Presentation given at the 12th International Symposium on Wireless Communication Systems (ISWCS) 2015 Special Session on Non-Circularity and Widely Linear Filtering in Radiocommunications http://www.iswcs2015.org/

1. ISWCS 2015
Real-Time Detection of
Rectilinear Sources for Wireless
Communication Signals
Sithan Kanna ssk08@ic.ac.uk
Min Xiang m.xiang13@ic.ac.uk
Danilo P. Mandic d.mandic@ic.ac.uk
1

2. 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

3. ISWCS 2015
Part 1:
Circularity & Rectilinearity
Tracker
3

4. ISWCS 2015
Definitions
4
Covariance
Consider a zero-mean random variable
sk
def
= E{|sk|2
}
def
= E{s2
k}cs
ps
⇢s
def
= cs
ps
Pseudo-Covariance
Circularity Quotient & Coefficient [Ollila ‘08]
|⇢s|
def
=
| |ps
cs
If r. v. is Rectilinear:
|⇢s| = 1

5. ISWCS 2015
Can we estimate the conjugate of a complex variable from
the variable itself?
5
Estimate of
the conjugate
Original random
variable
Linear Coefficient
skˆs⇤
k = w⇤
ek = s⇤
k ˆs⇤
k

6. ISWCS 2015 6
Estimate of
the conjugate
Estimation
Error
skˆs⇤
k = w⇤
ek = s⇤
k ˆs⇤
k
“True” conjugate
Can we estimate the conjugate of a complex variable from
the variable itself?

7. ISWCS 2015
What is the MMSE solution for the weight?
7
wopt = argmin E{|ek|2
}
w

8. ISWCS 2015 8
wopt = argmin E{|ek|2
}
w
=
E{s2
k}
E{|sk|2}
Pseudo-
Covariance
Covariance
What is the MMSE solution for the weight?

9. ISWCS 2015 9
wopt = argmin E{|ek|2
}
w
=
E{s2
k}
E{|sk|2}
Pseudo-
Covariance
Covariance
=
ps
cs
What is the MMSE solution for the weight?

10. ISWCS 2015 10
wopt = argmin E{|ek|2
}
w
=
E{s2
k}
E{|sk|2}
Circularity
Quotient !!!
=
ps
cs
= ⇢s
What is the MMSE solution for the weight?
[Kanna, Douglas
& Mandic ‘14]

11. ISWCS 2015 11
Idea: We can use an adaptive filter to track the circularity.
ˆ⇢k+1 = ˆ⇢k + µe⇤
ksk
ˆ⇢ksk
ˆs⇤
k
( )⇤
X
s⇤
k
ek
[Kanna, Douglas
& Mandic ‘14]
CLMS
Step-size

12. ISWCS 2015 12
0 500 1000 1500 2000 2500 3000
0
0.5
1
Real Part of the Circularity Quotient
Sample, k
Estimated
True
0 500 1000 1500 2000 2500 3000
−0.5
0
0.5
1
Sample, k
Imaginary Part of the Circularity Quotient
Estimated
True
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]

13. 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 1
0
0.002
0.004
0.006
0.008
0.01
Circularity Coefficient, | |
Misadjustment
Simulation
Theory
= µcs
1 |⇢s|2
2 |⇢s|2
2 µcs (2 + |⇢s|2)
lim
k!1
E{|⇢s ˆ⇢k|2
}
Steady State
Misadujstment
Inversely Proportional
to Circ. Coefficient

14. 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:
|ˆ⇢k|
ˆ⇢k
Rectilinear Signal
|ˆ⇢k| >

15. ISWCS 2015
Part 2:
Application to Wireless
Communication Signals
15

16. ISWCS 2015 16
Measurement Model – Receiver
N x 1
Measurements
From Receiver
Array
xk = ska + nk

17. ISWCS 2015 17
Measurement Model – Receiver
N x 1
Measurements
From Receiver
Array
xk = ska + nk
Signal of Interest
(SOI)

18. ISWCS 2015 18
Measurement Model – Receiver
N x 1
Measurements
From Receiver
Array
xk = ska + nk
Signal of Interest
(SOI)
N x 1
ChannelVector

19. ISWCS 2015 19
Measurement Model – Receiver
N x 1
Measurements
From Receiver
Array
xk = ska + nk
Signal of Interest
(SOI)
N x 1
ChannelVector
N x 1
Total NoiseVector:
Interference +
Background Noise

20. 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

21. ISWCS 2015 21
Measurement Model – Receiver
xk =
MX
`=1
s`,ka` + nb,k
N x 1
Measurements
Number of
Sources

22. 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

23. ISWCS 2015 23
Solution: Use Blind Source Separation
yk = Bkxk
Separate the Sources
[Chevalier et. al. ‘14]

24. ISWCS 2015 24
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk +
I g(yk)yH
k
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]

25. ISWCS 2015 25
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk +
I g(yk)yH
k
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]

26. ISWCS 2015 26
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk +
I g(yk)yH
k
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]

27. ISWCS 2015 27
Solution: Use Adaptive Blind Source Separation
Bk+1 = Bk +
I g(yk)yH
k
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]

28. ISWCS 2015 28
Proposed: Real Time Detection of Rectilinearity
xk...
Bk
yk
...
ˆ⇢M,k
ˆ⇢1,k
ˆ⇢2,k
Array
Measurements
Blind Source
Separation
Rectilinearity
Tracking

29. ISWCS 2015 29
Proposed: Real Time Detection of Rectilinearity
xk...
Bk
yk
...
ˆ⇢M,k
ˆ⇢1,k
ˆ⇢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

30. ISWCS 2015
Part 3:
Simulations
30

31. ISWCS 2015 31
Simulation Set-Up
• 4Transmitters
• 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)

32. 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 1000
0
0.5
0.9
1
Circ. Coefficient − Source 1
Sample
||
32
Simulation Results
Circularity
Estimates
Separated
Sources

33. 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 1000
0
0.5
0.9
1
Circ. Coefficient − Source 2
Sample
||
Simulation Results
Circularity
Estimates
Separated
Sources

34. 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 1000
0
0.5
0.9
1
Circ. Coefficient − Source 3
Sample
||

35. 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 1000
0
0.5
0.9
1
Circ. Coefficient − Source 4
Sample
||

36. ISWCS 2015
Part 4:
Conclusions
36

37. ISWCS 2015
Quick Recap
§ Part 1 : Principles of the Circularity Tracker
§ Exploit theVariance Result
§ Part 2 : MIMO application
§ Adaptive BSS + Online Circularity Tracker
§ Part 3 : Simulations
37
Future Work
§ What about M > N?
§ Does the additional complexity justify the benefit?
§ Can we perhaps only run it in certain intervals?

38. ISWCS 2015
Part 5:
Literature
38

39. 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.
39

40. ISWCS 2015
Book
40

41. ISWCS 2015 41
Thank you