This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions which are eigenfunctions of the continuous Fourier transform and by performing a novel error removal procedure. Then, the result of the eigendecomposition of the DFT matrix...

The spectrum of the convolution of two continuous functions can be determined as the continuous Fourier transform of the cross-correlation function. The same can be said about the spectrum of the convolution of two infinite discrete sequences, which can be determined as the discrete time Fourier transform of the cross-correlation function of the two sequences. In current digital signal processi...

In quantum mechanics, the Fourier Transform commonly converts from position space to momentum. For finite dimensional Hilbert spaces, the analog is the discrete (or quantum) Fourier transform, which has many applications in quantum information theory. We explore applications of this discrete Fourier transform to the elementary particle generations, and then present a related and elegant new par...

This paper is part 3 in a series of papers about the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). The focus of this paper is on computing the Power Spectral Density (PSD) of the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). The implementation is based on a well-known algorithm, called the decimation in time Radix 2 FFT, and re...

The aim of this paper is to examine a set of wavelet functions (wavelets) used in image watermarking system. Several wavelet functions are used to transform the component detail of the image in frequency domain. Discrete wavelet transform (DWT), discrete Fourier transform (DFT) and discrete cosine transform (DCT) are some commonly used transform domain functions out of which discrete wavelet tr...