Showing papers on "Split-radix FFT algorithm published in 1974"
TL;DR: The CORDIC iteration is applied to several Fourier transform algorithms and a new, especially attractive FFT computer architecture is presented as an example of the utility of this technique.
Abstract: The CORDIC iteration is applied to several Fourier transform algorithms. The number of operations is found as a function of transform method and radix representation. Using these representations, several hardware configurations are examined for cost, speed, and complexity tradeoffs. A new, especially attractive FFT computer architecture is presented as an example of the utility of this technique. Compensated and modified CORDIC algorithms are also developed.
304 citations
TL;DR: The Kronecker product notation and the ideal shuffle base r permutation operator form the basis for a unifying theory through which the various versions of the FFT can be viewed.
Abstract: The different versions of the fast Fourier transform (FFT) are described here for arbitrary base in terms of the matrix factors of the discrete Fourier transform matrix T_{N} . The Kronecker product notation and the ideal shuffle base r permutation operator form the basis for a unifying theory through which the various versions of the FFT can be viewed. The properties of the ideal shuffle base r permutation operator are used to arrive at FFT versions with such desirable properties as in-place computation or identical geometry from stage to stage. The FFT versions previously described in the literature are derived here. At the same time, algorithms for the sorting of FFT data in digit-reversed order are generated. These are explored and new sorting versions amenable to hardware implementation with sequential memory are presented. As an example of how the unifying theory is used, a number of FFT versions with identical geometry from stage to stage are derived. The hardware necessary for these algorithms is described for the base 4 case with N = 1024 data points.
32 citations
TL;DR: This work proposes an implementation whereby two radix-N1/2passes are carried out in parallel and in which each N 1/2-point transform is carried out via a serial input parallel output transform circuit.
Abstract: Different fast Fourier transform (FFT) algorithms for hardware implementation have been considered. We propose an implementation whereby two radix-N1/2passes are carried out in parallel and in which each N1/2-point transform is carried out via a serial input parallel output transform circuit. The processing rate is one clock cycle per input point for the N-point transform regardless of the value of N chosen. The circuit is being implemented with TTL logic and will be used to perform spatial frequency domain filtering on two dimensional infrared camera images in real time; real time meaning processing between frame display.
27 citations
01 Oct 1974
TL;DR: Dolph-Chebyshev weights, which realize a minimum side-Lobe level for a specified main-lobe width, can be generated by a single fast Fourier transform (FFT) for an even number of elements 2H.
Abstract: Dolph-Chebyshev weights, which realize a minimum side-lobe level for a specified main-lobe width, can be generated by a single fast Fourier transform (FFT). For an even number of elements 2H, the size of the FFT is H. This result has utility for spectral analysis as well as for array processing.
11 citations
TL;DR: In this paper, the analysis of rounding error in the one-dimensional fast Fourier transform (FFT) is extended to a class of generalized orthogonal transforms with a common fast algorithm similar to the FFT algorithm.
Abstract: The analysis of rounding error in the one-dimensional fast Fourier transform (FFT) is extended to a class of generalized orthogonal transforms [1] with a common fast algorithm similar to the FFT algorithm. This class includes the BInary FOurier REpresentation (BIFORE) transform (BT) [2], the complex BT (CBT) [3], and the discrete Fourier transform (DFT). Expressions for the mean square error (MSE) in the two-dimensional BT, CBT, and FFT are derived. In the case of white input data, the mean square error-to-signal ratio is derived for the multidimensional generalized transforms. The error-to-signal ratio for the one-dimensional FFT derived by Kaneko and Liu is modified with improvement. Some comparisons among BIFORE, DFT, and Haar transforms are also included. The theoretical results for the two-dimensional FFT and BIFORE have been verified experimentally. The experimental results are in good agreement with the theoretical results for lower order sequences, but deviate as the order increases due to the actual manner of rounding in the digital computer.
8 citations
TL;DR: This paper presents an algorithm for making use of symmetry in the fast Fourier transform in a simple and general way which is applicable to nearly all space groups and makes possible economical forward and reverse transforms of large unit cells in core.
Abstract: This paper presents an algorithm for making use of symmetry in the fast Fourier transform in a simple and general way which is applicable to nearly all space groups. This allows one to reduce storage requirements to approximately what is needed for an asymmetric unit of the electron-density function, and hence makes possible economical forward and reverse transforms of large unit cells in core.
7 citations
TL;DR: It is shown that a fast Fourier processor having an attractive cost performance ratio can be built by employing serial arithmetic in the implementation of the algorithm developed.
Abstract: An algorithm is presented that introduces two degrees of parallelism into the implementation of fast Fourier transform (FFT) processors. That is, both the radix of factorization and the number of arithmetic units may be selected to achieve the required processing speed. A serial vector multiplier that is ideally suited to the implementation of a general radix arithmetic unit is described. It is subsequently shown that a fast Fourier processor having an attractive cost performance ratio can be built by employing serial arithmetic in the implementation of the algorithm developed.
6 citations
TL;DR: In this paper, a simple procedure is presented to develop a fast Fourier transform (FFT) program for PQ points starting from a program for Q points, with emphasis on Q = 2M.
Abstract: A simple procedure is presented to develop a fast Fourier transform (FFT) program for PQ points starting from a program for Q points, with emphasis on Q = 2M. The transformation with respect to the factor P is followed by a transformation of P groups of Q points each using the existing subroutine, then the array is unscrambled with respect to P.
1 citations