Abstract
Determining orthonormal eigenvectors of the DFT matrix, which is closer to the samples of Hermite-Gaussian functions, is crucial in the definition of the discrete fractional Fourier transform. In this work, we disclose eigenvectors of the DFT matrix inspired by the ideas behind bilinear transform. The bilinear transform maps the analog space to the discrete sample space. As jω in the analog s-domain is mapped to the unit circle one-to-one without aliasing in the discrete z-domain, it is appropriate to use it in the discretization of the eigenfunctions of the Fourier transform. We obtain Hermite-Gaussian-like eigenvectors of the DFT matrix. For this purpose we propose three different methods and analyze their stability conditions. These methods include better conditioned commuting matrices and higher order methods.We confirm the results with extensive simulations.
Highlights
Discretization of the fractional Fourier transform (FrFT) is vital in many application areas including signal and image processing, filtering, sampling, and time-frequency analysis [1,2,3]
Candan et al [6] use the S matrix, which has been introduced earlier by Dickinson and Steiglitz [12] to find the eigenvectors of the DFT matrix in order to generate a discrete fractional Fourier transform (DFrFT) matrix
Candan et al [6] replace the derivative operator with the second-order discrete Taylor approximation to second derivative and the Fourier operator with the DFT matrix
Summary
Discretization of the fractional Fourier transform (FrFT) is vital in many application areas including signal and image processing, filtering, sampling, and time-frequency analysis [1,2,3]. Santhanam and McClellan [5] define a DFrFT as a linear combination of powers of the DFT matrix This definition is not satisfactory, since it does do not mimic the properties of the continuous FrFT. The S matrix commutes with the DFT matrix, which ensures that both of these matrices share at least one eigenvector set in common This approach is based on the second-order Hermite-Gaussian generating differential equation. Candan introduces Sk [8] matrices whose eigenvectors are higher-order approximations to the Hermite-Gaussian functions.
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