Abstract
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.
Highlights
Uncertainty principle holds in analog signals, and in discrete signals [1,2]
With the development of fractional Fourier transform (FRFT), analog generalized uncertainty relations associated with FRFT have been carefully explored in some papers such as [3,4,16], which effectively enrich the ensemble of FRFT
Up till there has been no reported article covering the discrete generalized uncertainty relations associated with discrete fractional Fourier transform (DFRFT)
Summary
Uncertainty principle holds in analog signals, and in discrete signals [1,2]. With the development of fractional Fourier transform (FRFT), analog generalized uncertainty relations associated with FRFT have been carefully explored in some papers such as [3,4,16], which effectively enrich the ensemble of FRFT. Up till there has been no reported article covering the discrete generalized uncertainty relations associated with discrete fractional Fourier transform (DFRFT). In this article we will discuss the entropic uncertainty relations [7,8] associated with DFRFT. The second contribution is that we derived the Shannon entropic uncertainty principle in FRFT domain for discrete case, based on which we derived the conditions when these uncertainty relations have the equalities via Lagrange optimi-. The third contribution is that we derived the Renyi entropic uncertainty principle in FRFT domain for discrete case. There have been no reported papers covering these generalized discrete entropic uncertainty relations on FRFT
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