Fading Modeling, MIMO Channel Generation, and
Spectrum Sensing for Wireless Communications
Wei-Ho Chung
Electrical Engineering
University of California, Los Angeles
April 2009
whc@ee.ucla.edu
Electrical Engineering. University of California, Los Angeles
Outline
• 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
2
Electrical Engineering. University of California, Los Angeles
Fading Effects
• In wireless communications, the
signals traverse from the
transmitter to the receiver
• The medium (channel) physically
influences the signals, including
– Reflection
Reflection
Scatter
• 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
Receiver
Transmitter
Diffraction
– Scatter
• Signals impinge a rough surface
• The roughness at the order of the
wavelength or less
Electrical Engineering. University of California, Los Angeles
[B. Sklar, IEEE Communications Magazine, 1997.]
3
6
7
• The fading channel model is the
mathematical description of the
fading channel
5
– Stochastic
3
4
• Mobile communication systems
• Mobile node moving, various
fading effects
2
5%
1
– Quasi-deterministic
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Probability Density
2.5
• Applications of fading channel
model include
Electrical Engineering. University of California, Los Angeles
1.8
2
0
• Transmitter and receiver are
relatively static
• Fading effects can be
approximately deterministic
2
Envelope
– 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]
Envelope
8
9
10
Fading Models
and Applications
1.5
1
0.5
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Time(second)
0.14
0.16
0.18
0.2
4
Flat Fading
• Fading channel is modeled as a linear time-variant system with the
impulse response c(t , )
–
–

t
represents the time index of filter response
represents the time dependence of the filter response




r (t )   c(t; )u(t   )d   C (t; f )U ( f )e j 2ft df

– Delay spread of the filter response m , the coherence bandwidth Bcoh  1/  m
– Flat fading channel, bandwidth of input signal smaller than coherence
bandwidth

r (t )  C (t;0) U ( f )e j 2 ft df  C (t )u(t )  x(t )e j (t )u(t )

– Multiplicative effect on the transmitted signal x(t )
• The fading channel model is focused on modeling the statistical
properties of x(t )
[S. Stein, IEEE JSAC, 1987]
5
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Related Work
Imaginary
• 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
Real
Imaginary
– Rician model
• Dominant impinging signal
• Larger amount of scattered signals
Real
• 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
6
Electrical Engineering. University of California, Los Angeles
Outline
• 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
7
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Multi-Modal Observations
•
Envelope PDFs can have multi-modes
–
–
–
LOS v.s. NLOS
Time-Variant Fading Conditions
Simulation
–
Experiment
0.14
4000
Data Set 1
0.12
2000
Envelope
0.1
0.08
0
0
0.06
4000
0.02
1.5
2
2.5
1
1.5
2
2.5
Data Set 2
2000
100
200
300
400
500
Time(second)
•
1
6000
0.04
0
0
0.5
0
0
0.5
New mechanism to describe the dynamics
8
Electrical Engineering. University of California, Los Angeles
Modified Hidden Semi-Markov Model
• Amplitude-based Finite-State Markov Chains Model
a
a
(AFSMCM)
11
22
– Output channel amplitude
a12
S2
S1
a 21
a11
x2
x1
• Hidden Markov Model (HMM)
a12
– Output channel amplitude probabilistically
S1
S2
S1
PS1 ( x )
a 21
• Hidden Semi-Markov Model (HSMM)
– State duration probability
a 22
x1
a12
PS1 ( x ) P
(t )
dur ,S 1
PS 2 ( x )
x2
S2
Pdur ,S 2 ( t ) PS 2 ( x )
a 21
9
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x1
x2
Properties
• AFSMCM
–
–
–
–
Output vector 
Transition matrix P
1  m T
R
(
m
)

 (P ) 
ACF
N
PDF: Steady-state probability
• HMM
– Independent samples
– PDF: Mixtures of steady-state prob. and output PDF
• HSMM
– Independent samples
10
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Modified 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
q1=S2
q2=S1
2
1
q3=S4
3
...
...
0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 . . .
...
x x x x ...
1
2
3
4
[W. Chung and K. Yao, "Modified Hidden Semi-Markov Model for Modelling the
Flat Fading Channel," IEEE Tran. Comm., June, 2008.]
11
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Parameter Estimation of MHSMM
Observe Channel Gains
(1) Compute Local Mean
 t ,  t ,  t ....,  t
1
x1 , x 2 , x 3 ...., x n
2
n
3
X t1 ( f ), X t 2 ( f ), ...., X t n ( f )
{T1 , T 2 , T3 , ...., T ( k   k e ) }
T  ,1 , T  ,2 T  ,3 , ...., T  , k 
(4) Compute Spectrum
Entropies of Segments
(2) Short-Time Fourier
Transform
(6) Union of Segmentation Points to Obtain
(3) Segmentation by
Sliding Windows
e t 1 , e t 2 , e t 3 ...., e tn
(8) Perform k-means Clustering on the
(Mean, Entropy) pairs of Segments. Obtain
States S 1 , S 2 , ..., S Mˆ
States of Segments { q 1 , q 2 , q 3 , ..., q ( k   k  1) }
Te ,1 , Te , 2 , Te ,3 , ...., Te , k e
(7) Compute (Mean,Entropy) pair
of each Segments
{  x1 , e x1 },{  x 2 , e x 2 }, ...,{  x ( k
{ x 1 , x 2 , x 3 , ...., x ( k   k e  1) }
(5) Segmentation by
Sliding Windows
  k e 1 )
, ex( k
  k e 1 )
}
State Parameter Estimation Step
e
12
Electrical Engineering. University of California, Los Angeles
Clustering 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
13
Electrical Engineering. University of California, Los Angeles
Parameter estimation
Data from the Sequence Segmentation Step
{ x 1 , x 2 , x 3 , ...., x k } { q 1 , q 2 , q 3 , ...., q k }{ D 1 , D 2 , D 3 , ...., D k }
(1) Estimate State
Transition Matrix
a ij  N ( S i  S j )
 k  1
(2) Estimate Steady
State Probability
  A
(3) Estimate State (6) Compute the Conditional
Duration pdf
Envelope pdf
P Si ( x )
P dur , S i ( t )
(4) Compute Mean
of State Duration
 dur ( S ) 
i
tP
dur , S i
( t ) dt
(5) Estimate ACF R S
i
(t )
(7) Compute Overall Envelope pdf
• ACF: Sample ACF, i.e.,
• PDF:
Px ( x) 
 S dur ( S )
1

