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ISWCS 2015
Real-Time Detection of Rectilinear Sources for Wireless Communication Signals Sithan Kanna [email protected] Min Xiang [email protected] Danilo P. Mandic [email protected]
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ISWCS 2015
Outline § Part 1 : Circularity & Rectilinearity Tracker § Part 2 : Application to Wireless Communication Signals § Part 3 : Simulations § Part 4 : Conclusions & Further Work § Part 5 : Literature
2
ISWCS 2015
Part 1: Circularity & Rectilinearity Tracker
3
ISWCS 2015
Definitions
4
Covariance Consider a zero-mean random variable sk
def= E{|sk|2}
def= E{s2k}cs
ps⇢sdef= cs
ps
Pseudo-Covariance
Circularity Quotient & Coefficient [Ollila ‘08]
|⇢s|def=
| |pscs
If r. v. is Rectilinear : |⇢s| = 1
ISWCS 2015
Can we estimate the conjugate of a complex variable from the variable itself?
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Estimate of the conjugate
Original random variable
Linear Coefficient
sks⇤k = w⇤
ek = s⇤k� s⇤k
ISWCS 2015 6
Estimate of the conjugate
Estimation Error
sks⇤k = w⇤
ek = s⇤k� s⇤k
“True” conjugate
Can we estimate the conjugate of a complex variable from the variable itself?
ISWCS 2015
What is the MMSE solution for the weight?
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wopt
= argmin E{|ek|2}w
ISWCS 2015 8
wopt
= argmin E{|ek|2}w
=E{s2k}
E{|sk|2}
Pseudo-Covariance
Covariance
What is the MMSE solution for the weight?
ISWCS 2015 9
wopt
= argmin E{|ek|2}w
=E{s2k}
E{|sk|2}Pseudo-Covariance
Covariance =
pscs
What is the MMSE solution for the weight?
ISWCS 2015 10
wopt
= argmin E{|ek|2}w
=E{s2k}
E{|sk|2}
Circularity Quotient !!!
=pscs
= ⇢s
What is the MMSE solution for the weight? [Kanna, Douglas & Mandic ‘14]
ISWCS 2015 11
Idea: We can use an adaptive filter to track the circularity.
⇢k+1 = ⇢k + µe⇤ksk
⇢ksks⇤k
�
( )⇤
X
s⇤k
ek
[Kanna, Douglas & Mandic ‘14]
CLMS
Step-size
ISWCS 2015 12
0 500 1000 1500 2000 2500 30000
0.5
1
Real Part of the Circularity Quotient
Sample, k
EstimatedTrue
0 500 1000 1500 2000 2500 3000−0.5
0
0.5
1
Sample, k
Imaginary Part of the Circularity Quotient
EstimatedTrue
LMS based circularity tracker tracking the Circularity Quotient of a non-circular white Gaussian noise process
Idea: We can use an adaptive filter to track the circularity. [Kanna, Douglas & Mandic ‘14]
ISWCS 2015 13
Can we exploit the statistical properties of the Circularity Tracker?
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.002
0.004
0.006
0.008
0.01
Circularity Coefficient, |l|
Mis
adju
stm
ent
Simulation Theory
= µcs
�1� |⇢s|2
� �2� |⇢s|2
�
2� µcs (2 + |⇢s|2)limk!1
E{|⇢s � ⇢k|2}
Steady State Misadujstment
Inversely Proportional to Circ. Coefficient
ISWCS 2015 14
Proposed Rectlinearity Detector
At each time instant: • Track Circularity Quotient at Each time Instant: • Compute circularity coefficient: • Compare coefficient with threshold to detect rectilinearity:
|⇢k|
⇢k
Rectilinear Signal |⇢k| > �
ISWCS 2015
Part 2: Application to Wireless Communication Signals
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ISWCS 2015 16
Measurement Model – Receiver
N x 1 Measurements From Receiver
Array
xk = ska+ nk
ISWCS 2015 17
Measurement Model – Receiver
N x 1 Measurements From Receiver
Array
xk = ska+ nk
Signal of Interest (SOI)
ISWCS 2015 18
Measurement Model – Receiver
N x 1 Measurements From Receiver
Array
xk = ska+ nk
Signal of Interest (SOI)
N x 1 Channel Vector
ISWCS 2015 19
Measurement Model – Receiver
N x 1 Measurements From Receiver
Array
xk = ska+ nk
Signal of Interest (SOI)
N x 1 Channel Vector
N x 1 Total Noise Vector:
Interference + Background Noise
ISWCS 2015 20
• To reveal type of Modulation e.g. BPSK vs QPSK
• To choose type of receiver e.g. Widely Linear vs Strictly Linear
• Useful in Adaptive Modulation Schemes
Why? [Chevalier et. al. ‘14]
xk = ska+ nk
Goal: Track + Detect Rectilinearity of Source
Measurement Model – Receiver
ISWCS 2015 21
Measurement Model – Receiver
xk =MX
`=1
s`,ka` + nb,k
N x 1 Measurements
Number of Sources
ISWCS 2015 22
“Problem”: Multiple sources
xk =MX
`=1
s`,ka` + nb,k
N x 1 Measurements
