Showing papers on "Discrete-time Fourier transform published in 1975"

A new algorithm in spectral analysis and band-limited extrapolation

Abstract: If only a segment of a function f (t) is given, then its Fourier spectrum F(\omega) is estimated either as the transform of the product of f(t) with a time-limited window w(t) , or by certain techniques based on various a priori assumptions. In the following, a new algorithm is proposed for computing the transform of a band-limited function. The algorithm is a simple iteration involving only the fast Fourier transform (FFT). The effect of noise and the error due to aliasing are determined and it is shown that they can be controlled by early termination of the iteration. The proposed method can also be used to extrapolate bandlimited functions.

Fourier analysis with unequally-spaced data*

Abstract: The general problems of Fourier and spectral analysis are discussed. A discrete Fourier transformF N (v) of a functionf(t) is presented which (i) is defined for arbitrary data spacing; (ii) is equal to the convolution of the true Fourier transform off(t) with a spectral window. It is shown that the ‘pathology’ of the data spacing, including aliasing and related effects, is all contained in the spectral window, and the properties of the spectral windows are examined for various kinds of data spacing. The results are applicable to power spectrum analysis of stochastic functions as well as to ordinary Fourier analysis of periodic or quasiperiodic functions.

A restriction theorem for the Fourier transform

Abstract: Jl/(0)ld0 = ƒƒ* f(x)fà(x)dx = fmdè*f(x)dx<\\\\f\\\\p\\\\âd *f\\\\p, for conjugate indices p and p . Thus it suffices to prove that the operator given by convolution with *3$ is bounded from LP to LP for p in the appropriate range. Let K(x) be a radial Schwartz function with K(x) = 1 for \\x\\ < 100, and let Tk(x) = [K(x/2 ) -K(xl2-)] $)(*). It suffices to show there exists e = e(p) > 0 such that \\\\Tk * ƒ \\\\p, < C2~ || ƒ ||p. This follows from interpolating the estimates \\\\Tk * ƒ IL < C2\"~>*/|| f\\\\x and ||rfc *f\\\\2 < 2\\\\f\\\\2. Professor E. M. Stein has extended the range of this result to include p = 2(n + l)/(n + 3). His proof uses complex interpolation of the operators given by convolution with the functions Ba(x) = J0(27t\\x\\)/\\x\\°. Then

Number theoretic transforms to implement fast digital convolution

Abstract: Transforms using number theoretic concepts are developed as a method for fast and error-free calculation of finite digital convolution. The transforms are defined on finite fields and rings of integers with arithmetic carried out modulo an integer and it is shown that under certain conditions this gives the same results as conventional digital convolution. Because of these characteristics they are ideally suited to digital computation by taking into account quantization of amplitude as well as time in their definition. When the modulus is chosen as a Fermat number a transform results that requires only on the order of N log N additions and word shifts but no multiplications. In addition to being efficient, they have no roundoff error and do not require storage of basis functions. There is a restriction on sequence length imposed by word length and a problem with overflow but methods for overcoming these are presented. Results of an implementation on an IBM 370/155 are presented and compared with the fast Fourier transform showing a substantial improvement in efficiency and accuracy. Variations on the basic number theoretic transforms are also presented.

An Analysis of the Electromyogram by Fourier, Simulation and Experimental Techniques

Abstract: The electromyogram of a single motor unit is studied by considering it as a time function defined by a convolution integral where a point process input passes through a filter whose impulse response is the shape of a single motor unit action potential. The interspike intervals are assumed to be normally distributed, independent random variables. Simulation is performed on a digital computer. The theoretical analysis shows that the absolute value of the ensemble average of the Fourier transform of the simulated EMG approaches the absolute value of the Fourier transform of the motor unit potential. This has been confirmed by simulation except at the very low end of the spectrum. These results are compared with the Fourier transforms of the recorded surface EMG data from human muscles.

