This thesis is concerned with various aspects of stationary and nonstationary time series analysis. In the nonstationary case, we study estimation of the Wold-Cram'er evolutionary spectrum, which is a time-dependent analogue of the spectrum of a stationary process. Existing estimators of the Wold-Cram'er evolutionary spectrum suffer from several problems, including bias in boundary regions of the time-frequency plane, poor frequency resolution, and an inability to handle the presence of purely harmonic frequencies. We propose techniques to handle all three of these problems. We propose a new estimator of the Wold-Cram'er evolutionary spectrum (the BCMTFSE) which mitigates the first problem. Our estimator is based on an extrapolation of the Wold-Cram'er evolutionary spectrum in time, using an estimate of its time derivative. We apply our estimator to a set of simulated nonstationary processes with known Wold-Cram'er evolutionary spectra to demonstrate its performance. We also propose an estimator of the Wold-Cram'er evolutionary spectrum, valid for uniformly modulated processes (UMPs). This estimator mitigates the second problem, by exploiting the structure of UMPs to improve the frequency resolution of the BCMTFSE. We apply this estimator to a simulated UMP with known Wold-Cram'er evolutionary spectrum. To deal with the third problem, one can detect and remove purely harmonic frequencies before applying the BCMTFSE. Doing so requires a consideration of the aliasing problem. We propose a frequency-domain technique to detect and unalias aliased frequencies in bivariate time series, based on the observation that aliasing manifests as nonlinearity in the phase of the complex coherency between a stationary process and a time-delayed version of itself. To illustrate this ``unaliasing'' technique, we apply it to simulated data and a real-world example of solar noon flux data. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-02-05 10:18:13.476

Spatio-temporal variation in activation rates during ventricular fibrillation (VF)provides insight into mechanisms of sustained re-entry during VF. This study had three objectives related to spatio-temporal dynamics in activation rates during VF. The first objective was to quantify spatio-temporal variability in activation rates,that is, in dominant frequencies, computed from epicardial electrograms recorded during VF in swine. Results showed that temporally and spatially, dominant frequencies variedas much as 20% of the mean dominant frequency, and the mean dominant frequencies increased during first 30 sec of VF. These results suggest that activation rates are nonstationary during VF. The second objective of the study was to develop a new stimulation protocol for quantifying restitution of action potential duration (APD) by independently controlling diastolic intervals (DI). A property of cardiac cells that determines spatio-temporal variability in dominant frequencies is restitution of APD, which relates APD to the previous DI. Independent control of DI permits explicit determination of the role of memory in restitution. Restitution functions quantified using mathematical models of activation and our stimulation protocol, showed significant hysteresis. That is, for adiastolic interval, the action potential durations were as much as 15% longer during periods when the DI were decreasing than when the DI were increasing. We verified the feasibility of implementing our protocol experimentally in isolated and perfused rat hearts with action potentials recorded using floating glass microelectrodes. The third objective of our study was to verify that spatio-temporal variability in dominant frequencies during VF could be modified using spatially distributed pacing strength stimuli. Simulated VF was induced in 400x400 and 400x800 matrices of cells. Electrical function of cells was simulated using the Luo-Rudy model. Stimulators were arranged in the matrices such that there were 5 rows of line stimulators. Results showed that it was possible to modify activations in almost 54% of the area and to modify spatio-temporal variability in activation during VF into a desired pattern by the use of synchronized pacing from multiple sites. These results support further exploration of distributed stimulation approach for potential improvements in defibrillation therapy.

There are around 10,426 bird species around the world. Recognizing the bird species for an untrained person is almost impossible either by watching or listening them. In order to identify the bird species from their sounds, there is a need for an application that can detect the bird species from its sound. Time-frequency domain analysis techniques are used to implement the application. We implemented two time-frequency domain feature extraction methods. In feature extraction, a signature matrix which consist of extracted features is created for bird sound signals. A database of signature matrix is created with bird chirps extracted features. We implemented two feature classification methods. They are auto-correlation feature classification method and reference difference feature classification method. An unknown bird chirp is compared with the database to detect the species name. The main aim of the research is to implement the time-frequency domain feature extraction method, create a signature matrix database, implement two feature classification methods and compare them. At last, bird species were identified in the research and the auto-correlation classification method detects the bird species better than the reference difference classification method.

