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Sparse Modeling of Harmonic Signals

Elvander, Filip LU (2017)
Abstract
This thesis considers sparse modeling and estimation of multi-pitch signals, i.e., signals whose frequency content can be described by superpositions of harmonic, or close-to-harmonic, 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 over-complete 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 multi-pitch signals, i.e., signals whose frequency content can be described by superpositions of harmonic, or close-to-harmonic, 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 over-complete 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 multi-pitch 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 time-recursive formulation of the multi-pitch estimation problem, allowing for the exploiting of longer-term correlations of the signal, as well as fundamental frequency estimates with a sample-level time resolution. Also presented is a signal-adaptive 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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author
supervisor
organization
publishing date
type
Thesis
publication status
published
subject
keywords
multi-pitch 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
1f08439c-fe97-4e07-a392-65c2aa15b9ef
date added to LUP
2017-06-14 11:09:00
date last changed
2017-06-21 13:41:42
@misc{1f08439c-fe97-4e07-a392-65c2aa15b9ef,
  abstract     = {This thesis considers sparse modeling and estimation of multi-pitch signals, i.e., signals whose frequency content can be described by superpositions of harmonic, or close-to-harmonic, 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 over-complete 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 multi-pitch 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 time-recursive formulation of the multi-pitch estimation problem, allowing for the exploiting of longer-term correlations of the signal, as well as fundamental frequency estimates with a sample-level time resolution. Also presented is a signal-adaptive 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},
  keyword      = {multi-pitch estimation,sparse modeling,convex optimisation,inharmonicity,sparse recursive least squares,adaptive signal processing,optimal transport distance},
  language     = {eng},
  note         = {Licentiate Thesis},
  pages        = {139},
  publisher    = {Lund University / Centre for Mathematical Sciences /LTH},
  title        = {Sparse Modeling of Harmonic Signals},
  year         = {2017},
}