Tracking Rectilinear Sources in Wireless Communications

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

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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

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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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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