Signal sampling without equal bins and thus a simple Nyquist Theorem. It turns out that this is not fatal for many signals.

Reviews in BaSt10, PiPe04, Eng07, Unse00.

This is sometimes extended to include signal reconstructions with *uncertain* sampling times.

For FFT of unevenly sampled points, you can try the Non uniform FFT (“NuFFT”).

Implementations:

TODO: Lomb–Scargle normalized periodogram and its uses.

## Refs

- UTAK14: M. Unser, P. D. Tafti, A. Amini, H. Kirshner (2014) A unified formulation of Gaussian vs sparse stochastic processes - Part II: Discrete-domain theory.
*IEEE Transactions on Information Theory*, 60(5), 3036–3051. DOI - UnTS14: M. Unser, P. D. Tafti, Q. Sun (2014) A unified formulation of Gaussian vs sparse stochastic processes—part i: continuous-domain theory.
*IEEE Transactions on Information Theory*, 60(3), 1945–1962. DOI - UnTa14: Michael A. Unser, Pouya Tafti (2014)
*An introduction to sparse stochastic processes*. New York: Cambridge University Press - AmUM11: Arash Amini, Michael Unser, Farokh Marvasti (2011) Compressibility of deterministic and random infinite sequences.
*IEEE Transactions on Signal Processing*, 59(11), 5193–5201. - LaFR04: E. Lahalle, G. Fleury, A. Rivoira (2004) Continuous ARMA spectral estimation from irregularly sampled observations. In Proceedings of the 21st IEEE Instrumentation and Measurement Technology Conference, 2004. IMTC 04 (Vol. 2, pp. 923-927 Vol.2). DOI
- TaGo08: V. Y. F. Tan, V. K. Goyal (2008) Estimating Signals With Finite Rate of Innovation From Noisy Samples: A Stochastic Algorithm.
*IEEE Transactions on Signal Processing*, 56(10), 5135–5146. DOI - BrBo06: P. M. T. Broersen, R. Bos (2006) Estimating time-series models from irregularly spaced data. In IEEE Transactions on Instrumentation and Measurement (Vol. 55, pp. 1124–1131). DOI
- Jone81: Richard H. Jones (1981) Fitting a continuous time autoregression to discrete data. In Applied time series analysis II (pp. 651–682).
- Jone84: Richard H. Jones (1984) Fitting multivariate models to unequally spaced data. In Time series analysis of irregularly observed data (pp. 158–188). Springer
- LaSö02: Erik K. Larsson, Torsten Söderström (2002) Identification of continuous-time AR processes from unevenly sampled data.
*Automatica*, 38(4), 709–718. DOI - SuUn12: Qiyu Sun, Michael Unser (2012) Left-inverses of fractional Laplacian and sparse stochastic processes.
*Advances in Computational Mathematics*, 36(3), 399–441. - LiMa92: Keh-Shin Lii, Elias Masry (1992) Model fitting for continuous-time stationary processes from discrete-time data.
*Journal of Multivariate Analysis*, 41(1), 56–79. DOI - Wolf82: Stephen James Wolfe (1982) On a continuous analogue of the stochastic difference equation Xn=[rho]Xn-1+Bn.
*Stochastic Processes and Their Applications*, 12(3), 301–312. DOI - SöMo00: T. Söderström, M. Mossberg (2000) Performance evaluation of methods for identifying continuous-time autoregressive processes.
*Automatica*, 1(36), 53–59. DOI - MaVB06: P. Marziliano, M. Vetterli, T. Blu (2006) Sampling and exact reconstruction of bandlimited signals with additive shot noise.
*IEEE Transactions on Information Theory*, 52(5), 2230–2233. DOI - MaVe05: I. Maravic, M. Vetterli (2005) Sampling and reconstruction of signals with finite rate of innovation in the presence of noise.
*IEEE Transactions on Signal Processing*, 53(8), 2788–2805. DOI - Unse15: M. Unser (2015) Sampling and (sparse) stochastic processes: A tale of splines and innovation. In 2015 International Conference on Sampling Theory and Applications (SampTA) (pp. 221–225). DOI
- BKNU13: E. Bostan, U. S. Kamilov, M. Nilchian, M. Unser (2013) Sparse Stochastic Processes and Discretization of Linear Inverse Problems.
*IEEE Transactions on Image Processing*, 22(7), 2699–2710. DOI - Scar81: Jeffrey D Scargle (1981) Studies in astronomical time series analysis I-Modeling random processes in the time domain.
*The Astrophysical Journal Supplement Series*, 45, 1–71. - Broe05: Piet M. T. Broersen (2005) Time Series Analysis for Irregularly Sampled Data.
*IFAC Proceedings Volumes*, 38(1), 154–159. DOI