# Uncertainty principles and signal recovery

@article{Donoho1989UncertaintyPA, title={Uncertainty principles and signal recovery}, author={David L. Donoho and Philip B. Stark}, journal={Siam Journal on Mathematical Analysis}, year={1989} }

The uncertainty principle can easily be generalized to cases where the “sets of concentration” are not intervals. Such generalizations are presented for continuous and discrete-time functions, and for several measures of “concentration” (e.g., $L_2 $ and $L_1 $ measures). The generalizations explain interesting phenomena in signal recovery problems where there is an interplay of missing data, sparsity, and bandlimiting.

## 883 Citations

Uncertainty principles for signal concentrations

- Mathematics, Computer Science2006 Australian Communications Theory Workshop
- 2006

It is shown that an uncertainty principle exists for concentration of single-frequency signals for regions in space and that the uncertainty is related to the volumes of the spatial regions.

Recovery of a Sparse Signal When the Low Frequency Information is Missing By

- 2008

We develop inequalities for the concentration of bandlimited functions to sets of small density. The L2 inequality implies that wideband signals concentrated on sets of density < 1/6 can be…

Discrete Uncertainty Principles and Sparse Signal Processing

- Mathematics, Computer ScienceArXiv
- 2015

New discrete uncertainty principles in terms of numerical sparsity, which is a continuous proxy for the 0-norm, are developed and certain consequences in a number of sparse signal processing applications are identified.

Recovery of Signals with Low Density

- Mathematics, Computer ScienceArXiv
- 2015

A novel signal-density measure is introduced that extends the common notion of sparsity to non-sparse signals whose entries' magnitudes decay rapidly and derives a more general and less restrictive kernel result and uncertainty relation.

Signal recovery and the large sieve

- Mathematics
- 1992

Inequalities are developed for the fraction of a bandlimited function’s $L_p $ norm that can be concentrated on any set of small “Nyquist density.” Two applications are mentioned. First, that a…

Support-Limited Generalized Uncertainty Relations on Fractional Fourier Transform

- Mathematics
- 2015

This paper investigates the generalized
uncertainty principles of fractional Fourier transform (FRFT) for concentrated
data in limited supports. The continuous and discrete generalized uncertainty…

1 Uncertainty Relations and Sparse Signal Recovery

- 2019

This chapter provides a principled introduction to uncertainty relations underlying sparse signal recovery. We start with the seminal work by Donoho and Stark, 1989, which defines uncertainty…

A survey of uncertainty principles and some signal processing applications

- Computer Science, MathematicsAdv. Comput. Math.
- 2014

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, highlight their mutual connections and investigate practical consequences. The discussion…

Sparse signal recovery in Hilbert spaces

- Computer Science, Mathematics2012 IEEE International Symposium on Information Theory Proceedings
- 2012

This paper reports an effort to consolidate numerous coherence-based sparse signal recovery results available in the literature and presents a single theory that applies to general Hilbert spaces with the sparsity of a signal defined as the number of subspaces participating in the signal's representation.

General uncertainty relations and sparse signal recovery

- Computer Science
- 2011

Improved sparsity thresholds guaranteeing perfect recovery of signals that are sparse in general dictionaries are derived, which can improve on the well-known (1 + 1/μ)/2-threshold for dictionaries with coherence μ by up to a factor of two.

## References

SHOWING 1-10 OF 27 REFERENCES

The uncertainty principle on groups

- Mathematics
- 1990

The classical uncertainty principle asserts that both a function and its Fourier transform cannot be largely concentrated on intervals of small measure. Donoho and Stark [SIAM J. App. Math., 49…

Constrained iterative restoration algorithms

- Computer ScienceProceedings of the IEEE
- 1981

It is shown that by predistorting the signal (and later removing this predistortion) it is possible to achieve spectral extrapolation, to broaden the class of signals for which these algorithms achieve convergence, and to improve their performance in the presence of broad-band noise.

Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution

- Mathematics
- 1981

An algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its Fourier components. It is shown that as little as 20–25 percent of the Fourier spectrum is…

The recovery of distorted band-limited signals

- Mathematics
- 1961

Abstract In this paper we devise a method for recovering band limited signals which have been subjected to a distorting procedure called companding. A band limited signal is a function in L 2 whose…

Super-resolution through Error Energy Reduction

- Physics
- 1974

A new view of the problem of continuing a given segment of the spectrum of a finite object is presented. Based on this, the problem is restated in terms of reducing a defined ‘error energy’ which is…

Linear inversion of ban limit reflection seismograms

- Mathematics
- 1986

We present a method for the linear inversion (deconvolution) of band-limited reflection seismograms. A large convolution problem is first broken up into a sequence of smaller problems. Each small…

Eigenvalue distribution of time and frequency limiting

- Mathematics
- 1980

The operator in (1) consists of restricting $k(~) to the set cS, restricting the Fourier transform of the function so obtained to the set T, and viewing the result again on cS. We can therefore…

On support properties of Lp-functions and their Fourier transforms

- Mathematics
- 1977

We give a criterion for the intersection of two projections in Hilbert space to be a projection of finite-dimensional range. This criterion is applied to Schrodinger operators in L2(Rn) and to the…

Prolate spheroidal wave functions, fourier analysis and uncertainty — II

- Mathematics
- 1961

The theory developed in the preceding paper1 is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite-energy signal is given, the possible proportions…

Improvement of Range Resolution by Spectral Extrapolation

- Mathematics
- 1979

Under various simplifying assumptions, the reflected signal y(t) in the interrogation of a substance by an ultrasonic wave is a convolution of the transmitted signal x(t) with a function h(t) that is…