1
Si
dur ( S )
i
PS1 ( x) 
1
RSi ( ) 
N
 S dur ( S
2

Si
2)
dur ( S )
i
N
x x
t 1
t t 
PS2 ( x)  ... 
 S dur ( S
M

Si
M
)
dur ( S )
PSM ( x)
i
14
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(a)
(d)
Segmentation
Example
Estimated Transition Point
Estimated Transition Point
True Transition Point
True Transition Point
(b)
(e)
Estimated Transition Point
Estimated Transition Point
True Transition Point
True Transition Point
(f)
(c)
Estimated Transition Point
Estimated Transition Point
True Transition Point
True Transition Point
15
Electrical Engineering. University of California, Los Angeles
Clustering
(a)
(c) 0.8
Cluster Center
X (Mean,Entropy)
Representation of Segments
Davis-Bouldin Index
0.7
0.6
0.5
0.4
0.3
0.2
0.1
2
(b)
4
6
8
Number of Clusters
10
12
4
6
8
Number of Clusters
10
12
(d) 0.1
Dunn Index
0.08
0.06
0.04
0.02
0
2
Electrical Engineering. University of California, Los Angeles
16
Estimated pdf
• Kolgomorov D-statistic
MHSMM
0.02
AFSMCM
0.06
HMM
0.17
AFSMCM
A F S M C ppdf
df
17
Electrical Engineering. University of California, Los Angeles
Estimated ACFs
(f)
0.11
Estimated
ACF
Estimated
MHSMM
ACF
0.105
0.115
0.11
State 1
Estimated
ACF ACF
Estimated
MHSMM
(d)
0.1
0.095
0.09
0.085
0.08
-0.4
0.1
0.095
0.09
-0.2
0
Time(second)
0.2
0.08
-1
0.4
-0.5
0
Time(second)
0.5
1
0
Time(second)
0.5
1
(g)
1.05
State 4
State 2
Estimated
ACFACF
Estimated
MHSMM
Estimated
ACF ACF
Estimated
MHSMM
0.105
0.085
(e)
1.02
1.01
State 3
1
0.99
0.98
0.97
1
0.96
0.95
-0.4
-0.2
0
Time(second)
0.2
0.4
0.95
-1
-0.5
18
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Experiment
TX
Antenna
RX
Channel
Antenna
2
LowPass
C( t )
Oscillator/
Oscilloscope
Amplifier
HP 8350B
Sweep Oscillator
Pow er( t ) | c( t ) |  P( t )  V ( t )
Crystal
Detector
Amplifier
HP 8473C
Detector
Signal
Sampler
Output:
Floppy Disc
Tektronix TDS 724D
Oscilloscope
19
Electrical Engineering. University of California, Los Angeles
Experimental Data-Hallway
–
–
–
–
Hallway inside building
Rush hours
Numerous reflectors and scatters
Non-cooperative disturbances
10000
5000
0
0
0 .5
1
1 .5
2
15000
Frequencies
10000
D ata S et 2
5000
0
0
1
1 .5
Amplitude
2
D a ta S e t 2 p d f
M HS M M p d f
A
FSM
AFSMCM
M
C p dCf pdfpdf
4
HM M p d f
3
2
1
0
0 .5
D a ta S e t 1 p d f
5
D ata S et 1
P ro b a b ility D e n sity F u n ctio n
Frequencies
0
0 .5
1
E nve lo p e
1 .5
2
20
Electrical Engineering. University of California, Los Angeles
Summary
• Model Nonstationary Fading Processes
– Various channel conditions
– Piece-wise stationary processes
• Model the PDFs and ACFs
• Model Estimation Scheme
– Channel segmentation
– Parameter estimation
21
Electrical Engineering. University of California, Los Angeles
Outline
• 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
22
Electrical Engineering. University of California, Los Angeles
Generating 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
23
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Multiple Channels
• Space-Time correlation model
– Jake’s model by Bessel function [Jakes 94]
1
x
+
o
*
●
0.8
0.6
Theoretical ACF
Empirical  1 1
Empirical  2 2
Empirical  3 3
Empirical  4 4
ACF
0.4
0.2
0
-0.2
-0.4
-0.6
0
0.2
0.4
0.6
0.8
1
fD 
1.2
1.4
1.6
1.8
2
– Spatial and temporal correlations
• Multiple mobile fading channels [Abidi 02]
• MIMO channel for non-isotropic scattering environment
• MIMO channel for omnidirectional antennas [Rad 05]
24
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Example of Time-Space model
[Rad and Gazor, 05]
25
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Channel Generation by Autoregressive Processes
• Channels and Covariance Matrix
x[n]  [ x1[n] x2 [n]
xM [n]]T
 Rx , x [ j ] Rx , x [ j ]

 Rx , x [ j ] Rx , x [ j ]
Rxx [ j ]  E[ x[n  j ]x[n]H ]  



 Rx , x [ j ]
1
1
1
2
2
1
2
2
M
1
• AR processes
Rx1 , xM [ j ] 

Rx2 , xM [ j ] 




RxM , xM [ j ]
P
x[n]   A[k ]x[n  k ]  w[n]
k 1
[Baddour and Beaulieu, "Accurate Simulation of multiple cross-correlated
Rician fading channels," IEEE Tran. on Comm., Nov. 2004.]
26
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Solve AR parameters
• AR Coefficient
Rx , x [1]
 Rx , x [0]
 R [1]
Rx , x [0]
 x, x


 Rx , x [ p  1] Rx , x [ p  2]
Rx , x [ p  1]   AH [1] 
 Rxx [1] 
 R [2] 
Rx , x [ p  2]  AH [2] 
   xx 








Rx , x [0]   AH [ p]
R
[
p
]
 xx 
• Covariance of Noise
p
Q  Rxx [0]   Rxx [k ] AH [k ]
k 1
27
Electrical Engineering. University of California, Los Angeles
Simulation by Autoregressive Processes
28
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Generating 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 x N (0, Rx )
y
GM (m, Ry )
z
NK (m, Rz )
[Q. T. Zhang, "A decomposition technique for
efficient generation of correlated Nakagami fading
channels,“ IEEE JSAC, 2000.]
– Process x  y  z
– Problem
?
?
Rx  Ry  Rz
29
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Generating Correlated Nakagami
2m
y u
z  y
• Generate:
?
?
• Relating covariance matrices Rx  Ry  Rz
u  x
2
1/ 2
k
k 1
1 

2

(
m

)
1  1
2

1 

2 m  m  2 ( m) 


