Eigenvectors of the Discrete Fourier Transform Based on the Bilinear Transform
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Research Article Eigenvectors of the Discrete Fourier Transform Based on the Bilinear Transform Ahmet Serbes and Lutfiye Durak-Ata (EURASIP Member) Department of Electronics and Communications Engineering, Yildiz Technical University, Yildiz, Besiktas, 34349, Istanbul, Turkey Correspondence should be addressed to Ahmet Serbes, [email protected] Received 19 February 2010; Accepted 24 June 2010 Academic Editor: L. F. Chaparro Copyright © 2010 A. Serbes and L. Durak-Ata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.
1. Introduction Discretization of the fractional Fourier transform (FrFT) is vital in many application areas including signal and image processing, filtering, sampling, and time-frequency analysis [1–3]. As FrFT is related to the Wigner distribution [1], it is a powerful tool for time-frequency analysis, for example, chirp rate estimation [4]. There have been numerous discrete fractional Fourier transform (DFrFT) definitions [5–11]. Santhanam and McClellan [5] define a DFrFT simply as a linear combination of powers of the DFT matrix. However, this definition is not satisfactory, since it does do not mimic the properties of the continuous FrFT. 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 DFrFT matrix. 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 et al. [6] simply replace the derivative operator with the second-order discrete Taylor
approximation to second derivative and the Fourier operator with the DFT matrix. Pei et al. [7] define a commuting T matrix inspired by the work of Gr¨unbaum [13], whose eigenvectors approximate the samples of continuous Hermite-Gaussian functions better than the eigenvectors of S. Furthermore, the authors use linear combinations of S and T mat
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