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Tracking Rectilinear Sources in Wireless Communications

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Sithan Kanna
Sithan Kanna

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/

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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
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{s2 k}cs ps ?s def = cs ps Pseudo-Covariance Circularity Quotient & Coef?cient [Ollila ｀08] |?s| def = | |ps cs If r. v. is Rectilinear: |?s| = 1
ISWCS 2015 Can we estimate the conjugate of a complex variable from the variable itself? 5 Estimate of the conjugate Original random variable Linear Coef?cient sk?s? k = w? ek = s? k ?s? k
ISWCS 2015 6 Estimate of the conjugate Estimation Error sk?s? 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? 7 wopt = argmin E{|ek|2 } w
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?
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?
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]
ISWCS 2015 11 Idea: We can use an adaptive filter to track the circularity. ??k+1 = ??k + ?e? ksk ??ksk ?s? k ( )? X s? k ek [Kanna, Douglas & Mandic ｀14] CLMS Step-size
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]
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 = ?cs 1 |?s|2 2 |?s|2 2 ?cs (2 + |?s|2) lim k!1 E{|?s ??k|2 } Steady State Misadujstment Inversely Proportional to Circ. Coef?cient
ISWCS 2015 14 Proposed Rectlinearity Detector At each time instant: ?? Track Circularity Quotient at Each time Instant: ?? Compute circularity coef?cient: ?? Compare coef?cient with threshold to detect rectilinearity: |??k| ??k Rectilinear Signal ?|??k| >
ISWCS 2015 Part 2: Application to Wireless Communication Signals 15
ISWCS 2015 16 Measurement Model C Receiver N x 1 Measurements From Receiver Array xk = ska + nk
ISWCS 2015 17 Measurement Model C Receiver N x 1 Measurements From Receiver Array xk = ska + nk Signal of Interest (SOI)
ISWCS 2015 18 Measurement Model C Receiver N x 1 Measurements From Receiver Array xk = ska + nk Signal of Interest (SOI) N x 1 ChannelVector
ISWCS 2015 19 Measurement Model C 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
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 C Receiver
ISWCS 2015 21 Measurement Model C 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)yH k 1 + yH k g(yk) Bk yk = Bkxk Separate the Sources Update the De-mixing matrix Modi?ed EASI Algorithm [Cardoso & Laheld ｀96] [Li & Adali ｀10]
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 Modi?ed EASI Algorithm [Cardoso & Laheld ｀96] [Li & Adali ｀10]
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]
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]
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 30
ISWCS 2015 31 Simulation Set-Up ?? 4Transmitters ?? Sources: QPSK (non-rectilinear) or BPSK (rectilinear) ?? Type of modulation changes after ?rst 500 samples ?? 4 Receivers ?? Receiving a mixture of these signals ?? DOA = {C 45＜, 8＜, C 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 1000 0 0.5 0.9 1 Circ. Coefficient ? Source 1 Sample || 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 1000 0 0.5 0.9 1 Circ. Coefficient ? Source 2 Sample || 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 1000 0 0.5 0.9 1 Circ. Coefficient ? Source 3 Sample ||
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 ||
ISWCS 2015 Part 4: Conclusions 36
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?
ISWCS 2015 Part 5: Literature 38
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. 129C132. 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. 269C281, 2014. 3.? J.-F. Cardoso and B. Laheld, ＾Equivariant adaptive source separation,￣ IEEE Trans. on Signal Process., vol. 44, no. 12, pp. 3017C 3030, Dec 1996. 39
ISWCS 2015 Book 40
ISWCS 2015 41 Thank you

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Tracking Rectilinear Sources in Wireless Communications

  • 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 & Coef?cient [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 Coef?cient sk?s? k = w? ek = s? k ?s? k
  • 6. ISWCS 2015 6 Estimate of the conjugate Estimation Error sk?s? k = w? ek = s? k ?s? 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. ??k+1 = ??k + ?e? ksk ??ksk ?s? 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 = ?cs 1 |?s|2 2 |?s|2 2 ?cs (2 + |?s|2) lim k!1 E{|?s ??k|2 } Steady State Misadujstment Inversely Proportional to Circ. Coef?cient
  • 14. ISWCS 2015 14 Proposed Rectlinearity Detector At each time instant: ?? Track Circularity Quotient at Each time Instant: ?? Compute circularity coef?cient: ?? Compare coef?cient with threshold to detect rectilinearity: |??k| ??k Rectilinear Signal ?|??k| >
  • 15. ISWCS 2015 Part 2: Application to Wireless Communication Signals 15
  • 16. ISWCS 2015 16 Measurement Model C Receiver N x 1 Measurements From Receiver Array xk = ska + nk
  • 17. ISWCS 2015 17 Measurement Model C Receiver N x 1 Measurements From Receiver Array xk = ska + nk Signal of Interest (SOI)
  • 18. ISWCS 2015 18 Measurement Model C 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 C 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 C Receiver
  • 21. ISWCS 2015 21 Measurement Model C 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 Modi?ed 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 Modi?ed 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 ... ??M,k ??1,k ??2,k Array Measurements Blind Source Separation Rectilinearity Tracking
  • 29. 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 ?
  • 31. ISWCS 2015 31 Simulation Set-Up ?? 4Transmitters ?? Sources: QPSK (non-rectilinear) or BPSK (rectilinear) ?? Type of modulation changes after ?rst 500 samples ?? 4 Receivers ?? Receiving a mixture of these signals ?? DOA = {C 45＜, 8＜, C 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 ||
  • 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?
  • 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. 129C132. 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. 269C281, 2014. 3.? J.-F. Cardoso and B. Laheld, ＾Equivariant adaptive source separation,￣ IEEE Trans. on Signal Process., vol. 44, no. 12, pp. 3017C 3030, Dec 1996. 39