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Uncertainty principles for discrete signals associated with the fractional Fourier and linear canonical transforms

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Abstract

The fractional Fourier transform (FRFT), which generalizes the classical Fourier transform, has gained much popularity in recent years because of its applications in many areas, including optics, radar, and signal processing. There are relations between duration in time and bandwidth in fractional frequency for analog signals, which are called the uncertainty principles of the FRFT. However, these relations are only suitable for analog signals and have not been investigated in discrete signals. In practice, an analog signal is usually represented by its discrete samples. The purpose of this paper is to propose an equivalent uncertainty principle for the FRFT in discrete signals. First, we define the time spread and the fractional frequency spread for discrete signals. Then, we derive an uncertainty relation between these two spreads. The derived results are also extended to the linear canonical transform, which is a generalized form of the FRFT.

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Acknowledgments

This work was completed in part, while J. Shi was visiting the Electrical Engineering Department, University of California, Los Angeles, CA, USA. The work was supported in part by the National Natural Science Foundation of China under Grant 61501144, the Fundamental Research Funds for the Central Universities under Grant 01111305, and the National Basic Research Program of China under Grant 2013CB329003.

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Correspondence to Jun Shi.

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Shi, J., Han, M. & Zhang, N. Uncertainty principles for discrete signals associated with the fractional Fourier and linear canonical transforms. SIViP 10, 1519–1525 (2016). https://doi.org/10.1007/s11760-016-0965-7

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  • DOI: https://doi.org/10.1007/s11760-016-0965-7

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