Title: Fading Modeling, MIMO Channel Generation, and Spectrum Sensing for Wireless Communications
1Fading Modeling, MIMO Channel Generation, and
Spectrum Sensing for Wireless Communications
- Wei-Ho Chung
- Electrical Engineering
- University of California, Los Angeles
- April 2009
- whc_at_ee.ucla.edu
2Outline
- Introduction-Fading Channels
- Fading Channel Model by Modified Hidden
Semi-Markov Model - Generating Correlated MIMO Fading Channel
- Detection and Decision Fusion in Fading
Environment - Sequential Likelihood Ratio Test for Spectrum
Sensing - Future Work
- Conclusions
3Fading Effects
- In wireless communications, the signals traverse
from the transmitter to the receiver - The medium (channel) physically influences the
signals, including - Reflection
- Signal impinges on the smooth surface
- Surface dimension much larger
- than Wavelength
- Diffraction
- Signals impinges the edge or corner of the dense
entity - Secondary signals spread out from the impinged
edge - Non-line-of-sight (NLOS) communications
- Scatter
- Signals impinge a rough surface
- The roughness at the order of the wavelength or
less
B. Sklar, IEEE Communications Magazine, 1997.
4Fading Models and Applications
- The fading channel model is the mathematical
description of the fading channel - Stochastic
- Mobile communication systems
- Mobile node moving, various fading effects
- Quasi-deterministic
- Transmitter and receiver are relatively static
- Fading effects can be approximately deterministic
- Applications of fading channel model include
- Performance analyses, e.g. bit error rate
- Channel capacity Biglieri 98
- Outage probability
- Power control Caire 99
- Channel coding Hall 98
- Adaptive modulation Goldsmith 98
5Flat Fading
- Fading channel is modeled as a linear
time-variant system with the impulse response - represents the time index of filter
response - represents the time dependence of the
filter response - Delay spread of the filter response , the
coherence bandwidth - Flat fading channel, bandwidth of input signal
smaller than coherence bandwidth - Multiplicative effect on the transmitted signal
- The fading channel model is focused on modeling
the statistical properties of
S. Stein, IEEE JSAC, 1987
6Related Work
- Rayleigh, Rice, and Nakagami distributions have
been investigated to model flat fading channel
P. Beckman, 1967 - Rayleigh model
- Large amount of scattered signals
- Central limit theorem
- Rician model
- Dominant impinging signal
- Larger amount of scattered signals
- Markov Chain Tan and Beaulieu, IEEE Tran. Comm.,
2000 - Gilbert-Elliott model
- Two states, the good (high SNR) and bad states
(low SNR). - The Ray-Tracing Model Rizk 97
- Trace the geometry in signal propagation
- Trace reflections, diffractions, and scatters
- Site-specific information
7Outline
- Introduction-Fading Channels
- Fading Channel Model by Modified Hidden
Semi-Markov Model - Generating Correlated MIMO Fading Channel
- Detection and Decision Fusion in Fading
Environment - Sequential Likelihood Ratio Test for Spectrum
Sensing - Conclusions
8Multi-Modal Observations
- Envelope PDFs can have multi-modes
- LOS v.s. NLOS
- Time-Variant Fading Conditions
- Simulation
- Experiment
9Modified Hidden Semi-Markov Model
- Amplitude-based Finite-State Markov Chains Model
(AFSMCM) - Output channel amplitude
- Hidden Markov Model (HMM)
- Output channel amplitude probabilistically
- Hidden Semi-Markov Model (HSMM)
- State duration probability
10Properties
- AFSMCM
- Output vector
- Transition matrix P
- ACF
- PDF Steady-state probability
- HMM
- Independent samples
- PDF Mixtures of steady-state prob. and output
PDF - HSMM
- Independent samples
11Modified Hidden Semi-Markov Model
- Model scenarios
- LOS v.s. NLOS
- High speed v.s. Low speed
- Segmentation by features
- Channel gain
- Entropy of energy distribution
W. Chung and K. Yao, "Modified Hidden
Semi-Markov Model for Modelling the Flat Fading
Channel," IEEE Tran. Comm., June, 2008.
