Signal sampling

Discrete representation of continuous signals and converse

DSP is all about when you can approximate discrete systems with continuous ones and vice versa. Sampling theorems. Nyquist rates, Compressive sampling, nonuniform signal sampling, stochastic signal sampling, signatures of rough paths, etc.

There are a few ways to frame this. Traditionally we talk about Shannon sampling theorems, Nyquist rates and so on. To be frank, I haven’t actually read Shannon, because the setup is not useful for the types of problems I face in my work, although I’m sure it boils down to some similar results.

The received-wisdom version of the Shannon theorem is that you can reconstruct a signal if you know it has frequencies in it that are “too high”. Specifically, if you sample a continuous time signal at intervals of \(T\) seconds, then you had better have no frequencies of period shorter than \(2T\). I’m playing fast-and-loose with definitions here - the spectrum here is the continuous Fourier spectrogram. If you do much non-trivial signal processing, (in my case I constantly need to do things like multiplying signals) it rapidly becomes impossible to maintain bounds on the support of the spectrogram (TODO explain this with diagrams)

This doesn’t tell us much about more bizarre nonuniform sampling regimes, mild violations of frequency constraints, or whether other sets of (perhaps more domain-appropriate) constraints on our signals will lead to a sensible reconstruction theory.

Let’s talk about the modern, abstract and fashionable Hilbert-space framing of this problem This way is general, and based on projections between Hilbert spaces. Nice works in this tradition are, e.g. (Vetterli, Marziliano, and Blu 2002) that observes that you don’t care about Fourier spectrogram support, but rather the rate of degrees of freedom to construct a coherent sampling theory. Also accessible is (M. Unser 2000), which constructs the problem of discretising signals as a minimal-loss projection/reconstruction problem.

More recently you have fancy persons such as Adcock and Hansen unifying compressed sensing and signal sampling (Adcock et al. 2014; Adcock and Hansen 2016) with more or less the same framework, so I’ll dive into their methods here.



