Gravitational waves from BH-NS binaries: Effective Fisher matrices and parameter estimation using higher harmonics

Inspiralling black hole-neutron star (BH-NS) binaries emit a complicated gravitational wave signature, produced by multiple harmonics sourced by their strong local gravitational field and further modulated by the orbital plane's precession. Some features of this complex signal are easily accessible to ground-based interferometers (e.g., the rate of change of frequency); others less so (e.g., the polarization content); and others unavailable (e.g., features of the signal out of band). For this reason, an ambiguity function (a diagnostic of dissimilarity) between two such signals varies on many parameter scales and ranges. In this paper, we present a method for computing an approximate, effective Fisher matrix from variations in the ambiguity function on physically pertinent scales which depend on the relevant signal to noise ratio. As a concrete example, we explore how higher harmonics improve parameter measurement accuracy. As previous studies suggest, for our fiducial BH-NS binaries and for plausible signal amplitudes, we see that higher harmonics at best marginally improve our ability to measure parameters. For non-precessing binaries, these Fisher matrices separate into intrinsic (mass, spin) and extrinsic (geometrical) parameters; higher harmonics principally improve our knowledge about the line of sight. For the precessing binaries, the extra information provided by higher harmonics is distributed across several parameters. We provide concrete estimates for measurement accuracy, using coordinates adapted to the precession cone in the detector's sensitive band.Comment: 19 pages, 11 figure

Multiple multidimensional morse wavelets

This paper defines a set of operators that localize a radial image in space and radial frequency simultaneously. The eigenfunctions of the operator are determined and a nonseparable orthogonal set of radial wavelet functions are found. The eigenfunctions are optimally concentrated over a given region of radial space and scale space, defined via a triplet of parameters. Analytic forms for the energy concentration of the functions over the region are given. The radial function localization operator can be generalised to an operator localizing any L-2(R-2) function. It is demonstrated that the latter operator, given an appropriate choice of localization region, approximately has the same radial eigenfunctions as the radial operator. Based on a given radial wavelet function a quaternionic wavelet is defined that can extract the local orientation of discontinuous signals as well as amplitude, orientation and phase structure of locally oscillatory signals. The full set of quaternionic wavelet functions are component by component orthogonal; their statistical properties are tractable, and forms for the variability of the estimators of the local phase and orientation are given, as well as the local energy of the image. By averaging estimators across wavelets, a substantial reduction in the variance is achieved

Precession during merger 1: Strong polarization changes are observationally accessible features of strong-field gravity during binary black hole merger

The short gravitational wave signal from the merger of compact binaries encodes a surprising amount of information about the strong-field dynamics of merger into frequencies accessible to ground-based interferometers. In this paper we describe a previously-unknown "precession" of the peak emission direction with time, both before and after the merger, about the total angular momentum direction. We demonstrate the gravitational wave polarization encodes the orientation of this direction to the line of sight. We argue the effects of polarization can be estimated nonparametrically, directly from the gravitational wave signal as seen along one line of sight, as a slowly-varying feature on top of a rapidly-varying carrier. After merger, our results can be interpreted as a coherent excitation of quasinormal modes of different angular orders, a superposition which naturally "precesses" and modulates the line-of-sight amplitude. Recent analytic calculations have arrived at a similar geometric interpretation. We suspect the line-of-sight polarization content will be a convenient observable with which to define new high-precision tests of general relativity using gravitational waves. Additionally, as the nonlinear merger process seeds the initial coherent perturbation, we speculate the amplitude of this effect provides a new probe of the strong-field dynamics during merger. To demonstrate the ubiquity of the effects we describe, we summarize the post-merger evolution of 104 generic precessing binary mergers. Finally, we provide estimates for the detectable impacts of precession on the waveforms from high-mass sources. These expressions may identify new precessing binary parameters whose waveforms are dissimilar from the existing sample.Comment: 11 figures; v2 includes response to referee suggestion

