adaptive multitaper spectral detector for wideband...
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Adaptive Multitaper Spectral Detector for Wideband Spectrum Sensing in Strong Interference
EnvironmentStudent: Jung-Mao Lin
Advisor: Danijela Cabric and Hsi-Pin Ma
March 12, 2010
Jung-Mao Lin
Outlines
• Introduction• System Design
– Wideband Sensing– Narrowband Sensing
• Simulation Results• Future Works• Conclusions
Jung-Mao Lin
Introduction
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• Spectrum sensing – Matched-filter detection [1], [2]
• Resistant to noise but prior knowledge of primary user (PU) is required
– Energy detection [3], [4]• Simplest method but vulnerable to noise
– Autocorrelation- and cyclostationary-based detection [5]−[7]
• Without knowing noise level but periodic or cyclic features are required in the transmitted signal
Motivation (1/2)
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Motivation (2/2)
-2005 Haykin (Guidelines & Concepts) [10]
-2007 Erpek, Leu, and Mark (TV bands) [11]
-1982 Thomson (Multitapter Spectral Estimation, MTSE)
[9]
-1978 Slepian (Discrete Prolate Spheroidal Sequence,
DPSS) [8]
?
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Discrete Prolate Spheroidal Sequence (DPSS)N=256, NW=4
1st order
3rd order
2nd order
4th order
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Strong Interference in Wideband Environment
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System Design
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System Block Diagram
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Adaptive Multitaper Spectral Estimation (AMTSE)
∑
∑
=
== K
tt
K
t
mttt
fd
fSfdfS
1
2
1
)(2
)(
)(ˆ)()(ˆ
{ })()(ˆ)(ˆ
)(fBfS
fSfd
kk
kt E+
=λ
λ
21
0
2)( ][][)(ˆ ∑−
=
−⋅⋅=N
n
nfjt
mtt enhnxfS π
( )∑−
=
−1
0
2 1][N
nknx λ 1
1
42
1
2 )()(2−
==⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡= ∑∑
K
tt
K
tt fdfdν
DPSS windows
Input time sequence
Degree of Freedom (DoF)
∑=
=K
t
mtt
ini fSK
fS1
)()( )(ˆ1)(ˆ
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Wideband Locally Most Powerful Detector (LMPD) (1/6)
• Let Y1, Y2, … , YN be a stationary m-dependent sequence with finite variance
where
Then,
( )[ ]ilkmt
i
N
iiN fSYYS +
−
=
==∑ ,)(
1
0
ˆln ,
( ) 2,,
)( )(~ˆ
νχνlk
lkmt fS
fS
Sampled PSD
Chi-Square
( )2 ,~ σμ NNS YN Ν
( )[ ]{ } ( )[ ] ( ) ( )
( )[ ] ( )ν
νψνμ
gfS
fSfS
k
kkmt
Y
+=
⎟⎠⎞
⎜⎝⎛++−==
ln 2
2lnlnlnˆlnE )(
Digamma Function
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20
201
200
2 22 mσσσσ ++≈
{ } { }( ) ( ) ( ) ( )
( ) ( ){ } ( ) ( )[ ] ( )νρννχχ
μχν
χν
μσ
νν
νν
gg
fSfS
YYYYCov
iill
ilk
lk
illilli
Y
Y
′−=+−⋅=
−⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡+⎟⎠⎞
⎜⎝⎛
⎥⎦
⎤⎢⎣
⎡+⎟⎠⎞
⎜⎝⎛=
−⋅==
+
+
++
22,
2,
22,
2,
220
lnlnlnE
lnlnlnlnE
E,
( )[ ]{ } ( ) ( ) ⎟⎠⎞
⎜⎝⎛′=
⎭⎬⎫
⎩⎨⎧
+⎟⎠⎞
⎜⎝⎛==
2lnlnVarˆlnVar 2
,)(2
00νψχ
νσ ν l
kk
mt fSfS
TrigammaFunction
( )[ ] ( )[ ]νρνρνψσσσσ gg mm ′−+′−+⎟
⎠⎞
⎜⎝⎛′=++≈ 22
222 12
0201
200
2
Wideband Locally Most Powerful Detector (LMPD) (2/6)
If for simplicity, assume mρρρρ ≈≈= 21
( )[ ] νηρνρνψσ ,2 22
2 mmgm +=′−+⎟⎠⎞
⎜⎝⎛′≈⇒
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• Binary hypothesis test :
• Log Likelihood Ratio Test (LLRT) :
Wideband Locally Most Powerful Detector (LMPD) (3/6)
( ) ( ) ( )
( ) ( ) ( ) 2,
)(1
2,
)(0
)(~ˆ ; :
)(~ˆ ; :
ν
ν
χν
χν
kRxlk
mtkRxk
knlk
mtknk
fSfSfSfSH
fSfSfSfSH
=
=
( ) ( )( )
( )[ ]( )[ ]⎩
⎨⎧
====
=⇒kRxkRx
knkn
N
NNG SfS
SfSHSpHSpSL
,1
,0
00
11
lnln
,;,;
θθ
ξθθ
( ) [ ] ( )
( )[ ] ( ) ( )[ ]( ){ }( )ν
ν ηρνηρπ
μσ
πσθ
,
2
,
22
2
22ln22ln
21
212ln
21,;ln
mi
kiNmi
iNi
iiiN
mNfSgNSm
NSN
NHSp
++−
−+−≈
−−−=⇒
