# Widely Linear Adaptive Filters

**Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models**. Read more and download the supplementary MATLAB toolbox.

**You can download from here some Wind Data used in some of the simulations in the work below. These are 2D (complex) data, for three different wind regimes of 'low', 'medium', and 'high' dynamics. **

**We often use the term 'augmented' for 'widely linear', e.g. augmented CLMS (ACLMS) instead of widely linear CLMS, referring to the use of Augmented Statistics. This also makes the naming of the algorithms more elegant, e.g. 'augmented nonlinear gradient descent' instead of 'widely linear nonlinear gradient descent' **

** Below is some of our work on augmented linear, nonlinear and Kalman adaptive filters **

**Legend**:
MATLAB code,
PDF files,
Supplements and data.

**Linear Augmented Complex Adaptive Filters (Widely Linear Adaptive Filters)**:

- Y. Xia, M. Xiang, Z. Li, and D. P. Mandic,
**"Echo state networks for multidimensional data: Exploiting noncircularity and widely linear models",**in D. Comminiello and J. Principe (editors), Adaptive Learning Methods for Nonlinear System Modelling, pp. 267-288, Elsevier, 2018. [pdf] - Z. Li, Y. Xia, W. Pei, and D. P. Mandic,
**"A cost-effective nonlinear self-interference canceller in full-duplex direct-conversion transceivers",**Signal Processing, vol. 158, no. 5, pp. 4-14, 2019. [pdf] - H. Cheng, Y. Xia, Y. Huang, L. Yang, and D. P. Mandic,
**"A normalized complex LMS based blind I/Q imbalance compensator for GFDM receivers and its full second order performance analysis",**IEEE Transactions on Signal Processing, vol. 66, no. 17, pp. 4701-4712, 2018. [pdf] - A. Stott, S. Kanna, and D. P. Mandic,
**"Widely linear complex partial least squares for latent subspace regression",**Signal Processing, vol. 152, pp. 350-362, 2018. [pdf] - Z. Li, Y. Xia, W. Pei, K. Wang, and D. P. Mandic,
**"An augmented nonlinear LMS for digital self-interference cancellation in full-duplex direct-conversion transceivers",**IEEE Transactions on Signal Processing, vol. 66, no. 15, pp. 4065-4078, 2018. [pdf] - Y. Xia and D. P. Mandic,
**"Augmented performance bounds on strictly linear and widely linear estimators with complex data",**IEEE Transactions on Signal Processing, vol. 66, no. 2, pp. 507-514, 2018. [pdf] - Y. Xia, S. C. Douglas, and D. P. Mandic,
**"Performance analysis of the deficient length augmented CLMS algorithm for second order noncircular complex signals",**Signal Processing, vol. 144, pp. 214-225, 2018. [pdf] - Z. Li, W. Pei, Y. Xia, K. Wang, and D. P. Mandic,
**"Widely Linear CLMS based cancelation of nonlinear self-interference in full-duplex direct-conversion transceivers",**Proceedings of ICASSP'18, pp. 4329--4333, 2018. [pdf] - Y. Xia, S. C. Douglas, and D. P. Mandic,
**"A perspective on CLMS as a deficient length augmented CLMS: Dealing with second order noncircularity",**Signal Processing}, vol. 149, pp. 236-245, 2018. [pdf] - B. Scalzo Dees, S. C. Douglas, and D. P. Mandic,
**"Complementary complex-valued spectrum for real-valued data: Real time estimation of the panorama through circularity-preserving DFT",**Proceedings of ICASSP'18, pp. 3999-4003, 2018. [pdf] - B. Scalzo Dees, Y. Xia, S. C. Douglas, and D. P. Mandic,
**"Correntropy-based adaptive filtering of noncircular complex data",**Proceedings of ICASSP'18, pp. 4339-4343, 2018. [pdf] - Y. Xia and D. P. Mandic,
**"A full mean square analysis of CLMS for second order noncircular inputs",**IEEE Transactions on Signal Processing, vol. 65, no. 21, pp. 5578-5590, 2017. [pdf] - Y. Xia and D. P. Mandic,
**"Complementary mean square analysis of Augmented CLMS