 ( a, b) 
b
 ( a ) ( a  b )   2 ( a  )
2
1 1


Rz (i, j )   (m,1)  2 F1 ( ,  ; m; Ry (i, j ))  1
2 2


1
 var[ Rz (k , k )], k  l


Rx (k , l )  
1/ 2

var[
R
(
k
,
k
)]
var[
R
(
l
,
l
)]
var[
R
(
k
,
l
)]
,k  l


z
z
y


Rx
b
 2 (a  )
2
Ry
1 

2

(
m

)
Rz (i, i )  1
2
Ry (i, i ) 
1 

m  m  2 ( m) 


2
Rz
30
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Heterogeneous MIMO channel generation
• Previous works focus on PDFs of the same
family, e.g., Rayleigh
, Nakagami [Zhang 2000]
• Fading environment causes channels of various
properties-channels of different families
• Generate multiple channels that have specific
[Baddour 2004]
– Auto-correlation function (ACF)
– Cross-correlation function (CCF)
– Heterogeneous envelope PDFs
31
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Illustration of the problem
Desired fading
envelope processes
y1,n
yi ,n
?
Fi (.)
ij ,
ij
y j ,n
F1 (.)

Fj (.)
yM , n
FM (.)
32
Electrical Engineering. University of California, Los Angeles
Inverse Transform Sampling
• Framework
– Probability density functions
– Correlations
• Inverse Transform Sampling
– Generate x with CDF  ( x )
– y has CDF F ( y )
 ( x)
F ( y)
u   ( x)
y F
1
(u )
x
33
Electrical Engineering. University of California, Los Angeles
Proposed approach—
Inverse Transform Sampling
Gaussian vector
AR process
x1, n
xi ,n
x M ,n
Desired fading
envelope processes
Inverse Transform Sampling
 (.)
 (.)
 (.)
 ( x1, n )
 ( xi ,n )
 ( x M ,n )
y1, n
1
yi ,n
1
y M ,n
y1, n  F1 (  ( x1, n ))
1
y i , n  Fi (  ( x i , n ))
F1 (.)
Fi (.)
F
1
1
1
M
(.)
y M , n  F M (  ( x M , n ))
[W. Chung, K. Yao, and R. E. Hudson, “The Unified Approach for Generating Multiple Crosscorrelated and Auto-correlated Fading Envelope Processes.” Accepted. IEEE Tran. Comm., 2009.]
34
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Sketch of derivation
• Definition of correlation
ij , 
1
 i j

  y
i , n 
 i   y j ,n   j  fi , j ( yi ,n  , y j ,n )dyi ,n  dy j ,n
0 0
• Jacobian
fi , j ( yi ,n  , y j ,n ) 
fi , j ( xi ,n  , x j ,n )
J Y , X 
xi ,n  yi ,n
x j , n  y j ,n
• Correlations of input and output
ij , 
1
 i j

  y
i , n 
 i   y j , n   j 
0 0


  1 F ( y )
 i i,n 
 exp 

2



dyi ,n  dy j ,n .

f i ( yi ,n  ) f j ( y j ,n )
 1
r
 ji ,
rij , 
1 
1/ 2
 1  Fi ( yi ,n  )    1

 
1
   Fj ( y j ,n )    rji ,


T
     F ( y )
2
1
j
2
Electrical Engineering. University of California, Los Angeles
j ,n
2
rij , 
1 
2
1
  1  Fi ( yi ,n  )   


1
   F j ( y j ,n )   






35
Example- Heterogeneous channels of
Nakagami, Rician, and Rayleigh pdfs
• Three channels
–
–
m 2
2mm
m 2
2 m 1
F
(
y
)


(
m
,
y )
f
(
y
)

y
exp(

y
)
Naka
m
Nakagami Naka

(m)

vRi y
( y 2  vRi2 )
yvRi
y
F
(
y
)

