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Discrete Uncertainty Principles and Virial Identities  [PDF]
Aingeru Fernández-Bertolin
Mathematics , 2014,
Abstract: In this paper we review the Heisenberg uncertainty principle in a discrete setting and, as in the classical uncertainty principle, we give it a dynamical sense related to the discrete Schr\"odinger equation. We study the convergence of the relation to the classical uncertainty principle, and, as a counterpart, we also obtain another discrete uncertainty relation that does not have an analogous form in the continuous case. Moreover, in the case of the Discrete Fourier Transform, we give a inequality that allows us to relate the minimizer to the Gaussian.
Discrete Entropic Uncertainty Relations Associated with FRFT  [PDF]
Guanlei Xu, Xiaotong Wang, Lijia Zhou, Xiaogang Xu, Limin Shao
Journal of Signal and Information Processing (JSIP) , 2013, DOI: 10.4236/jsip.2013.43B021

Based on the definition and properties of discrete fractional Fourier transform (DFRFT), we introduced the discrete Hausdorff-Young inequality. Furthermore, the discrete Shannon entropic uncertainty relation and discrete Rényi entropic uncertainty relation were explored. Also, the condition of equality via Lagrange optimization was developed, as shows that if the two conjugate variables have constant amplitudes that are the inverse of the square root of numbers of non-zero elements, then the uncertainty relations reach their lowest bounds. In addition, the resolution analysis via the uncertainty is discussed as well.

An Uncertainty Principle for Discrete Signals  [PDF]
Sangnam Nam
Computer Science , 2013,
Abstract: By use of window functions, time-frequency analysis tools like Short Time Fourier Transform overcome a shortcoming of the Fourier Transform and enable us to study the time- frequency characteristics of signals which exhibit transient os- cillatory behavior. Since the resulting representations depend on the choice of the window functions, it is important to know how they influence the analyses. One crucial question on a window function is how accurate it permits us to analyze the signals in the time and frequency domains. In the continuous domain (for functions defined on the real line), the limit on the accuracy is well-established by the Heisenberg's uncertainty principle when the time-frequency spread is measured in terms of the variance measures. However, for the finite discrete signals (where we consider the Discrete Fourier Transform), the uncertainty relation is not as well understood. Our work fills in some of the gap in the understanding and states uncertainty relation for a subclass of finite discrete signals. Interestingly, the result is a close parallel to that of the continuous domain: the time-frequency spread measure is, in some sense, natural generalization of the variance measure in the continuous domain, the lower bound for the uncertainty is close to that of the continuous domain, and the lower bound is achieved approximately by the 'discrete Gaussians'.
Logarithm of the Discrete Fourier Transform  [PDF]
Michael Aristidou,Jason Hanson
International Journal of Mathematics and Mathematical Sciences , 2007, DOI: 10.1155/2007/20682
Abstract: The discrete Fourier transform defines a unitary matrix operator. The logarithm of this operator is computed, along with the projection maps onto its eigenspaces. A geometric interpretation of the discrete Fourier transform is also given.
On the diagonalization of the discrete Fourier transform  [PDF]
Shamgar Gurevich,Ronny Hadani
Mathematics , 2008,
Abstract: The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing the discrete oscillator transform in certain cases.
Approximating the Analytic Fourier Transform with the Discrete Fourier Transform  [PDF]
Jeremy Axelrod
Mathematics , 2015,
Abstract: The Fourier transform is approximated over a finite domain using a Riemann sum. This Riemann sum is then expressed in terms of the discrete Fourier transform, which allows the sum to be computed with a fast Fourier transform algorithm more rapidly than via a direct matrix multiplication. Advantages and limitations of using this method to approximate the Fourier transform are discussed, and prototypical MATLAB codes implementing the method are presented.
Discrete Weierstrass Fourier Transform and Experiments  [PDF]
Sheng Zhang,Michael F. Barnsley,Brendan Harding
Mathematics , 2015,
Abstract: We establish a new method called Discrete Weierstrass Fourier Transform to approximate data sets, which is a generalization of Discrete Fourier Transform. The theory of this method as well as some experiments are analyzed in this paper. In some examples, this method has a faster convergent speed than Discrete Fourier Transform.
Support-Limited Generalized Uncertainty Relations on Fractional Fourier Transform  [PDF]
Xiaotong Wang, Guanlei Xu
Journal of Signal and Information Processing (JSIP) , 2015, DOI: 10.4236/jsip.2015.63021
Abstract: This paper investigates the generalized uncertainty principles of fractional Fourier transform (FRFT) for concentrated data in limited supports. The continuous and discrete generalized uncertainty relations, whose bounds are related to FRFT parameters and signal lengths, were derived in theory. These uncertainty principles disclose that the data in FRFT domains may have?much higher concentration than that in traditional time-frequency domains, which will enrich the ensemble of generalized uncertainty principles.
Sparse Eigenvectors of the Discrete Fourier Transform  [PDF]
William F. Bradley
Mathematics , 2009,
Abstract: We construct a basis of sparse eigenvectors for the N-dimensional discrete Fourier transform. The sparsity differs from the optimal by at most a factor of four. When N is a perfect square, the basis is orthogonal.
Green's function of a finite chain and the discrete Fourier transform  [PDF]
S. Cojocaru
Physics , 2007, DOI: 10.1142/S0217979206033401
Abstract: A new expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Comparison to Bethe ansatz clarifies the relation between the two approaches.
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