for second order noncircular Gaussian signals",**IEEE Signal Processing Letters, vol. 24, no. 98, p. 1413-1417, 2017. [pdf] - Y. Xia, Y. He, K. Wang, W. Pei, Z. Blazic, and D. P. Mandic,
**"A complex least squares enhanced smart DFT technique for power system frequency estimation",**IEEE Transactions on Power Delivery, vol. 32, no. 3, pp. 1270-1278, 2017. [pdf] - S. Kanna and D. P. Mandic,
**"Self-stabilising adaptive three-phase transforms via widely linear modelling",**Electronics Letters, vol. 53, no. 13, pp. 875-877, 2017. [pdf] - Z. Li, Y. Xia, K. Wang, W. Pei, and D. P. Mandic,
**"Noncircular measurement and mitigation of I/Q imbalance in OFDM-based WLAN transmitters",**IEEE Transactions on Instrumentation and Measurement, vol. 66, no. 3, pp. 383-393, 2017. [pdf] - Y. Xia, L. Qiao, Q. Yang, W. Pei, and D. P. Mandic,
**"Widely linear adaptive frequency estimation for unbalanced three-phase power systems with multiple noisy measurements",**Proceedings of the 22nd International Conference on Digital Signal Processing (DSP'17), pp. 1-5, 2017. [pdf] - Z. Li, Y. Xia, W. Pei, K. Wang, and D. P. Mandic,
**"A DFT enhanced complex LMS for digital adaptive spur cancellation",**Proceedings of the 22nd International Conference on Digital Signal Processing (DSP'17), pp. 1-5, 2017. [pdf] - S. C. Douglas and D. P. Mandic,
**"Performance advantage of quaternion widely linear estimation: An approximate uncorrelating transform approach",**Proceedings of ICASSP'16, pp. 4935--4939, 2016. [pdf] - Y. Kim, J. Ryu, K. Kim, C. C. Took, D. P. Mandic, and C. Park,
**"Motor imagery classification using mu and beta rhythms of EEG with strong uncorrelating transform based complex common spatial patterns",**Computational Intelligence and Neuroscience, vol. 2016, article ID 1489692, 13 pages, doi:10.1155/2016/1489692, 2016. [pdf] - S. Kanna and D. P. Mandic,
**"Steady-state behaviour of general complex-valued diffusion LMS strategies",**IEEE Signal Processing Letters, vol. 23, no. 5, pp. 722-726, 2016. [pdf] - H. Zhang and D. P. Mandic,
**"Is a complex-valued stepsize advantageous in complex-valued gradient learning algorithms?",**IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 12, pp. 2730-2735, 2016. [pdf] - S. Kanna, D. Dini, Y. Xia, R. Hui, and D. P. Mandic,
**"Distributed widely linear Kalman filtering for frequency estimation in power networks",**IEEE Transactions on Signal and Information Processing over Networks, vol. 1, no. 1, pp. 45-57, 2015. [pdf] - D. Xu, H. Zhang, and D. P. Mandic,
**"Convergence analysis of an augmented algorithm for fully complex-valued neural networks",**Neural Networks, vol. 69, pp. 44-50, 2015. [pdf] - S. Kanna, M. Xiang, and D. P. Mandic,
**"Real-time detection of rectilinear sources for wireless communication signals",**Proc. of IEEE International Symposium on Wireless Communication Systems (ISWCS), pp. 541-545, 2015. [pdf] - D. P. Mandic, S. Kanna, and S. C. Douglas,
**"Mean square analysis of the CLMS and ACLMS for non-circular signals: The approximate uncorrelating transform approach",**Proceedings of ICASSP'15, pp. 3531-3535, 2015. [pdf] - C. Cheong Took, S. C. Douglas, and D. P. Mandic,
**"Maintaining the integrity of sources in complex learning systems: Intraference and the correlation preserving transform",**IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 3, pp. 500-509, 2015. [pdf] - C. Jahanchahi, S. Kanna, and D. P. Mandic,
**"Complex dual channel estimation: Cost effective widely linear adaptive filtering",**Signal Processing, vol. 104, pp. 33-42, 2014. [pdf] - S. Kanna, P. Talebi, and D. P. Mandic,
**"Diffusion widely linear adaptive estimation of system frequency in distributed