1

Q
(
,
)
f Rician ( y)  2 exp(
)
I
(
)
Rician
Rician
0
2
2


Ri
Ri
 Ri
2 Ri
 Ri
– Rayleigh
f Ray ( y ) 
• Correlations
– ACF
– CCF
 y2
y exp( 2 )
2 Ray
2
 Ray
 y2
FRay ( y)  1  exp( 2 )
2 Ray
11,   22,  33,  exp( f D  |  |)
12,  21,  13,  31,  23,  32,   exp( f D  |  |)
36
Electrical Engineering. University of California, Los Angeles
Results
1
1
10
+
ACF
Nakagami
Rician 1 3
Rayleigh
x Theoretical 1 311
o Empirical  11
0.5
0
0
1
2
3
4
5
6
7
8
9
10
1
x Theoretical 1 322
o Empirical  22
0
ACF
10
0.5
0
0
1
2
3
4
5
6
7
8
9
10
1
ACF
x Theoretical 1 333
o Empirical  33
-1
0.5
10
0
20
40
60
80
100
120
140
160
180
200
0
0
1
2
3
4
5
Time lag
6
7
8
9
10
37
Electrical Engineering. University of California, Los Angeles
Example- Heterogeneous Channels of
Nakagami, Rician, and Rayleigh pdfs
(a)
1
0.8
cdf
1
CCF
x Theoretical
13
o Empirical 12
+ Empirical  21
0.5
0.6
0.4
Theoretical Nakagami cdf
Empirical Nakagami cdf
0.2
0
0
0
0
1
2
3
4
5
6
7
8
9
10
(b)
1
1.5
2
2
3
4
5
6
7
8
9
10
0
CCF
(c)
cdf
1
2
3
4
5
6
Time lag
7
8
9
0.5
1
1.5
2
2.5
3
3.5
4
1
0.8
0
0
4
Theoretical Rician cdf
Empirical Rician cdf
0
x Theoretical
13
o Empirical 23
+ Empirical  32
0.5
3.5
0.4
0
1
3
1
0.2
0
1
2.5
0.6
cdf
CCF
1
0.8
x Theoretical
13
o Empirical 13
+ Empirical  31
0.5
0.5
0.6
0.4
Theoretical Rayleigh cdf
Empirical Rayleigh cdf
10
0.2
0
0
0.5
1
1.5
2
2.5
Envelope
3
3.5
4
38
Electrical Engineering. University of California, Los Angeles
Example- 2x2 Rayleigh MIMO Channels
• PDF
f Ray ( y ) 
 y2
y exp( 2 )
2 Ray
2
 Ray
• ACFs 11,   22,  33,   44,  J 0 (2 f D  |  |)
1
1
0.9
x
+
o
*●
0.8
0.8
0.6
Theoretical ACF
Empirical  11
Empirical  22
Empirical  33
Empirical  44
0.7
0.4
cdf
ACF
0.6
0.5
0.2
0.4
0
0.3
-0.2
x Theoretical cdf
0.2
Empirical cdf(channel 1)
Empirical cdf(channel 2)
Empirical cdf(channel 3)
Empirical cdf(channel 4)
0.1
-0.4
0
0
0.5
1
1.5
Envelope
2
2.5
3
-0.6
0
10
20
30
40
50
60
70
80
90
Time Lag
39
Electrical Engineering. University of California, Los Angeles
Example- 2x2 Rayleigh MIMO Channels
• CCFs
–
–
–
1
2
12,  34,  J 0 ({a  b  2ab cos(    )} ) 1
13,   24,  J 0 ({a 2  c 2 2 2ab sin( ) sin( )}2 )
2
2
14,  32,  J 0 ({a  b  c
2
2
2
Theoretical  12 , 34
x Empirical  12
o Empirical  34
2
1
2
2ab cos(    )  2c sin( )[a sin( )  b sin(  )]} )
Theoretical  13 ,  24
x Empirical  13
o Empirical  24
Theoretical  14 ,  32
x Empirical  14
o Empirical  32
0.4
0.3
0.2
CCF
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
10
20
30
40
50
Time lag
Electrical Engineering. University of California, Los Angeles
60
70
80
90
40
Example-Single Nakagami Channel
1
vf c
v
– fD  
 3 108
– f c  900 MHz
– v  27.78 m/s
– High sampling rate
– Low sampling rate
0.8
0.6
ACF
• ACF
Naka,  J 02 (2 f D |  |)
Theoretical acf (Low Sampling Rate)
Empirical acf (Low Sampling Rate)
xxx Theoretical acf (High Sampling Rate)
ooo Empirical acf (High Sampling Rate)
0.4
  1/ 24300
  1/ 9600
0.2
0
0
10
20
30
40
50
60
Time Lag
70
80
90
100
1
0.9
0.8
0.7
cdf
0.6
0.5
0.4
0.3
0.2
Theoretical cdf
Empirical cdf( low sampling rate)
Empirical cdf( high sampling rate)
0.1
41
0
Electrical Engineering. University of California, Los Angeles
0
0.5
1
1.5
Envelope
2
2.5
Outline
• 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
42
Electrical Engineering. University of California, Los Angeles
Detection 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
43
Electrical Engineering. University of California, Los Angeles
Hypothesis Test
• Hypothesis Test applications
– Surveillance
– Target Detection
– Spectrum Sensing
Hypothesis
Sensor
H1 or H0
Decision=f(x)
x
• Example-Matched Filter Detection
– Signal model
S  N , H 0
X  0
 S1  N , H1
–
–
  X , ( S1  S0 )

 S , ( S  S0 )  N , ( S1  S0 ) , H 0
 0 1

 S1 , ( S1  S0 )  N , ( S1  S0 ) , H1

 Hˆ ,  
Decision   0
ˆ

 H1 ,  
0.5
pdf
P ( | H 1 )
P ( | H 0 )
0.4
0.3
0.2
PD
0.1
PF A
0
-6
-4
-2

0
2
4
6
44
Electrical Engineering. University of California, Los Angeles
Receiver Operating Curve
• Receiver Operating Curve: PFA v.s. PD
• Setting Threshold

• Criteria
1
Bayes Criterion
0.9
S lope=
0.8
– Neyman-Pearson
• PF A upper-bounded
– Bayes
• Priors and costs
P0 ( c10  c 00 )
P1 ( c 01  c11 )


0.7
0.6
PD
0.5

0.4
0.3
0.2
0.1
Neyman-Pearson
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PF A
45
Electrical Engineering. University of California, Los Angeles
Spectrum 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.]
46
Electrical Engineering. University of California, Los Angeles
Detection 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
47
Electrical Engineering. University of California, Los Angeles
Decision Fusion Framework
• Sensors make binary decisions.
– Many applications require binary decisions.
– Accuracy of a single sensor is limited.
– Fusion of multiple decisions increases accuracies.
Sensor
Hypothesis
H1 or H0
D1
Fusion Center
D2
D3
Decision=f({Di,i=1~N})
D3
[R. Viswanathan and P. K. Varshney, "Distributed Detection with
Multiple sensors I. Fundamentals," Proceedings of the IEEE, 1997.]
48
Electrical Engineering. University of California, Los Angeles
Decision Fusion Framework
• N sensors make binary
decisions.
– Probability of False Alarm {PFAi | i  1, 2,..., N}
– Probability of Detection {PDi | i  1, 2,..., N}
– Sensor decisions {Di | i  1, 2,..., N}
• The fusion center makes final
decision.
– Fusion Rule: f 0 : {1,1}N  {1,1}
– Fusion Rule with random strategy:
– Solve the parameters of the
Rule: S0 , S1 , r ,  .
Sensors
Fusion Center
{1,1}N
f0
{1,1}
1, when {Di }  S0

1, when {Di }  S1
f 0 ({Di })  
+1 with probability 
 -1 with probability 1- , when {Di }  r.
Fusion 
49
Electrical Engineering. University of California, Los Angeles
Algorithm for Computing the Fusion Rule
• For each element 
 {1,1} ,
N
j
we denote
– P ( j | H 0 ) by P |H
– P ( j | H 1 ) by P j | H1
j
0
• The likelihood ratio, associated with  j , is defined as  j  P |H P |H
j
1
j
0
else
[W. Chung and K. Yao, “Decision Fusion in Sensor Networks for Spectrum
Sensing based on Likelihood Ratio Tests,” Proceedings of SPIE, 2008.]
50
Electrical Engineering. University of California, Los Angeles
Fusion of Two Sensors
• Two Sensors
Sensor 1
Sensor 2
PD
PF A
0.2
0.9
0.1
0.7
– Operating points
– Goal: (Lower-Bounded Probability of Detection Criterion) Minimizing PFAfc while PDfc
is lower bounded by 0.91
• Result
P
|H 0
 P ( D 2  0 | H 0 ) P ( D1  0 | H 0 )
 (1  PF A2 )(1  PF A1 )
S0
P
D 2 D1
1
j
|H 1
P
j
|H 0
 j  P
j
|H 1
1
00
0.08
0.27
0.29
2
01
0.02
0.03
0.66
3
10
0.72
0.63
1.14
4
11
0.18
0.07
2.57
50%
S1
 D fc  0, 50%
r 
 D fc  1, 50%
Electrical Engineering. University of California, Los Angeles
P
j
|H 0
50%
PF A fc  0.07  0.63  0.03 * 50%  0.715
PD fc  0.18  0.72  0.02 * 50%  0.91
51
Fusion of Two Sensors
Fusion Center
1
X
0.9
Sensor 2
0.8
0.7
0.6
PD fc
0.5
0.4
0.3
0.2
Sensor 1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PF A fc
52
Electrical Engineering. University of California, Los Angeles
1
Examples
3 sensors
0.9
0.8
0.7
0.6
• Proposed algorithm
• K out of N
PD fc 0.5
0.4
0.3
0.2
– FC declares H1 if k or
more than k sensors
declare H1. Otherwise,
FC declares H0.
• Decision Space
search f0 : {1,1}N
Proposed Algorithm
0.1
k out of N
Decision Space Search
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PF A fc
4 sensors
 {1,1}
– All possible
combinations of
decision fusion rule
PD fc
53
Electrical Engineering. University of California, Los Angeles
PF A fc
Detection under Fading--Likelihood Ratio Test with Fading Statistics (LRFS)
N , H0