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• LMP Test
Wideband Locally Most Powerful Detector (LMPD) (4/6)
( ) ( ) ( ) ( ) ( )
00
2
2
001
LMP;lnE , ;ln
θθθθ θθθθ
θθ
=
−
=⎥⎦
⎤⎢⎣
⎡∂
∂−=⋅
∂∂
= NNN
SpIISpST
( ) ( )[ ]νηρ
ν
,2;ln
m
kN
k
kN
mSgNS
SSSp
++−
=∂
∂
( )νηρ ,
2
2
2;ln
mk
kN
mN
SSSp
+−
=∂
∂
( ) ( )[ ]N
mm
SgNSST m
m
knNN
ν
ν
ηρηρ
ν ,
,
,LMP
22
++++−
=⇒
Test Statistic
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Fisher Information Matrix
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Wideband Locally Most Powerful Detector (LMPD) (5/6)
• Assume (Asymptotically Performance)∞→N
( ) ( )( )( )( )
, 1 ,IN , 1 0,N
~1010
0LMP
⎩⎨⎧
−⇒
HH
ST N θθθ
( ){ } ( ) ( )FA1
0LMPFA Pr PQQHSTP N−=⇒=>=⇒ ξξξ
( ){ } ( )( )
( ) ( )[ ] ( )[ ]( )
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=>=⇒
−
−
SNR1ln2
lnln2
1I
Pr
,FA
1
,FA
1
0101LMPD
ν
ν
ηρ
ηρ
θθθξξ
m
knkRxm
N
mNPQQ
fSfSm
NPQQ
QHSTP
( )( ) SNR1 Assume +=
kn
kRx
fSfS
( )ννψη ν gmm ′−⎟⎠⎞
⎜⎝⎛′= 2
2, 15
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• Detection performance
• Minimum required observations
( )( )[ ]
( )⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+′−+⎟
⎠⎞
⎜⎝⎛′
−= − SNR1ln2
2
FA1
D
νρνψ gm
NPQQP
Monotonically Decreasing
Monotonically Decreasing
( )[ ] ( ) ( )[ ]( )[ ]2
2D
1FA
1
SNR1ln2
2 +−
⎭⎬⎫
⎩⎨⎧ ′−+⎟
⎠⎞
⎜⎝⎛′=
−− PQPQgmN νρνψ
Wideband Locally Most Powerful Detector (LMPD) (6/6)
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NW=3, K=1~5FFT=1024,
BW=500 MHz( ) ( ){ }( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )( )
21
0
22
2
21
0
222
22
2*
,0,
assume and ,0 Since
,
*
ˆ,ˆcov
∑
∑
∫
−
=
−
−
=
−
=≡
+≈=
≡=
≈
⋅−+⋅−≈
+
N
t
tjt
N
t
tjt
(mt)(mt)
ehfRfRR
fSfSfSfR
fRehfS
HHfS
duuSufHufH
fSfS
πη
πη
ηη
η
η
η
η
η
Decimation in Frequency
Narrowband Sensing (1/4)
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• Setand assume all the are independent and identically distributed (i.i.d.)
• Assume TLRT is Gaussian distributed according to Central Limit Theorem
Test Statistic
Narrowband Sensing (2/4)
( )( ) ( )[ ] ξ ˆln fˆ1
0
)()(LRT ∑
−
=
=N
kk
mtmt fST S
( )( ) ( )( )
, ,N , ,N
~fˆ1
211
02
0)(LRT
0
⎩⎨⎧
⇒HH
T mt
σμσμ
S
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Jung-Mao Lin
Narrowband Sensing (3/4)
( )( ){ } ( ) ( ) ( )[ ]∑−
=
+⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++−==
1
00
)(LRT0 ln
22lnln ;fˆ E
N
kkn
mt fSNHT νψνμ S
( )( ){ } ( ){ } ⎟⎠⎞
⎜⎝⎛′⋅=== ∑
−
= 2lnV ;fˆ V
1
0
20
)(LRT
20
νψχσ ν NarHTarN
k
mtS
( )( ){ } ( ) ( ) ( )[ ]∑−
=
+⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++−==
1
01
)(LRT1 ln
22lnln ;fˆ E
N
kkRx
mt fSNHT νψνμ S
( )( ){ } ( ){ } ⎟⎠⎞
⎜⎝⎛′⋅=== ∑
−
= 2lnV ;fˆ V
1
0
21
)(LRT
21
νψχσ ν NarHTarN
k
mtS
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Jung-Mao Lin
( )[ ]( ) ( )[ ]
( )[ ]22
D1
FA1
2 SNR1ln2
SNR1ln2
+
⎟⎠⎞
⎜⎝⎛′
∝−+
⎟⎠⎞
⎜⎝⎛′
= −−
νψνψPQPQN
• Detection performance
• Minimum required observations
( ) ( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+⋅⋅
⎟⎠⎞
⎜⎝⎛′
−= − SNR1ln
2
1FA
1D NPQQP
νψ
Narrowband Sensing (4/4)
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• The noise uncertainty model [12]
• Detection performance
• Minimum required observations
Noise Uncertainty
SNR Wall
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Simulation Results
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Narrowband DetectionPU = WGN
N=8, SNR=0dB PFA=10-3
[5][4]
[4]
[5]
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Wideband DetectionPU = QPSK, Fs=64MHz, FFT=128
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability of False Alarm (PFA)
Pro
babi
lity
of D
etec
tion
(PD
)
AMTSE Det. (theo.), NW=3, K=2AMTSE Det. (sim.), NW=3, K=2AMTSE Det. (theo.), NW=3, K=3AMTSE Det. (sim.), NW=3, K=3