Adcock, Ben, and Anders C. Hansen. 2016. “Generalized Sampling and Infinite-Dimensional Compressed Sensing.” Foundations of Computational Mathematics 16 (5, 5): 1263–323.
Adcock, Ben, Anders C. Hansen, and Bogdan Roman. 2015. “The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing.” In Compressed Sensing and Its Applications: MATHEON Workshop 2013, edited by Holger Boche, Robert Calderbank, Gitta Kutyniok, and Jan Vybíral, 143–67. Applied and Numerical Harmonic Analysis. Cham: Springer International Publishing.
Adcock, Ben, Anders Hansen, Bogdan Roman, and Gerd Teschke. 2014. “Generalized 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 Gröchenig. 2001. “Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces.” SIAM Review 43 (4): 585–620.
Amini, Arash, and Farokh Marvasti. 2008. “Convergence 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. “Spectral Analysis of Nonuniformly Sampled Data – a Review.” Digital Signal Processing 20 (2): 359–78.
Baisch, Stefan, and Götz H. R. Bokelmann. 1999. “Spectral Analysis with Incomplete Time Series: An Example from Seismology.” Computers & Geosciences 25 (7): 739–50.
Bartlett, M. S. 1946. “On the Theoretical Specification and Sampling Properties of Autocorrelated Time-Series.” Supplement to the Journal of the Royal Statistical Society 8 (1): 27–41.
Bostan, E., U. S. Kamilov, M. Nilchian, and M. Unser. 2013. “Sparse Stochastic Processes and Discretization of Linear Inverse Problems.” IEEE Transactions on Image Processing 22 (7): 2699–2710.
Bretó, Carles, Daihai He, Edward L. Ionides, and Aaron A. King. 2009. “Time Series Analysis via Mechanistic Models.” The Annals of Applied Statistics 3 (1): 319–48.
Brémaud, Pierre, Laurent Massoulié, and Andrea Ridolfi. 2005. “Power Spectra of Random Spike Fields and Related Processes.” Advances in Applied Probability 37 (4): 1116–46.
Broersen, P. M. T., and R. Bos. 2006. “Estimating Time-Series Models from Irregularly Spaced Data.” In IEEE Transactions on Instrumentation and Measurement, 55:1124–31.
Broersen, Petrus MT. 2006. Automatic Autocorrelation and Spectral Analysis. Secaucus, NJ, USA: Springer.
Broersen, Piet M. T. 2005. “Time Series Analysis for Irregularly Sampled Data.” IFAC Proceedings Volumes, 16th IFAC World Congress, 38 (1): 154–59.
Broersen, Piet M. T., Stijn de Waele, and Robert Bos. 2004. “Autoregressive Spectral Analysis When Observations Are Missing.” Automatica 40 (9): 1495–1504.
Cauchemez, Simon, and Neil M. Ferguson. 2008. “Likelihood-Based Estimation of Continuous-Time Epidemic Models from Time-Series Data: Application to Measles Transmission in London.” Journal of The Royal Society Interface 5 (25): 885–97.
Cochran, W. T., James W. Cooley, D. L. Favin, H. D. Helms, R. A. Kaenel, W. W. Lang, Jr. Maling G. C., D. E. Nelson, C. M. Rader, and Peter D. Welch. 1967. “What Is the Fast Fourier Transform?” Proceedings of the IEEE 55 (10): 1664–74.
Coulaud, Benjamin, and Frédéric JP Richard. 2018. “A Consistent Framework for a Statistical Analysis of Surfaces Based on Generalized Stochastic Processes.”
Dumitrescu, Bogdan. 2017. Positive Trigonometric Polynomials and Signal Processing Applications. Second edition. Signals and Communication Technology. Cham: Springer.
Eldar, Y. C., and A. V. Oppenheim. 2000. “Filterbank Reconstruction of Bandlimited Signals from Nonuniform and Generalized Samples.” IEEE Transactions on Signal Processing 48 (10): 2864–75.
Feichtinger, Hans G., and Karlheinz Gröchenig. 1989. “Multidimensional 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 / Série Internationale d’Analyse Numérique. Birkhäuser Basel.
———. 1992. “Iterative Reconstruction of Multivariate Band-Limited Functions from Irregular Sampling Values.” SIAM Journal on Mathematical Analysis 23 (1): 244–61.
———. 1994. “Theory and Practice of Irregular Sampling.” Wavelets: Mathematics and Applications 1994: 305–63.
Feichtinger, Hans G., Karlheinz Gröchenig, and Thomas Strohmer. 1995. “Efficient Numerical Methods in Non-Uniform Sampling Theory.” Numerische Mathematik 69 (4): 423–40.
Feichtinger, Hans G., and Thomas Strohmer. 1992. “Fast 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. “Improved Locality for Irregular Sampling Algorithms.” In IEEE International Conference on Acoustics, Speech, and Signal Processing, 2000. ICASSP ’00. Proceedings, 6:3834–3837 vol.6.
Fessler, Jeffrey A., and Bradley P. Sutton. 2003. “Nonuniform Fast Fourier Transforms Using Min-Max Interpolation.” IEEE Transactions on Signal Processing 51 (2).