12Parameter Estimation of MHSMM
13Clustering in the Feature Domain
- Perform k-means Clustering
- For a specific k, generate clusters and centers
- Detect Number of States
- Davis-Bouldin Bezdek 98
- Within-cluster scatter/Between-cluster
separation - Pursue small Davis-Bouldin Index for good
clustering - Dunn Dunn 73
- Min inter-cluster distance/Cluster diameter
- Pursue large Dunn Index for good clustering
- Percentage of residual explained Aldenderfer 84
- Between-group variance/Total variance
- Elbow rule
14Parameter estimation
- ACF Sample ACF, i.e.,
- PDF
15Segmentation Example
16Clustering
17Estimated pdf
MHSMM 0.02
AFSMCM 0.06
HMM 0.17
18Estimated ACFs
19Experiment
20Experimental Data-Hallway
- Hallway inside building
- Rush hours
- Numerous reflectors and scatters
- Non-cooperative disturbances
21Summary
- Model Nonstationary Fading Processes
- Various channel conditions
- Piece-wise stationary processes
- Model the PDFs and ACFs
- Model Estimation Scheme
- Channel segmentation
- Parameter estimation
22Outline
- Introduction-Fading Channels
- Fading Channel Model by Modified Hidden
Semi-Markov Model - Generating Correlated MIMO Fading Channel
- Detection and Decision fusion in Fading
Environment - Sequential Likelihood Ratio Test for Spectrum
Sensing - Future Work
- Conclusions
23Generating Correlated MIMO Channels
- Motivations
- Channels Codes
- Modulations
- Diversity Combining
- MIMO systems
- Generate multiple channels that have specific
- Auto-correlation function (ACF)
- Cross-correlation function (CCF)
- Envelope pdfs
24Multiple Channels
- Space-Time correlation model
- Jakes model by Bessel function Jakes 94
- Spatial and temporal correlations
- Multiple mobile fading channels Abidi 02
- MIMO channel for non-isotropic scattering
environment - MIMO channel for omnidirectional antennas Rad
05
25Example of Time-Space model
Rad and Gazor, 05
26Channel Generation by Autoregressive Processes
- Channels and Covariance Matrix
- AR processes
Baddour and Beaulieu, "Accurate Simulation of
multiple cross-correlated Rician fading
channels," IEEE Tran. on Comm., Nov. 2004.
27Solve AR parameters
- AR Coefficient
- Covariance of Noise
28Simulation by Autoregressive Processes
29Generating Correlated Nakagami
- Nakagami channels
- Measurements M. Nakagami,1960 H. Suzuki, 1977
- Modulation Alouini and Goldsmith, 2000
- Diversity Combining Beaulieu and Abu-Dayya,
1991 - Gaussian Random Variable is Well Researched
- Operate on Gaussian RV
- Notation
- Process
- Problem
Q. T. Zhang, "A decomposition technique for
efficient generation of correlated Nakagami
fading channels, IEEE JSAC, 2000.
30Generating Correlated Nakagami
- Generate
- Relating covariance matrices
31Heterogeneous MIMO channel generation
- Previous works focus on PDFs of the same family,
e.g., Rayleigh Baddour 2004 , Nakagami Zhang
2000 - Fading environment causes channels of various
properties-channels of different families - Generate multiple channels that have specific
- Auto-correlation function (ACF)
- Cross-correlation function (CCF)
- Heterogeneous envelope PDFs
32Illustration of the problem
33Inverse Transform Sampling
- Framework
- Probability density functions
- Correlations
- Inverse Transform Sampling
- Generate x with CDF
- y has CDF
34Proposed approachInverse Transform Sampling
W. Chung, K. Yao, and R. E. Hudson, The Unified
Approach for Generating Multiple Cross-correlated
and Auto-correlated Fading Envelope Processes.
Accepted. IEEE Tran. Comm., 2009.
35Sketch of derivation
- Definition of correlation
- Jacobian
- Correlations of input and output
36Example- Heterogeneous channels of Nakagami,
Rician, and Rayleigh pdfs
- Three channels
- Nakagami
- Rician
- Rayleigh
- Correlations
- ACF
- CCF
37Results
38Example- Heterogeneous Channels of Nakagami,
Rician, and Rayleigh pdfs
39Example- 2x2 Rayleigh MIMO Channels
40Example- 2x2 Rayleigh MIMO Channels
41Example-Single Nakagami Channel
- ACF
-
-
-
- High sampling rate
- Low sampling rate
42Outline
- Introduction-Fading Channels
- Fading Channel Model by Modified Hidden
Semi-Markov Model - Generating Correlated MIMO Fading Channel
- Detection and Decision Fusion in Fading
Environment - Sequential Likelihood Ratio Test for Spectrum
Sensing - Future Work
- Conclusions
43Detection and Decision Fusion in Fading
Environment
- Detection by Single sensor
- Hypothesis test
- Cognitive radio
- Decision Fusion using Multiple Sensors
- Detection by Single Sensor under Fading
- Multi-Sensor Decision Fusion under Fading
44Hypothesis Test
- Hypothesis Test applications
- Surveillance
- Target Detection
- Spectrum Sensing
- Example-Matched Filter Detection
- Signal model
-
-
45Receiver Operating Curve
- Receiver Operating Curve v.s.
