# Collaborative Adaptive Filtering Approach for the Identification of Complex-Valued Improper Signals

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## Abstract

This paper proposes a novel hybrid filter for data-adaptive optimal identification and modeling of complex-valued real-world signals based on the convex combination approach. It is equipped with different complex domain characteristics of subfilter algorithm. The proposed hybrid filter takes advantage of the complex nonlinear gradient descent (CNGD) algorithm that exhibits fast convergence and the steady state of the augmented complex nonlinear gradient descent (ACNGD) algorithm. The output of CNGD and ACNGD was combined to work in parallel, feeding each individual subfilter output into a mixing algorithm, which in the end produced a single hybrid filter output. The mixing parameter \( \lambda \left( k \right) \) within the hybrid filter architecture was made gradient adaptive in order to preserve the nature of inherent characteristics of the subfilters and to show its optimal performance in identifying and tracking second-order properness (circular) and improperness (noncircular) of the complex signals in real time. Further analysis was made on the properties of the algorithms, and the relationship between fast convergence and steady-state error was discussed. This analysis is supported by the complex-valued synthetic simulation and real-world application dataset as applied in renewable energy (wind).

## Keywords

Widely nonlinear modeling Augmented complex statistics Augmented (CNGD) Nonlinear systems Collaborative filters Wind modeling## Notes

### Funding

Funding was provided by Fundamental Research Grant Scheme. Ministry of Higher Education Malaysia (Grant No. 08-01-13-1178FR).

