Multi-pitch estimation using semidefinite programming

Multi-pitch estimation concerns the problem of estimating the fundamental frequencies (pitches) and amplitudes/phases of multiple superimposed harmonic signals with application in music, speech, vibration analysis, and other fields. In this paper we formulate a complex-valued multi-pitch estimator via a semidefinite programming method for continuous sparse optimization over an infinite dictionary of vectors of complex exponentials and extend this to real-valued data via a real semidefinite program with the same dimensions (i.e. half the size). We further impose a continuous frequency constraint naturally occurring from assuming a Nyquist sampled signal by adding an additional semidefinite constraint. In our numerical experiments, the proposed estimator shows superior performance compared to state-of-the-art methods for separating two closely spaced fundamentals and approximately achieves the asymptotic Cramér-Rao lower bound.