Non-uniform signal sampling

Discrete sample representation of continuous signals without a grid

Signal sampling without a uniform grid and thus a simple Nyquist Theorem. It turns out that this generalisation is not necessarily fatal for the theory.

Reviews in a functional analysis setting are given in (Piroddi and Petrou 2004; Babu and Stoica 2010; Unser 2000; Adcock et al. 2014; Adcock and Hansen 2016).

This problem AFAICT becomes much easier if one can use priors to provide a theoretically tractable model of the nonuniformly sampled signal. The Gaussian process formalism for probabilistic spectral analysis, is one such method is even computationally tractable using the methods of e.g. Saatçi (2012).

For FFT of unevenly sampled points, you can try the Non uniform FFT. (β€œNuFFT”)

Implementations of non-uniform sampling methods.

πŸ— Lombβ€”Scargle periodogram and its uses.


Adcock, Ben, and Anders C. Hansen. 2016. β€œGeneralized Sampling and Infinite-Dimensional Compressed Sensing.” Foundations of Computational Mathematics 16 (5): 1263–323.
Adcock, Ben, Anders Hansen, Bogdan Roman, and Gerd Teschke. 2014. β€œGeneralized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, 182:187–279. Elsevier.
Aldroubi, Akram, and Karlheinz GrΓΆchenig. 2001. β€œNonuniform Sampling and Reconstruction in Shift-Invariant Spaces.” SIAM Review 43 (4): 585–620.
Amini, Arash, and Farokh Marvasti. 2008. β€œConvergence Analysis of an Iterative Method for the Reconstruction of Multi-Band Signals from Their Uniform and Periodic Nonuniform Samples.” Sampling Theory in Signal & Image Processing 7 (2).
Babu, Prabhu, and Petre Stoica. 2010. β€œSpectral Analysis of Nonuniformly Sampled Data – a Review.” Digital Signal Processing 20 (2): 359–78.
Benedetto, John J. 1992. β€œIrregular Sampling and Frames.” In Wavelets: A Tutorial in Theory and Applications, edited by Charles K. Chui, 2:445–507. Wavelet Analysis and Its Applications. San Diego: Academic Press.
Broersen, P. M. T., and R. Bos. 2006. β€œEstimating Time-Series Models from Irregularly Spaced Data.” In IEEE Transactions on Instrumentation and Measurement, 55:1124–31.
Broersen, Piet M. T. 2005. β€œTime Series Analysis for Irregularly Sampled Data.” IFAC Proceedings Volumes, 16th IFAC World Congress, 38 (1): 154–59.
Eldar, Y. C., and A. V. Oppenheim. 2000. β€œFilterbank Reconstruction of Bandlimited Signals from Nonuniform and Generalized Samples.” IEEE Transactions on Signal Processing 48 (10): 2864–75.
Feichtinger, Hans G., and Karlheinz GrΓΆchenig. 1989. β€œMultidimensional Irregular Sampling of Band-Limited Functions in Lp-Spaces.” In Multivariate Approximation Theory IV, 135–42. International Series of Numerical Mathematics / Internationale Schriftenreihe Zur Numerischen Mathematik / SΓ©rie Internationale d’Analyse NumΓ©rique. BirkhΓ€user Basel.
β€”β€”β€”. 1992. β€œIterative Reconstruction of Multivariate Band-Limited Functions from Irregular Sampling Values.” SIAM Journal on Mathematical Analysis 23 (1): 244–61.
β€”β€”β€”. 1994. β€œTheory and Practice of Irregular Sampling.” Wavelets: Mathematics and Applications 1994: 305–63.
Feichtinger, Hans G., Karlheinz GrΓΆchenig, and Thomas Strohmer. 1995. β€œEfficient Numerical Methods in Non-Uniform Sampling Theory.” Numerische Mathematik 69 (4): 423–40.
Feichtinger, Hans G., and Thomas Strohmer. 1992. β€œFast Iterative Reconstruction of Band-Limited Images from Non-Uniform Sampling Values.” In SpringerLink, 231:82–89. Springer Berlin Heidelberg.
Feichtinger, Hans G., and Thomas Werther. 2000. β€œImproved Locality for Irregular Sampling Algorithms.” In IEEE International Conference on Acoustics, Speech, and Signal Processing, 2000. ICASSP ’00. Proceedings, 6:3834–3837 vol.6.
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Greengard, L., and J. Lee. 2004. β€œAccelerating the Nonuniform Fast Fourier Transform.” SIAM Review 46 (3): 443–54.
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Haris, Asad, Ali Shojaie, and Noah Simon. n.d. β€œWavelet Regression and Additive Models for Irregularly Spaced Data,” 11.
Jones, Richard H. 1984. β€œFitting Multivariate Models to Unequally Spaced Data.” In Time Series Analysis of Irregularly Observed Data, 158–88. Springer.
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Larsson, Erik K., and Torsten SΓΆderstrΓΆm. 2002. β€œIdentification of Continuous-Time AR Processes from Unevenly Sampled Data.” Automatica 38 (4): 709–18.
Maravic, I., and M. Vetterli. 2005. β€œSampling and Reconstruction of Signals with Finite Rate of Innovation in the Presence of Noise.” IEEE Transactions on Signal Processing 53 (8): 2788–2805.
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Martin, R. J. 1998. β€œAutoregression and Irregular Sampling: Filtering.” Signal Processing 69 (3): 229–48.
β€”β€”β€”. 1999. β€œAutoregression and Irregular Sampling: Spectral Estimation.” Signal Processing 77 (2): 139–57.
Marvasti, F. A., and L. Chuande. 1990. β€œParseval Relationship of Nonuniform Samples of One- and Two-Dimensional Signals.” IEEE Transactions on Acoustics, Speech, and Signal Processing 38 (6): 1061–63.
Marvasti, F., M. Analoui, and M. Gamshadzahi. 1991. β€œRecovery of Signals from Nonuniform Samples Using Iterative Methods.” IEEE Transactions on Signal Processing 39 (4): 872–78.
Marvasti, Farokh. 2012. Nonuniform Sampling: Theory and Practice. Springer Science & Business Media.
Mobli, Mehdi, and Jeffrey C. Hoch. 2014. β€œNonuniform Sampling and Non-Fourier Signal Processing Methods in Multidimensional NMR.” Progress in Nuclear Magnetic Resonance Spectroscopy 83 (November): 21–41.
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