The optimal transform for the discrete Hirschman uncertainty principle

Przebinda, Tomasz; DeBrunner, V.; Özaydın, Murad

doi:10.1109/18.930948

Cited by 62 publications

(30 citation statements)

References 9 publications

(34 reference statements)

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“…With this new measure in place, we proceed to determine the effect of replacing the DFT with the discrete FRT. As in the original discrete Hirschman Uncertainty, we find that the picket fence function, and not any Gaussian waveform, is an optimizer [7].…”

Section: Introductionmentioning

confidence: 99%

A study on the impact of the fourier transform on Hirschman Uncertainty

Ghuman

DeBrunner

2015

2015 49th Asilomar Conference on Signals, Systems and Computers

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The Hirschman Uncertainty [1] is defined by the average of the Shannon entropies of a discrete-time signal and its Fourier transform. The picket fence functions has been found to be the optimal basis for the Hirschman Uncertainty, as given in a previous paper of ours [2]. We have seen that a basis can be constructed from signals with minimum Hirschman Uncertainty in that paper, leading to a transform we called the Hirschman Optimal Transform. In [3] and [4], we showed that the minimizers, and thus the uncertainty of these minimizers, are invariant to the Rényi entropy order. This characteristic strongly suggests that Hirschman Uncertainty is a fundamental characteristic of digital signals. In this paper, we study the effect of incorporating the discrete fractional Fourier transfom (dFRT) instead of the DFT in the predefined Hirschman Unceratinty and develop a new uncertainty measure Hirschman uncertainty using the dFRT denoted by U a 1 2 (x). We explore this new uncertainty measure using the discrete signals and study how the transfer order variations affects the uncertainty of different discrete signals. To help verify our theory, we study the effect of transfer order value variations on the classification rate in an image classification experiment.

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Section: Introductionmentioning

confidence: 99%

A study on the impact of the fourier transform on Hirschman Uncertainty

Ghuman

DeBrunner

2015

2015 49th Asilomar Conference on Signals, Systems and Computers

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“…Our entropy-based measure was used to show that discretized Gaussian pulses may not be the most compact basis with respect to joint timefrequency resolution. In [1], we found a basis (HOT transform) that is orthonormal and uniquely minimizes the discrete-time, discrete-frequency Hirschman uncertainty principle. For comparison, we considered a discretized Gaussian pulse, which is comparable to the HOT basis.…”

Section: Introductionmentioning

confidence: 99%

Spectral estimation with the Hirschman optimal transform filter bank and compressive sensing

DeBrunner

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

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The traditional Heisenberg-Weyl measure quantifies the joint localization, uncertainty, or concentration of a signal in the phase plane based on a product of energies expressed as signal variances in time and in frequency. Unlike the Heisenberg-Weyl measure, the Hirschman notion of joint uncertainty is based on the entropy rather than the energy [1]. Furthermore, as we noted in [2], the Hirschman optimal transform (HOT) is superior to the discrete Fourier transform (DFT) and discrete cosine transform (DCT) in terms of its ability to resolve two limiting cases of localization in frequency, viz pure tones and additive white noise. We found in [3] that the HOT has a superior resolution to the DFT when two pure tones are close in frequency. In this paper, we improve on that method to present a more complete spectral analysis tool. Here, we implement a stationary spectral estimation method using compressive sensing (in particular, Iterative Hard Thresholding) on HOT filterbanks. We compare its frequency resolution to that of a DFT filterbank using compressive sensing. In particular, we compare the performance of the HF with that of the DFT in resolving two close frequency components in additive white Gaussian noise (AWGN). We find the HF method to be superior to the DFT method in frequency estimation, and ascribe the difference to the HOT's relationship to entropy.

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“…1 The average time constant of the DFT block LMS filter is [10] 2 The misadjustment of the DFT block LMS algorithm is [10] …”

Section: Convergence Analysis In the Time Domainmentioning

confidence: 99%

“…The HOT is a recently developed discrete unitary transform that uses the orthonormal minimizers of the entropy-based Hirschman uncertainty measure [2]. This measure is different from the energy-based Heisenberg uncertainty measure that is only suited for continuous time signals.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, it has been shown recently that a thresholding algorithm using the HOT yields superior frequency resolution of a pure tone in additive white noise to a similar algorithm based on the DFT [4]. The main theorem in [2] describes a method to generate an N = K 2 -point orthonormal HOT basis, where K is an integer. A HOT basis sequence of length K 2 is the most compact bases in the time-frequency plane.…”

Section: Introductionmentioning

confidence: 99%

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