http://www.norbertwiener.umd.edu/Research/Duhamel_Vetterli_FFT_90.pdf

Fast fourier transforms: a tutorial review and a state of the art

Note: V. K. Madisetti, D. B. Williams, Eds. Reference LCAV-CHAPTER-2005-009 Record created on 2005-06-27, modified on 2017-05-12

Fast Fourier transforms: a tutorial review and a state of the art

Master the basic concepts and methodologies of digital signal processing with this systematic introduction, without the need for an extensive mathematical background The authors lead the reader through the fundamental mathematical principles underlying the operation of key signal processing techniques, providing simple arguments and cases rather than detailed general proofs Coverage of practical implementation, discussion of the limitations of particular methods and plentiful MATLAB illustrations allow readers to better connect theory and practice A focus on algorithms that are of theoretical importance or useful in real-world applications ensures that students cover material relevant to engineering practice, and equips students and practitioners alike with the basic principles necessary to apply DSP techniques to a variety of applications Chapters include worked examples, problems and computer experiments, helping students to absorb the material they have just read Lecture slides for all figures and solutions to the numerous problems are available to instructors

Applied Digital Signal Processing: Theory and Practice

Many versions of the fast Fourier transform require a reordering of either the input or the output data that corresponds to reversing the order of the bits in the array index. There has been a surprisingly large number of papers on this subject in the recent literature.This paper collects 30 methods for bit reversing an array. Each method was recoded into a uniform style in Fortran and its performance measured on several different machines, each with a different memory system. This paper includes a description of how the memories of the machines operate to motivate two new algorithms that perform substantially better than the others.

Bit reversal on uniprocessors

Known in the signal processing literature mainly as a computational technique, the Hartley transform has proven to possess attributes in the physical world of experiment, especially in optics and microwaves. The Fourier transform, which has constituted the standard approach to spectral analysis, is understood both in the domain of numerical analysis and in the world of natural phenomena, gaining its central significance from the ubiquity of sinusoidal natural behavior; it also gains appeal from the convenience offered by complex variable analysis, which is a basic part of technical education. Computers, however, prefer real numbers, an attribute that first opened a niche for the Hartley transform. As applications have multiplied, and others open up, it will be helpful to understand the Hartley idea from more than one point of view. Several topics, including multidimensional transforms, the complex generalization, and microwave phase measurement by amplitude measurement only, are discussed with the intention of throwing light on the numerical and physical relationships between the Hartley and Fourier transforms. >

Aspects of the Hartley transform

All radix-B fast Fourier transforms (FFT) or fast Hartley transforms (FHT) performed "in-place" require at some point that the sequence elements he permuted such that, indexing the elements 0 to N - 1, the element with index i is swapped with the element whose index is j. The permutation is called digit-reversing, because if i is represented as a string of digits, base B, then j is that index whose representation is the same string of digits written in reverse order. N is a power of B and B \geq 2 . An elegant algorithm has been found that Performs this "perfect shuffle" more efficiently and, according to timing experiments, runs about eight times faster than the fastest other algorithm known to the author. The algorithm is of order O(N) and led, for example, to a saving of 7 percent in the total (radix-2) FFT running time for N = 1024.

An improved digit-reversal permutation algorithm for the fast Fourier and Hartley transforms

Based on three previously published theorems and an algorithm for the digit-reversal permutation required by fast transform algorithms, a fourth theorem is given, indicating an alternate order of generating the index pairs for swapping, and a second permutation algorithm results. This algorithm uses the same principles as the first and differs principally in that in its innermost loop, one of the pair of indexes is usually generated by an integer increment (i:=i+n). This will result in slightly faster execution on most computers. >

D.M.W. Evans

Papers

An improved digit-reversal permutation algorithm for the fast Fourier and Hartley transforms

A second improved digit-reversal permutation algorithm for fast transforms