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# Fundamental frequency estimation

The fundamental frequency (F0) is central in describing speech signals whereby we need methods for estimating the F0 from speech signals. In speech analysis applications, it can be informative to study the absolute value of the fundamental frequency as such, but more commonly, extraction of the F0 is usually a pre-processing step. For example, in recognition tasks, F0 is often used as a feature for machine learning methods. A voice activity detector could, for instance, set a lower and higher threshold on the F0, such that sounds with an F0 outside the valid range would be classified as non-speech.

The fundamental frequency is visible in multiple different domains:

• In an acoustic time-signal, the F0 is visible as a repetition after every T samples.
• In the autocovariance or -correlation, the F0 is visible as a peak at lag T as well as its integer multiples kT.
• In the magnitude, power or log-magnitude spectrum, the F0 is visible as a peak at the frequency F0=Fs/T, as well as its integer multiples, where Fs is the sampling frequency.
• In the cepstrum, the F0 is visible as a peak at quefrency T as well as its integer multiples kT.

Consequently, we can use any of these domains to estimate the fundamental frequency F0. A typical approach applicable in all domains except the time-domain, is peak-picking. The fundamental frequency corresponds to a peak in each domain, such that we can determine the F0 by finding the highest peak. For better robustness to spurious peaks and for computational efficiency, we naturally limit our search to the range of valid F0's, such as  $$80\leq F_0\leq 450.$$

The harmonic structure however poses a problem for peak-picking. Peaks appear at integer multiples of either F0 or lag T, such that sometimes, by coincidence or due to estimation errors, the harmonic peaks can be higher than the primary peak. Such estimation errors are known as octave errors, because the error in F0 corresponds to the musical interval of an octave. A typical post-processing step is therefore to check for octave jumps. We can check whether F0/2 or F0/3 would correspond to a sensible F0. Alternatively, we can check whether the previous analysis frame had an F0 which an octave or two octaves off. Depending on application, we can then fix errors or label problematic zones for later use.

Another problem with peak-picking is that peak locations might not align with the samples. For example in the autocorrelation domain, the true length of the period could be 100.3 samples. However, the peak in the autocorrelation would then appear at lag 100. One approach would then be to use quadratic interpolation between samples in the vicinity of a peak and use the location of the maximum of the interpolated peak as an estimate of the peak location. Interpolation makes the estimate less sensitive to noise. For example, background noise could happen to have a peak at lag 102, such that desired maximum of 100 is lower than the peak at 102. By using data from more samples, as in the interpolation approach, we can therefore reduce the likelihood that a single corrupted data point would cause an error.

An alternative approach to peak-picking is to use all distinctive peaks to jointly estimate the F0. That is, if you find N peaks at frequencies pk, which approximately correspond to harmonic peaks kF0, then you can approximate $$F0 \approx \frac1N \sum_{k=1}^N p_k/k.$$ Another alternative is to calculate the distance between consecutive peaks an estimate $$F0 \approx \frac1{N-1} \sum_{k=1}^{N-1} (p_{k+1}-p_k).$$ We can also combine these methods as we like.

In many other applications, we use discrete Fourier transform (DFT) or cosine transforms (DCT) to resolve frequency components. It would therefore be tempting to apply the same approach also here. In such a domain we would also already have a joint estimate which does not rely on single data points. However, note that the spectrum is already the DFT of the time signal and the cepstrum is the DCT or DFT of the log-spectrum. Additional transforms therefore usually do not resolve the ambiguity between harmonics.

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