The Fast Fourier Transform

This tutorial discusses the fast Fourier transform, which has numerous applications in signal and image processing. The FFT computes the frequency components of a signal that has been sampled at n points in O( n log n) time. We explain the FFT and illustrate it by examples and Pascal algorithms. We assume that you are familiar with elementary calculus

The Fast Fourier Transform Telescope

We propose an all-digital telescope for 21 cm tomography, which combines key advantages of both single dishes and interferometers. The electric field is digitized by antennas on a rectangular grid, after which a series of Fast Fourier Transforms recovers simultaneous multifrequency images of up to half the sky. Thanks to Moore's law, the bandwidth up to which this is feasible has now reached about 1 GHz, and will likely continue doubling every couple of years. The main advantages over a single dish telescope are cost and orders of magnitude larger field-of-view, translating into dramatically better sensitivity for large-area surveys. The key advantages over traditional interferometers are cost (the correlator computational cost for an N-element array scales as N log N rather than N^2) and a compact synthesized beam. We argue that 21 cm tomography could be an ideal first application of a very large Fast Fourier Transform Telescope, which would provide both massive sensitivity improvements per dollar and mitigate the off-beam point source foreground problem with its clean beam. Another potentially interesting application is cosmic microwave background polarization.Comment: Replaced to match accepted PRD version. 21 pages, 9 fig

An investigation of pulsar searching techniques with the Fast Folding Algorithm

Here we present an in-depth study of the behaviour of the Fast Folding Algorithm, an alternative pulsar searching technique to the Fast Fourier Transform. Weaknesses in the Fast Fourier Transform, including a susceptibility to red noise, leave it insensitive to pulsars with long rotational periods (P > 1 s). This sensitivity gap has the potential to bias our understanding of the period distribution of the pulsar population. The Fast Folding Algorithm, a time-domain based pulsar searching technique, has the potential to overcome some of these biases. Modern distributed-computing frameworks now allow for the application of this algorithm to all-sky blind pulsar surveys for the first time. However, many aspects of the behaviour of this search technique remain poorly understood, including its responsiveness to variations in pulse shape and the presence of red noise. Using a custom CPU-based implementation of the Fast Folding Algorithm, ffancy, we have conducted an in-depth study into the behaviour of the Fast Folding Algorithm in both an ideal, white noise regime as well as a trial on observational data from the HTRU-S Low Latitude pulsar survey, including a comparison to the behaviour of the Fast Fourier Transform. We are able to both confirm and expand upon earlier studies that demonstrate the ability of the Fast Folding Algorithm to outperform the Fast Fourier Transform under ideal white noise conditions, and demonstrate a significant improvement in sensitivity to long-period pulsars in real observational data through the use of the Fast Folding Algorithm.Comment: 19 pages, 15 figures, 3 table

Parallel Processing of the Fast Fourier Transform

The first FFT algorithms were reported by Runge and Konig in 1924, and by Danielson and Lanczos in 1942. However, the FFT didn\u27t receive much attention at all until Cooley and Tukey published their algorithm in 1965. The Cooley-Tukey algorithm is simple and widely used in many application software packages. Winograd developed his FFT in 1976, which is based upon the prime factor theory. It is typically faster than the Cooley-Tukey Algorithm, if the computer system has no multiplication instructions. According to the book prepared by the Digital Signal Processing Committee of the IEEE in 1979, the speed difference among these FFT algorithms is around 40%. My objective in this paper is to choose a proper algorithm, establish the appropriate programming techniques, and determine the sequence of steps required to implement a FFT both on a conventional IBM-PC and a Vector Processor (VP) system. I will demonstrate how to vectorize a FFT so that the algorithm can be performed under a VP system. The analysis of data dependence in an algorithm is another important part of this paper. The paper includes the analysis of the Cooley-Tukey and Winograd FFT algorithms. The Prime factor method will be used in these two FFTs. It will be seen that the Cooley-Tukey Algorithm can be more easily implemented on a vector system and needs fewer memory locations. The details of\u27 the Winograd FFT algorithm can be found in. In addition, this paper has two Cooley-Tukey FFTs and one DFT program written in Assembly Language. One of two FFT programs has been tested and executed on a conventional IBM-PC which has an Intel-8088 processor as the Central Processing Unit, and one Intel-8087 Numeric Data Processor. The 8087 is specially designed to perform real number operations efficiently and quickly. Because of the special architecture of the 8087, single or double precision can be easily processed. The tested program was compiled and linked by Microsoft Assembly Language version 5.0 and the required results of both the FFT and Inverse FFT were obtained