In this paper, we construct discrete fractional Fourier transforms (DFrFT) using recently introduced closed-form Hermite–Gaussian-like (HGL) eigenvectors. With respect to such eigenvectors, we discuss the convergence of their components to samples of the corresponding continuous Hermite–Gaussian functions and propose solutions to deal with some restrictions related to their construction. This allows us to give new procedures for obtaining orthonormal bases of HGL eigenvectors, which are used to fractionalize the discrete Fourier transform. We illustrate the application of the resulting DFrFT in the scenario of filtering in the fractional domain and compare the results with existing DFrFT approaches.

Discrete Fractional Fourier Transforms Based on Closed-Form Hermite–Gaussian-Like DFT Eigenvectors

http://ece-research.unm.edu/bsanthan/publications/SP-08.pdf

On discrete Gauss-Hermite functions and eigenvectors of the discrete Fourier transform

Efficient computation of DFT commuting matrices by a closed-form infinite order approximation to the second differentiation matrix

We present a novel method to generate centered discrete dilated Hermite functions by utilizing Hermite-Gaussian differential equation and DFT commuting matrix proposed by Pei Compared with the conventional methods, our method provides much higher accuracy and wider working range. Based on expansion of compact support signals into discrete dilated Hermite basis, several examples of signal approximation and scaling are demonstrated. Our results in ECG signal compression can achieve lesser approximation error in comparison with discrete Daubechies wavelet transform. Also, our proposed discrete dilated Hermite functions have better ability to do signal scaling than the existing method.

Signal Scaling by Centered Discrete Dilated Hermite Functions

The Modified Discrete Cosine Transform (MDCT) is widely used in audio signals compression, but mostly limited to representing audio signals. This is because the MDCT is a real transform: Phase information is missing and spectral power varies frame to frame even for pure sine waves. We have a key observation concerning the structure of the MDCT spectrum of a sine wave: Across frames, the complete spectrum changes substantially, but if separated into even and odd subspectra, neither changes except scaling. Inspired by this observation, we find that the MDCT spectrum of a sine wave can be represented as an envelope factor times a phase-modulation factor. The first one is shift-invariant and depends only on the sine wave's amplitude and frequency, thus stays constant over frames. The second one has the form of sinθ for all odd bins and cosθ for all even bins, leading to subspectra's constant shapes. But this θ depends on the start point of a transform frame, therefore, changes at each new frame, and then changes the whole spectrum. We apply this formulation of the MDCT spectral structure to frequency estimation in the MDCT domain, both for pure sine waves and sine waves with noises. Compared to existing methods, ours are more accurate and more general (not limited to the sine window). We also apply the spectral structure to stereo coding. A pure tone or tone-dominant stereo signal may have very different left and right MDCT spectra, but their subspectra have similar shapes. One ratio for even bins and one ratio for odd bins will be enough to reconstruct the right from the left, saving half bitrate. This scheme is simple and at the same time more efficient than the traditional Intensity Stereo (IS).

MDCT Sinusoidal Analysis for Audio Signals Analysis and Processing

Existing approaches tofurnishing abasis ofeigenvectors for thediscrete Fourier transform (DFT)arebasedupondefiningtridiagonal operators thatcommutewiththeDFT.In this paper, motivated byideas fromquantummechanics in finite dimensions, wedefine asymmetric matrixthatcommuteswiththecentered DFT,thereby furnishing abasis of eigenvectors fortheDFT.We showthatthese eigenvectors inthelimit converge toGauss-Hermite (G-H)functions and thattheeigenvalue spectrum ofthecommutorprovides a verygooddiscrete approximation tothatofthecontinuous G-Hdifferential operator.

Thalanayar S. Santhanam

Papers

DISCRETE GAUSS-HERMITE FUNCTIONS AND EIGENVECTORS OF THE CENTERED DISCRETE FOURIER TRANSFORM BaluSanthanam