Widely Linear vs. Conventional Subspace-Based Estimation of SIMO Flat-Fading Channels: Mean-Squared Error Analysis

We analyze the mean-squared error (MSE) performance of widely linear (WL) and conventional subspace-based channel estimation for single-input multiple-output (SIMO) flat-fading channels employing binary phase-shift-keying (BPSK) modulation when the covariance matrix is estimated using a finite number of samples. The conventional estimator suffers from a phase ambiguity that reduces to a sign ambiguity for the WL estimator. We derive closed-form expressions for the MSE of the two estimators under four different ambiguity resolution scenarios. The first scenario is optimal resolution, which minimizes the Euclidean distance between the channel estimate and the actual channel. The second scenario assumes that a randomly chosen coefficient of the actual channel is known and the third assumes that the one with the largest magnitude is known. The fourth scenario is the more realistic case where pilot symbols are used to resolve the ambiguities. Our work demonstrates that there is a strong relationship between the accuracy of ambiguity resolution and the relative performance of WL and conventional subspace-based estimators, and shows that the less information available about the actual channel for ambiguity resolution, or the lower the accuracy of this information, the higher the performance gap in favor of the WL estimator.Comment: 20 pages, 7 figure

Distributed adaptive signal processing for frequency estimation

It is widely recognised that future smart grids will heavily rely upon intelligent communication and signal processing as enabling technologies for their operation. Traditional tools for power system analysis, which have been built from a circuit theory perspective, are a good match for balanced system conditions. However, the unprecedented changes that are imposed by smart grid requirements, are pushing the limits of these old paradigms. To this end, we provide new signal processing perspectives to address some fundamental operations in power systems such as frequency estimation, regulation and fault detection. Firstly, motivated by our finding that any excursion from nominal power system conditions results in a degree of non-circularity in the measured variables, we cast the frequency estimation problem into a distributed estimation framework for noncircular complex random variables. Next, we derive the required next generation widely linear, frequency estimators which incorporate the so-called augmented data statistics and cater for the noncircularity and a widely linear nature of system functions. Uniquely, we also show that by virtue of augmented complex statistics, it is possible to treat frequency tracking and fault detection in a unified way. To address the ever shortening time-scales in future frequency regulation tasks, the developed distributed widely linear frequency estimators are equipped with the ability to compensate for the fewer available temporal voltage data by exploiting spatial diversity in wide area measurements. This contribution is further supported by new physically meaningful theoretical results on the statistical behavior of distributed adaptive filters. Our approach avoids the current restrictive assumptions routinely employed to simplify the analysis by making use of the collaborative learning strategies of distributed agents. The efficacy of the proposed distributed frequency estimators over standard strictly linear and stand-alone algorithms is illustrated in case studies over synthetic and real-world three-phase measurements. An overarching theme in this thesis is the elucidation of underlying commonalities between different methodologies employed in classical power engineering and signal processing. By revisiting fundamental power system ideas within the framework of augmented complex statistics, we provide a physically meaningful signal processing perspective of three-phase transforms and reveal their intimate connections with spatial discrete Fourier transform (DFT), optimal dimensionality reduction and frequency demodulation techniques. Moreover, under the widely linear framework, we also show that the two most widely used frequency estimators in the power grid are in fact special cases of frequency demodulation techniques. Finally, revisiting classic estimation problems in power engineering through the lens of non-circular complex estimation has made it possible to develop a new self-stabilising adaptive three-phase transformation which enables algorithms designed for balanced operating conditions to be straightforwardly implemented in a variety of real-world unbalanced operating conditions. This thesis therefore aims to help bridge the gap between signal processing and power communities by providing power system designers with advanced estimation algorithms and modern physically meaningful interpretations of key power engineering paradigms in order to match the dynamic and decentralised nature of the smart grid.Open Acces

The Augmented Synthetic Control Method

The synthetic control method (SCM) is a popular approach for estimating the impact of a treatment on a single unit in panel data settings. The "synthetic control" is a weighted average of control units that balances the treated unit's pre-treatment outcomes as closely as possible. A critical feature of the original proposal is to use SCM only when the fit on pre-treatment outcomes is excellent. We propose Augmented SCM as an extension of SCM to settings where such pre-treatment fit is infeasible. Analogous to bias correction for inexact matching, Augmented SCM uses an outcome model to estimate the bias due to imperfect pre-treatment fit and then de-biases the original SCM estimate. Our main proposal, which uses ridge regression as the outcome model, directly controls pre-treatment fit while minimizing extrapolation from the convex hull. This estimator can also be expressed as a solution to a modified synthetic controls problem that allows negative weights on some donor units. We bound the estimation error of this approach under different data generating processes, including a linear factor model, and show how regularization helps to avoid over-fitting to noise. We demonstrate gains from Augmented SCM with extensive simulation studies and apply this framework to estimate the impact of the 2012 Kansas tax cuts on economic growth. We implement the proposed method in the new augsynth R package

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

Widely Linear State Space Filtering of Improper Complex Signals

Complex signals are the backbone of many modern applications, such as power systems, communication systems, biomedical sciences and military technologies. However, standard complex valued signal processing approaches are suited to only a subset of complex signals known as proper, and are inadequate of the generality of complex signals, as they do not fully exploit the available information. This is mainly due to the inherent blindness of the algorithms to the complete second order statistics of the signals, or due to under-modelling of the underlying system. The aim of this thesis is to provide enhanced complex valued, state space based, signal processing solutions for the generality of complex signals and systems. This is achieved based on the recent advances in the so called augmented complex statistics and widely linear modelling, which have brought to light the limitations of conventional statistical complex signal processing approaches. Exploiting these developments, we propose a class of widely linear adaptive state space estimation techniques, which provide a unified framework and enhanced performance for the generality of complex signals, compared with conventional approaches. These include the linear and nonlinear Kalman and particle filters, whereby it is shown that catering for the complete second order information and system models leads to significant performance gains. The proposed techniques are also extended to the case of cooperative distributed estimation, where nodes in a network collaborate locally to estimate signals, under a framework that caters for general complex signals, as well as the cross-correlations between observation noises, unlike earlier solutions. The analysis of the algorithms are supported by numerous case studies, including frequency estimation in three phase power systems, DIFAR sonobuoy underwater target tracking, and real-world wind modeling and prediction.Open Acces

