The Fractional Discrete Fourier Transform

A fractional version of the Discrete Fourier Transform or DFT, denoted by the Fractional Discrete Fourier Transform or FDFT for short, is discussed here. First, results of a fractional version of the continuous-time Fourier Transform or CTFT are explored and then parallels are made between the DFT and the CTFT. Using the method of spectral decomposition [1], an expression for the FDFT is then derived which satisfies properties analogous to the fractional CTFT. Afterwards, properties of the FDFT are discovered and proven, and an example of an FDFT pair is given. Finally, various applications of the FDFT in signal processing in areas such as allpass filter networks and the M-channel maximally decimated filter bank are discussed.