Closed-form discrete fractional and affine Fourier transforms

• Published in: IEEE transactions on signal processing 2000-05, Vol.48 (5), p.1338-1353
• Main Authors: PEI, S.-C, DING, J.-J
• Format: Article
• Language: English
• Subjects: Additives; Applied sciences; Digital signal processing; Discrete Fourier transforms; Exact sciences and technology; Fourier transforms; Information, signal and communications theory; Optical filters; Optical signal processing; Pattern analysis; Pattern recognition; Signal and communications theory; Signal processing; Signal, noise; Studies; Telecommunications and information theory
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/78.839981

The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.