29 O(b log(b)) operations (using standard multiplication). As there are O(b= log(b)) primes in total, the running time of this stage of the algorithm is O(b 2 L), even using the \grammar school" method of integer multiplication. At this stage of the algorithm we have obtained a vector of length L whose entries are integral linear combinations of powers of with coeecients bounded by M in absolut...

This paper shows that there are fast Fourier transform (FFT) algorithms that work, for a fixed number of points, independent of the dimension. Changing the dimension is achieved by relabeling the input and the output and changing the “twiddle factors.” An important consequence of this result, is that a program designed to compute the 1-dimensional Fourier transform can be easily modified to com...

The fuzzy transform (F -transform for short) is a universal tool for a fuzzy modeling with convincing applications to image processing. The aim of this contribution is to explain the effect of the F transform in image processing. With this purpose, we investigate properties of the Fourier transform over the F -transform components. We prove that the direct F -transform is a low-pass filter. Thi...

nowadays, radar systems have many applications and radar imaging is one of the most important of these applications. inverse synthetic aperture radar (isar) is used to form an image from moving targets. conventional methods use fourier transform to retrieve doppler information. however, because of maneuvering of the target, the doppler spectrum becomes time-varying and the image is blurred. joi...

Synopsis Iterative reconstructions of undersampled non-Cartesian data are computationally expensive because non-Cartesian Fourier transforms are much less e cient than Cartesian Fast Fourier Transforms. Here, we introduce an algorithm that does not require non-uniform Fourier transforms during optimization iterations, resulting in large reductions in computation times with no impairment of imag...

It was shown [ 21 that any SFF can be decomposed in this manner. Thus, F(x) is an SFF if, and only if, it can be expressed as the sum of four functions in the form of the above equation. Additional SFF studies are reported in refs. [ 3-51. Another issue that has been recently investigated is the fractional Fourier transform [ 6-91. Two distinct definitions of the fractional Fourier transform ha...

We now start considering discrete–time signals. A discrete–time signal is a function (real or complex valued) whose argument runs over the integers, rather than over the real line. We shall use square brackets, as in x[n], for discrete–time signals and round parentheses, as in x(t), for continuous–time signals. This is the notation used in EECE 359 and EECE 369. Discrete–time signals arise in t...

We now start considering discrete–time signals. A discrete–time signal is a function (real or complex valued) whose argument runs over the integers, rather than over the real line. We shall use square brackets, as in x[n], for discrete–time signals and round parentheses, as in x(t), for continuous–time signals. This is the notation used in EECE 359 and EECE 369. Discrete–time signals arise in t...