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...

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...

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...

The fractional Fourier transform is a time–frequency distribution and an extension of the classical Fourier transform. There are several known applications of the fractional Fourier transform in the areas of signal processing, especially in signal restoration and noise removal. This paper provides an introduction to the fractional Fourier transform and its applications. These applications deman...

We estimate discrete Fourier transform, ambiguity, and Hamming-autocorrelation of m-ary sequences in terms of their (periodic) correlation measure of order 4. Roughly speaking, we show that every pseudorandom sequence, that is, any sequence with small correlation measure up to a sufficiently large order, cannot have a large discrete Fourier transform, ambiguity, or Hamming-autocorrelation. Conv...

s is a time-frequency transform based on the concept of short-time fourier transform and wavelet transform. the main property of this transform that separate it from the discrete short-time fourier transform is variable frequency resolution. but in this transform the change of resolution of high frequencies to low frequencies is very high. so another transform as a modified s transform is intro...

All algorithms for impedance calculation use an analog-to-digital converter. The high accuracy of the impedance seen by a distance relay is an important factor in the correct isolation of the faulty part of power systems. To achieve this, a novel&#10technique based on third order interpolation is used in this paper. According to this technique, the times and the values of the obtained samples a...

We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of “operations” of the original Cooley-Tukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies (l/x)Z,,,2n A(n) (n2/9)(x2/log x). The average is not a good indication of the number of operations. For examp...