power grids",**Proceedings of ENERGYCON'14, pp. 772-778, 2014. [pdf] - C. Cheong Took, S. C. Douglas, and D. P. Mandic,
**"The correlation preserving transform resolves phase ambiguity in complex decorrelating transforms",**Proceedings of the 8th IEEE Workshop on Sensor Array and Multichannel Signal Processing (SAM'14), pp. 73-76, 2014. [pdf] - C. Park, C. Cheong Took, and D. P. Mandic,
**"Augmented complex common spatial patterns for classification of noncircular EEG from motor imagery tasks",**IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 22, no. 1, pp. 1-10, 2014. [pdf] - Y. Xia and D. P. Mandic,
**"A widely linear least mean phase algorithm for adaptive frequency estimation of unbalanced power system",**International Journal of Electrical Power and Energy Systems, vol. 54, pp. 367-375, 2014. [pdf] - S. Kanna, S. C. Douglas, and D. P. Mandic,
**"A real time tracker of complex circularity",**Proceedings of the 8th IEEE Workshop on Sensor Array and Multichannel Signal Processing (SAM'14), pp. 129-132, 2014. [pdf] - S. C. Douglas and D. P. Mandic,
**"Autoconvolution and panorama: Augmenting second-order signal analysis",**Proceedings of ICASSP 2014, pp. 384-388, 2014. [pdf] - C. Cheong Took, S. C. Douglas, and D. P. Mandic,
**"On intraference and its implications in complex-valued signal processing",**Proceedings of ICASSP'13, pp. 3934-3937, 2013. [pdf] - D. Dini, P. Djuric, and D. P. Mandic,
**"The augmented complex particle filter",**IEEE Transactions on Signal Processing, vol. 61, no. 17, pp. 4341-4346, 2013. [pdf] - D. Dini and D. P. Mandic,
**"Exploring sparsity in widely linear estimation",**Proceedings of the IEEE International Symposium on Wireless Communication Systems, pp. 1-5, 2013. [pdf] - D. Dini and D. P. Mandic,
**"Widely linear modeling for frequency estimation in unbalanced three-phase power systems",**IEEE Transactions on Instrumentation and Measurement, vol. 62, no. 2, pp. 353-363, 2013. [pdf] - D. Dini, C. Jahanchahi, and D. P. Mandic,
**"Widely linear complex and quaternion valued bearings only tracking",**IET Signal Processing (invited), vol. 6, no. 5, pp. 435-445, 2012. [pdf] - S. C. Douglas and D. P. Mandic,
**"A family of least-squares magnitude phase algorithms",**Proceedings of ICASSP'12, 3769-3772, 2012. [pdf] - Y. Xia and D. P. Mandic,
**"Widely linear adaptive frequency estimation of unbalanced three-phase power systems",**Transactions on Instrumentation and Measurement, vol. 61, no. 1, pp. 74-83, 2012. [pdf] - D. Dini and D. P. Mandic,
**"Class of Widely Linear Kalman Filters",**IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 5, pp. 775-786, 2012. [pdf] - C. Cheong Took, S. C. Douglas, and D. P. Mandic,
**"On Approximate Diagonalization of Correlation Matrices in Widely Linear Signal Processing",**IEEE Transactions on Signal Processing, vol. 60, no. 3, pp. 1469-1473, 2012. [pdf] - B. Jelfs, D. P. Mandic, and S. C. Douglas
**"An adaptive approach for the identification of improper complex signals",**Signal Processing, vol. 92, pp. 335-344, 2012. [pdf] - D. Dini and D. P. Mandic,
**"An enhanced bearing estimation technique for DIFAR sonobuoy underwater target tracking",**Proceedings of the SSPD conference, pp. 1-4, 2012. [pdf] - D. Dini and D. P. Mandic,
**"Widely linear collaborative estimation",**Proceedings of the Asilomar Conference on Signals and Systems, pp. 1518-1522, 2012. [pdf] - Y. Xia, D. P. Mandic, and A. H. Sayed,
**"An Adaptive Diffusion Augmented CLMS Algorithm for Distributed Filtering of Noncircular Complex Signals",**IEEE Signal Processing Letters, vol. 18, no. 11, pp. 659-662, 2011. [pdf] - S. Javidi, D. P. Mandic, C. Cheong Took, and