tracking rectilinear sources in wireless communications
TRANSCRIPT
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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Outline § Part 1 : Circularity & Rectilinearity Tracker § Part 2 : Application to Wireless Communication Signals § Part 3 : Simulations § Part 4 : Conclusions & Further Work § Part 5 : Literature
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Part 1: Circularity & Rectilinearity Tracker
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Definitions
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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
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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
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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?
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What is the MMSE solution for the weight?
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wopt
= argmin E{|ek|2}w
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wopt
= argmin E{|ek|2}w
=E{s2k}
E{|sk|2}
Pseudo-Covariance
Covariance
What is the MMSE solution for the weight?
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wopt
= argmin E{|ek|2}w
=E{s2k}
E{|sk|2}Pseudo-Covariance
Covariance =
pscs
What is the MMSE solution for the weight?
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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]
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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
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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]
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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
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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| > �
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Part 2: Application to Wireless Communication Signals
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Measurement Model – Receiver
N x 1 Measurements From Receiver
Array
xk = ska+ nk
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Measurement Model – Receiver
N x 1 Measurements From Receiver
Array
xk = ska+ nk
Signal of Interest (SOI)
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Measurement Model – Receiver
N x 1 Measurements From Receiver
Array
xk = ska+ nk
Signal of Interest (SOI)
N x 1 Channel Vector
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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
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• 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
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Measurement Model – Receiver
xk =MX
`=1
s`,ka` + nb,k
N x 1 Measurements
Number of Sources
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“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
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Solution: Use Blind Source Separation
yk = Bkxk
Separate the Sources
[Chevalier et. al. ‘14]
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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]
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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]
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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]
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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]
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Proposed: Real Time Detection of Rectilinearity
xk...Bk
yk...
⇢M,k
⇢1,k
⇢2,k
Array Measurements
Blind Source Separation
Rectilinearity Tracking
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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
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Part 3: Simulations
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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)
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−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|
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Simulation Results
Circularity Estimates
Separated Sources
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−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
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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|
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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|
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Part 4: Conclusions
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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?
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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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Book
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Thank you