Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Linear combination-type DFRFT <s> A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Linear combination-type DFRFT <s> We define a discrete version of the angular Fourier transform and present the properties of the transform that show it to be a rotation in time-frequency space, this new transform is a generalization of the DFT. Efficient algorithms for its computation can then be based on the FFT and the eigenstructure of the DFT. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Linear combination-type DFRFT <s> This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Linear combination-type DFRFT <s> The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform. <s> BIB004
One of the first approaches for the DFRFT was proposed by Dickinson and Steiglitz BIB001 . In their paper, the DFRFT was derived by using the linear combination of identity operation (F 0 where for 0 r k r3. In particular, it has been shown that the operator defined by (49) is unitary BIB002 . In other words, Furthermore, the operator satisfies the angle additivity property BIB002 : and angle multiciplity property BIB002 : It has also been noticed that the operator is periodic in the parameter a with a fundamental period 2p BIB002 . This type of the DFRFT corresponds to a completely distinct definition of the fractional Fourier transform. However, the main problem is that the transform results will not match to the continuous FRFT BIB004 . In other words, it is not the discrete version of the continuous transform BIB003 .