Accurate Estimation of Low Fundamental Frequencies From Real-Valued Measurements

Abstract

In this paper, the difficult problem of estimating low fundamental frequencies from real-valued measurements is addressed. The methods commonly employed do not take the phenomena encountered in this scenario into account and thus fail to deliver accurate estimates. The reason for this is that they employ asymptotic approximations that are violated when the harmonics are not well-separated in frequency, something that happens when the observed signal is real-valued and the fundamental frequency is low. To mitigate this, we analyze the problem and present some exact fundamental frequency estimators that are aimed at solving this problem. These estimators are based on the principles of nonlinear least-squares, harmonic fitting, optimal filtering, subspace orthogonality, and shift-invariance, and they all reduce to already published methods for a high number of observations. In experiments, the methods are compared and the increased accuracy obtained by avoiding asymptotic approximations is demonstrated.

Estimating acoustic speech features in low signal-to-noise ratios using a statistical framework

“…A further benefit of parametric methods is that they have been shown to estimate accurately low fundamental frequencies which non-parametric methods tend to be less effective at doing (Christensen, 2013a).…”

Fast and statistically efficient fundamental frequency estimation

“…1, examples of the NLS and HS cost functions are shown. Clearly, HS is inaccurate when the fundamental frequency is low (see much more on this in [12]), but this is currently the price to pay to reduce the computational complexity. In the next section, however, we show how the NLS cost function can be computed with the same order of complexity as HS.…”

“…• When the fundamental frequency is very low, the columns of Z l (ω0) are nearly linearly dependent [12]. The matrix A l (ω0) is therefore ill-conditioned and may give numerical problems.…”

“…The former is going to play an important role later. For the model in (2), the ML estimate can be derived as in [12] tô…”

Default Bayesian Estimation of the Fundamental Frequency

“…Note that a realvalued signal of the form in (2) can be cast into the form of a complex-valued signal in (2) by computing its downsampled analytic signal [8]. Provided that the frequencies of the first and last harmonics are not too close to zero and the Nyquist frequency (relative to N ), respectively, the parameter estimates based on the down-sampled analytic signal are nearly identical to the corresponding parameter estimates based on the real-valued signal [2], [9]. In this paper, the focus is on the complex-valued signal model since it leads to simpler notation and faster algorithms [2], [10].…”