Sparse Modeling of Harmonic Signals

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.