A Fourier series method for numerical Kramers-Kronig analysis

Abstract: Using a time domain method the Kramers-Kronig integrals are derived without recourse to complex analysis (except in evaluating the Fourier transform of sgn (t)). From the time domain result, a Fourier series method for numerical evaluation of causality relations is derived. This method eliminates the need to use numerical integration, the use of logarithms in evaluating the function and the consideration of Cauchy principal parts. Through the use of the fast Fourier transform algorithm the calculation can be vary rapid. The accuracy of the technique is considered.

A Fourier method for the numerical solution of Poisson’s equation

Abstract: A method for the solution of Poisson's equation in a rectangle, based on the relation between the Fourier coefficients for the solution and those for the right-hand side, is developed. The Fast Fourier Transform is used for the computation and its influence on the accuracy is studied. Error estimates are given and the method is shown to be second order accurate under certain general conditions on the smoothness of the solution. The accuracy is found to be limited by the lack of smoothness of the periodic extension of the inhomogeneous term. Higher order methods are then derived with the aid of special solutions. This reduces the problem to a case with sufficiently smooth data. A comparison of accuracy and efficiency is made between our Fourier method and the Buneman algorithm for the solution of the standard finite difference formulae.

System identification using pseudorandom signals and the discrete Fourier transform

Abstract: The theory of estimation of the frequency response of a system as the ratio of the discrete Fourier transforms of its sampled output and input, when the input is a pseudorandom signal, is developed. The principal sources of error are identified, their effects on the estimates are determined, and methods of error correction and reduction are described. Properties of the discrete Fourier transforms of pseudorandom sequences derived from binary and ternary m sequences are obtained, and the suitability of the corresponding pseudorandom signals for use as test signals in this application is established. The use of fast Fourier-transform techniques for the reduction of computation time is discussed, and the relative performance of these techniques and the crosscorrelation method for the estimation of both frequency and impulse responses of systems is evaluated.

A 500-point fourier transform using charge-coupled devices

Abstract: A CCD transversal filter chip, which performs a 500-point discrete Fourier transform using the chirp z-transform algorithm, will be described. Performance characteristics will be demonstrated, new operational modes presented, and system applications discussed.

A Parallel Radix-4 Fast Fourier Transform Computer

Abstract: The organization and functional design of a parallel radix-4 fast Fourier transform (FFT) computer for real-time signal processing of wide-band signals is introduced.

Apparatus for the generation of a two-dimensional discrete fourier transform

Abstract: An apparatus for the generation of a two-dimensional discrete Fourier transform of an input signal developed from data within a data block, or field, having a size of N1 by N2 data points, N1 and N2 being relatively prime with respect to each other, the transforms being in a form suitable for subsequent electronic processing. The apparatus comprises an input scan apparatus connectable to the N1 by N2 two-dimensional data field comprising the input signal, such as a television viewing field. The apparatus receives and scans in proper order, or sequence, the N1 by N2 input data so that the subsequently generated one-dimensional Fourier transform of the length N1 N2 serial data string is identical to an N1 by N2 two-dimensional discrete Fourier transform of the N1 by N2 input data samples. A serial-access one-dimensional discrete Fourier transform (DFT) device, connected to the input scan generator, generates a one-dimensional discrete Fourier transform of the length N1 N2 serial data string.

Relation between the radiation pattern of an array and the two-dimensional discrete Fourier transform

Abstract: A linear wideband array with each element followed by a tapped-delay line may be considered as a two-dimensional distal filter. Accordingly, the radiation pattern of the processor may be expressed as the product of a pair of two-dimensional discrete Fourier transforms (DFT's).

Discrete Fourier transform via cross correlation charge transfer device

Abstract: A circuit for generating a discrete Fourier transform in real time employs a digital and analog shift register, each cell of which is tapped to feed an analog switch. The outputs of the individual analog switches are fed to a summing bus where the switched analog signals combine to form the desired Fourier transform.

Spectral analysis using Fourier Transform techniques

Abstract: The present study is intended as a guide for those who wish to undertake spectral analyses using Discrete Fast Fourier Transforms. Points of particular difficulty in using Fourier Transforms are derived in some detail. Experimental results are offered to illustrate the mathematical derivations. Finally the case of a signal with discontinuities along the time-axis is discussed.

