[hasley]

FFT/DFT is not precise if you do not have the exact harmonic in you signal. If you are also (or only) interested in phases you might use a maximum likelihood estimator (which brings other problems though).

And as the previous answer said: compressed sensing (or compressive sensing) can help as well for some non-standard cases.

[geokon]

Do you have any good reference for compressed sensing?

The high level description on wikipedia seems very compelling.. And would you say it'd be a huge task to really grok it?

[ghtbircshotbe]

Article by Terrence Tao: https://terrytao.wordpress.com/2007/04/13/compressed-sensing...

Paper by Stan Osher et al: https://arxiv.org/abs/1104.0262

[adinisom]

A while back I looked at matching pursuit. At first it seemed very complicated, but after staring at it a bit realized it's simple.

- Start with a list of basis functions and your signal.

- Go through the list and find the basis function that best correlates with the signal. This gives you a basis function and a coefficient.

- Subtract out the basis function (scaled by the coefficient) from your signal, and then repeat with this new residual signal.

The Fourier transform is similar using sine wave basis functions.

The key that makes this work in situations where the Nyquist theorem says we don't have a high enough sampling rate is ensuring our sampling (possibly random) is un-correlated with the basis functions and our basis functions are good approximations for the signal. That lowers the likelihood that our basis functions correlating well with our samples is by chance and raises likelihood it correlates well with the actual signal.