Abstract
The uncertainty principle plays an important role in mathematics, physics, signal processing, and so on. Firstly, based on definition of the linear canonical transform (LCT) and the traditional Pitt's inequality, one novel Pitt's inequality in the LCT domains is obtained, which is connected with the LCT parameters and Then one novel logarithmic uncertainty principle is derived from this novel Pitt's inequality in the LCT domains, which is associated with parameters of the two LCTs. Secondly, from the relation between the original function and LCT, one entropic uncertainty principle and one Heisenberg's uncertainty principle in the LCT domains are derived, which are associated with the LCT parameters and The reason why the three lower bounds are only associated with LCT parameters and and independent of and is presented. The results show it is possible that the bounds tend to zeros.
Highlights
The uncertainty principle is one elementary principle in signal processing [1,2,3,4,5,6,7,8,9,10] and physics [11,12,13]
Why do not the parameters c, d have relation with the entropic uncertainty principle in the linear canonical transform (LCT) domains? From definition (3) of the LCT, we find that the parameters c, d only play the role of scaling and modulation
Three uncertainty principles associated with the LCT are presented in this paper
Summary
The uncertainty principle is one elementary principle in signal processing [1,2,3,4,5,6,7,8,9,10] and physics [11,12,13]. In this paper we will give three uncertainty principles in the LCT domains: one logarithmic uncertainty principle based on Pitt’s inequality [14,15,16]; one entropic uncertainty principle; one Heisenberg’s uncertainty principle. The results of this paper and most of the derivation are different and novel. In [19], Pitt’s inequality and logarithmic uncertainty principle on LCT have not been involved. As a generalization of the traditional FT and the FRFT, the LCT has some properties with its transformed parameter.
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