Showing papers on "Non-uniform discrete 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.

The use of finite fields to compute convolutions

Abstract: A transform is defined in the Galois field of q^2 elements GF(q^2) , a finite field analogous to the field of complex numbers, when q is a prime such that (--1) is not a quadratic residue. It is shown that the action of this transform over GF(q^2) is equivalent to the discrete Fourier transform of a sequence of complex integers of finite dynamic range. If q is a Mersenne prime, one can utilize the fast Fourier transform (FFT) algorithm to yield a fast convolution without the usual roundoff problem of complex numbers.

Electrocardiographic Data Compression Via Orthogonal Transforms

Abstract: Electrocardiographic data compression via orthogonal transform processing is studied using canine ECG data. The Haar transform and the discrete cosine transform are considered. While the basis vectors for the Haar transform are sampled rectangular waves, those for the discrete cosine transform are sampled sinusoids.

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.

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.

Modular discrete cosine transform system

Abstract: Apparatus for performing an even discrete cosine transform (EDCT) on an it signal of size 2N, with components for an EDCT of size N, which comprises four similar modules, for performing an extended discrete Fourier transform (EDFT) on their input signals. Each EDFT module comprises means for generating the signal ##EQU1## s = - (N- 1), . . . , (N- 1), an input multiplier, a transversal filter, and an output multiplier. The apparatus further comprises a switching means, means for generating a signal (- 1)n, two input multipliers, two complex attenuators, two other means for generating a signal, four output multipliers, two signal summers, and two circuits for taking the real part of an input signal.

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.

Programable filter using chirp-Z transform

Abstract: A transform structure employing surface wave devices performs Fourier transform operations on a time-limited substantially band limited signal. The output signal of the transform structure is related to the Fourier transform of the input signal in a manner which permits forming the product of this output with a similar output of a second transform structure. This product is inverted by a third transform structure so as to provide the convolution of the two original input functions. The overall structure comprises a filter apparatus readily programable in either the time or frequency domain.

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).

Natural-abundance carbon-13 Fourier transform n.m.r. studies of large molecules

Abstract: The resolution in proton-decoupled natural-abundance i3 NMRt spectra of large organic and biological molecules is normally considerably greater than in proton NMR spectra, as a result of the large range of '3C chemical shifts 'and the absence of' 3C—' 3C coupling. The Fourier transform NMR technique has partly overcome the problem of very poor sensitivity of natural-abundance l3 NMR. As examples of structural applications, 13C NMR has been used to determine the detailed anomeric composition of aqueous fructose, and to measure the extent and type of branching of various dextrans. Some problems that arise when l3( Fourier transform NMR is used for quantitative analysis are discussed. Fourier transform NMR instrumentation provides a built-in capability of measuring the spin—lattice relaxation time (T,) of individual-carbon resonances. The meaning of T, and how spin—lattice relaxation occurs are briefly discussed. Examples of T, determinations are presented. The use of T, measurements for studying rotational motions of molecules and internal rotations of sidechains is discussed. PRFT NMR spectra can be used in some cases for assigning resonances to specific carbons. The development of a probe for sample tubes of 20 mm outside diameter has increased the sensitivity of natural-abundance '3C Fourier transform nuclear magnetic resonance to the point that single-carbon resonances of proteins can be studied. Numerous narrow single-carbon resonances are observed in the aromatic region of the '3C spectrum of native hen egg-white lysozyme. Theoretical and experimental evidence shows that these narrow resonances are those of the 28 non-protonated aromatic carbons. The 59 protonated aromatic carbons give rise to a background of broad peaks. Significant chemical shift variations occur upon folding of the protein into its * Contribution No. 2613. t Abbreviations: NMR, nuclear magnetic resonance PRFT, partially relaxed Fourier transform NOE, nuclear Overhauser enhancement VLDL, very-low-density lipoprotein LDL, low-density lipoprotein HDL, high-density lipoprotein 247

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.

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.

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.

A fast Gaussian method for Fourier transform evaluation

Abstract: A Gaussian method for fast evaluation of approximations to Fourier integral transforms is presented. This method is faster than the FFT for transforms of functions that require considerable computer time to compute. It is especially useful when transforms of high accuracy are needed.

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.

Fourier Representation of Signals

Abstract: The purpose of this chapter is twofold. First, it presents a review of the Fourier methods of representing signals. Second, it provides the foundation for a systematic transition from the Fourier representation of analog signals to that of digital signals.

Binary windows for the discrete Fourier transform

Abstract: An efficient structure is suggested for the frequency-domain windowing of discrete Fourier transforms. In this scheme, multiplications are replaced by shifts in the position of the binary point. Three new window functions are described which can be realized by the suggested structure.

How to compute two complex even Fourier transforms with one transform step

Abstract: A procedure is given for calculating the discrete Fourier transforms (DFT) of two separate complex even N-point functions simultaneously using only a single N-point DFT plus N additional multiplications.

Real Time Discrete Fourier Transforms Using a Programmable Diode-Convolver Module

Abstract: A new surface acoustic wave delay line module is described which provides, for the first time, the ability to compute the real time Discrete Fourier Transform (DFT) with electronically variable bandwidth. Initial experiments with 12 and 32 tap PDC modules are described which demonstrated a 40 dB tap dynamic range and Fourier bandwidth variable from zero to beyond 10 MHz.

Factorization of Fourier and Laplace transforms as a basis of the fast Fourier transform

Abstract: A mechanistic method of finding the next step in an arbitrarily large fast Fourier transform at an arbitrary stage is outlined. The method starts by writing down two different forms for the discrete Fourier transform of a sampled function and converting one into the other by decomposition using partial fractions. The process demonstrated corresponds to decimation in time. The alternative of decimation in frequency is demonstrated through an analogous procedure using the Laplace transform of a sampled function. It is suggested that the method should help in understanding the organization of a large-scale FFT.