32 research outputs found
Adaptive filtering algorithms for quaternion-valued signals
Advances in sensor technology have made possible the recoding of three and four-dimensional signals which afford a better representation of our actual three-dimensional world than the ``flat view'' one and two-dimensional approaches. Although it is straightforward to model such signals as real-valued vectors, many applications require unambiguous modeling of orientation and rotation, where the division algebra of quaternions provides crucial advantages over real-valued vector approaches.
The focus of this thesis is on the use of recent advances in quaternion-valued signal processing, such as the quaternion augmented statistics, widely-linear modeling, and the HR-calculus, in order to develop practical adaptive signal processing algorithms in the quaternion domain which deal with the notion of phase and frequency in a compact and physically meaningful way. To this end, first a real-time tracker of quaternion impropriety is developed, which allows for choosing between strictly linear and widely-linear quaternion-valued signal processing algorithms in real-time, in order to reduce computational complexity where appropriate. This is followed by the strictly linear and widely-linear quaternion least mean phase algorithms that are developed for phase-only estimation in the quaternion domain, which is accompanied by both quantitative performance assessment and physical interpretation of operations. Next, the practical application of state space modeling of three-phase power signals in smart grid management and control systems is considered, and a robust complex-valued state space model for frequency estimation in three-phase systems is presented. Its advantages over other available estimators are demonstrated both in an analytical sense and through simulations. The concept is then expanded to the quaternion setting in order to make possible the simultaneous estimation of the system frequency and its voltage phasors. Furthermore, a distributed quaternion Kalman filtering algorithm is developed for frequency estimation over power distribution networks and collaborative target tracking. Finally, statistics of stable quaternion-valued random variables, that include quaternion-valued Gaussian random variables as a special case, is investigated in order to develop a framework for the modeling and processing of heavy-tailed quaternion-valued signals.Open Acces
Asymptotic regime for impropriety tests of complex random vectors
Impropriety testing for complex-valued vector has been considered lately due
to potential applications ranging from digital communications to complex media
imaging. This paper provides new results for such tests in the asymptotic
regime, i.e. when the vector dimension and sample size grow commensurately to
infinity. The studied tests are based on invariant statistics named impropriety
coefficients. Limiting distributions for these statistics are derived, together
with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in
the Gaussian case. This characterization in the asymptotic regime allows also
to identify a phase transition in Roy's test with potential application in
detection of complex-valued low-rank subspace corrupted by proper noise in
large datasets. Simulations illustrate the accuracy of the proposed asymptotic
approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS
Spectral analysis of stationary random bivariate signals
A novel approach towards the spectral analysis of stationary random bivariate
signals is proposed. Using the Quaternion Fourier Transform, we introduce a
quaternion-valued spectral representation of random bivariate signals seen as
complex-valued sequences. This makes possible the definition of a scalar
quaternion-valued spectral density for bivariate signals. This spectral density
can be meaningfully interpreted in terms of frequency-dependent polarization
attributes. A natural decomposition of any random bivariate signal in terms of
unpolarized and polarized components is introduced. Nonparametric spectral
density estimation is investigated, and we introduce the polarization
periodogram of a random bivariate signal. Numerical experiments support our
theoretical analysis, illustrating the relevance of the approach on synthetic
data.Comment: 11 pages, 3 figure
Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets
Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components.
For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper.
We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles.
Quaternions are isomorphic to an algebra of structured 4x4 real matrices.
This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms.
A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance.
Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs.
Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces
Tapering promotes propriety for Fourier transforms of real-valued time series
We examine Fourier transforms of real-valued stationary time series from the point of view of the statistical propriety. Processes with a large dynamic range spectrum have transforms that are very significantly improper for some frequencies; the real and imaginary parts can be highly correlated, and the periodogram will not have the standard chi-square distribution at these frequencies, nor have two degrees of freedom. Use of a taper reduces impropriety to just frequencies close to zero and Nyquist only, and frequency ranges where the propriety breaks down can be quite accurately and easily predicted by half the autocorrelation width of |H * H(2f)|, denoted by c, where H(f) is the Fourier transform of the taper and * denotes convolution. For vector time series we derive an improved distributional approximation for minus twice the log of the generalized likelihood ratio test statistic for testing for propriety of the Fourier transform at any frequency, and compare frequency range cutoffs for propriety determined by the hypothesis test with those determined by c
A frequency domain test for propriety of complex-valued vector time series
This paper proposes a frequency domain approach to test the hypothesis that a stationary complexvalued
vector time series is proper, i.e., for testing whether the vector time series is uncorrelated with its
complex conjugate. If the hypothesis is rejected, frequency bands causing the rejection will be identified
and might usefully be related to known properties of the physical processes. The test needs the associated
spectral matrix which can be estimated by multitaper methods using, say, K tapers. Standard asymptotic
distributions for the test statistic are of no use since they would require K → ∞, but, as K increases
so does resolution bandwidth which causes spectral blurring. In many analyses K is necessarily kept
small, and hence our efforts are directed at practical and accurate methodology for hypothesis testing for
small K. Our generalized likelihood ratio statistic combined with exact cumulant matching gives very
accurate rejection percentages. We also prove that the statistic on which the test is based is comprised of
canonical coherencies arising from our complex-valued vector time series. Frequency specific tests are
combined using multiple hypothesis testing to give an overall test. Our methodology is demonstrated on
ocean current data collected at different depths in the Labrador Sea. Overall this work extends results
on propriety testing for complex-valued vectors to the complex-valued vector time series setting
Adaptive signal processing algorithms for noncircular complex data
The complex domain provides a natural processing framework for a large class of signals
encountered in communications, radar, biomedical engineering and renewable
energy. Statistical signal processing in C has traditionally been viewed as a straightforward
extension of the corresponding algorithms in the real domain R, however,
recent developments in augmented complex statistics show that, in general, this leads
to under-modelling. This direct treatment of complex-valued signals has led to advances
in so called widely linear modelling and the introduction of a generalised
framework for the differentiability of both analytic and non-analytic complex and
quaternion functions. In this thesis, supervised and blind complex adaptive algorithms
capable of processing the generality of complex and quaternion signals (both
circular and noncircular) in both noise-free and noisy environments are developed;
their usefulness in real-world applications is demonstrated through case studies.
The focus of this thesis is on the use of augmented statistics and widely linear modelling.
The standard complex least mean square (CLMS) algorithm is extended to
perform optimally for the generality of complex-valued signals, and is shown to outperform
the CLMS algorithm. Next, extraction of latent complex-valued signals from
large mixtures is addressed. This is achieved by developing several classes of complex
blind source extraction algorithms based on fundamental signal properties such
as smoothness, predictability and degree of Gaussianity, with the analysis of the existence
and uniqueness of the solutions also provided. These algorithms are shown
to facilitate real-time applications, such as those in brain computer interfacing (BCI).
Due to their modified cost functions and the widely linear mixing model, this class of
algorithms perform well in both noise-free and noisy environments. Next, based on a
widely linear quaternion model, the FastICA algorithm is extended to the quaternion
domain to provide separation of the generality of quaternion signals. The enhanced
performances of the widely linear algorithms are illustrated in renewable energy and
biomedical applications, in particular, for the prediction of wind profiles and extraction
of artifacts from EEG recordings