This thesis develops a new method for time-frequency signal processing and examines the relevance of the new representation in studies of neural coding in songbirds. The method groups together associated regions of the time-frequency plane into objects defined by time-frequency contours. By combining information about structurally stable contour shapes over multiple time-scales and angles, a signal decomposition is produced that distributes resolution adaptively. As a result, distinct signal components are represented in their own most parsimonious forms.  Next, through neural recordings in singing birds, it was found that activity in song premotor cortex is significantly correlated with the objects defined by this new representation of sound. In this process, an automated way of finding sub-syllable acoustic transitions in birdsongs was first developed, and then increased spiking probability was found at the boundaries of these acoustic transitions. Finally, a new approach to study auditory cortical sequence processing more generally is proposed. In this approach, songbirds were trained to discriminate Morse-code-like sequences of clicks, and the neural correlates of this behavior were examined in primary and secondary auditory cortex. It was found that a distinct transformation of auditory responses to the sequences of clicks exists as information transferred from primary to secondary auditory areas. Neurons in secondary auditory areas respond asynchronously and selectively -- in a manner that depends on the temporal context of the click. This transformation from a temporal to a spatial representation of sound provides a possible basis for the songbird's natural ability to discriminate complex temporal sequences.

A scaling model based on an effective Gol'dberg number is proposed for predicting the presence of cumulative nonlinear distortions in the acoustic waveforms produced by high-speed jets. Two acoustic length scales, the shock formation distance and the absorption length are expressed in terms of jet exit parameters. This approach allows one to compute the degree of cumulative nonlinear distortion in a full-scale scenario, from laboratory-scale observations, or vice versa. Surveys of the acoustic pressure waveforms emitted by a laboratory-scale, shock-free and unheated Mach 3 jet are used to support the findings of the model. These acoustic waveforms are acquired on a planar grid in an acoustically treated and range-restricted environment. Various statistical metrics are employed to examine the degree of local and cumulative nonlinearity in the measured waveforms and their temporal derivatives. This includes skewness, kurtosis, the number of zero crossings in the waveform, a wave steepening factor, the Morfey-Howell nonlinearity indicator and an application of the generalized Burgers equation. It is advocated that in order for the Morfey-Howell indicator to be used as an investigative tool for the presence of cumulative nonlinear waveform distortion, that it be applied as a multi-point indicator. Based on findings of the model and the spatial topography of the metrics, it is concluded that cumulative nonlinear steepening effects are absent in the current data set. This implies that acoustic shock-structures in the waveforms are generated by local mechanisms in, or in close vicinity to, the jet's hydrodynamic region. Furthermore, these shock-structures induce the crackle noise component. The research aims to quantify crackle in a temporal and spectral fashion, and is motivated by the fact that (1) it is perceived as the most annoying component of jet noise, (2) no unique measures of crackle exist, and (3) significant reductions in jet noise will be achieved when crackle can be controlled. A unique detection algorithm is introduced which isolates the shock-structures in the temporal waveform that are responsible for crackle. Ensemble-averages of the identified waveform sections are employed to gain an in-depth understanding of the crackling structures. Moreover, PDF's of the temporal intermittence of these shocks reveal modal trends and show evidence that crackling shock-structures are present in groups of multiple shocks. A spectral measure of crackle is considered by using wavelet-based time-frequency analyses. The increase in sound energy is computed by considering the global pressure spectra of the waveforms and the ones that represent the spectral behavior during instances of crackle. This energy-based metric is postulated to be an appropriate metric for the level of crackle. / text

Time-frequency analysis for non-linear and non-stationary signals is extraordinarily challenging. To capture the changes in these types of signals, it is necessary for the analysis methods to be local, adaptive and stable. In recent years, decomposition based analysis methods were developed by different researchers to deal with non-linear and non-stationary signals. These methods share the feature that a signal is decomposed into finite number of components on which the time-frequency analysis can be applied. Differences lie in the strategies to extract these components: by iteration or by optimization. However, considering the requirements of being local, adaptive and stable, neither of these decompositions are perfectly satisfactory. Motivated to find a local, adaptive and stable decomposition of a signal, this thesis presents Adaptive Local Iterative Filtering (ALIF) algorithm. The adaptivity is obtained having the filter lengths being determined by the signal itself. The locality is ensured by the filter we designed based on a PDE model. The stability of this algorithm is shown and the convergence is proved. Moreover, we also propose a local definition for the instantaneous frequency in order to achieve a completely local analysis for non-linear and non-stationary signals. Examples show that this decomposition really helps in both simulated data analysis and real world application.