X 
 S  N , H1
• Signal Model
• Fading Gains
 2
pRay ( ; R )   exp( 2 )  R2
– Rayleigh
2 R

( 2  v 2 )
v
p
(

;

,
v
)

exp(
)
I
[
]
Rician
Ri
0
– Rician
 Ri2
2 Ri2
 Ri2
• Test Statistic
 LRFS ( X ) 
p ( X | H1 )

p( X | H 0 )


0
1
e ( X  S )
T
(2 )
2 m/2
n
1
(2 )
2 m/2
n

 e
(2 X T S  2 S T S ) /(2 n2 )
0
e
( X  S ) /(2 n2 )
p( )d
 X T X /(2 n2 )
p( )d
54
Electrical Engineering. University of California, Los Angeles
LRFS
• Explicit Expressions of the test statistics
– Rayleigh:
T
 2R (2X T S ) 4
1
S
S
T
 LRFS ( X )  e 2( R n S S  n ) 2 X S  2 n2


2
2



R
n

2
T


T
X S
T

2 X S Erf 
T
2
2
S
S
2


n

 R2
 n2
– Rician:
 LRFS ( X ) 
1
 Ri2


0
e
2


T
2
2
 R (X S) 
 2( R2 n2 S T S  n4 ) 

e





 v2  2 2 X T S  2 S T S

2
2 Ri
 n2

2  S S 
I0[
2
R
T
2
n

1
 R2
T

S S
 n2
v
]d
2
 Ri
55
Electrical Engineering. University of California, Los Angeles
Multi-Sensor Decision Fusion under Fading
• Signal Model: yi   iui  ni
• Likelihood Ratio: (Y )  p(Y | H ) / p(Y | H )
• Reformulate by fading statistics
1
– Under H1:
– Under H0:

p(Y | H1 ) 

p(Y | H 0 ) 
 p(Y |{ },{u }, H ) p({u }| H ) p({ })d d ...d
i
{ui }{1, 1}
N
i
1
i
1
i
0
i
0
• Test Statistics
(Y ) 

 p(Y |{ },{u }, H ) p({u }| H ) p({ })d d ...d

 p(Y |{i },{ui }, H 0 ) p({ui }| H 0 ) p({i })d1d 2 ...d N
{ui }{1, 1}N
{ui }{1, 1}N
i
i
1
i
1
2
N
 p(Y |{ },{u }, H ) p({u }| H ) p({ })d d ...d
i
{ui }{1, 1}
N
0
i
1
i
1
2
N
u fc  1


u fc  1
i
1
2
 fc
56
Electrical Engineering. University of California, Los Angeles
N
1
0.9
Numerical Examples
1
d
10
5d
B
B 0 dB
5
dB
0.8
10 dB
B
5d
B
0d
LRFS
Rician 0 dB
MF
0
dB
0.7
LRFS
Rayleigh
0.6
• LRT with Fading Statistics P
– LRFS under Rayleigh
– LRFS under Rician
– Matched Filter
• Decision Fusion with Fading Statistics
– 3 Sensors
– 2 Sensors
0.5
D
0.4
0.3
LRFS (Rician)
LRFS (Rayleigh)
MF
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PF A
1

2 detectors
0.
R
0.

2
0.
R


R
PDfc 0.5


0.6
2


0.5

8
0
.8
R
R
0.7

R
0.8
0.
5
0.9
0.4
0.3
3 detectors
0.2
3 detectors
2 detectors
0.1
57
0
0
Electrical Engineering. University of California, Los Angeles
0.1
0.2
0.3
0.4
0.5
PFAfc
0.6
0.7
0.8
0.9
1
Summary
• 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
58
Electrical Engineering. University of California, Los Angeles
Outline
• 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
59
Electrical Engineering. University of California, Los Angeles
Sequential Likelihood Ratio Test
for 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,” ]
60
Electrical Engineering. University of California, Los Angeles
Formulation
• Signal Model
nt , H 0
– Received x  
t
 t  nt , H1
– Signal (primary user)  t
– Signal follows AR model
 t  a1 t 1  a2 t 2  a3 t 3  ...  aq t q  wt
• Received signal follows ARMA
xt  a1 xt 1  a2 xt 2  a3 xt 3  ...  aq xt q  wt  nt  a1nt 1  a2 nt 2  ...  aq nt  q
61
Electrical Engineering. University of California, Los Angeles
Sequential Decision
• Decision at time t
 H1

Dt   H 0
continue

• Log LR
P( x1 , x2 ,..., xt | H1 )
 t ( x1 , x2 ,..., xt )  log
P( x1 , x2 ,..., xt | H 0 )
• Sequential decision
t ( xt | xt 1 , xt  2 ,..., xt  p 1 )  log
P( xt | xt 1 , xt  2 ,..., xt  p 1 , H1 )
P( xt | xt 1 , xt  2 ,..., xt  p 1 , H 0 )
t  t 1  t
62
Electrical Engineering. University of California, Los Angeles
Decision and Thresholds
• Decision
H1
if t  log A