- Setting Threshold
- Criteria
- Neyman-Pearson
- upper-bounded
- Bayes
- Priors and costs
46Spectrum Sensing in Cognitive Radio
- Wireless communications rely on spectra.
- Current usage model frequency bands are
licensed. - The licensed bands are often vacant- low
utilizations. - Cognitive Radio-to increase the spectrum
utilization. - Allows secondary user to access the spectrum when
it is vacant. - Secondary users sense the spectrum before
accessing. - Accuracies of the spectrum sensing is crucial.
- Formulated as binary hypothesis test problem
- H0 Spectrum Vacant
- H1 Spectrum Occupied
S. Haykin, "Cognitive radio brain-empowered
wireless communications," IEEE JSAC. 2005.
47Detection Criteria and Implications in Cognitive
Radio
- Interpretations of PD and PFA in cognitive radio
- Detection performed by the secondary users
- H1 Spectrum used by the primary users
- Secondary users access the spectrum if decision
is H0 - Channel conflict Decision H0 under the truth H1
- Miss of the spectrum opportunity Decision H1
under the truth H0 - Neyman-Pearson
- Upper-bound probability of false alarm while
maximizing probability of detection - Protect the spectrum opportunities of the
secondary users while minimizing the channel
conflicts - Lower-Bounded Probability of Detection (LBPD)
Chung 08 - Lower-bound probability of detection while
minimizing probability of false alarm - Protect the primary users while maximizing the
spectrum opportunities for the secondary users
48Decision Fusion Framework
- Sensors make binary decisions.
- Many applications require binary decisions.
- Accuracy of a single sensor is limited.
- Fusion of multiple decisions increases
accuracies.
R. Viswanathan and P. K. Varshney,
"Distributed Detection with Multiple sensors I.
Fundamentals," Proceedings of the IEEE, 1997.
49Decision Fusion Framework
- N sensors make binary decisions.
- Probability of False Alarm
- Probability of Detection
- Sensor decisions
- The fusion center makes final decision.
- Fusion Rule
- Fusion Rule with random strategy
- Solve the parameters of the Fusion Rule
50Algorithm for Computing the Fusion Rule
- For each element , we denote
- by
- by
- The likelihood ratio, associated with , is
defined as
W. Chung and K. Yao, Decision Fusion in Sensor
Networks for Spectrum Sensing based on Likelihood
Ratio Tests, Proceedings of SPIE, 2008.
51Fusion of Two Sensors
- Two Sensors
- Operating points
- Goal (Lower-Bounded Probability of Detection
Criterion) Minimizing while
is lower bounded by 0.91 - Result
52Fusion of Two Sensors
53Examples
3 sensors
- Proposed algorithm
- K out of N
- FC declares H1 if k or more than k sensors
declare H1. Otherwise, FC declares H0. - Decision Space search
- All possible combinations of decision fusion rule
4 sensors
54Detection under Fading---Likelihood Ratio Test
with Fading Statistics (LRFS)
- Signal Model
- Fading Gains
- Rayleigh
- Rician
- Test Statistic
55LRFS
- Explicit Expressions of the test statistics
- Rayleigh
- Rician
56Multi-Sensor Decision Fusion under Fading
- Signal Model
- Likelihood Ratio
- Reformulate by fading statistics
- Under H1
- Under H0
- Test Statistics
57Numerical Examples
- LRT with Fading Statistics
- LRFS under Rayleigh
- LRFS under Rician
- Matched Filter
- Decision Fusion with Fading Statistics
- 3 Sensors
- 2 Sensors
58Summary
- Explicit Algorithms
- Neyman-Pearson
- Lowered-Bounded Probability of Detection
- Test Statistics under Rayleigh and Rician Fading.