Finzi, Marc, Roberto Bondesan, and Max Welling. 2020. “Probabilistic Numeric Convolutional Neural Networks.” October 21, 2020.
García, Antonio G. 2002. “A Brief Walk Through Sampling Theory.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, 124:63–137. Elsevier.
Gray, R. 1984. “Vector Quantization.” IEEE ASSP Magazine 1 (2): 4–29.
Greengard, L., and J. Lee. 2004. “Accelerating the Nonuniform Fast Fourier Transform.” SIAM Review 46 (3): 443–54.
Gröchenig, Karlheinz. 1992. “Reconstruction Algorithms in Irregular Sampling.” Mathematics of Computation 59 (199): 181–94.
———. 1993. “A Discrete Theory of Irregular Sampling.” Linear Algebra and Its Applications 193 (November): 129–50.
Jones, Richard H. 1981. “Fitting a Continuous Time Autoregression to Discrete Data.” In Applied Time Series Analysis II, 651–82.
———. 1984. “Fitting Multivariate Models to Unequally Spaced Data.” In Time Series Analysis of Irregularly Observed Data, 158–88. Springer.
Kazhdan, Michael, Matthew Bolitho, and Hugues Hoppe. 2006. “Poisson Surface Reconstruction.” In Sgp06: Eurographics Symposium on Geometry Processing, 1:0. The Eurographics Association.
Lahalle, E., G. Fleury, and 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, 2:923–927 Vol.2.
Landau, H. J. 1967. “Necessary Density Conditions for Sampling and Interpolation of Certain Entire Functions.” Acta Mathematica 117 (1): 37–52.
Larsson, Erik K., and Torsten Söderström. 2002. “Identification of Continuous-Time AR Processes from Unevenly Sampled Data.” Automatica 38 (4): 709–18.
Lii, Keh-Shin, and Elias Masry. 1992. “Model Fitting for Continuous-Time Stationary Processes from Discrete-Time Data.” Journal of Multivariate Analysis 41 (1): 56–79.
Maravic, I., and 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.
Margolis, E., and Y. C. Eldar. 2008. “Nonuniform Sampling of Periodic Bandlimited Signals.” IEEE Transactions on Signal Processing 56 (7): 2728–45.
Marple, S. Lawrence, Jr. 1987. Digital Spectral Analysis with Applications.
Martin, R. J. 1998. “Autoregression and Irregular Sampling: Filtering.” Signal Processing 69 (3): 229–48.
———. 1999. “Autoregression and Irregular Sampling: Spectral Estimation.” Signal Processing 77 (2): 139–57.
Marvasti, F. A., and L. Chuande. 1990. “Parseval 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. “Recovery 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.
Marziliano, P., M. Vetterli, and T. Blu. 2006. “Sampling and Exact Reconstruction of Bandlimited Signals with Additive Shot Noise.” IEEE Transactions on Information Theory 52 (5): 2230–33.
Matheron, G. 1973. “The Intrinsic Random Functions and Their Applications.” Advances in Applied Probability 5 (3): 439–68.
McCrorie, J. Roderick. 2002. “The Likelihood of the Parameters of a Continuous Time Vector Autoregressive Model.” Statistical Inference for Stochastic Processes 5 (3): 273–86.
Mishali, M., and Y. C. Eldar. 2009. “Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals.” IEEE Transactions on Signal Processing 57 (3): 993–1009.
Mishali, Moshe, and Yonina C. Eldar. 2010. “From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals.” IEEE Journal of Selected Topics in Signal Processing 4 (2): 375–91.
Mobli, Mehdi, and Jeffrey C. Hoch. 2014. “Nonuniform Sampling and Non-Fourier Signal Processing Methods in Multidimensional NMR.” Progress in Nuclear Magnetic Resonance Spectroscopy 83 (November): 21–41.
Murray-Smith, Roderick, and Barak A. Pearlmutter. 2005. “Transformations of Gaussian Process Priors.” In Deterministic and Statistical Methods in Machine Learning, edited by Joab Winkler, Mahesan Niranjan, and Neil Lawrence, 110–23. Lecture Notes in Computer Science. Springer Berlin Heidelberg. barak/papers/MLW-Jul-2005.pdf.
O’Callaghan, Simon Timothy, and Fabio T. Ramos. 2011. “Continuous Occupancy Mapping with Integral Kernels.” In Twenty-Fifth AAAI Conference on Artificial Intelligence.
Papavasiliou, Anastasia, and Kasia B. Taylor. 2016. “Approximate Likelihood Construction for Rough Differential Equations.” December 8, 2016.
Piroddi, Roberta, and Maria Petrou. 2004. “Analysis of Irregularly Sampled Data: A Review.” In Advances in Imaging and Electron Physics, 132:109–65. Advances in Imaging and Electron Physics. Elsevier.
Särkkä, Simo. 2007. “On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems.” IEEE Transactions on Automatic Control 52 (9): 1631–41.
Scargle, Jeffrey D. 1981. “Studies in Astronomical Time Series Analysis. I-Modeling Random Processes in the Time Domain.” The Astrophysical Journal Supplement Series 45: 1–71.