Number of Sources
[Chevalier et. al. ‘14]
• Conventional Case:
• Rectilinear Sources:
M N
M > N
ISWCS 2015 23
Solution: Use Blind Source Separation
yk = Bkxk
Separate the Sources
[Chevalier et. al. ‘14]
ISWCS 2015 24
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk + �I� g(yk)y
Hk
1 + ���yH
k g(yk)��Bk
yk = Bkxk
Separate the Sources
Update the De-mixing matrix Modified EASI Algorithm
[Cardoso & Laheld ‘96] [Li & Adali ‘10]
ISWCS 2015 25
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk + �I� g(yk)y
Hk
1 + ���yH
k g(yk)��Bk
yk = Bkxk
Separate the Sources
Update the De-mixing matrix
N x 1 Measurements
M x N De-mixing Matrix
M x 1 Separated Sources
Modified EASI Algorithm
[Cardoso & Laheld ‘96] [Li & Adali ‘10]
ISWCS 2015 26
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk + �I� g(yk)y
Hk
1 + ���yH
k g(yk)��Bk
yk = Bkxk
Separate the Sources
Update the De-mixing matrix
Step-size
Non-linearity
[Cardoso & Laheld ‘96] [Li & Adali ‘10]
ISWCS 2015 27
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk + �I� g(yk)y
Hk
1 + ���yH
k g(yk)��Bk
yk = Bkxk
Separate the Sources
Update the De-mixing matrix
Step-size
Non-linearity : gi(yi) = yi|yi|2
[Cardoso & Laheld ‘96] [Li & Adali ‘10]
ISWCS 2015 28
Proposed: Real Time Detection of Rectilinearity
xk...Bk
yk...
⇢M,k
⇢1,k
⇢2,k
Array Measurements
Blind Source Separation
Rectilinearity Tracking
ISWCS 2015 29
Proposed: Real Time Detection of Rectilinearity
xk...Bk
yk...
⇢M,k
⇢1,k
⇢2,k
Schreier et. al. ‘06 Ollila et. al. ‘09 Walden et. al. ‘09 Delmas et. al. ‘10 Hellings et. al. ‘15
Cardoso et. al. ‘96 Cichocki et al. ‘02 Li et. al. ‘10
ISWCS 2015
Part 3: Simulations
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ISWCS 2015 31
Simulation Set-Up
• 4 Transmitters • Sources: QPSK (non-rectilinear) or BPSK (rectilinear) • Type of modulation changes after first 500 samples
• 4 Receivers • Receiving a mixture of these signals • DOA = {– 45°, 8°, – 13°, 30°} • Corrupted by circular WGN, 10 dB (SNR)
ISWCS 2015
−2 0 2−2
−1
0
1
2
Real
Imag
Source 1
−2 0 2−2
−1
0
1
2
Real
Imag
Source 1
0 250 500 750 10000
0.5
0.91
Circ. Coefficient − Source 1
Sample
|l|
32
Simulation Results
Circularity Estimates
Separated Sources
ISWCS 2015 33
−2 0 2−2
−1
0
1
2
Real
Imag
Source 2
−2 0 2−2
−1
0
1
2
Real
Imag
Source 2
0 250 500 750 10000
0.5
0.91
Circ. Coefficient − Source 2
Sample
|l|
Simulation Results
Circularity Estimates
Separated Sources
ISWCS 2015 34
Simulation Results
Circularity Estimates
Separated Sources
−2 0 2−2
−1
0
1
2
Real
Imag
Source 3
−2 0 2−2
−1
0
1
2
Real
Imag
Source 3
0 250 500 750 10000
0.5
0.91
Circ. Coefficient − Source 3
Sample
|l|
ISWCS 2015 35
Simulation Results
Circularity Estimates
Separated Sources
−2 0 2−2
−1
0
1
2
Real
Imag
Source 4
−2 0 2−2
−1
0
1
2
Real
Imag
Source 4
0 250 500 750 10000
0.5
0.91
Circ. Coefficient − Source 4
Sample
|l|
ISWCS 2015
Part 4: Conclusions
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ISWCS 2015
Quick Recap
§ Part 1 : Principles of the Circularity Tracker § Exploit the Variance Result
§ Part 2 : MIMO application § Adaptive BSS + Online Circularity Tracker
§ Part 3 : Simulations
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Future Work § What about M > N? § Does the additional complexity justify the benefit?
§ Can we perhaps only run it in certain intervals?
ISWCS 2015
Part 5: Literature
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ISWCS 2015
Selected Literature 1. S. Kanna, S. Douglas, and D. Mandic, “A real time tracker of
complex circularity,” in Proc. of the 8th IEEE Sensor Array and Multichannel Signal Process. Workshop (SAM), June 2014, pp. 129–132.
2. P. Chevalier, J. P. Delmas, and A. Oukaci, “Properties, performance
and practical interest of the widely linear MMSE beamformer for nonrectilinear signals,” Signal Processing, vol. 97, pp. 269–281, 2014.
3. J.-F. Cardoso and B. Laheld, “Equivariant adaptive source separation,” IEEE Trans. on Signal Process., vol. 44, no. 12, pp. 3017–3030, Dec 1996.
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ISWCS 2015
Book
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ISWCS 2015 41
Thank you