den
sity
Fun
ctio
n
Measured PDF
Est. -Stable PDF
Est. Class A PDFEst. Gaussian PDF
Measured PDF
Gaussian PDF
Middleton Class A PDF
Alpha Stable PDF
14
0 5 10 15 20 250
0.05
0.1
0.15
0.2
0.25
Measurement Set Number
Kul
lbac
k-Le
ible
r (K
L) D
iver
genc
e
Estimated Alpha Stable modelEstimated Class A modelEstimated Gaussian model
14
Results on Measured RFI Data
Best fit for 25 data sets under different platform RFI conditions
KL divergence plotted for three candidate distributions vs. data set number
Smaller KL value means closer fit
Gaussian
Class A
Alpha Stable
15
Video over Impulsive Channels
Video demonstration for MPEG II video stream 10.2 MB compressed stream from camera (142 MB uncompressed) Compressed file sent over additive impulsive noise channel Binary phase shift keying
Raised cosine pulse10 samples/symbol10 symbols/pulse length
Composite of transmitted and received MPEG II video streamshttp://www.ece.utexas.edu/~bevans/projects/rfi/talks/
video_demo19dB_correlation.wmv Shows degradation of video quality over impulsive channels with
standard receivers (based on Gaussian noise assumption)Wireless Networking and Communications Group
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Additive Class A Noise
Value
Overlap index (A) 0.35
Gaussian factor () 0.001
SNR 19 dB
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Wireless Networking and Communications Group
Filtering and Detection16
Pulse Shaping Pre-Filtering Matched
FilterDetection
Rule
Impulsive Noise
Middleton Class A noise Symmetric Alpha Stable noise
Filtering Wiener Filtering (Linear)
Detection Correlation Receiver (Linear) Bayesian Detector
[Spaulding & Middleton, 1977] Small Signal Approximation to
Bayesian detector[Spaulding & Middleton, 1977]
Filtering Myriad Filtering
Optimal Myriad [Gonzalez & Arce, 2001]
Selection Myriad Hole Punching
[Ambike et al., 1994]
Detection Correlation Receiver (Linear) MAP approximation
[Kuruoglu, 1998]
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AssumptionMultiple samples of the received signal are available• N Path Diversity [Miller, 1972]• Oversampling by N [Middleton, 1977]
AssumptionMultiple samples of the received signal are available• N Path Diversity [Miller, 1972]• Oversampling by N [Middleton, 1977]
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Wireless Networking and Communications Group
Results: Class A Detection17
Pulse shapeRaised cosine
10 samples per symbol10 symbols per pulse
ChannelA = 0.35
= 0.5 × 10-3
Memoryless
-35 -30 -25 -20 -15 -10 -5 0 5 10 1510
-5
10-4
10-3
10-2
10-1
100
SNR
Bit
Err
or R
ate
(BE
R)
Correlation ReceiverWiener FilteringBayesian DetectionSmall Signal Approximation
Communication Performance Binary Phase Shift Keying
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Wireless Networking and Communications Group
Results: Alpha Stable Detection18
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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10-2
10-1
100
Generalized SNR (in dB)
Bit
Err
or R
ate
(BE
R)
Matched FilterHole PunchingMAPMyriad
Communication Performance Same transmitter settings as previous slide
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Video over Impulsive Channels #2
Video demonstration for MPEG II video stream revisited 5.9 MB compressed stream from camera (124 MB uncompressed) Compressed file sent over additive impulsive noise channel Binary phase shift keying
Raised cosine pulse10 samples/symbol10 symbols/pulse length
Composite of transmitted video stream, video stream from a correlation receiver based on Gaussian noise assumption, and video stream for a Bayesian receiver tuned to impulsive noise
http://www.ece.utexas.edu/~bevans/projects/rfi/talks/video_demo19dB.wmv
Wireless Networking and Communications Group
19
Additive Class A Noise
Value
Overlap index (A) 0.35
Gaussian factor () 0.001
SNR 19 dB
20
Video over Impulsive Channels #2
Structural similarity measure [Wang, Bovik, Sheikh & Simoncelli, 2004]
Score is [0,1] where higher means better video quality
Frame number
Bit error rates for ~50 million bits sent:
6 x 10-6 for correlation receiver
0 for RFI mitigating receiver (Bayesian)
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Wireless Networking and Communications Group
Our Contributions22
2 x 2 MIMO receiver design in the presence of RFI[Gulati, Chopra, Heath, Evans, Tinsley & Lin, Globecom 2008]
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-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 23
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)
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Wireless Networking and Communications Group
Results: RFI Mitigation in 2 x 2 MIMO 24
Complexity Analysis
Complexity Analysis for decoding M-level 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)
2525
Co-Channel Interference Modeling
Wireless Networking and Communications Group
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Region of interferer locations determines interference model [Gulati, Chopra, Evans & Tinsley, Globecom 2009]
Symmetric Alpha Stable Middleton Class A
2626
Co-Channel Interference Modeling
Propose unified framework to derive narrowband interference models for ad-hoc and cellular network environments Key result: tail probabilities (one minus cumulative distribution function)
Wireless Networking and Communications Group
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Case 1: Ad-hoc network Case 3-a: Cellular network (mobile user)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-4
10-3
10-2
10-1
100
Interference threshold (a)
P (
Inte
rfe
rnc
e a
mp
litu
de
> a
)
Simulated Symmetric Alpha Stable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10-10
10-5
100
Interference threshold (a)
P (
Inte
rfe
ren
ce
am
plit
ud
e >
a)
SimulatedSymmetric Alpha StableGaussianMiddleton Class A
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Wireless Networking and Communications Group
Conclusions27
Radio Frequency Interference from computing platform Affects wireless data communication transceivers Models include Middleton and alpha stable distributions