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability of False Alarm (PFA)
Pro
babi
lity
of D
etec
tion
(PD
)
AMTSE Det. (theo.), NW=3, K=2AMTSE Det. (sim.), NW=3, K=2AMTSE Det. (theo.), NW=3, K=3AMTSE Det. (sim.), NW=3, K=3
N=16, SNR=0dB, BW=1M, m=1
N=16, SNR=0dB, BW=2M, m=1
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Minimum Required Observations (W/ Noise Uncertainty)
-40 -35 -30 -25 -20 -15 -10 -5 010
0
102
104
106
108
1010
1012
1014
1016
SNR (dB)
Min
imum
Req
uire
d S
ampl
e P
oint
s (lo
g10
N)
Energy.(anal.)Prop.(anal.), K=2Prop.(anal.), K=4Prop.(anal.), K=2, noise uncertaintyProp.(anal.), K=4, noise uncertaintyEnergy.(anal.), noise uncertainty
x=1dB
x=0.001dB
x=0.1dB
PD=PFA=10-3
Jung-Mao Lin
Future Works
• Fine sensing– Mixer, low pass filter, and AMTSE narrowband detector– Strong interference suppression
• Whole wideband sensing system construction– Two stages detection– Performance evaluation
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• A wideband AMTSE detector based on LMP test and a narrowband AMTSE detector, including mathematical formulation and numerical analysis, is presented.
• By providing a higher order degree of freedom (DoF) from multitaper gain, the provided AMTSE detector is more reliable in detection performance or has less required observations compared to others.
• By taking advantage of multitaper estimation, the provided detector can achieve a non-parametric detection without the prior knowledge of PU.
Conclusions
Jung-Mao Lin
References
[1] D. Middleton, “On the detection of stochastic signals in additive normal noise -Part I,” IEEE Trans. Inf. Theory, vol. 3, pp. 86–121, June 1957.
[2] T. Yucek and H. Arslan, “Spectrum characterization for opportunistic cognitiveradio systems,” in Proc. IEEE MILCOM ’06, Washington, DC, Oct. 2006, pp.1–6.
[3] T. Ikuma and M. Naraghi-Pour, “A comparison of three classes of spectrumsensing techniques,” in Proc. IEEE GLOBECOM ’08, New Orleans, LO, Nov.2008, pp. 1–5.
[4] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory.Englewood Cliffs, NJ: Prentice-Hall PTR, 1998.
Jung-Mao Lin
[5] S. Chaudhari, V. Koivunen, and H. V. Poor, “Distributed autocorrelation-basedsequential detection of OFDM signals in cognitive radios,” in Proc. IEEECrownCom ’07, Singapore, May 2008, pp. 1–6.
[6] W. Gardner, “Signal interception: A unifying theoretical framework for featuredetection,” IEEE Trans. Commun., vol. 36, pp. 897–906, Aug. 1988.
[7] J. Lunden, V. Koivnen, A. Huttunen, and H. V. Poor, “CollaborativeCyclostationary Spectrum Sensing for Cognitive Radio Systems,” IEEE Trans.Signal Process., vol. 57, no. 11, pp. 4182–4195, Dec. 1992.
[8] D. Slepian, “Prolate spheroidal wave functions, Fourier analysis anduncertainty – V: The discrete case,” Bell Syst. Tech. J., vol. 57, pp. 1371–1429, 1978.
[9] D. J. Thomson, “Spectrum estimation and harmonic analysis,” Proc. IEEE,vol. 20, pp. 1055–1096, Sep. 1982.
[10] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,”IEEE J. Sel. Areas Commun., vol. 23, no. 3, pp. 201–205, Feb. 2005.
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[11] T. Erpek, A. Leu, and B. L. Mark, “Spectrum sensing performance in TV bands using the multitaper method,” in Proc. IEEE 15th Signal Processing and Communication Applications Conf., Eskisehir, Turkey, Jun. 2007, pp. 1–4.
[12] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 4–17, Feb. 2008.
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Appendix
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Noise Uncertainty
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