Sampling Expansions with Derivatives for Finite Hankel and Other Transforms

Abstract: A sampling expansion involving the samples of a function represented by a finite Hankel transform and the samples of the derivative of the function is derived. Also, the general procedure for obtaining sampling expansions with derivatives for functions represented by other finite integral transforms is outlined. It is shown that in parallel to the known special case of the finite Fourier transform that the advantage of sampling with N derivatives is to increase by $(N + 1)$-fold the asymptotic spacing between the sampling points. The importance of such an advantage for the Hankel transform can be realized in a time-varying or spatial-varying system.Finally, an extension to two dimensions of the sampling theorem with N derivatives for a function having a finite double Fourier transform is stated.

Fast convolution with finite field fast transforms

Abstract: The fast convolution procedure for processing discrete data requires that a transform of the data and the filter pulse response be formed, followed by the inverse transform of their (complex) product. The finite field fast transform eliminates any roundoff error due to internal multiplication, eliminates truncation of irrational coefficients, and requires only real arithmetic (addition and multiplication). This note develops a realization scheme for such a transform using the Chinese remainder theorem.

Fast Fourier Transform

Abstract: The main objective of this chapter is to develop a fast algorithm for efficient computation of the DFT. This algorithm, called the fast Fourier transform (FFT), significantly reduces the number of arithmetic operations and memory required to compute the DFT (or its inverse). Consequently, it has accelerated the application of Fourier techniques in digital signal processing in a number of diverse areas. A detailed development of the FFT is followed by some numerical examples which illustrate its applications.

Fourier Transformation in Partially Coherent Light

Abstract: Fourier transformation in partially coherent light has been theoretically studied. It is shown that the intensity distribution at the back focal plane of a lens when a transparency located at the front focal plane and illuminated by a partially space coherent light is given by the convolution of the Fourier transform of mutual intensity function and the autocorrelation function of the object. A similarity between this result and Schell's theorem has been identified.

Pseudonoise test signals and the fast Fourier transform

Abstract: Recent published work has indicated a growing interest in the combined use of periodic pseudonoise (p.n.) test signals and the fast Fourier transform (f.f.t.). However, it has been noted by Barker and Davy that the period of pseudonoise sequences are such that it is not possible to use the most efficient form of the f.f.t. This letter points out ways round the problem by drawing attention to a previously published result in this area. In addition, a little known algorithm that directly exploits the f.f.t. to generate pseudonoise is given.

On approximating Fourier integral transforms by their discrete counterparts in certain geophysical applications

Abstract: In the past, fast Fourier transforms (FFT's) have been used in geophysical data analysis and, with some success, in theoretical analysis. However, in the general situation when neither the function nor its transform image is of compact support, the digital transforms inevitably introduce aliasing errors in both the domain and range space. A technique to assess and minimize this error is given for a restricted set of functions.

Roundoff error in fast Fourier transforms

Abstract: The finite word length used in the computer causes round-off error in the calculation of Fourier coefficients. When the fast Fourier transform method is used, the statistical mean-square error has been previously determined [3] for the case of the decimation-infrequency algorithm. This letter treats the same problem for the decimation-in-time algorithm.

Continuous Time Regressions with Discrete Data

Abstract: A general continuous time distributed lag model is considered. The problem is that of estimating the parameters of the kernel when, as is often the case, the available data consist not of a continuous record but of discrete observations recorded at regular intervals of time. Fourier transformation of the model and insertion of the computable, discrete Fourier transforms of the variables produce an approximate model which is of non-linear regression type and is relatively easy to handle. Estimators are proposed and their asymptotic properties established, assuming principally that the variables are stationary and ergodic and that an "aliasing" condition on the independent variable is satisfied. The results of the paper imply a theory for the estimation of rather general continuous time systems, involving the operations of differentiation, integration and translation through time.