On uncertainty principle for signal concentrations with fractional Fourier transform

• Published in: Signal processing 2012-12, Vol.92 (12), p.2830-2836
• Main Authors: Shi, Jun, Liu, Xiaoping, Zhang, Naitong
• Format: Article
• Language: English
• Subjects: Applied sciences; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; Fractional domain; Fractional Fourier transform; Information, signal and communications theory; Signal and communications theory; Signal concentration; Signal, noise; Telecommunications and information theory; Uncertainty principle
• ISSN: 0165-1684; 1872-7557
• DOI: 10.1016/j.sigpro.2012.04.008

The fractional Fourier transform (FRFT) – a generalized form of the classical Fourier transform – has been shown to be a powerful analyzing tool in signal processing. This paper investigates the uncertainty principle for signal concentrations associated with the FRFT. It is shown that if the fraction of a nonzero signal's energy on a finite interval in one fractional domain with a certain angle α is specified, then the fraction of its energy on a finite interval in other fractional domain with any angle β(β≠α) must remain below a certain maximum. This is a generalization of the fact that any nonzero signal cannot have arbitrarily large proportions of energy in both a finite time duration and a finite frequency bandwidth. The signals which are the best in achieving simultaneous concentration in two arbitrary fractional domains are derived. Moreover, some applications of the derived theory are presented. ► We derive an uncertainty principle for signal concentrations in fractional domains. ► A nonzero signal's energy cannot be arbitrarily large in any two fractional domains. ► We present signals that are best concentrated in any two fractional domains. ► Applications of the derived result in signal and filter design are presented.