Sparse Modeling of Harmonic Signals
(2017) Abstract
 This thesis considers sparse modeling and estimation of multipitch signals, i.e., signals whose frequency content can be described by superpositions of harmonic, or closetoharmonic, structures, characterized by a set of fundamental frequencies. As the number of fundamental frequencies in a given signal is in general unknown, this thesis casts the estimation as a sparse reconstruction problem, i.e., estimates of the fundamental frequencies are produced by finding a sparse representation of the signal in a dictionary containing an overcomplete set of pitch atoms. This sparse representation is found by using convex modeling techniques, leading to highly tractable convex optimization problems from whose solutions the estimates of the... (More)
 This thesis considers sparse modeling and estimation of multipitch signals, i.e., signals whose frequency content can be described by superpositions of harmonic, or closetoharmonic, structures, characterized by a set of fundamental frequencies. As the number of fundamental frequencies in a given signal is in general unknown, this thesis casts the estimation as a sparse reconstruction problem, i.e., estimates of the fundamental frequencies are produced by finding a sparse representation of the signal in a dictionary containing an overcomplete set of pitch atoms. This sparse representation is found by using convex modeling techniques, leading to highly tractable convex optimization problems from whose solutions the estimates of the fundamental frequencies can be deduced.
In the first paper of this thesis, a method for multipitch estimation for stationary signal frames is proposed. Building on the heuristic of spectrally smooth pitches, the proposed method produces estimates of the fundamental frequencies by minimizing a sequence of penalized least squares criteria, where the penalties adapt to the signal at hand. An efficient algorithm building on the alternating direction method of multipliers is proposed for solving these least squares problems.
The second paper considers a timerecursive formulation of the multipitch estimation problem, allowing for the exploiting of longerterm correlations of the signal, as well as fundamental frequency estimates with a samplelevel time resolution. Also presented is a signaladaptive dictionary learning scheme, allowing for smooth tracking of frequency modulated signals.
In the third paper of this thesis, robustness to deviations from the harmonic model in the form of inharmonicity is considered. The paper proposes a method for estimating the fundamental frequencies by, in the frequency domain, mapping each found spectral line to a set of candidate fundamental frequencies. The optimal mapping is found as the solution to a minimimal transport problem, wherein mappings leading to sparse pitch representations are promoted. The presented formulation is shown to yield robustness to varying degrees of inharmonicity without requiring explicit knowledge of the structure or scope of the inharmonicity.
In all three papers, the performance of the proposed methods are evaluated using simulated signals as well as real audio.
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Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1f08439cfe974e07a39265c2aa15b9ef
 author
 Elvander, Filip ^{LU}
 supervisor

 Andreas Jakobsson ^{LU}
 organization
 publishing date
 2017
 type
 Thesis
 publication status
 published
 subject
 keywords
 multipitch estimation, sparse modeling, convex optimisation, inharmonicity, sparse recursive least squares, adaptive signal processing, optimal transport distance
 pages
 139 pages
 publisher
 Lund University / Centre for Mathematical Sciences /LTH
 language
 English
 LU publication?
 yes
 id
 1f08439cfe974e07a39265c2aa15b9ef
 date added to LUP
 20170614 11:09:00
 date last changed
 20181121 21:32:45
@misc{1f08439cfe974e07a39265c2aa15b9ef, abstract = {This thesis considers sparse modeling and estimation of multipitch signals, i.e., signals whose frequency content can be described by superpositions of harmonic, or closetoharmonic, structures, characterized by a set of fundamental frequencies. As the number of fundamental frequencies in a given signal is in general unknown, this thesis casts the estimation as a sparse reconstruction problem, i.e., estimates of the fundamental frequencies are produced by finding a sparse representation of the signal in a dictionary containing an overcomplete set of pitch atoms. This sparse representation is found by using convex modeling techniques, leading to highly tractable convex optimization problems from whose solutions the estimates of the fundamental frequencies can be deduced.<br/><br/>In the first paper of this thesis, a method for multipitch estimation for stationary signal frames is proposed. Building on the heuristic of spectrally smooth pitches, the proposed method produces estimates of the fundamental frequencies by minimizing a sequence of penalized least squares criteria, where the penalties adapt to the signal at hand. An efficient algorithm building on the alternating direction method of multipliers is proposed for solving these least squares problems.<br/><br/>The second paper considers a timerecursive formulation of the multipitch estimation problem, allowing for the exploiting of longerterm correlations of the signal, as well as fundamental frequency estimates with a samplelevel time resolution. Also presented is a signaladaptive dictionary learning scheme, allowing for smooth tracking of frequency modulated signals.<br/><br/>In the third paper of this thesis, robustness to deviations from the harmonic model in the form of inharmonicity is considered. The paper proposes a method for estimating the fundamental frequencies by, in the frequency domain, mapping each found spectral line to a set of candidate fundamental frequencies. The optimal mapping is found as the solution to a minimimal transport problem, wherein mappings leading to sparse pitch representations are promoted. The presented formulation is shown to yield robustness to varying degrees of inharmonicity without requiring explicit knowledge of the structure or scope of the inharmonicity.<br/><br/>In all three papers, the performance of the proposed methods are evaluated using simulated signals as well as real audio.<br/>}, author = {Elvander, Filip}, language = {eng}, note = {Licentiate Thesis}, publisher = {Lund University / Centre for Mathematical Sciences /LTH}, title = {Sparse Modeling of Harmonic Signals}, url = {https://lup.lub.lu.se/search/ws/files/26854757/thesis_FilipElvander.pdf}, year = {2017}, }