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If there is an N-point sequence, it has a Fourier Transform. DFT of this sequence is the sampling of the Fourier Transform at N equally spaced points on the unit circle. What happens if we take some M non-uniformly spaced samples on the unit circle with M < N, M = N and M > N. Any studies done on this. If yes, what are the applications?

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  • $\begingroup$ Unevenly-Spaced frequency samples are pretty common. However, I'm not sure there is any real inherent benefit to them. The data are usually resampled to an evenly spaced grid for computational efficiency. $\endgroup$ – AnonSubmitter85 Apr 11 '16 at 16:09
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Uniform sampling is the definite standard as it simplifies not only mathematical derivations but also practical applications. On the other hand, there are some applications (such as array of sensors located in a nonuniform grid) in which taking uniformly distributed samples from a signal of interest is impossible. In such cases nonuniform sampling in either time or space is performed and the usual approach is to convert those nonuniform samples to uniform ones. Then follows the standard operations on those precomputed uniform samples.

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You might like to have a look at Wikipedia here. There are non-uniform Fourier Transformation applications. The article also gives some fields as example that use this technique.

One more application that I am aware of is magnetic resonance imaging (MRI). In MRI, image data is sampled in Fourier space, hence you simply do a FFT of your data and you have your image - as long as the imaging sequence samples the data on a Cartesian grid. However, there are techniques that do not obey this restriction, e.g. sample the data radially. By this, the measured data is distributed on concentric circles around the DC frequency and a simple FFT is therefore not suitable. It is possible to interpolate this data to a Cartesian grid and use FFT for the imaging process (this process is called regridding). Another technique is using a non-uniform FFT algorithm - which answers your question where it is used.

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  • $\begingroup$ :The link is talking about non-uniform sampling in time domain. I am talking about what happens if we sample non uniformly in the frequency domain (but still on the unit circle) for a uniformly sampled in time signal. $\endgroup$ – Seetha Rama Raju Sanapala Mar 12 '16 at 10:52
  • $\begingroup$ Sorry if this is not relevant for your question. However, I think that my MRI example still is valid in this case. There, or course, a 2D/3D frequency spectrum is sampled. $\endgroup$ – M529 Mar 12 '16 at 12:11
  • $\begingroup$ @SeethaRamaRaju There is no theoretical difference, so any useful information you find for the time domain will be applicable to the frequency domain as well. $\endgroup$ – AnonSubmitter85 Apr 11 '16 at 16:07

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