An easy way to deal with this is to resample at a uniform rate first.
Ford, C.; Etter, D.M., "Wavelet basis reconstruction of nonuniformly sampled data," in Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on , vol.45, no.8, pp.1165-1168, Aug 1998
doi: 10.1109/82.718832
keywords: {sampled data systems;signal reconstruction;signal resolution;signal sampling;wavelet transforms;global information;multiresolutional setting;nonuniformly sampled data;signal processing technique;wavelet basis reconstruction;Discrete wavelet transforms;Multiresolution analysis;Nonuniform sampling;Sampling methods;Signal analysis;Signal processing;Signal reconstruction;Signal resolution;Signal sampling;Wavelet analysis},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=718832&isnumber=15534
There is no need to re-sample.
The discrete Fourier transform (DFT) may be derived from linear regression, where a complete set of N complex sinusoids are the basis functions to be fitted to N points, using
b = inv(X'*X)*X'*Y, (1)
where Y is a column vector of length N containing the sampled signal; X is an NxN matrix, containing the basis functions along its columns, evaluated at the sample times; and b is your complex spectrum - a column vector of length N .
For a uniform sampling rate and a uniform error-weighting function, the basis set is orthonormal, so X'*X = I (the identity matrix). This tends to hide this fundamental fact. In this case, we simply get
b = X'*Y. (2)
So for uniform sampling and a uniform error weight, we use (2), perhaps without realizing that we are actually doing linear regression.
For non uniform sampling (and a uniform weight), you will need to use (1), because the basis set will not be orthogonal.
- Badges
- Science topic
- Similar topics
- Engineering
- Electrical Engineering