Estimation and Classification of NonStationary Processes : Applications in TimeFrequency Analysis
(2019) Abstract
 This thesis deals with estimation and classification problems of nonstationary 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 nonstationary signals where the spectral power distribution may change with respect to time. Here, we are interested in finding timefrequency representations (TFR) of the signal which can depict how the... (More)  This thesis deals with estimation and classification problems of nonstationary 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 nonstationary signals where the spectral power distribution may change with respect to time. Here, we are interested in finding timefrequency 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 timefrequency 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 piecewise 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 onedimensional nonstationary processes.
In Paper D we consider the problem of estimating the parameters of a multicomponent 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 nonlinear least squares step to avoid the bias which is introduced by the spacing of the dictionary.
In Paper E we propose that the WignerVille 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 WignerVille 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.
(Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/b961aec1d3484a7a84a483b4b15647da
 author
 Brynolfsson, Johan ^{LU}
 supervisor

 Maria Sandsten ^{LU}
 opponent

 Associate Professor Karlsson, Johan, KTH Royal Institute of Technology, Stockholm, Sweden.
 organization
 publishing date
 20190517
 type
 Thesis
 publication status
 published
 subject
 keywords
 TimeFrequency Estimation, Parameter Estimation, Reassignment method, NonStationary Processes, Smooth spectral estimation, Neural Networks
 pages
 224 pages
 publisher
 Centre for Mathematical Sciences, Lund University
 defense location
 MH:Rieszsalen, Matematikhuset, SÃ¶lvegatan 18, Lund University, Faculty of Engineering LTH
 defense date
 20190614 09:15
 ISBN
 9789178951345
 9789178951338
 language
 English
 LU publication?
 yes
 id
 b961aec1d3484a7a84a483b4b15647da
 date added to LUP
 20190521 10:05:36
 date last changed
 20190522 15:03:38
@phdthesis{b961aec1d3484a7a84a483b4b15647da, abstract = {This thesis deals with estimation and classification problems of nonstationary processes in a few special cases.<br/>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.<br/>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.<br/>All papers handle nonstationary signals where the spectral power distribution may change with respect to time. Here, we are interested in finding timefrequency representations (TFR) of the signal which can depict how the frequencies and corresponding amplitudes change.<br/><br/>In Paper A, we consider the estimation of the shape parameter detailing time and frequency translated Gaussian bell functions.<br/>The algorithm is based on the scaled reassigned spectrogram, where the spectrogram is calculated using a unit norm Gaussian window.<br/>The spectrogram is then reassigned using a large set of candidate scaling factors.<br/>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.<br/><br/>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.<br/>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 timefrequency domain if scaled reassignment is applied.<br/>Given a parametric model of an observed signal, one may tune the parameter(s) to minimize the entropy of the matched reassigned spectrogram.<br/>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. <br/><br/>In Paper C, we consider the problem of estimating the spectral content of signals where the spectrum is assumed to have a smooth structure.<br/>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.<br/>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 piecewise linear spectrum.<br/>Additionally, we allow for adding constraints that make the solution obey common assumptions of spectral representations.<br/>We apply the algorithm to stationary signals in one and two dimensions, as well as to onedimensional nonstationary processes. <br/><br/>In Paper D we consider the problem of estimating the parameters of a multicomponent chirp signal, where a harmonic structure may be imposed.<br/>The algorithm is based on a group sparsity with sparse groups framework where a large dictionary of candidate parameters is constructed.<br/>An optimization scheme is formulated such as to find harmonic groups of chirps that also punish the number of harmonics within each group.<br/>Additionally, we form a nonlinear least squares step to avoid the bias which is introduced by the spacing of the dictionary. <br/><br/>In Paper E we propose that the WignerVille distribution should be used as input to convolutional neural networks, as opposed to the often used spectrogram.<br/>As the spectrogram may be expressed as a convolution between a kernel function and the WignerVille distribution, we argue that the kernel function should not be chosen manually.<br/>Instead, said convolutional kernel should be optimized together with the rest of the kernels that make up the neural network.<br/>}, author = {Brynolfsson, Johan}, isbn = {9789178951345}, keyword = {TimeFrequency Estimation,Parameter Estimation,Reassignment method,NonStationary Processes,Smooth spectral estimation,Neural Networks}, language = {eng}, month = {05}, pages = {224}, publisher = {Centre for Mathematical Sciences, Lund University}, school = {Lund University}, title = {Estimation and Classification of NonStationary Processes : Applications in TimeFrequency Analysis}, year = {2019}, }