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  "body": "\n## Summary\n\nThe non-uniform discrete Fourier transform (NUDFT) is a variant of the [[discrete Fourier transform|TST0GH66A5]].\nIn it the sample points, the frequencies, or both are not equally spaced.\nIt evaluates finite sums $\\sum_n x_n e^{-2\\pi i p_n f_k}$ with arbitrary real $p_n$ and $f_k$.\nSuch data arise in medical imaging, astronomy and spectral methods, where measurements are not taken on a regular grid.\nFast approximate algorithms for it are called non-uniform FFTs.\n\n## Explanations\n\n### L1 human\n\nThe ordinary discrete Fourier transform assumes the signal was sampled at evenly spaced times.\nReal measurements often come at irregular times or positions, for instance along spiral paths in an MRI scanner.\nThe non-uniform transform uses the same kind of sum, but with the actual sample positions in the exponent.\nFor example, samples at times $0$, $0.3$ and $0.7$ give the coefficient $X_k = x_0 + x_1 e^{-0.6\\pi i k} + x_2 e^{-1.4\\pi i k}$ at frequency $k$.\nThe loss of regular spacing means the clever factorizations of the FFT no longer apply directly.\n",
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