A. Cichocki,
**"Kurtosis Based Blind Source Extraction of Complex Noncircular Signals with Application in EEG Artifact Removal in Real Time",**Frontiers in Neuroscience, vol. 5, no. 105, pp. 118, 2011. [pdf] - S. Javidi, C. Cheong Took, C. Jahancahi, N. Le Bihan, and D. P. Mandic,
**"Blind Extraction of Improper Quaternion Sources,"**in Proceedings of ICASSP'11, pp. 3708-3711, 2011. [pdf] - S. C. Douglas and D. P. Mandic,
**"The least mean magnitude phase algorithm with applications to communications systems,"**in Proceedings of ICASSP'11}, pp. 4152-4155, 2011. [pdf] - D. Looney and D. P. Mandic
**"Augmented Complex Matrix Factorisation,"**in Proceedings of ICASSP'11, pp. 4072-4075, 2011. [pdf] - S. Javidi, D. P. Mandic, and A. Cichocki,
**"Complex Blind Source Extraction From Noisy Mixtures Using Second-Order Statistics,"**IEEE Transactions on Circuits and Systems I, vol. 57, no. 7, pp. 1404-1416, 2010. [pdf] - Y. Xia, C. Cheong-Took, and D. P. Mandic,
**"An Augmented Affine Projection Algorithm for the Filtering of Noncircular Complex Signals",**Signal Processing, vol. 90, pp. 1788-1799, 2010. [pdf] - S. Javidi, D. P. Mandic and A. Kuh,
**"Optimisation of Real Functions of Complex Matrices for the Adaptive Estimation of Complex Sources",**in Proceedings of the International Conference on Green Circuits and Systems (ICGCS), pp. 30-35, 2010. [pdf] - D. P. Mandic, Y. Xia, and S. C. Douglas,
**"Steady State Analysis of CLMS and Augmented CLMS Algorithms for Noncircular Complex Signals",**Proceedings of the 44th Asilomar Conference on Signals, Systems, and Computers, pp. 1635-1639, 2010. [pdf] - S. C. Douglas and D. P. Mandic,
**"Performance Analysis of the Conventional Complex LMS and Augmented Complex LMS Algorithms",**in Proceedings of IEEE ICASSP'10, pp. 3794-3797, 2010. [pdf] - Y. Xia, S. Javidi, and D. P. Mandic,
**"A Regularized Normalized Augmented Complex Least Mean Square Algorithm",**Proceedings of the Internation Symposium on Wireless Communication Systems, pp. 355-359, 2010. [pdf] - S. Javidi and D. P. Mandic,
**"A Fast Algorithm for Blind Extraction of Smooth Complex Sources with Applications in EEG Conditioning",**Proceedings of the IEEE MLSP Workshop, pp. 397-402, 2010. [pdf] - D. P. Mandic, S. Javidi, S. L. Goh, A. Kuh and K. Aihara,
**"Complex Valued Prediction of Wind Profile Using Augmented Complex Statistics",**Renewable Energy, vol. 34, no. 1, pp. 196-201, 2009. [pdf] - D. P. Mandic, S. Still, and S. C. Douglas,
**"Duality between widely linear and dual channel adaptive filtering",**in Proceedings of ICASSP'09, pp. 1729-1732, 2009. [pdf] - S. C. Douglas and D. P. Mandic
**"Mean and Mean Square Analysis of the Complex LMS Algorithm for Non-Circular Gaussian Signals,"**Proceedings of the Digital Signal Processing Workshop and the 5th IEEE Signal Processing Education Workshop, pp. 101-106, 2009. [pdf] - C. Cheong-Took and D. P. Mandic,
**"Adaptive IIR Filtering of Noncircular Complex Signals",**IEEE Transactions on Signal Processing, vol. 57, no. 10, pp. 4111-4118, 2009. [pdf] - S. Javidi, M. Pedzisz, S. L. Goh, and D. P. Mandic
**"The Augmented Complex Least Mean Square Algorithm with Application to Adaptive Prediction Problems,"**Proceedings of the Cognitive Information Processing Conference, pp. 54-57, 2008. [pdf] - B. Jelfs, Y. Xia, D. P. Mandic, and K. Aihara,
**"Collaborative Adaptive Filtering in the Complex Domain",**Proceedings of ICASSP, pp. 421-424, 2008. [pdf] - D. P. Mandic, S. Javidi, G. Souretis, and V. S. L. Goh,
**"Why a Complex Valued Solution for a Real Domain Problem",**in Proceedings of the IEEE Workshop on Machine Learning for Signal Processing (MLSP07) 384389, 2007. [pdf]