Time-frequency analysis is a powerful tool for signal analysis and processing. The Fourier transform and wavelet transforms are used extensively as is the Short-Time Fourier Transform (or Gabor transform). In 1996 the Stockwell transform was introduced to maintain the phase of the Fourier transform, while also providing the progressive resolution of the wavelet transform. The discrete orthonormal Stockwell transform is a more efficient, less redundant transform with the same properties. There has been little work on mathematical properties of the Stockwell transform, particularly how it behaves under operations such as translation and modulation. Previous results do discuss a resolution of the identity, as well as some of the function spaces that may be associated with it [2]. We extend the resolution of the identity results, and behaviour under translation, modulation, convolution and differentiation. boundedness and continuity properties are also developed, but the function spaces associated with the transform are unrelated to the focus of this thesis. There has been some work on image processing using the Stockwell transform and discrete orthonormal Stockwell transform. The tests were quite preliminary. In this thesis, we explore some of the mathematics of the Stockwell transform, examining properties, and applying it to various continuous examples. The discrete orthonormal Stockwell transform is compared directly with Newland’s harmonic wavelet transform, and we extend the definition to include varitions, as well as develop the discrete cosine based Stockwell transform. All of these discrete transforms are tested against current methods for image compression.

Time-frequency methods are widely used tools to diagnose planetary gearbox fault under non-stationary conditions. However, the existing time-frequency methods still have some problems, such as smearing effect and cross-term interference, and these problems limit the effectiveness of the existing time-frequency methods in planetary gearbox fault diagnosis under non-stationary conditions. To address the aforementioned problems, four time-frequency methods are proposed in this thesis. As nowadays a large portion of the industrial equipment is equipped with tachometers, the first three methods are for the cases that the shaft rotational speed is easily accessible and the last method is for the cases of shaft rotational speed is not easily accessible. The proposed methods are itemized as follows: (1) The velocity synchronous short-time Fourier transform (VSSTFT), which is a type of linear transform based on the domain mappings and short-time Fourier transform to address the smear effect of the existing linear transforms under known time-varying speed conditions; (2) The velocity synchrosqueezing transform (VST), which is a type of remapping method based on the domain mapping and synchrosqueezing transform to address the smear effect of existing remapping methods under known time-varying speed conditions; (3) The velocity synchronous bilinear distribution (VSBD), which is a type of bilinear distribution based on the generalized demodulation and Cohen’s class bilinear distribution to address the smear effect and cross-term interference of existing bilinear distributions under known time-varying speed conditions and (4) The velocity synchronous linear chirplet transform (VSLCT), which is a non-parametric combined approach of linear transform and concentration-index-guided parameter determination to provide a smear-free and cross-term-free TFR under unknown time-varying speed conditions. In this work, simple algorithms are developed to avoid the signal resampling process required by the domain mappings or demodulations of the first three methods (i.e., the VSSTFT, VST and VSBD). They are designed to have different resolutions, readabilities, noise tolerances and computational efficiencies. Therefore, they are capable to adapt different application conditions. The VSLCT, as a kind of linear transform, is designed for unknown rotational speed conditions. It utilizes a set of shaft-rotational-speed-synchronous bases to address the smear problem and it is capable to dynamically determine the signal processing parameters (i.e., window length and normalized angle) to provide a clear TFR with desirable time-frequency resolution in response to condition variations. All of the proposed methods in this work are smear-free and cross-term-free, the TFRs generated by the methods are clearer and more precise compared with the existing time-frequency methods. The faults of planetary gearboxes, if any, can be diagnosed by identifying the fault-induced components from the obtained TFRs. The four methods are all newly applied to fault diagnosis. The effectiveness of them has been validated using both simulated and experimental vibration signals of planetary gearboxes collected under non-stationary conditions.

This thesis presents results on the description of plasma waves in terms of wavepackets. The wave field is decomposed into a distribution of wavepackets in a space of position, wavevector, time, and frequency. A complex structure joining each pair of Fourier conjugate variables into a single complex coordinate allows the efficient derivation of equations of motion for the phase space distribution by exploiting its analytic properties. The Wick symbol calculus, a mathematical tool generalizing many convenient properties of the Fourier transform to a local setting, is used to derive new exact phase space equations which maintain full information on the phase of the waves and include effects nonlocal in phase space such as harmonic generation. A general purpose asymptotic expansion of the Wick symbol product formula is used to treat dispersion, refraction, photon acceleration, and ponderomotive forces. Examples studied include the nonlinear Schrödinger equation, mode conversion, and the Vlasov equation. The structure of partially coherent wave fields is understood in terms of zeros in the phase space distribution caused by dislocations in its complex phase which are shown to be correlated with the field entropy. Simulations of plasma heating by crossing electron beams are understood by representing the resulting plasma waves in phase space. The local coherence properties of the beam driven Langmuir waves are studied numerically.

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