Soroush Javidi

Fast Independent Component Analysis Algorithm for Quaternion Valued Signals

An extension of the fast independent component analysis algorithm is proposed for the blind separation of both \BBQ-proper and \BBQ-improper quaternion-valued signals. This is achieved by maximizing a negentropy-based cost function, and is derived rigorously using the recently developed \mbi\BBH\BBR calculus in order to implement Newton optimization in the augmented quaternion statistics framework. It is shown that the use of augmented statistics and the associated widely linear modeling provides theoretical and practical advantages when dealing with general quaternion signals with noncircular (rotation-dependent) distributions. Simulations using both benchmark and real-world quaternion-valued signals support the approach.

Complex Blind Source Extraction From Noisy Mixtures Using Second-Order Statistics

A class of second-order complex domain blind source extraction algorithms is introduced to cater for signals with noncircular probability distributions, which is a typical case in real-world scenarios. This is achieved by employing the so-called augmented complex statistics and based on the temporal structures of the sources, thus permitting widely linear (WL) predictability to be the extraction criterion. For rigor, the analysis of the existence and uniqueness of the solution is provided based on both the covariance and the pseudocovariance and for both noise-free and noisy cases, and serves as a platform for the derivation of the algorithms. Both direct solutions and those requiring prewhitening are provided based on a WL predictor, thus making the methodology suitable for the generality of complex signals (both circular and noncircular). Simulations on synthetic noncircular sources support the uniqueness and convergence study, followed by a real-world example of electrooculogram artifact removal from electroencephalogram recordings in real time.

Why a Complex Valued Solution for a Real Domain Problem

An insight into the potential benefits of using complex valued models for real valued data is provided. The problem itself is not new; it is however timely and important to revisit this issue, due to a plethora of modern applications based on multidimensional and multichannel measurements which can be cast into an equivalent problem in the field of complex numbers C. The analysis and simulations highlight the duality between several classes of real domain problems and their complex valued representations. This is supported by case studies on image processing, modelling of point processes for brain prosthetics, and forecasting of vector fields.

Kurtosis-Based Blind Source Extraction of Complex Non-Circular Signals with Application in EEG Artifact Removal in Real-Time

A new class of complex domain blind source extraction algorithms suitable for the extraction of both circular and non-circular complex signals is proposed. This is achieved through sequential extraction based on the degree of kurtosis and in the presence of non-circular measurement noise. The existence and uniqueness analysis of the solution is followed by a study of fast converging variants of the algorithm. The performance is first assessed through simulations on well understood benchmark signals, followed by a case study on real-time artifact removal from EEG signals, verified using both qualitative and quantitative metrics. The results illustrate the power of the proposed approach in real-time blind extraction of general complex-valued sources.

Blind extraction of improper quaternion sources

Blind extraction of quaternion-valued latent sources is addressed based on their local temporal properties. The extraction criterion is based on the minimum mean square widely linear prediction error, thus allowing for the extraction of both proper and improper quaternion sources. The use of the widely linear adaptive predictor is justified by the relationship between the mean square prediction error and the crosscorrelation and cross-pseudocorrelations of the source signals. Simulations on benchmark improper quaternion sources together with a real-world example of EEG artifact removal illustrate the usefulness of the proposed methodology.

Characterisation of Signal Modality: Exploiting Signal Nonlinearity in Machine Learning and Signal Processing

A novel method for online tracking of the changes in the nonlinearity within both real-domain and complex–valued signals is introduced. This is achieved by a collaborative adaptive signal processing approach based on a hybrid filter. By tracking the dynamics of the adaptive mixing parameter within the employed hybrid filtering architecture, we show that it is possible to quantify the degree of nonlinearity within both real- and complex-valued data. Implementations for tracking nonlinearity in general and then more specifically sparsity are illustrated on both benchmark and real world data. It is also shown that by combining the information obtained from hybrid filters of different natures it is possible to use this method to gain a more complete understanding of the nature of the nonlinearity within a signal. This also paves the way for building multidimensional feature spaces and their application in data/information fusion.

A Regularised Normalised Augmented Complex Least Mean Square algorithm

A Regularised Normalised Augmented Complex Least Mean Square (RNACLMS) algorithm is proposed for widely linear adaptive filtering in the complex domain C. Based on augmented complex statistics, the RNACLMS is shown to utilise complete second order information in C, thus being suitable to deal with both circular and noncircular complex signals. Furthermore, a gradient adaptive regularisation term makes the proposed algorithm exhibit enhanced robustness and convergence over the NACLMS algorithm. Simulations on circular and noncircular benchmark signals and on real-world noncircular wind signals support the analysis.

Online tracking of the degree of nonlinearity within complex signals

A novel method for online tracking of the changes in the non- linearity within complex-valued signals is introduced. This is achieved by a collaborative adaptive signal processing approach by means of a hybrid filter. By tracking the dynamics of the adaptive mixing parameter within the employed hybrid filtering architecture, we show that it is possible to quantify the degree of nonlinearity within complex-valued data. Simulations on both benchmark and real world data support the approach.

Blind extraction of noncircular complex signals using a widely linear predictor

Real valued blind source extraction based on a linear predictor is extended to the complex domain using recent advances in complex domain statistics. It is shown that, in general, the mean square prediction error of the algorithm depends both on the covariance matrix and the pseudo-covariance matrix of the source signals. To fully utilise the available information, it is thus natural to adopt a widely linear predictor to extract the latent sources from the observed mixture. This way, we derive a new algorithm for the extraction of general complex signals and provide simulation results using benchmark complex data.

A widely linear affine projection algorithm

A widely linear affine projection algorithm (WL-APA) utilising the full second order statistical information in the complex domain ℂ is proposed. It is based on recent developments in augmented complex statistics, ℂℝ calculus, and the widely linear modelling in ℂ, making it suitable for the processing of noncircular complex valued signals. Simulations on circular and noncircular benchmark and real-world noncircular wind signals show the effectiveness of WL-APA as compared to standard APA.