- Performance Improvements by Incorporating Fading
Statistics - Single-Sensor Detection under Fading
- Decision Fusion under Fading
59Outline
- Introduction-Fading Channels
- Fading Channel Model by Modified Hidden
Semi-Markov Model - Generating Correlated MIMO Fading Channel
- Detection and Decision Fusion in Fading
Environment - Sequential Likelihood Ratio Test for Spectrum
Sensing - Future Work
- Conclusions
60Sequential Likelihood Ratio Testfor Spectrum
Sensing
- Problem Spectrum Sensing under Fading
- Goal
- Faster decision
- Allow setting both probability of false alarm and
probability of detection - Conventional Approaches
- Collect fixed amount of data
- Uncertain signal strength in fading
- Can we reach faster decision when signal is
strong? - Sequential Likelihood Ratio Test
W. Chung and K. Yao, Sequential Likelihood
Ratio Test under Incomplete Signal Model for
Spectrum Sensing,
61Formulation
- Signal Model
- Received
- Signal (primary user)
- Signal follows AR model
- Received signal follows ARMA
62Sequential Decision
- Decision at time t
- Log LR
- Sequential decision
63Decision and Thresholds
- Decision
- Thresholds
- Expected termination time can be derived as a
function of accuracy and SNR
64Example 1-Scenario
- SNR uniformly distributed between -20 dB to 10 dB
- Prior 0.5 for H1 and H0
- Jakes ACF
65Example 1-results
66Example 2fixed SNR at -20 dB
67Conclusion
- Modified Hidden Semi-Markov Model
- ACFs and Durations
- Channel Segmentations
- Parameter Estimation
- Multiple Channel Generation
- Correlated Heterogeneous Channels
- Detection and Decision Fusion under Fading
- Single Detector
- Sensor Network
- Sequential LRT allows faster decision while
maintaining targeted detection accuracies
68Future Works
- Detection, Estimation, and Learning
- Demodulation under Correlated Heterogeneous
Channels - Joint Detection and Estimation of Information and
Environment - Cognitive Radio
- Sequential Detection
- Quickest Detection
- Protocol Enforcement
- Game-Theoretical
- Sniffer
69 70Publications
- Wei-Ho Chung, Kung Yao, and Ralph E. Hudson, "The
Unified Approach for Generating Multiple
Cross-correlated and Auto-correlated Fading
Envelope Processes." Accepted for publication in
IEEE Transactions on Communications, Oct. 2008.
To appear. - Wei-Ho Chung and Kung Yao, "Decision Fusion in
Sensor Networks for Spectrum Sensing based on
Likelihood Ratio Tests," Proceedings of SPIE,
Vol. 7074, No. 70740H, Aug. 2008. - Wei-Ho Chung and Kung Yao, "Modified Hidden
Semi-Markov Model for Modelling the Flat Fading
Channel," Accepted for publication in IEEE
Transactions on Communications, Feb. 2008. To
appear. - Wei-Ho Chung and Kung Yao, "Empirical
Connectivity for Mobile Ad Hoc Networks under
Square and Rectangular Covering Scenarios," IEEE
Proc. International Conference on Communications,
Circuits, and Systems, Vol. 3, pp. 1482-1486,
June 2006. - Wei-Ho Chung, "Probabilistic Analysis of Routes
on Mobile Ad Hoc Networks," IEEE Communications
Letters, Vol.8, Issue 8, pp.506-508, Aug. 2004. - Wei-Ho Chung, Sy-Yen Kuo, and Shih-I Chen,
"Direction-Aware Routing Protocol for Mobile Ad
Hoc Networks," Proceedings of IEEE International
Conference on Communications, Circuits and
Systems, Vol. 1, pp. 165-169, June 2002.
71References
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channels information-theoretic and
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mobile digital communication systems. I.
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Engineering," IEEE Journal on Selected Areas in
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Simulation of multiple cross-correlated Rician
fading channels," IEEE Transactions on
Communications, Vol. 52, Issue 11, pp. 1980-1987,
Nov. 2004. - A. Abdi and M. Kaveh, "A space-time correlation
model for multielement antenna systems in mobile
fading channels," IEEE Journal on Selected Areas
in Communications, Vol. 20, Issue 3, pp. 550-560,
Apr. 2002. - H. S. Rad and S. Gazor, "A cross-correlation MIMO
channel model for non-isotropic scattering
environment and non-omnidirectional antennas,"
Canadian Conference on Electrical and Computer
Engineering, pp. 25-28, May 2005. - M. Alouini and A. J. Goldsmith, "Adaptive
Modulation over Nakagami Fading Channels,"
Wireless Personal Communications, Vol. 13, pp.
119-143, Springer Netherlands, May 2000. - E. K. Hall and S. G. Wilson, "Design and analysis
of turbo codes on Rayleigh fading channels," IEEE
Journal on Selected Areas in Communications, Vol.
