Discrete Convolution First of all we need to introduce what we might call the “wraparound” convention. Because the complex numbers wj = e i 2πj N have the property wj±N = wj, which readily extends to wj+mN = wj for any integer m, and since in the discrete Fourier context we represent all N -dimensional vectors as linear combinations of the Fourier vectors Wk whose components are wkj , we make t...

einstein, möbius, and proper velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper lorentz group in the real minkowski space-time $bbb{r}^n$. using the gyrolanguage we study their gyroharmonic analysis. although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them. our study focus ...

The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. It is equivalent to a DFT of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), where in some variants the input and/or output data are shifted by half a sampl...

A new algorithm is presented for the determination of the generalized inverse and the drazin inverse of a polynomial matrix. The proposed algorithms are based on the discrete Fourier transform and thus are computationally fast in contrast to other known algorithms. The above algorithms are implemented in the Mathematica programming language and illustrated via examples.

This paper gives an overview on the techniques needed to implement the discrete Fourier transform (DFT) efficiently on current multicore systems. The focus is on Intel compatible multicores but we also discuss the IBM Cell, and briefly, graphics processing units (GPUs). The performance optimization is broken down into three key challenges: parallelization, vectorization, and memory hierarchy op...

This paper will look at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of discrete Fourier transform (DFT). We will develop an algorithm, called the quick Fourier transform (QFT), that will reduce the number of oating point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for p...