A fixed-point fast Fourier transform error analysis
IBM1
TL;DR: In this article, an analysis of the fixed-point accuracy of the power of two, fast Fourier transform algorithm is presented, which leads to approximate upper and lower bounds on the root-mean-square error.
Abstract: This paper contains an analysis of the fixed-point accuracy of the power of two, fast Fourier transform algorithm. This analysis leads to approximate upper and lower bounds on the root-mean-square error. Also included are the results of some accuracy experiments on a simulated fixed-point machine and their comparison with the error upper bound.
Citations
More filters
TL;DR: Note: V. Madisetti, D. B. Williams, Eds.
862 citations
01 Aug 1972
TL;DR: The groundwork is set through a discussion of the relationship between the binary representation of numbers and truncation or rounding, and a formulation of a statistical model for arithmetic roundoff, to illustrate techniques of working with particular models.
Abstract: When digital signal processing operations are implemented on a computer or with special-purpose hardware, errors and constraints due to finite word length are unavoidable. The main categories of finite register length effects are errors due to A/D conversion, errors due to roundoffs in the arithmetic, constraints on signal levels imposed by the need to prevent overflow, and quantization of system coefficients. The effects of finite register length on implementations of linear recursive difference equation digital filters, and the fast Fourier transform (FFT), are discussed in some detail. For these algorithms, the differing quantization effects of fixed point, floating point, and block floating point arithmetic are examined and compared. The paper is intended primarily as a tutorial review of a subject which has received considerable attention over the past few years. The groundwork is set through a discussion of the relationship between the binary representation of numbers and truncation or rounding, and a formulation of a statistical model for arithmetic roundoff. The analyses presented here are intended to illustrate techniques of working with particular models. Results of previous work are discussed and summarized when appropriate. Some examples are presented to indicate how the results developed for simple digital filters and the FFT can be applied to the analysis of more complicated systems which use these algorithms as building blocks.
333 citations
TL;DR: A signed logarithmic number system, which is capable of representing negative as well as positive numbers is described, and it is shown that negative numbers can be represented in the sign/logarithm number system.
Abstract: A signed logarithmic number system, which is capable of representing negative as well as positive numbers is described. A number is represented in the sign/logarithm number system by a sign bit and the logarithm of the absolute value of the number (scaled to avoid negative logarithms).
273 citations
TL;DR: Two concurrent error detection (CED) schemes are proposed for N-point fast Fourier transform (FFT) networks that consists of log/sub 2/N stages with N/2 two-point butterfly modules for each stage to achieve both error detection and location.
Abstract: Two concurrent error detection (CED) schemes are proposed for N-point fast Fourier transform (FFT) networks that consists of log/sub 2/N stages with N/2 two-point butterfly modules for each stage. The method assumes that failures are confined to a single complex multiplier or adder or to one input or output set of lines. Such a fault model covers a broad class of faults. It is shown that only a small overhead ratio, O(2/log/sub 2/N) of hardware, is required for the networks to obtain fault-secure results in the first scheme. A novel data retry technique is used to locate the faulty modules. Large roundoff errors can be detected and treated in the same manner as functional errors. The retry technique can also distinguish between the roundoff errors and functional errors that are caused by some physical failures. In the second scheme, a time-redundancy method is used to achieve both error detection and location. It is sown that only negligible hardware overhead is required. However, the throughput is reduced to half that of the original system, without both error detection and location, because of the nature of time-redundancy methods. >
257 citations
233 citations
References
More filters
TL;DR: Good generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series, applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices.
Abstract: An efficient method for the calculation of the interactions of a 2' factorial ex- periment was introduced by Yates and is widely known by his name. The generaliza- tion to 3' was given by Box et al. (1). Good (2) generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices, where m is proportional to log N. This results inma procedure requiring a number of operations proportional to N log N rather than N2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of N. It is also shown how special advantage can be obtained in the use of a binary computer with N = 2' and how the entire calculation can be performed within the array of N data storage locations used for the given Fourier coefficients. Consider the problem of calculating the complex Fourier series N-1 (1) X(j) = EA(k)-Wjk, j = 0 1, * ,N- 1, k=0
11,795 citations
07 Nov 1966
TL;DR: The "Fast Fourier Transform" has had a major effect on several areas of computing, the most striking example being techniques of numerical convolution, which have been completely revolutionized.
Abstract: The "Fast Fourier Transform" has now been widely known for about a year. During that time it has had a major effect on several areas of computing, the most striking example being techniques of numerical convolution, which have been completely revolutionized. What exactly is the "Fast Fourier Transform"?
493 citations
TL;DR: A digital processor capable of computing the discrete Fourier transform for a range of audio signals in real time has been built as part of a facility to conduct research in signal processing.
Abstract: —A digital processor capable of computing the discrete Fourier transform for a range of audio signals in real time has been built as part of a facility to conduct research in signal processing. The digitized sample values can be complex. The arithmetic unit is configured to perform complex connectives, and automatic array scaling is used to make numerical accuracy independent of signal level. The Cooley–Tukey "fast Fourier transform" is the algorithm used.
25 citations
TL;DR: The class of digital filters which have an impulse response of finite duration and are implemented by means of circular convolutions performed using the discrete Fourier transform is considered and a least upper bound is obtained for the maximum possible output of a circular convolution for the general case of complex input sequences.
Abstract: When implementing a digital filter, it is important to utilize in the design a bound or estimate of the largest output value which will be obtained. Such a bound is particularly useful when fixed point arithmetic is to be used since it assists in determining register lengths necessary to prevent overflow. In this paper we consider the class of digital filters which have an impulse response of finite duration and are implemented by means of circular convolutions performed using the discrete Fourier transform. A least upper bound is obtained for the maximum possible output of a circular convolution for the general case of complex input sequences. For the case of real input sequences, a lower bound on the least upper bound is obtained. The use of these results in the implementation of this class of digital filters is discussed.
8 citations