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Engineering. New York Harcourt Brace and World,
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Vishwanath, "Capacity limits of MIMO channels,"
IEEE Journal on Selected Areas in Communications,
Vol. 21, Issue 5, pp. 684-702, June 2003. - Q. T. Zhang, "A decomposition technique for
efficient generation of correlated Nakagami
fading channels," IEEE Journal on Selected Areas
in Communications, Vol. 18, Issue 11, pp.
2385-2392, Nov. 2000. - W. C. Jakes, Microwave mobile communication, 2nd
ed., IEEE Press, 1994. - S. Haykin,"Cognitive radio brain-empowered
wireless communications," IEEE Journal on
Selected Areas in Communications, Volume 23,
Issue 2, pp. 201-220, Feb. 2005.
73References
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formula of intensity distribution of rapid
fading, in Statistical Methods in Radio Wave
Propagation, W. G. Hoffman, Ed. Oxford, England
Pergamon, 1960. - H. Suzuki, A statistical model for urban radio
channel model, IEEE Trans. Commun., vol. 25, pp.
673680, July 1977.
74Cooperative Spectrum Sensing Scheme based on
Nash Equilibrium
- Wei-Ho Chung
- Electrical Engineering
- University of California, Los Angeles
- March 2009
- whc_at_ee.ucla.edu
75Decision Fusion using Nash equilibrium
- Detection by Single sensor
- Hypothesis test
- Cognitive radio
- Decision Fusion using Multiple Sensors
- Increase detection accuracy for spectrum sensing
- Nash equilibrium to enforce cooperative scheme
76Hypothesis Test
- Hypothesis Test applications
- Surveillance
- Target Detection
- Spectrum Sensing
- Example-Matched Filter Detection
- Signal model
-
-
77Receiver Operating Curve
- Receiver Operating Curve v.s.
- Setting Threshold
- Criteria
- Neyman-Pearson
- upper-bounded
- Bayes
- Priors and costs
78Spectrum Sensing in Cognitive Radio
- Wireless communications rely on spectra.
- Current usage model frequency bands are
licensed. - The licensed bands are often vacant- low
utilizations. - Cognitive Radio-to increase the spectrum
utilization. - Allows secondary user to access the spectrum when
it is vacant. - Secondary users sense the spectrum before
accessing. - Accuracies of the spectrum sensing is crucial.
- Formulated as binary hypothesis test problem
- H0 Spectrum Vacant
- H1 Spectrum Occupied
S. Haykin, "Cognitive radio brain-empowered
wireless communications," IEEE JSAC. 2005.
79Detection Criteria and Implications in Cognitive
Radio
- Interpretations of PD and PFA in cognitive radio
- Detection performed by the secondary users
- H1 Spectrum used by the primary users
- Secondary users access the spectrum if decision
is H0 - Channel conflict Decision H0 under the truth H1
- Miss of the spectrum opportunity Decision H1
under the truth H0 - Neyman-Pearson
- Upper-bound probability of false alarm while
maximizing probability of detection - Protect the spectrum opportunities of the
secondary users while minimizing the channel
conflicts - Lower-Bounded Probability of Detection (LBPD)
Chung 08 - Lower-bound probability of detection while
minimizing probability of false alarm - Protect the primary users while maximizing the
spectrum opportunities for the secondary users
80Decision Fusion Framework
- Sensors make binary decisions.
- Many applications require binary decisions.
- Accuracy of a single sensor is limited.
- Fusion of multiple decisions increases
accuracies.
R. Viswanathan and P. K. Varshney,
"Distributed Detection with Multiple sensors I.
Fundamentals," Proceedings of the IEEE, 1997.
81Decision Fusion Framework
- N sensors make binary decisions.
- Probability of False Alarm
- Probability of Detection
- Sensor decisions
- The fusion center makes final decision.
- Fusion Rule
- Fusion Rule with random strategy
- Solve the parameters of the Fusion Rule
82Algorithm for Computing the Fusion Rule
- For each element , we denote
- by
- by
- The likelihood ratio, associated with , is
defined as
W. Chung and K. Yao, Decision Fusion in Sensor
Networks for Spectrum Sensing based on Likelihood
Ratio Tests, Proceedings of SPIE, 2008.
