Fast algorithms for fundamental frequency estimation in autoregressive noise

Barry Gerard Quinn, Jesper Kjær Nielsen*, Mads Græsbøll Christensen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


Many signals can accurately be modelled as a periodic function in coloured noise. An important parameter of the periodic function is the fundamental frequency. Often, fundamental frequency estimators are either ad hoc or have been derived under a white Gaussian noise (WGN) assumption. In this paper, we first derive the joint maximum likelihood (ML) estimator of the fundamental frequency estimator in autoregressive noise. Since a naïve implementation of this ML estimator has a very high computational complexity, we derive three fast algorithms that produce either exact or asymptotically equivalent estimators for all candidate sinusoidal and AR-orders. Through experiments, we show that the fast algorithms are at least two orders of magnitude faster than the naïve implementation and that the two fast approximate algorithm are faster and have a worse time-frequency resolution than the fast exact algorithm. Moreover, we show that jointly estimating the fundamental frequency and AR-parameters using our fast, exact algorithm is both faster and more accurate than computing the estimates iteratively. Finally, we apply the estimator to real data to show examples of how modelling the noise to be coloured significantly reduces the number of outliers produced by the fundamental frequency estimator compared to modelling the noise as WGN.

Original languageEnglish
Article number107860
Pages (from-to)1-16
Number of pages16
JournalSignal Processing
Publication statusPublished - Mar 2021


  • Harmonic regression
  • Coloured noise estimation
  • Fundamental frequency estimation
  • Pitch estimation


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