Multitaper Estimation of the Coherence Spectrum in low SNR

A pseudo coherence estimate using multitapers is presented. The estimate has better localization for sinusoids and is shown to have lower variance for disturbances compared to the usual coherence estimator. This makes it superior in terms of finding coherent frequencies between two sinusoidal signals; even when observed in low SNR. Different sets of multitapers are investigated and the weights of the final coherence estimate are adjusted for a low-biased estimate of a single sinusoid. The proposed method is more computationally efficient than data dependent methods, and does still give comparable results

Sparse Semi-Parametric Chirp Estimator

In this work, we present a method for estimating the parameters detailing an unknown number of linear chirp signals, using an iterative sparse reconstruction framework. The proposed method is initiated by a re-weighted Lasso approach, and then use an iterative relaxation-based refining step to allow for high resolution estimates. The resulting estimates are found to be statistically efficient, achieving the Cramér-Rao lower bound. Numerical simulations illustrate the achievable performance, offering a notable improvement as compared to other recent approaches

Sparse Semi-Parametric Estimation of Harmonic Chirp Signals

In this work, we present a method for estimating the parameters detailing an unknown number of linear, possibly harmonically related, chirp signals, using an iterative sparse reconstruction framework. The proposed method is initiated by a re-weighted group-sparsity approach, followed by an iterative relaxation-based refining step, to allow for high resolution estimates. Numerical simulations illustrate the achievable performance, offering a notable improvement as compared to other recent approaches. The resulting estimates are found to be statistically efficient, achieving the corresponding Cram´er-Rao lower bound

Estimation and Classification of Non-Stationary Processes : Applications in Time-Frequency Analysis

This thesis deals with estimation and classification problems of non-stationary processes in a few special cases.In paper A and paper D we make strong assumptions about the observed signal, where a specific model is assumed and the parameters of the model are estimated.In Paper B, Paper C, and Paper E more general assumptions about the structure of the observed processes are made, and the methods in these papers may be applied to a wider range of parameter estimation and classification scenarios.All papers handle non-stationary signals where the spectral power distribution may change with respect to time. Here, we are interested in finding time-frequency representations (TFR) of the signal which can depict how the frequencies and corresponding amplitudes change.In Paper A, we consider the estimation of the shape parameter detailing time- and frequency translated Gaussian bell functions.The algorithm is based on the scaled reassigned spectrogram, where the spectrogram is calculated using a unit norm Gaussian window.The spectrogram is then reassigned using a large set of candidate scaling factors.For the correct scaling factor, with regards to the shape parameter, the reassigned spectrogram of a Gaussian function will be perfectly localized into one single point.In Paper B, we expand on the concept in Paper A, and allow it to be applied to any twice differentiable transient function in any dimension.Given that the matched window function is used when calculating the spectrogram, we prove that all energy is reassigned to one single point in the time-frequency domain if scaled reassignment is applied.Given a parametric model of an observed signal, one may tune the parameter(s) to minimize the entropy of the matched reassigned spectrogram.We also present a classification scheme, where one may apply multiple different parametric models and evaluate which one of the models that best fit the data. In Paper C, we consider the problem of estimating the spectral content of signals where the spectrum is assumed to have a smooth structure.By dividing the spectral representation into a coarse grid and assuming that the spectrum within each segment may be well approximated as linear, a smooth version of the Fourier transform is derived.Using this, we minimize the least squares norm of the difference between the sample covariance matrix of an observed signal and any covariance matrix belonging to a piece-wise linear spectrum.Additionally, we allow for adding constraints that make the solution obey common assumptions of spectral representations.We apply the algorithm to stationary signals in one and two dimensions, as well as to one-dimensional non-stationary processes. In Paper D we consider the problem of estimating the parameters of a multi-component chirp signal, where a harmonic structure may be imposed.The algorithm is based on a group sparsity with sparse groups framework where a large dictionary of candidate parameters is constructed.An optimization scheme is formulated such as to find harmonic groups of chirps that also punish the number of harmonics within each group.Additionally, we form a non-linear least squares step to avoid the bias which is introduced by the spacing of the dictionary. In Paper E we propose that the Wigner-Ville distribution should be used as input to convolutional neural networks, as opposed to the often used spectrogram.As the spectrogram may be expressed as a convolution between a kernel function and the Wigner-Ville distribution, we argue that the kernel function should not be chosen manually.Instead, said convolutional kernel should be optimized together with the rest of the kernels that make up the neural network

Time Frequency Analysis of EEG Measured When Performing the Flanker Task

This thesis handles time frequency analysis of EEG signals measured on participants performing the so-called flanker task. The analysis is done mainly using multitapering techniques on the quadratic class. Using multiple orthonormal windows when estimating the spectra of a process, one lowers the variance of estimate. A class of locally stationary processes (LSP) is presented to use as a model of EEG which can then be used to evaluate the different time-frequency methods that are presented. This LSP contains only one component is used to model only one part of the EEG signal. When analyzing the set of EEG signals of this thesis one is most interested in the so-called N2 event and the model is therefore applied to this event. Having this model one can then find the optimal multitapers in the mean square error sense. Abstract Different sets of multitapers are used to analyze the time-frequency representation of the EEG-signals. These are evaluated on LSPs where the true spectra are known. Abstract Spectra are then estimated for the EEG-signals. As there are multiple channels and different methods are used, only a selected set of these spectra are presented here

Parameter estimation of Oscillating Gaussian functions using the scaled reassigned spectrogram

In this paper we suggest an algorithm for estimation of the parameters detailing Oscillating Gaussian functions. The different components of the signal are first detected in the spectrogram. After this we exploit the fact that a Gaussian function may be perfectly reassigned into one single point given a correct scaling factor, where this scaling factor is a function of the unknown shape parameter of the Gaussian function. The scaled reassignment of the spectrogram is performed using a set of candidate scaling factors and the local Renyi entropy is used to measure the concentration of each component using every candidate scaling factor. The estimates are refined by using non-linear least squares. The algorithm is evaluated on both simulated and real data

Optimal Time-Frequency analysis of the multiple time-translated locally stationary processes

A previously proposed model for non-stationary signals is extended in this contribution. The model consists of mul- tiple time-translated locally stationary processes. The opti- mal Ambiguity kernel for the process in mean-square-error sense is computed analytically and is used to estimate the time-frequency distribution. The performance of the kernel is compared with other commonly used kernels. Finally the model is applied to electrical signals from the brain (EEG) measured during a concentration task

The Scaled Reassigned Spectrogram with Perfect Localization for Estimation of Gaussian Functions

The reassignment technique is used to increase localization for signal components in the time-frequency representation. The technique gives perfect localization for infinite linear chirp-signals, impulses and constant frequency signals but not for short non-stationary signals. In this paper, a scaled reassignment is proposed, based on the spectrogram using a Gaussian window. The resulting reassignment gives perfect localization for a Gaussian function when the window length matches the function length. Based on the scaled reassignment, an algorithm that estimates the Gaussian function length is also proposed

Classification of bird song syllables using wigner-ville ambiguity function cross-terms

A novel feature extraction method for lowdimensional signal representation is presented. The features are useful for classification of non-stationary multi-component signals with stochastic variation in amplitudes and time-frequency locations. Using a penalty function to suppress the Wigner-Ville ambiguity function auto-terms, the proposed feature set is based on the cross-term doppler- and lag profiles. The investigation considers classification where strong similar components appear in all signals and where the differences between classes are related to weaker components. The approach is evaluated and compared with established methods for simulated data and bird song syllables of the great reed warbler. The results show that the novel feature extraction method gives a better classification than established methods used in bird song analysis