83Fusion of Two Sensors
- Two Sensors
- Operating points
- Goal (Lower-Bounded Probability of Detection
Criterion) Minimizing while
is lower bounded by 0.91 - Result
84Fusion of Two Sensors
85Costs structure and game formulation
- Utility from accessing the channel
- 0 for channel conflict
- for a successful access
- Costs
- for a channel conflict
- for a successful access
- Game
- Users
- Actions
- user perform detection -
incurs cost - user not perform detection -
incurs cost 0 - Utility
86Operating point and Costs
- Operating point by Bayes criterion--the point on
the ROC with slope of its tangent line equal to - Expected Costs
- Individual user
- Overall cost
87Action profile by Nash equilibrium
- For the action profiles
- Compute costs of each action profile
- Compute Nash equilibrium
88Results - influences of charges from primary users
- Cost of conflict
- Cost of successful access
89Results-influence of detection cost
90Conclusions
- Propose Decision fusion framework using Nash
equilibrium - Increase accuracies of spectrum sensing
- Protocol enforcement by Nash equilibrium
- Framework allows analyzing interactions among
- Prices
- Cost of detection
- Probability of false alarm
- Probability of detection
- Utilities
91References
- S. Haykin, "Cognitive Radio Brain-Empowered
Wireless Communications," IEEE Journal on
Selected Areas in Communications, Volume 23,
Issue 2, pp. 201-220, Feb. 2005. - D. Cabric, A. Tkachenko, and R. W. Brodersen,
"Experimental Study of Spectrum Sensing Based on
Energy Detection and Network Cooperation,"
Proceedings of the first international workshop
on Technology and policy for accessing spectrum,
Article No. 12, 2006. - H. P. Shiang and M. van der Schaar, "Distributed
Resource Management in Multi-hop Cognitive Radio
Networks for Delay Sensitive Transmission," IEEE
Trans. Veh. Tech., to appear. - H. Park and M. van der Schaar, "Coalition based
Resource Negotiation for Multimedia Applications
in Informationally Decentralized Networks," IEEE
Trans. Multimedia, to appear. - W. Chung and K. Yao, "Decision Fusion in Sensor
Networks for Spectrum Sensing based on Likelihood
Ratio Tests," Proceedings of SPIE, Vol. 7074, No.
70740H, Aug. 2008. - Q. Zhao, L. Tong, A. Swami, and Y. Chen,
"Decentralized Cognitive MAC for Opportunistic
Spectrum Access in Ad Hoc Networks A POMDP
Framework," IEEE Journal on Selected Areas in
Communications, Vol. 25, No. 3, pp. 589-600,
April 2007. - Z. Ji and K. J. R. Liu, "Dynamic Spectrum
Sharing A Game Theoretical Overview," IEEE
Communications Magazine, pp. 88-94, May 2007. - R. Etkin, A. Parekh, and D. Tse, "Spectrum
Sharing for Unlicensed Bands," IEEE Journal on
Selected Areas in Communications, Vol. 25, No. 3,
pp. 517-528, April 2007.
92Detecting Number of Coherent Signals in Array
Processing by Ljung-Box Statistic
- Wei-Ho Chung
- Electrical Engineering
- University of California, Los Angeles
- April 2009
- whc_at_ee.ucla.edu
93Array Signal Processing
- Estimate
- Direction of arrival in far-field
- Localizations near-field
- Likelihood formulation
- Grid search
- Newton method
- Genetic Algorithm
- Detect number of signals
94Signal Model
- Signals
- Source signal
- Received signal
- Unknown parameters
- DOAs
- Amplitudes
- Maximum Likelihood Estimation
95Estimate Number of Signals
- Estimations of DOAs are based on the assumed
number of signals-need to detect number of
signals. - Information-theoretical approaches
- Minimum description length (MDL)
- Akaike information criterion (AIC)
- Exploit rank of the signal covariance matrix
- Not applicable to coherent signals
96Whiteness of the residue
- Residue
- Residuereceived signal-estimated signal
- Residue is approximately white
- Measure whiteness of the residue
- Use whiteness as the goodness of fit of the model
order (number of signals)
97Detection statistic
- Sample Autocorrelations
- Ljung-Box statistic
-
- The whiter, the smaller LB statistic
- Large values for the model order k smaller than
the true number of signals - Small values for the model order k equal or
larger than the true number of signals - Difference of LB statistic is the good indication
of true model order - Q-statistic
- Detection by Q-statistic criterion (QSC)
98Procedure of detection
99Examples
- Scenario
- 7 sensors (ULA)
- Half wavelength
- separation
- 100 samples
- Examples
- 2 signals
- 3 signals
- 4 signals
100Detection results-2 signals
101Detection results-3 signals
102Detection results-4 signals
103References
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