Dt  
H0
if t  log B
continue if log A    log B
t

• Thresholds
1  PM
PFA
PM
B
1  PFA
A
• Expected termination time can be derived as a
function of accuracy and SNR
63
Electrical Engineering. University of California, Los Angeles
Example 1-Scenario
• SNR uniformly distributed between -20 dB
to 10 dB
• Prior 0.5 for H1 and H0
• Jake’s ACF
64
Electrical Engineering. University of California, Los Angeles
Example 1-results
Decision Threshold H 1
1
4
x 10
Test Statistic  t
0.75
14
0.5
0.25
12
0
-0.25
10
-0.5
-0.75
Decision Threshold H 0
-1
0
0.5
1
1.5
2
Time (Number of Samples)
2.5
3
4
x 10
(b)
Decision Threshold H 1
1
0.75
Test Statistic  t
Average Number of Samples
(a)
0.5
8
6
En
er
4
Propo
sed A
0.25
2
0
Propose
d Appro
pproa
ch-Th
eo
ach-Em
pir
-0.25
0
0.1
-0.5
-0.75
0
2
4
6
8
Time (Number of Samples)
10
0.2
De
tec
tor
retica
l
ical
0.25
0.3
0.35
0.4
PF A
Decision Threshold H 0
-1
0.15
gy
12
4
x 10
65
Electrical Engineering. University of California, Los Angeles
Example 2—fixed SNR at -20 dB
(a)
Decision Threshold H 1
1
4
x 10
14
0.5
12
0
-0.25
-0.5
10
-0.75
Decision Threshold H 0
-1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (Number of Samples)
(b)
5
4
x 10
Decision Threshold H 1
1
0.75
Test Statistic  t
o
x
*
x
*
0.25
Average Number of Samples
Test Statistic  t
0.75
0.5
0.25
Energy Detector
Proposed Approach at SNR(-20 dB) -Theoretical
Proposed Approach at SNR(-20 dB) -Empirical
Proposed Approach at SNR(0 dB) -Theoretical
Proposed Approach at SNR(0 dB) -Empirical
8
6
4
0
2
-0.25
-0.5
0
0.1
-0.75
Decision Threshold H 0
-1
0
1
2
3
4
5
6
Time (Number of Samples)
7
0.15
0.2
0.25
0.3
0.35
0.4
PF A
4
x 10
66
Electrical Engineering. University of California, Los Angeles
Conclusion
• 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
67
Electrical Engineering. University of California, Los Angeles
Future 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
68
Electrical Engineering. University of California, Los Angeles
Thank you
69
Electrical Engineering. University of California, Los Angeles
Publications
•
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.
70
Electrical Engineering. University of California, Los Angeles
References
•
•
•
•
•
•
•
•
•
•
P. Beckman, Probability In Communication Engineering. New York: Harcourt Brace and World,
1967.
E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and
communications aspects," IEEE Transactions on Information Theory, vol. 44, issue 6, pp.
2619-2692, Oct. 1998.
Bernard Sklar, "Rayleigh fading channels in mobile digital communication systems. I.
Characterization, "IEEE Communications Magazine,Vol. 35, Issue 7, pp.90-100,Jul. 1997.
Seymour Stein, "Fading Channel Issues in System Engineering," IEEE Journal on Selected
Areas in Communications, Vol. 5, Issue 2, pp.68-89, Feb. 1987.
C. C. Tan, N. C. Beaulieu, "On first-order Markov modeling for the Rayleigh fading channel,"
IEEE Transactions on Communications,Vol. 48, Issue 12,pp. 2032-2040, Dec 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. 16, issue 2, pp. 160-174, Feb. 1998.
A. J. Goldsmith, S. G. Chua, "Adaptive coded modulation for fading channels," IEEE
Transactions on Communications, vol.46, issue 5, pp. 595-602, May 1998.
J. C. Bezdek, N.R. Pal, "Some new indexes of cluster validity," IEEE Transactions on Systems,
Man and Cybernetics, Part B, Volume 28, Issue 3, pp. 301-315, Jun. 1998.
M. S. Aldenderfer and R.K. Blashfield, Cluster Analysis, Newbury Park, California U.S.A., Sage
Press, 1984.
J. C. Dunn, "A fuzzy relative of the ISODATA process and its use in detecting compact wellseparated clusters," Journal of Cybernetics, vol. 3, no. 3, pp. 32–57, 1973.
71
Electrical Engineering. University of California, Los Angeles
References
•
•
•
•
•
•
•
•
•
•
K. E. Baddour and N. C. Beaulieu, "Accurate 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. 16, Issue 2, pp. 160-174, Feb. 1998.
P. Beckman, Probability In Communication Engineering. New York: Harcourt Brace and World,
1967.
A. Goldsmith, S. A. Jafar, N. Jindal, and S. 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. 23852392, 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.
72
Electrical Engineering. University of California, Los Angeles
References
•
•
M. Nakagami, “The m-distribution, a general 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.
673–680, July 1977.
73
Electrical Engineering. University of California, Los Angeles
Cooperative Spectrum Sensing Scheme based on
Nash Equilibrium
Wei-Ho Chung
Electrical Engineering
University of California, Los Angeles
March 2009
whc@ee.ucla.edu
Electrical Engineering. University of California, Los Angeles
Decision 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
75
Electrical Engineering. University of California, Los Angeles
Hypothesis Test
• Hypothesis Test applications
– Surveillance
– Target Detection
– Spectrum Sensing
Hypothesis
Sensor
H1 or H0
Decision=f(x)
x
• Example-Matched Filter Detection
– Signal model
S  N , H 0
X  0
 S1  N , H1
–
–
  X , ( S1  S0 )

 S , ( S  S0 )  N , ( S1  S0 ) , H 0
 0 1

 S1 , ( S1  S0 )  N , ( S1  S0 ) , H1

 Hˆ ,  
Decision   0
ˆ

 H1 ,  
0.5
pdf
P ( | H 1 )
P ( | H 0 )
0.4
0.3
0.2
PD
0.1
PF A
0
-6
-4
-2

0
2
4
6
76
Electrical Engineering. University of California, Los Angeles
Receiver Operating Curve
• Receiver Operating Curve: PFA v.s. PD
• Setting Threshold

• Criteria
1
Bayes Criterion
0.9
S lope=
0.8
– Neyman-Pearson
• PF A upper-bounded
– Bayes
• Priors and costs
P0 ( c10  c 00 )
P1 ( c 01  c11 )