RFI mitigation can improve communication performance Single carrier, single antenna systems
Linear and non-linear filtering/detection methods explored Single carrier, multiple antenna systems
Optimal and sub-optimal receivers designed Bounds on communication performance in presence of RFI
Results extend to co-channel interference modeling
28
RFI Mitigation Toolbox
Provides a simulation environment for RFI generation Parameter estimation
algorithms Filtering and detection methods Demos for communication
performance analysis
Latest Toolbox ReleaseVersion 1.3, Aug 26th 2009
Wireless Networking and Communications Group
28
Snapshot of a demo
http://users.ece.utexas.edu/~bevans/projects/rfi/software/index.html
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Wireless Networking and Communications Group
Other Contributions29
Publications[Journal Articles]M. Nassar, K. Gulati, M. R. DeYoung, B. L. Evans and K. R. Tinsley, “Mitigating Near-Field Interference in
Laptop Embedded Wireless Transceivers”, J. of Signal Proc. Systems, Mar 2009, invited paper. [Conference Papers]M. Nassar, K. Gulati, A. K. Sujeeth, N. Aghasadeghi, B. L. Evans and K. R. Tinsley, “Mitigating Near-field
Interference in Laptop Embedded Wireless Transceivers”, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., Mar. 30-Apr. 4, 2008, Las Vegas, NV USA.
K. Gulati, A. Chopra, R. W. Heath Jr., B. L. Evans, K. R. Tinsley, and X. E. Lin, “MIMO Receiver Design in the Presence of Radio Frequency Interference”, Proc. IEEE Int. Global Communications Conf., Nov. 30-Dec. 4th, 2008, New Orleans, LA USA.
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, accepted.
K. Gulati, A. Chopra, B. L. Evans and K. R. Tinsley, “Statistical Modeling of Co-Channel Interference”, Proc. IEEE Int. Global Communications Conf., Nov. 30-Dec. 4, 2009, Honolulu, HI USA, accepted.
Project Websitehttp://users.ece.utexas.edu/~bevans/projects/rfi/index.html
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Future Work30
Extend RFI modeling for Adjacent channel interference Multi-antenna systems Temporally correlated interference
Multi-input multi-output (MIMO) single carrier systems RFI modeling and receiver design
Multicarrier communication systems Coding schemes resilient to RFI System level techniques to reduce computational platform
generated RFI
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Wireless Networking and Communications Group
References32
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
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Wireless Networking and Communications Group
References (cont…)33
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.
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Wireless Networking and Communications Group
Backup Slides34
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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Wireless Networking and Communications Group
Common Spectral Occupancy35
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
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Wireless Networking and Communications Group
Impact of RFI36
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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37
Impact of LCD clock on 802.11g
Pixel clock 65 MHz LCD Interferers and 802.11g center frequencies
Wireless Networking and Communications Group
37
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Wireless Networking and Communications Group
Middleton Class A, B and C Models38
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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Wireless Networking and Communications Group
Middleton Class B Model39
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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Wireless Networking and Communications Group
Middleton Class B Model (cont…)40
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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Wireless Networking and Communications Group
Middleton Class B Model (cont…)41
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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Wireless Networking and Communications Group
Accuracy of Middleton Noise Models42
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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Wireless Networking and Communications Group
Symmetric Alpha Stable PDF43
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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Wireless Networking and Communications Group
Symmetric Alpha Stable Model44
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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Wireless Networking and Communications Group
Parameter Estimation: Middleton Class A45
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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Results Backup
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Wireless Networking and Communications Group
Results: EM Estimator for Class A47
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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Wireless Networking and Communications Group
Results: EM Estimator for Class A48
• 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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Wireless Networking and Communications Group
Parameter Estimation: Symmetric Alpha Stable49
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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Wireless Networking and Communications Group
Parameter Est.: Symmetric Alpha Stable Results50
• 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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51
Wireless Networking and Communications Group
Parameter Est.: Symmetric Alpha Stable Results51
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
53
Wireless Networking and Communications Group
Parameter Estimators for Alpha Stable53
0 < p < α
Return
54
Wireless Networking and Communications Group
Filtering and Detection54