Smith, Julius O. 2018. “Digital Audio Resampling Home Page.” Center for Computer Research in Music and Acoustics (CCRMA), Stanford University. jos/resample/.
Söderström, T., and M. Mossberg. 2000. “Performance Evaluation of Methods for Identifying Continuous-Time Autoregressive Processes.” Automatica 1 (36): 53–59.
Stark, Jaroslav. 2001. “Delay Reconstruction: Dynamics Versus Statistics.” In Nonlinear Dynamics and Statistics, edited by Alistair I. Mees, 81–103. Birkhäuser Boston.
Stoica, Petre, and Niclas Sandgren. 2006. “Spectral Analysis of Irregularly-Sampled Data: Paralleling the Regularly-Sampled Data Approaches.” Digit. Signal Process. 16 (6): 712–34.
Strohmer, T. 1997. “Computationally Attractive Reconstruction of Bandlimited Images from Irregular Samples.” IEEE Transactions on Image Processing 6 (4): 540–48.
Sun, Qiyu, and Michael Unser. 2012. “Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes.” Advances in Computational Mathematics 36 (3): 399–441.
Tan, V. Y. F., and V. K. Goyal. 2008. “Estimating Signals With Finite Rate of Innovation From Noisy Samples: A Stochastic Algorithm.” IEEE Transactions on Signal Processing 56 (10): 5135–46.
Tarczynski, A., and N. Allay. 2004. “Spectral Analysis of Randomly Sampled Signals: Suppression of Aliasing and Sampler Jitter.” IEEE Transactions on Signal Processing 52 (12): 3324–34.
Tobar, Felipe. 2019. “Band-Limited Gaussian Processes: The Sinc Kernel.” Advances in Neural Information Processing Systems 32: 12749–59.
Tropp, J., J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk. 2010. “Beyond Nyquist: Efficient Sampling of Bandlimited Signals.” IEEE Transactions on Information Theory 56: 1–26.
Unser, M. 1999. “Splines: A Perfect Fit for Signal and Image Processing.” IEEE Signal Processing Magazine 16 (6): 22–38.
———. 2000. “Sampling: 50 Years After Shannon.” Proceedings of the IEEE 88 (4, 4): 569–87.
———. 2015. “Sampling and (sparse) Stochastic Processes: A Tale of Splines and Innovation.” In 2015 International Conference on Sampling Theory and Applications (SampTA), 221–25.
Unser, M., A. Aldroubi, and M. Eden. 1992. “Polynomial Spline Signal Approximations: Filter Design and Asymptotic Equivalence with Shannon’s Sampling Theorem.” IEEE Transactions on Information Theory 38 (1): 95–103.
Unser, M., P. D. Tafti, A. Amini, and H. Kirshner. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” IEEE Transactions on Information Theory 60 (5): 3036–51.
Unser, M., P. D. Tafti, and Q. Sun. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes—Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory 60 (3): 1945–62.
Unser, Michael A. 1995. “General Hilbert Space Framework for the Discretization of Continuous Signal Processing Operators.” In Wavelet Applications in Signal and Image Processing III, 2569:51–62. International Society for Optics and Photonics.
Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press.
Unser, Michael, and Akram Aldroubi. 1992. “Polynomial Splines and Wavelets-A Signal Processing Perspective.” In Wavelets, edited by Charles K Chui, 2:91–122. Wavelet Analysis and Its Applications. San Diego: Academic Press.
———. 1994. “A General Sampling Theory for Nonideal Acquisition Devices.” IEEE Transactions on Signal Processing 42 (11): 2915–25.
Venkataramani, R., and Y. Bresler. 2000. “Perfect Reconstruction Formulas and Bounds on Aliasing Error in Sub-Nyquist Nonuniform Sampling of Multiband Signals.” IEEE Transactions on Information Theory 46 (6): 2173–83.
Vetterli, M., P. Marziliano, and T. Blu. 2002. “Sampling Signals with Finite Rate of Innovation.” IEEE Transactions on Signal Processing 50 (6): 1417–28.
Wolfe, Stephen James. 1982. “On a Continuous Analogue of the Stochastic Difference Equation Xn=[rho]Xn-1+Bn.” Stochastic Processes and Their Applications 12 (3): 301–12.
Yadrenko, Mikhail Iosifovich. 1983. Spectral Theory of Random Fields. Translation Series in Mathematics and Engineering. New York, NY: Optimization Software.
Yaglom, A. M. 1987. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics. New York, NY: Springer Science & Business Media.
Yaroslavsky, Leonid P., Gil Shabat, Benny G. Salomon, Ianir A. Ideses, and Barak Fishbain. 2009. “Non-Uniform Sampling, Image Recovery from Sparse Data and the Discrete Sampling Theorem.” Journal of the Optical Society of America A 26 (3): 566.
Yen, J. 1956. “On Nonuniform Sampling of Bandwidth-Limited Signals.” IRE Transactions on Circuit Theory 3 (4): 251–57.

Warning! Experimental comments system! If is does not work for you, let me know via the contact form.

No comments yet!

GitHub-flavored Markdown & a sane subset of HTML is supported.