0.7
0.6
PD
0.5

0.4
0.3
0.2
0.1
Neyman-Pearson
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PF A
77
Electrical Engineering. University of California, Los Angeles
Spectrum 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.]
78
Electrical Engineering. University of California, Los Angeles
Detection 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
79
Electrical Engineering. University of California, Los Angeles
Decision Fusion Framework
• Sensors make binary decisions.
– Many applications require binary decisions.
– Accuracy of a single sensor is limited.
– Fusion of multiple decisions increases accuracies.
Sensor
Hypothesis
H1 or H0
D1
Fusion Center
D2
D3
Decision=f({Di,i=1~N})
D3
[R. Viswanathan and P. K. Varshney, "Distributed Detection with
Multiple sensors I. Fundamentals," Proceedings of the IEEE, 1997.]
80
Electrical Engineering. University of California, Los Angeles
Decision Fusion Framework
• N sensors make binary
decisions.
– Probability of False Alarm {PFAi | i  1, 2,..., N}
– Probability of Detection {PDi | i  1, 2,..., N}
– Sensor decisions {Di | i  1, 2,..., N}
• The fusion center makes final
decision.
f 0 : {0,1}N  {0,1}
Fusion Center
– Fusion Rule:
– Fusion Rule with random strategy:
– Solve the parameters of the
Rule: S0 , S1 , r ,  .
{ 0 ,1}
N
f0
Final Decision
{ 0 ,1}
0, when {Di }  S0

1, when {Di }  S1
f 0 ({Di })  
 1 with probability 
 0 with probability 1- , when {Di }  r.
Fusion 
81
Electrical Engineering. University of California, Los Angeles
Algorithm for Computing the Fusion Rule
• For each element 
 {1,1} ,
N
j
we denote
– P ( j | H 0 ) by P |H
– P ( j | H 1 ) by P j | H1
j
0
• The likelihood ratio, associated with  j , is defined as  j  P |H P |H
j
1
j
0
else
[W. Chung and K. Yao, “Decision Fusion in Sensor Networks for Spectrum
Sensing based on Likelihood Ratio Tests,” Proceedings of SPIE, 2008.]
82
Electrical Engineering. University of California, Los Angeles
Fusion of Two Sensors
• Two Sensors
Sensor 1
Sensor 2
PD
PF A
0.2
0.9
0.1
0.7
– Operating points
– Goal: (Lower-Bounded Probability of Detection Criterion) Minimizing PFAfc while PDfc
is lower bounded by 0.91
• Result
P
|H 0
 P ( D 2  0 | H 0 ) P ( D1  0 | H 0 )
 (1  PF A2 )(1  PF A1 )
S0
P
D 2 D1
1
j
|H 1
P
j
|H 0
 j  P
j
|H 1
1
00
0.08
0.27
0.29
2
01
0.02
0.03
0.66
3
10
0.72
0.63
1.14
4
11
0.18
0.07
2.57
50%
S1
 D fc  0, 50%
r 
 D fc  1, 50%
Electrical Engineering. University of California, Los Angeles
P
j
|H 0
50%
PF A fc  0.07  0.63  0.03 * 50%  0.715
PD fc  0.18  0.72  0.02 * 50%  0.91
83
Fusion of Two Sensors
Fusion Center
1
X
0.9
Sensor 2
0.8
0.7
0.6
PD fc
0.5
0.4
0.3
0.2
Sensor 1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PF A fc
84
Electrical Engineering. University of California, Los Angeles
Costs structure and game formulation
• Utility from accessing the channel
– 0 for channel conflict
– us / N for a successful access
• Costs
– cb N for a channel conflict
– cq N for a successful access
• Game
– Users { i ,i  1 ~ N }
– Actions { Ai ,i  1 ~ N }
•
•
Ai  1 :user  i perform detection - incurs cost cd
Ai  0 :user  i not perform detection - incurs cost 0
– Utility { ui ,i  1 ~ N }
Electrical Engineering. University of California, Los Angeles
ui  Ai  cd  P0  [( cq  us ) / N ]  ( 1  PFAfc )
 P1  ( cb / N )  ( 1  PDfc )
85
Operating point and Costs
• Operating point by Bayes criterion--the
point on the ROC with slope of its tangent
line equal to
P0 ( us  cq ) P(
1 cb )
• Expected Costs
– Individual user
ui  Ai  cd  P0  [( cq  us ) / N ]  ( 1  PFAfc )
 P1  ( cb / N )  ( 1  PDfc )
– Overall cost
U  j ,total 
u
i
i
86
Electrical Engineering. University of California, Los Angeles
Action profile by Nash equilibrium
 j  {0, 1}
• For the action profiles
M
– Compute costs of each action profile
– Compute Nash equilibrium
Compute Costs
 j  { 0 , 1}
M
{( PF A i , PD i ) | i  I a }
Compute ROC
ROC
Compute Bayes Optimal ( PF A fc , PD fc )
Operating Point
u i  Ai  c d  P0  [( c q  u s ) / N ]  ( 1  PF A fc )
 P1  ( c b / N )  ( 1  PD fc )
U  j ,total 
u
i
i
P0 ( u s  c q ) P1 ( c b )
ˆ
Compute Nash Equilibrium
Pick the max utility in NE
ˆ  m ax{ U 

