A fast algorithm for maximum likelihood-based fundamental frequency estimation

Abstract

Periodic signals are encountered in many applications. Such signals can be modelled by a weighted sum of sinusoidal components whose frequencies are integer multiples of a fundamental frequency. Given a data set, the fundamental frequency can be estimated in many ways including a maximum likelihood (ML) approach. Unfortunately, the ML estimator has a very high computational complexity, and the more inaccurate, but faster correlation-based estimators are therefore often used instead. In this paper, we propose a fast algorithm for the evaluation of the ML cost function for complex-valued data over all frequencies on a Fourier grid and up to a maximum model order. The proposed algorithm significantly reduces the computational complexity to a level not far from the complexity of the popular harmonic summation method which is an approximate ML estimator.

Grid size selection for nonlinear least-squares optimisation in spectral estimation and array processing

“…. , L} must be found, and an efficient algorithm for obtaining this is given in [19] which reduces the computational complexity of the ML estimator to the same order as that of the HS method. We therefore evaluate the former here and report the total computation time of computing estimates for all orders up to L = 10.…”

Fast and statistically efficient fundamental frequency estimation

“…Although most periodic signals are real-valued, a complexvalued representation is often used instead for analytical and computational reasons. We recently demonstrated the latter point in [19] where we reduced the computational complexity of the ML estimator for the complex-valued signal model corresponding to (2). This was achieved by exploiting the Toeplitz structure of the complex analogue to Z T l (ω0)Z l (ω0).…”

“…This was achieved by exploiting the Toeplitz structure of the complex analogue to Z T l (ω0)Z l (ω0). In the real-valued case, however, Z T l (ω0)Z l (ω0) has a much more complicated block Toeplitz-plusHankel structure so the proposed algorithm in [19] cannot directly be applied to the real-valued case. Instead, we reformulate the problem so that an efficient and recursive Toeplitz-plus-Hankel solver can be used.…”