Capturing the zero: a new class of zero-augmented distributions and multiplicative error processes

We propose a novel approach to model serially dependent positive-valued variables which realize a non-trivial proportion of zero outcomes. This is a typical phenomenon in financial time series observed on high frequencies, such as cumulated trading volumes or the time between potentially simultaneously occurring market events. We introduce a flexible pointmass mixture distribution and develop a semiparametric specification test explicitly tailored for such distributions. Moreover, we propose a new type of multiplicative error model (MEM) based on a zero-augmented distribution, which incorporates an autoregressive binary choice component and thus captures the (potentially different) dynamics of both zero occurrences and of strictly positive realizations. Applying the proposed model to high-frequency cumulated trading volumes of liquid NYSE stocks, we show that the model captures both the dynamic and distribution properties of the data very well and is able to correctly predict future distributions. Keywords: High-frequency Data , Point-mass Mixture , Multiplicative Error Model , Excess Zeros , Semiparametric Specification Test , Market Microstructure JEL Classification: C22, C25, C14, C16, C5

Statistical signal processing of nonstationary tensor-valued data

Real-world signals, such as the evolution of three-dimensional vector fields over time, can exhibit highly structured probabilistic interactions across their multiple constitutive dimensions. This calls for analysis tools capable of directly capturing the inherent multi-way couplings present in such data. Yet, current analyses typically employ multivariate matrix models and their associated linear algebras which are agnostic to the global data structure and can only describe local linear pairwise relationships between data entries. To address this issue, this thesis uses the property of linear separability -- a notion intrinsic to multi-dimensional data structures called tensors -- as a linchpin to consider the probabilistic, statistical and spectral separability under one umbrella. This helps to both enhance physical meaning in the analysis and reduce the dimensionality of tensor-valued problems. We first introduce a new identifiable probability distribution which appropriately models the interactions between random tensors, whereby linear relationships are considered between tensor fibres as opposed to between individual entries as in standard matrix analysis. Unlike existing models, the proposed tensor probability distribution formulation is shown to yield a unique maximum likelihood estimator which is demonstrated to be statistically efficient. Both matrices and vectors are lower-order tensors, and this gives us a unique opportunity to consider some matrix signal processing models under the more powerful framework of multilinear tensor algebra. By introducing a model for the joint distribution of multiple random tensors, it is also possible to treat random tensor regression analyses and subspace methods within a unified separability framework. Practical utility of the proposed analysis is demonstrated through case studies over synthetic and real-world tensor-valued data, including the evolution over time of global atmospheric temperatures and international interest rates. Another overarching theme in this thesis is the nonstationarity inherent to real-world signals, which typically consist of both deterministic and stochastic components. This thesis aims to help bridge the gap between formal probabilistic theory of stochastic processes and empirical signal processing methods for deterministic signals by providing a spectral model for a class of nonstationary signals, whereby the deterministic and stochastic time-domain signal properties are designated respectively by the first- and second-order moments of the signal in the frequency domain. By virtue of the assumed probabilistic model, novel tests for nonstationarity detection are devised and demonstrated to be effective in low-SNR environments. The proposed spectral analysis framework, which is intrinsically complex-valued, is facilitated by augmented complex algebra in order to fully capture the joint distribution of the real and imaginary parts of complex random variables, using a compact formulation. Finally, motivated by the need for signal processing algorithms which naturally cater for the nonstationarity inherent to real-world tensors, the above contributions are employed simultaneously to derive a general statistical signal processing framework for nonstationary tensors. This is achieved by introducing a new augmented complex multilinear algebra which allows for a concise description of the multilinear interactions between the real and imaginary parts of complex tensors. These contributions are further supported by new physically meaningful empirical results on the statistical analysis of nonstationary global atmospheric temperatures.Open Acces

R-dimensional ESPRIT-type algorithms for strictly second-order non-circular sources and their performance analysis

High-resolution parameter estimation algorithms designed to exploit the prior knowledge about incident signals from strictly second-order (SO) non-circular (NC) sources allow for a lower estimation error and can resolve twice as many sources. In this paper, we derive the R-D NC Standard ESPRIT and the R-D NC Unitary ESPRIT algorithms that provide a significantly better performance compared to their original versions for arbitrary source signals. They are applicable to shift-invariant R-D antenna arrays and do not require a centrosymmetric array structure. Moreover, we present a first-order asymptotic performance analysis of the proposed algorithms, which is based on the error in the signal subspace estimate arising from the noise perturbation. The derived expressions for the resulting parameter estimation error are explicit in the noise realizations and asymptotic in the effective signal-to-noise ratio (SNR), i.e., the results become exact for either high SNRs or a large sample size. We also provide mean squared error (MSE) expressions, where only the assumptions of a zero mean and finite SO moments of the noise are required, but no assumptions about its statistics are necessary. As a main result, we analytically prove that the asymptotic performance of both R-D NC ESPRIT-type algorithms is identical in the high effective SNR regime. Finally, a case study shows that no improvement from strictly non-circular sources can be achieved in the special case of a single source.Comment: accepted at IEEE Transactions on Signal Processing, 15 pages, 6 figure