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 Shaping Pre-Filtering Matched
FilterDetection
Rule
Impulsive Noise
N samples per symbolN samples per symbol
55
Wireless Networking and Communications Group
Wiener Filtering55
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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56
Wireless Networking and Communications Group
Wiener Filter Design56
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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57
Wireless Networking and Communications Group
Results: Wiener Filtering57
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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58
Wireless Networking and Communications Group
MAP Detection for Class A58
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(
Return
59
Wireless Networking and Communications Group
MAP Detection for Class A: Small Signal Approx.59
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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60
Wireless Networking and Communications Group
Incoherent Detection60
Bayesian 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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61
Wireless Networking and Communications Group
Filtering for Alpha Stable Noise61
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
Return
62
Wireless Networking and Communications Group
Filtering for Alpha Stable Noise (Cont..)62
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
Return
63
Wireless Networking and Communications Group
Filtering for Alpha Stable Noise (Cont..)63
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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64
Wireless Networking and Communications Group
MAP Detection for Alpha Stable: PDF Approx.64
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
Return
66
Wireless Networking and Communications Group
Complexity Analysis for Alpha Stable Detection66
Return
67
Wireless Networking and Communications Group
Bivariate Middleton Class A Model67
Joint spatial distribution Return
68
Wireless Networking and Communications Group
Results on Measured RFI Data68
50,000 baseband noise samples represent broadband interference
-4 -3 -2 -1 0 1 2 3 40
0.2
0.4
0.6
0.8
1
1.2
1.4
Noise amplitude
Pro
ba
bili
ty D
en
sity
Fu
nct
ion
Measured PDFEstimated MiddletonClass A PDFEqui-powerGaussian PDF
Marginal PDFs of measured data compared with estimated model densities
Return
69
2 x 2 MIMO System
Maximum Likelihood (ML) receiver
Log-likelihood function
Wireless Networking and Communications Group
System Model69
Sub-optimal ML Receiversapproximate
Return
70
Wireless Networking and Communications Group
Sub-Optimal ML Receivers70
Two-piece linear approximation
Four-piece linear approximation
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
z
Ap
pro
xma
tion
of
(z)
(z)
1(z)
2(z)
chosen to minimizeApproximation of
Return
71
Wireless Networking and Communications Group
Results: Performance Degradation71
Performance degradation in receivers designed assuming additive Gaussian noise in the presence of RFI
-10 -5 0 5 10 15 2010
-5
10-4
10-3
10-2
10-1
100
SNR [in dB]
Vec
tor
Sym
bol E
rror
Rat
e
SM with ML (Gaussian noise)SM with ZF (Gaussian noise)Alamouti coding (Gaussian noise)SM with ML (Middleton noise)SM with ZF (Middleton noise)Alamouti coding (Middleton noise)
Simulation Parameters•4-QAM for Spatial Multiplexing (SM) transmission mode•16-QAM for Alamouti transmission strategy•Noise Parameters:A = 0.1, 1= 0.01, 2= 0.1, = 0.4
Severe degradation in communication performance in
high-SNR regimes
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72
Wireless Networking and Communications Group
Performance Bounds (Single Antenna)72
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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73
Wireless Networking and Communications Group
Performance Bounds (Single Antenna)73
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
Return
74
Wireless Networking and Communications Group
Performance Bounds (Single Antenna)74
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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75
Wireless Networking and Communications Group
Performance Bounds (Single Antenna)75
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
Return
77
Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)77
Channel capacity
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
System Model
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78
Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)78
Channel capacity in presence of RFI for 2x2 MIMO
System Model
Capacity
-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 = 0.01, 2 = 0.1, = 0.4
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79
Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)79
Probability of symbol error for uncoded transmissions
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
Return
80
Wireless Networking and Communications Group
Performance Bounds (2x2 MIMO)80
Chernoff factors for coded transmissions
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
Return
81
Performance Bounds (2x2 MIMO)
Cutoff rates for coded transmissions Similar measure as channel capacity Relates transmission rate (R) to Pe for a length T codes
Wireless Networking and Communications Group
81
Return
82
Performance Bounds (2x2 MIMO)
Wireless Networking and Communications Group
82
Cutoff rate
-30 -20 -10 0 10 20 30 400
0.5
1
1.5
2
2.5
3
3.5
4
SNR [in dB]
Cut
off R
ate
[bits
/tran
smis
sion
]
BPSK, Middleton noiseBPSK, Gaussian noiseQPSK, Middleton noiseQPSK, Gaussian noise16QAM, Middleton noise16QAM, Gaussian noise
Return
83
Wireless Networking and Communications Group
Extensions to Multicarrier Systems83
Impulse noise with impulse event followed by “flat” region Coding may improve communication performance In multicarrier modulation, impulsive event in time domain
spreads over all subcarriers, reducing 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 As number of subcarriers increase, impulsive noise case
approaches the Gaussian noise case [Haring 2003]
Return