The centered discrete fractional Fourier transform, properties, computation, and application to linear chirp signals

The Discrete Fourier Transform (DFT) has been the workhorse for discrete-time signal analysis for many years due to the existence of a fast and efficient algorithm for its computation, i.e., the FFT algorithm. In recent years, there has been an increasing interest in fractionalizing the DFT operator after observing some of the properties of its continuous counterpart such as its linear chirp basis and its relation with the Wigner-Ville distribution. Some versions of fractional DFTs based on the eigenvector-eigenvalue decomposition have been proposed, where the most commonly used are those which use eigenvectors that resemble sampled versions of the HermiteGauss functions. Most of the fractional DFTs that have been studied are based on the eigenvectors of the Dickinson-Steiglitz type of commuting matrix and they use the regular DFT as the starting point, resulting in a non-uniform distribution of eigenvalues.; In this dissertation, we define the Centered Fractional Fourier Transform (CDFRFT) based on the eigenvectors of the Grunbaum type of commuting matrix. This transform has a single definition for any size of the transform, and the associated eigenvectors are a closer match to the Hermite-Gauss functions. We study some of its properties that are specifically related to linear chirp signals, in particular, we observe that the angle of the transform has a relation to the chirp rate of the signals and empirical relations between the two parameters that will allow us to estimate the chirp rate of a linear chirp signal are obtained. A fast algorithm for the computation of the CDFRFT using the FFT is developed for the case when we compute a set of equally spaced angles. This results in the definition of the Multi-Angle CDFRFT (MA-CDFRFT), that is a two dimensional array that can be interpreted as a Chirp Rate-Frequency representation. The developed MA-CDFRFT is applied to the problem of chirp rate estimation of single and multi-component signals, and to the improvement of the spectrogram for the particular case of linear chirp signals.; Future work related to the application of the CDFRFT to the detection of linear chirp signals with noise, and to the separation of multi-component signals is discussed. The possibility to obtain another set of eigenvectors for the CDFRFT that is a closer match to the Hermite-Gauss functions using a perturbation of the DFT matrix is also discussed as future research.