Eigenvectors of Ordinary, Generalized, Centered and Offset Discrete Fourier Transforms Based on Lookup Table Methods: Efficiency and Approximation Uses

Abstract

In this paper, we first propose a methodology to construct an eigenvector of the N × N discrete Fourier transform (DFT) matrix from any eigenvector of the ( k 2 N ) × ( k 2 N ) DFT matrix, where k is a positive integer. Specifically, using the procedure to down-sample by k any ( k 2 N )-point DFT eigenvector with zero as the initial index and then fold modulus N and add the resultant down-sampled ( kN )-point vector, we obtain an N -point DFT eigenvector. Computing an N -point DFT eigenvector using the proposed method is very efficient, which requires only N ( k -1) additions and no multiplications with a pre-stored lookup table of ( k 2 N )-point DFT eigenvectors. Similar efficient methods are also developed to compute N -point eigenvectors of other DFT-related transforms, including the generalized DFT (GDFT), centered DFT (CDFT) and the offset DFT (ODFT), from ( k 2 N )-point eigenvectors of their corresponding transforms or the DFT. As application examples, we apply the proposed methods to compute N -point DFT, GDFT, CDFT and ODFT eigenvectors, which are closer to samples of the continuous Hermite-Gaussian function (HGF) than existing N -point eigenvectors, from their corresponding ( k 2 N )-point eigenvectors. These improved N -point Hermite-Gaussian-like eigenvectors can be used to define fractional versions of the various N × N DFT transforms whose outputs are closer to samples of the continuous fractional Fourier transform than outputs of the corresponding existing fractional transforms. Finally, we perform computer experiments to verify the effectiveness and demonstrate applications of our proposed methods.