j
,total
|  j  { A i }N E }
j
87
Electrical Engineering. University of California, Los Angeles
Results - influences of charges from primary users
– Cost of conflict cb
– Cost of successful access cq
1
1
0.9
0.9
0.8
0.8
pfa
Probability
0.7
Probability
0.7
0.6
0.6
0.5
0.5
0.4
PD fc
0.4
PF A fc
0.2
PD fc
0.3
0.3
PF A fc
0.2
0.1
0.1
0
0.5
0
0
0.1
0.2
0.3
cq
0.4
0.5
0.6
0.7
cq
0.6
0.7
0.8
0.9
cb
1
1.1
1.2
1.3
1.4
1.5
1.4
1.5
0.14
0.2
0.18
0.12
0.16
0.1
Utility
Utility
0.14
0.12
0.1
u m ax,N E
0.08
u m ax
cb
u m ax
0.08
0.06
0.06
u m ax,N E
0.04
0.04
0.02
0.02
0
0
0.1
0.2
0.3
cq
0.4
0.5
0.6
max
0.7
0
0.5
0.6
0.7
0.8
0.9
cb
1
1.1
1.2
1.3
88
Electrical Engineering. University of California, Los Angeles
Results-influence of detection cost
• Detection cost cd
1
0.9
Probability
0.8
0.7
PD fc
0.6
0.5
0.4
PF A fc
0.3
0.2
0
0.005
0.01
cd
0.015
0.02
0.025
0.02
0.025
0.14
0.12
u s ,m ax
Utility
0.1
0.08
u s ,m ax N E
0.06
0.04
0.02
0
0
0.005
0.01
cd
0.015
89
Electrical Engineering. University of California, Los Angeles
Conclusions
• 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
90
Electrical Engineering. University of California, Los Angeles
References
•
•
•
•
•
•
•
•
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.
91
Electrical Engineering. University of California, Los Angeles
Detecting Number of Coherent Signals in
Array Processing by Ljung-Box Statistic
Wei-Ho Chung
Electrical Engineering
University of California, Los Angeles
April 2009
whc@ee.ucla.edu
Electrical Engineering. University of California, Los Angeles
Array Signal Processing
• Estimate
– Direction of arrival in far-field
Signal Source 2
– Localizations near-field
• Likelihood formulation
– Grid search
– Newton method
– Genetic Algorithm
Signal Source 1
1
Sensor Array
(ULA)
d
Receive Signals
• Detect number of signals
93
Electrical Engineering. University of California, Los Angeles
Signal Model
• Signals
S j ( t;s j )  s j ei 2 f0t
– Source signal
– Received signal
K
x( t )   a(  j )S j ( t;s j )  n( t )
j 1
• Unknown parameters
– DOAs   {  | j  1 ~ K }
– Amplitudes S  { s | j  1 ~ K }
j
j
• Maximum Likelihood Estimation
N
k
2
ˆ }  min
{ ˆ ,S
 x( t )   a(  j )S j ( t;s j )
 j ,s j
t 1
j 1~k
j 1
l2
94
Electrical Engineering. University of California, Los Angeles
Estimate 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
95
Electrical Engineering. University of California, Los Angeles
Whiteness of the residue
• Residue
– Residue=received signal-estimated signal
k
k
ˆ t )  x( t )   a( ˆ j )S j ( t;sˆ j )
n(
j 1
– Residue is approximately white
– Measure whiteness of the residue
– Use whiteness as the goodness of fit of the
model order (number of signals)
96
Electrical Engineering. University of California, Los Angeles
Detection statistic
• Sample Autocorrelations
• Ljung-Box statistic
–
–
–
–
–
rk (  ) 
M N 
N   [ k nˆ h ( j ) k nˆ h ( j   )* ]
h 1 j 1
M
N
( N   ) [ k nˆ h ( j ) k nˆ h ( j )* ]
h 1 j 1
rk (  )rk (  )*
L( k )  N( N  2 )
N 
 1
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
m
• Q-statistic Q( k )  L( k  1 )  L( k ),k  1 ~ M  1
• Detection by Q-statistic criterion (QSC) K̂  max Q( k )
k
97
Electrical Engineering. University of California, Los Angeles
Procedure of detection
k 0
k : k  1
x(t)
ML Estimation
{ ˆ , ˆ }
Compute Residue
k
ˆ t )  x( t )   a( ˆ j )S j ( t ; ˆs j )
n(
k
j 1
Compute Ljung-Box Statistic
L( k )
Compute Q-Statistic
Q( k )
if k  M
if k  M
Detection
K̂  m ax Q ( k )
k
98
Electrical Engineering. University of California, Los Angeles
Examples
800
2 Signals Detected by QSC
600
• Scenario
Statistic
(Example of 2 Signals)
400
• Examples
0
-200
1
2
1000
3
4
5
6
Number of Signals
3 Signals Detected by QSC
(Example of 3 Signals)
500
Statistic
– 7 sensors (ULA)
– Half wavelength
separation
– 100 samples
200
0
-500
1
– 2 signals
– 3 signals 3   / 3
– 4 signals 4   / 2
3
4
5
6
Number of Signals
1000
4 Signals Detected by QSC
(Example of 4 Signals)
Statistic
1   / 9 2   / 4
2
500
0
-500
1
2
3
4
5
6
Number of Signals
99
Electrical Engineering. University of California, Los Angeles
Detection results-2 signals
100
Electrical Engineering. University of California, Los Angeles
Detection results-3 signals
101
Electrical Engineering. University of California, Los Angeles
Detection results-4 signals
102
Electrical Engineering. University of California, Los Angeles
References
•
•
•
•
•
M. Wax and T. Kailath, "Detection of signals by information theoretic criteria," IEEE
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387-392, Apr. 1985.
M. Wax and I. Ziskind, "Detection of the number of coherent signals by the MDL
principle," IEEE Transactions on Acoustics, Speech and Signal Processing,
Volume 37, Issue 8, pp. 1190-1196, Aug. 1989.
Q. T. Zhang, K. M. Wong, P. C. Yip, and J. P. Reilly, "Statistical analysis of the
performance of information theoretic criteria in the detection of the number of
signals in array processing," IEEE Transactions on Acoustics, Speech and Signal
Processing, Volume 37, Issue 10, pp. 1557-1567, Oct. 1989.
M. S. Barlett, “A note on the multiplying factors for various chi-square
approximations,” J. Royal Stat. Soc., Ser. B, vol. 16, pp. 296-298, 1954.
D. N. Lawley, “Tests of significance of the latent roots of the covariance and
correlation matrices,” Biometrika, vol. 43, pp. 128-136, 1956.
103
Electrical Engineering. University of California, Los Angeles
References
•
•
•
•
•
P. Stoica and K. C. Sharman, "Maximum likelihood methods for direction-of-arrival
estimation," IEEE Transactions on Acoustics, Speech and Signal Processing, Volume
38, Issue 7, pp. 1132-1143, Jul. 1990.
M. Pesavento and A. B. Gershman, "Maximum-likelihood direction-of-arrival
estimation in the presence of unknown nonuniform noise," IEEE Transactions on
Signal Processing, Volume 49, Issue 7, pp. 1310-1324, Jul. 2001.
J. C. Chen, R. E. Hudson, and K. Yao, "Maximum-likelihood source localization and
unknown sensor locationestimation for wideband signals in the near-field," IEEE
Transactions on Signal Processing, Volume 50, Issue 8, pp. 1843-1854, Aug. 2002.
H. Krim and M. Viberg, "Two decades of array signal processing research: the
parametric approach," IEEE Signal Processing Magazine, Volume 13, Issue 4, pp.
67-94, Jul. 1996.
G. M. Ljung and G. E. P. Box, "On a measure of lack of fit in time series models,"
Biometrika, Volume 65, Number 2, pp. 297-303, 1978.
104
Electrical Engineering. University of California, Los Angeles