Fast algorithms for fundamental frequency estimation in autoregressive noise

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.