Auditory Spectrum-Based Pitched Instrument Onset Detection

In this paper, a method for onset detection of music signals using auditory spectra is proposed. The auditory spectrogram provides a time-frequency representation that employs a sound processing model resembling the human auditory system. Recent work on onset detection employs DFT-based features describing spectral energy and phase differences, as well as pitch-based features. These features are often combined for maximizing detection performance. Here, the spectral flux and phase slope features are derived in the auditory framework and a novel fundamental frequency estimation algorithm based on auditory spectra is introduced. An onset detection algorithm is proposed, which processes and combines the aforementioned features at the decision level. Experiments are conducted on a dataset covering 11 pitched instrument types, consisting of 1829 onsets in total. Results indicate that auditory representations outperform various state-of-the-art approaches, with the onset detection algorithm reaching an F-measure of 82.6%

Data-driven multivariate and multiscale methods for brain computer interface

This thesis focuses on the development of data-driven multivariate and multiscale methods for brain computer interface (BCI) systems. The electroencephalogram (EEG), the most convenient means to measure neurophysiological activity due to its noninvasive nature, is mainly considered. The nonlinearity and nonstationarity inherent in EEG and its multichannel recording nature require a new set of data-driven multivariate techniques to estimate more accurately features for enhanced BCI operation. Also, a long term goal is to enable an alternative EEG recording strategy for achieving long-term and portable monitoring. Empirical mode decomposition (EMD) and local mean decomposition (LMD), fully data-driven adaptive tools, are considered to decompose the nonlinear and nonstationary EEG signal into a set of components which are highly localised in time and frequency. It is shown that the complex and multivariate extensions of EMD, which can exploit common oscillatory modes within multivariate (multichannel) data, can be used to accurately estimate and compare the amplitude and phase information among multiple sources, a key for the feature extraction of BCI system. A complex extension of local mean decomposition is also introduced and its operation is illustrated on two channel neuronal spike streams. Common spatial pattern (CSP), a standard feature extraction technique for BCI application, is also extended to complex domain using the augmented complex statistics. Depending on the circularity/noncircularity of a complex signal, one of the complex CSP algorithms can be chosen to produce the best classification performance between two different EEG classes. Using these complex and multivariate algorithms, two cognitive brain studies are investigated for more natural and intuitive design of advanced BCI systems. Firstly, a Yarbus-style auditory selective attention experiment is introduced to measure the user attention to a sound source among a mixture of sound stimuli, which is aimed at improving the usefulness of hearing instruments such as hearing aid. Secondly, emotion experiments elicited by taste and taste recall are examined to determine the pleasure and displeasure of a food for the implementation of affective computing. The separation between two emotional responses is examined using real and complex-valued common spatial pattern methods. Finally, we introduce a novel approach to brain monitoring based on EEG recordings from within the ear canal, embedded on a custom made hearing aid earplug. The new platform promises the possibility of both short- and long-term continuous use for standard brain monitoring and interfacing applications

Nonlinear approximation using Blaschke polynomials

This dissertation, entitled Nonlinear Approximation Using Blaschke Polynomials, is motivated by questions arising from Empirical Mode Decomposition (EMD). EMD is a signal processing method which decomposes input signals into components called intrinsic mode functions (IMFs). These IMFs often have the desirable property that the instantaneous frequency of their analytic signals is positive. However, this is not always the case.;The first two chapters are introductions to approximation in general, and Empirical Mode Decomposition, respectively.;The third chapter presents a characterization of which analytic signals have the property of non-negative instantaneous frequency. These \u27analytic signals with non-negative instantaneous frequency\u27 (ASNIFs) are described using Hardy spaces on the unit disc. Analogous results are also found for analytic signals which are boundary values fof elements of Hardy spaces on the half-plane.;The fourth chapter describes in general terms how one might construct an approximation method using ASNIFs, or possibly some restricted subclass of ASNIFs, as approximants.;The fifth chapter introduces a special set of ASNIFs: Blaschke polynomials. These are functions which are linear combinations of Blaschke products, which are a special kind of ASNIF. Blaschke polynomials are a natural extension of other classical approximating sets, and have some interesting properties of their own. Some of these properties are explored.;The sixth chapter contains an implementation of a signal decomposition method using Blaschke polynomials, and a discussion of the results.;The seventh chapter further explores the conclusions of this work, and lays out future research questions