**Widely Linear (Augmented) Complex Kalman Filters, Detection of Noncircularity and Signal Modality**:

- S. Kanna, D. Dini, Y. Xia, R. Hui, and D. P. Mandic,
**"Distributed widely linear Kalman filtering for frequency estimation in power networks",**IEEE Transactions on Signal and Information Processing over Networks, vol. 1, no. 1, pp. 45-57, 2015. [pdf] - D. Dini and D. P. Mandic,
**"Class of Widely Linear Kalman Filters",**IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 5, pp. 775-786, 2012. [pdf] - D. Dini, D. P. Mandic, and S. Julier,
**"A Widely Linear Complex Unscented Kalman Filter",**IEEE Signal Processing Letters, vol. 18, no. 11, pp. 623-626, 2011. [pdf] - D. H. Dini and D. P. Mandic,
**"Analysis of the Widely Linear Complex Kalman Filter",**Proceedings of the Sensor Signal Processing for Defense Conference, pp. 1-4, 2010. [pdf] - S. Javidi, D. P. Mandic and A. Kuh,
**"Optimisation of Real Functions of Complex Matrices for the Adaptive Estimation of Complex Sources",**in Proceedings of the International Conference on Green Circuits and Systems (ICGCS), pp. 30-35, 2010. [pdf] - B. Che Ujang, C. Cheong-Took, and D. P. Mandic,
**"Identification of improper processes by variable taplength complex-valued adaptive filters",**in Proceedings of IJCNN, pp. 2265-2270, 2010. [pdf] - D. P. Mandic, P. Vayanos, M. Chen, and S. L. Goh,
**"Online Detection of the Modality of Complex Valued Real World Signals,"**International Journal of Neural Systems, vol. 18, no. 2, pp. 67-74, 2008. [pdf] - S. L. Goh and D. P. Mandic,
**"An Augmented Extended Kalman Filter Algorithm for Complex-Valued Recurrent Neural Networks,"**Neural Computation, vol. 19, no. 4, pp. 1039-1055, 2007. [pdf] - T. Gautama, D. P. Mandic, and M. M. Van Hulle
**"A non-parametric test for detecting the complex-valued nature of time series",**International Journal of Knowledge-based and Intelligent Engineering Systems, vol. 8, pp. 99-106, 2004. [pdf]

**Nonlinear Widely Linear (Augmented) Complex Adaptive Filters**:

- Y. Xia, B. Jelfs, M. M. Van Hulle, J. Principe, and D. P. Mandic,
**"An Augmented Echo State Network for Adaptive Filtering of Complex Noncircular Signals",**IEEE Transactions on Neural Networks, vol. 22, no. 1, pp. 74-83, 2011. [pdf] - D. P. Mandic, P. Vayanos, S. Javidi, B. Jelfs, and K. Aihara,
**"Online Tracking of the Degree of Nonlinearity within Complex Signals",**in Proceedings of ICASSP08, pp. 20612065, 2008. [pdf] - S. L. Goh and D. P. Mandic,
**"An Augmented CRTRL For Complex-Valued Recurrent Neural Networks",**Neural Networks, vol. 20, no. 10, pp. 1061-1066, 2007. [pdf] - S. L. Goh and D. P. Mandic,
**"A General Complex RTRL Algorithm for Nonlinear Adaptive Filters",**Neural Computation, vol. 16, no. 12, pp. 2699-2713, 2004. [pdf]

**Some Applications (mostly wind prediction and power system/smart grid) and Other**:

- Y. Xia and D. P. Mandic
**"Widely Linear Adaptive Frequency Estimation of Unbalanced Three-Phase Power Systems,"**IEEE Transactions on Instrumentation and Measurement, vol. 61, no. 1, pp. 74-83, 2012. [pdf] - D. P. Mandic, S. Javidi, S. L. Goh, A. Kuh and K. Aihara,
**"Complex Valued Prediction of Wind Profile Using Augmented Complex Statistics",**Renewable Energy, vol. 34, no. 1, pp. 196-201, 2009. [pdf] - A. Kuh and D. P. Mandic,
**"Applications of Complex Augmented Kernels to Wind Profile Prediction,"**Proceedings of ICASSP 2009, pp. 3581-3584, 2009. [pdf] - S. L. Goh, M. Chen, D. H. Popovic, K. Aihara, D. Obradovic and D. P. Mandic,
**"Complex-Valued Forecasting of Wind Profile,"**Renewable Energy, vol. 31, pp. 1733-1750, 2006. [pdf]