Analysis of incremental augmented affine projection algorithm for distributed estimation of complex-valued signals

This paper considers the problem of distributed estimation in an incremental network when the measurements taken by the node follow a widely linear model. The proposed algorithm which we refer to it as incremental augmented affine projection algorithm (incAAPA) utilizes the full second order statistical information in the complex domain. Moreover, it exploits spatio-temporal diversity to improve the estimation performance. We derive steady-state performance metric of the incAAPA in terms of the mean-square deviation (MSD). We further derive sufficient conditions to ensure mean-square convergence. Our analysis illustrate that the proposed algorithm is able to process both second order circular (proper) and noncircular (improper) signals. The validity of the theoretical results and the good performance of the proposed algorithm are demonstrated by several computer simulations

Collaborative adaptive filtering for machine learning

Quantitative performance criteria for the analysis of machine learning architectures and algorithms have long been established. However, qualitative performance criteria, which identify fundamental signal properties and ensure any processing preserves the desired properties, are still emerging. In many cases, whilst offline statistical tests exist such as assessment of nonlinearity or stochasticity, online tests which not only characterise but also track changes in the nature of the signal are lacking. To that end, by employing recent developments in signal characterisation, criteria are derived for the assessment of the changes in the nature of the processed signal. Through the fusion of the outputs of adaptive filters a single collaborative hybrid filter is produced. By tracking the dynamics of the mixing parameter of this filter, rather than the actual filter performance, a clear indication as to the current nature of the signal is given. Implementations of the proposed method show that it is possible to quantify the degree of nonlinearity within both real- and complex-valued data. This is then extended (in the real domain) from dealing with nonlinearity in general, to a more specific example, namely sparsity. Extensions of adaptive filters from the real to the complex domain are non-trivial and the differences between the statistics in the real and complex domains need to be taken into account. In terms of signal characteristics, nonlinearity can be both split- and fully-complex and complex-valued data can be considered circular or noncircular. Furthermore, 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. To produce online tests for sparsity, adaptive filters for sparse environments are investigated and a unifying framework for the derivation of proportionate normalised least mean square (PNLMS) algorithms is presented. This is then extended to derive variants with an adaptive step-size. In order to create an online test for noncircularity, a study of widely linear autoregressive modelling is presented, from which a proof of the convergence of the test for noncircularity can be given. Applications of this method are illustrated on examples such as biomedical signals, speech and wind data

Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing

Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the non-commutative nature of the quaternion product has been prohibitive to the development of quaternion uncorrelating transforms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance and complementary covariance matrices in the quaternion domain, whereby the quaternion version of the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.Comment: 41 pages, single column, 10 figure

Variants of partial update augmented CLMS algorithm and their performance analysis

Naturally complex-valued information or those presented in complex domain are effectively processed by an augmented complex least-mean-square (ACLMS) algorithm. In some applications, the ACLMS algorithm may be too computationally and memory-intensive to implement. In this paper, a new algorithm, termed partial-update ACLMS (PU-ACLMS) algorithm is proposed, where only a fraction of the coefficient set is selected to update at each iteration. Doing so, two types of partial update schemes are presented referred to as the sequential and stochastic partial-updates, to reduce computational load and power consumption in the corresponding adaptive filter. The computational cost for full-update PU-ACLMS and its partial update implementations are discussed. Next, the steady-state mean and mean-square performance of PU-ACLMS for noncircular complex signals are analyzed and closed-form expressions of the steady-state excess mean-square error (EMSE) and mean-square deviation (MSD) are given. Then, employing the weighted energy-conservation relation, the EMSE and MSD learning curves are derived. The simulation results are verified and compared with those of theoretical predictions through numerical examples

Study of L0-norm constraint normalized subband adaptive filtering algorithm

Limited by fixed step-size and sparsity penalty factor, the conventional sparsity-aware normalized subband adaptive filtering (NSAF) type algorithms suffer from trade-off requirements of high filtering accurateness and quicker convergence behavior. To deal with this problem, this paper proposes variable step-size L0-norm constraint NSAF algorithms (VSS-L0-NSAFs) for sparse system identification. We first analyze mean-square-deviation (MSD) statistics behavior of the L0-NSAF algorithm innovatively in according to a novel recursion form and arrive at corresponding expressions for the cases that background noise variance is available and unavailable, where correlation degree of system input is indicated by scaling parameter r. Based on derivations, we develop an effective variable step-size scheme through minimizing the upper bounds of the MSD under some reasonable assumptions and lemma. To realize performance improvement, an effective reset strategy is incorporated into presented algorithms to tackle with non-stationary situations. Finally, numerical simulations corroborate that the proposed algorithms achieve better performance in terms of estimation accurateness and tracking capability in comparison with existing related algorithms in sparse system identification and adaptive echo cancellation circumstances.Comment: 15 pages,15 figure