## References

- 1.C.C. Amadi, B.C. Ujang, F. Hashim,
*Nonlinear System Modelling Utilizing Second Order Augmented Statistics Complex Value Algorithm*, in Advances in Machine Learning and Signal Processing. Vol. 387 of the series Lecture note in electrical engineering (Springer, 2016), pp. 223–235Google Scholar - 2.L.A. Azpicueta-Ruiz, J. Arenas-García, M.T.M. Silva, R. Candido, Chapter 11—Combined Filtering Architectures for Complex Nonlinear Systems, in Adaptive Learning Methods for Nonlinear System Modeling, ed. by D. Comminiello, J.C. Príncipe (Butterworth-Heinemann, Oxford, 2018), pp. 243–264Google Scholar
- 3.S.C. Douglas, D.P. Mandic, Mean and mean-square analysis of the complex LMS algorithm for non-circular Gaussian signals, in Proceedings of the IEEE Digital Signal Processing Workshop (2009), pp. 101–106Google Scholar
- 4.S.C. Douglas, W. Pan, Exact expectation analysis of the LMS adaptive filter. IEEE Trans. Signal Process.
**43**(12), 2863–2871 (1995)CrossRefGoogle Scholar - 5.S.L. Goh, M. Chen, D. Popovic, K. Aihara, D. Obradovic, D.P. Mandic, Complex-valued forecasting of wind profile. Renew. Energy
**31**(11), 1733–1750 (2006)CrossRefGoogle Scholar - 6.A.I. Hanna, D.P. Mandic, A fully adaptive normalized nonlinear gradient descent algorithm for complex-valued nonlinear adaptive filters. IEEE Trans. Signal Process.
**51**(10), 2540–2549 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 7.M.H. Hayes,
*Statistical Digital Signal Processing and Modeling*(Wiley, New York, 1996)Google Scholar - 8.S. Haykin,
*Adaptive Filter Theory*, fourth edn. (Prentice-Hall, Englewood Cliffs, 2002)zbMATHGoogle Scholar - 9.S. Javidi, M. Pedzisz, S.L. Goh, D.P. Mandic, The augmented complex least mean square algorithm with application to adaptive prediction problems, in Proceedings of the IAPR Workshop on Cognitive Information Processing (2008), pp. 54–57Google Scholar
- 10.C. Jahanchahi, S. Kanna, D.P. Mandic, Complex dual channel estimation: cost effective widely linear adaptive filtering. Signal Process.
**104**, 33–42 (2014)CrossRefGoogle Scholar - 11.B. Jelfs, S. Javidi, P. Vayanos, D.P. Mandic, Characterization of signal modality: exploiting signal nonlinearity in machine learning and signal processing. J. Signal Process. Syst.
**61**(1), 105–115 (2010)CrossRefGoogle Scholar - 12.B. Jelfs, D.P. Mandic, S.C. Douglas, Adaptive approach for the identification of improper complex signal. Signal Process.
**92**(2), 335–344 (2012)CrossRefGoogle Scholar - 13.B. Jelfs, Y. Xia, D.P. Mandic, S.C. Douglas, Collaborative adaptive filtering in the complex domain, in Proceedings IEEE International Workshop on Machine Learning for Signal Processing (2008), pp. 421–425Google Scholar
- 14.T. Kim, T. Adali, Approximation by fully complex multilayer perceptron. Neural Comput.
**15**(7), 1641–1666 (2003)CrossRefzbMATHGoogle Scholar - 15.L. Lu, H. Zhao, Z. He, B. Chen, A novel sign adaptation scheme for convex combination of two adaptive filters. AEU–Int. J. Electron. Commun.
**69**(11), 1590–1598 (2015)Google Scholar - 16.D.P. Mandic, S.L. Goh,
*Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models*(Wiley, New York, 2009)CrossRefGoogle Scholar - 17.D.P. Mandic, S. Javidi, G. Souretis, S.L. Goh, Why a complex valued solution for a real domain problem, in Proceedings IEEE International Workshop on Machine Learning for Signal Processing (2007), pp. 384–389Google Scholar
- 18.D.P. Mandic, P. Vayanos, M. Chen, S.L. Goh, Online detection of the modality of complex-valued real-world signals. Int. J. Neural Syst.
**18**(2), 67–74 (2008)CrossRefGoogle Scholar - 19.D.P. Mandic, P. Vayanos, C. Boukis, B. Jelfs, S.L. Goh, T. Gautama, T. Rutkowski, Collaborative adaptive learning using hybrid filters, in Proceeding IEEE International Conference on Acoustics, Speech and Signal Processing ICASSP, vol. 3 (2007), pp. 921–924Google Scholar
- 20.F. Neeser, J. Massey, Proper complex random processes with application to information theory. IEEE Trans. Inf. Theory
**39**(4), 1293–1302 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 21.T. Nitta, An extension of the back-propagation algorithm to complex number. Neural Netw.
**10**(8), 1391–1415 (1997)CrossRefGoogle Scholar - 22.E. Ollila, On the circularity of a complex random variable. IEEE Signal Process. Lett.
**15**, 841–844 (2008)CrossRefGoogle Scholar - 23.B. Picinbono,
*Random Signals and Systems*(Prentice-Hall, Englewood Cliffs, 1993)zbMATHGoogle Scholar - 24.B. Picinbono, On circularity. IEEE Trans. Signal Process.
**42**(12), 3473–3482 (1994)CrossRefGoogle Scholar - 25.A.H. Sayed,
*Fundamentals of Adaptive Filtering*(Wiley, New york, 2003)Google Scholar - 26.P.J. Schreier, L.L. Scharf,
*Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals*(Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar - 27.P.J. Schreier, L.L. Scharf, Second-order analysis of improper complex random vectors and processes. IEEE Trans. Signal Process.
**51**(3), 714–725 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 28.B.C. Ujang, C.C. Took, D.P. Mandic, Adaptive convex combination approach for the identification of improper quaternion processes. IEEE Trans. Neural Netw. Learn. Syst.
**25**(1), 172–182 (2014)CrossRefGoogle Scholar - 29.B. Widrow, S. Stearns,
*Adaptive Signal Processing*(Prentice-Hall, Englewood Cliffs, 1985)zbMATHGoogle Scholar - 30.B. Widrow, J. McCool, M. Ball, The complex LMS algorithm. Proc. IEEE
**63**(4), 719–720 (1975)CrossRefGoogle Scholar - 31.Wind Data: http://www.commsp.ee.ic.ac.uk/~mandic/research/wind.htm (2016). Accessed 26 May 2016
- 32.Y. Xia, B. Jelfs, M.M.V. Hulle, J.C. Principe, D.P. Mandic, An augmented echo state network for nonlinear adaptive filtering of complex noncircular signals. IEEE Trans. Neural Netw.
**22**(1), 74–83 (2011)CrossRefGoogle Scholar