A survey of uncertainty principles and some signal processing applications

Abstract-Compressed sensing is a signal acquisition scheme that measures signals at sub-Nyquist rate amenable to sparse recovery, with high probability, from a reduced set of measurements. One of the main requirements of compressive sensing is the sparsity of the class of signals of interest in some basis. A method to construct a sparsifying basis for a class of signals using information theoretic measures is proposed in this paper. The algorithm constructs the sparsifying basis from a known nonsparsifying basis by concentrating the probability distribution of the basis in the representation of a class of signals. Simulation studies using speech and image signals confirm that the basis constructed using the proposed method results in an improved sparsity of the signals with thresholded coefficients but without degrading the signal quality.

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“…The relation between the entropies of representation of a signal in these bases is given by the entropy uncertainty relation [19] - [21].…”

Section: Probability Of Representation Basedmentioning

confidence: 99%

Transform learning algorithm based on the probability of representation of signals

Parthasarathy

Abhilash

2017

2017 25th European Signal Processing Conference (EUSIPCO)

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“…Considerable attention has been devoted recently to discovering new mathematical formulations and new contexts for the uncertainty principle (see the surveys [4,9,20] and the book [13] for other forms of the uncertainty principle). This paper will adopt the broader view that the uncertainty principle can be seen not only as a statement about the phase space (or time-frequency) localization of a single function but also as a statement on the degradation of localization when one considers successive elements of an orthonormal basis.…”

Section: Introductionmentioning

confidence: 99%

“…Then the Heisenberg-Pauli-Weil inequality (see e.g. [9,20]) leads to the following classical formulation of the uncertainty principle in form of the lower bound of the product of the dispersions of a unit-norm function in L 2 (R d ) and its Fourier transform:…”

Section: Introductionmentioning

confidence: 99%

Phase space localization of orthonormal sequences in Lα2(R+)

Ghobber

2015

Journal of Approximation Theory

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The aim of this paper is to prove a quantitative extension of Shapiro's result on the time-frequency concentration of orthonormal sequences in L 2 α (R + ). More precisely, we prove that, if {ϕ n } +∞ n=0 is an orthonormal sequence in L 2 α (R + ), then for all N ≥ 0and the equality is attained for the sequence of Laguerre functions. Particularly if the elements of an orthonormal sequence and their Fourier-Bessel transforms (or Hankel transforms) have uniformly bounded dispersions then the sequence is finite. Moreover we prove the following strong uncertainty principle for bases for L 2 α (R + ), that is if {ϕ n } +∞ n=0 is an orthonormal basis for L 2 α (R + ) and s > 0, then

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“…As elegant as the existing T-F analysis techniques are, however, there are intrinsic difficulties toward a deeper insight into the TEOAE. For the widely applied linear-type T-F analysis, like STFT, CWT and S-transform, the uncertainty principle Flandrin (1999); Ricaud and Torresani (2014) is inevitable. A direct consequence of the uncertainty principle is a blurring of the spectrum, depending on the chosen window and its length.…”

Section: Introductionmentioning

confidence: 99%

Analyzing transient-evoked otoacoustic emissions by concentration of frequency and time

Liu

2018

The Journal of the Acoustical Society of America

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The linear part of transient evoked otoacoustic emission (TEOAE) is thought to be generated via coherent reflection near the characteristic place of constituent wave components. Because of the tonotopic organization of the cochlea, high frequency emissions return earlier than low frequencies; however, due to the random nature of coherent reflection, the instantaneous frequency (IF) and amplitude envelope of TEOAEs both fluctuate. Multiple reflection components and synchronized spontaneous emissions can further make it difficult to extract the IF by linear transforms. This paper proposes to model TEOAEs as a sum of intrinsic mode-type functions and analyze it by a nonlinear-type time-frequency (T-F) analysis technique called concentration of frequency and time (ConceFT). When tested with synthetic otoacoustic emission signals with possibly multiple oscillatory components, the present method is able to produce clearly visualized traces of individual components on the T-F plane. Further, when the signal is noisy, the proposed method is compared with existing linear and bilinear methods in its accuracy for estimating the fluctuating IF. Results suggest that ConceFT outperforms the best of these methods in terms of optimal transport distance, reducing the error by 10% to 21% when the signal to noise ratio is 10 dB or below.

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A survey of uncertainty principles and some signal processing applications

Cited by 82 publications

References 28 publications

Transform learning algorithm based on the probability of representation of signals

Transform learning algorithm based on the probability of representation of signals

Phase space localization of orthonormal sequences in Lα2(R+)

Analyzing transient-evoked otoacoustic emissions by concentration of frequency and time

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