Showing papers on "Discrete-time Fourier transform published in 1972"

Fourier Descriptors for Plane Closed Curves

Abstract: A method for the analysis and synthesis of closed curves in the plane is developed using the Fourier descriptors FD's of Cosgriff [1]. A curve is represented parametrically as a function of arc length by the accumulated change in direction of the curve since the starting point. This function is expanded in a Fourier series and the coefficients are arranged in the amplitude/phase-angle form. It is shown that the amplitudes are pure form invariants as well as are certain simple functions of phase angles. Rotational and axial symmetry are related directly to simple properties of the Fourier descriptors. An analysis of shape similarity or symmetry can be based on these relationships; also closed symmetric curves can be synthesized from almost arbitrary Fourier descriptors. It is established that the Fourier series expansion is optimal and unique with respect to obtaining coefficients insensitive to starting point. Several examples are provided to indicate the usefulness of Fourier descriptors as features for shape discrimination and a number of interesting symmetric curves are generated by computer and plotted out.

Discrete Convolutions via Mersenne Transrorms

Abstract: A transform analogous to the discrete Fourier transform is defined in the ring of integers with a multiplication and addition modulo a Mersenne number. The arithmetic necessary to perform the transform requires only additions and circular shifts of the bits in a word. The inverse transform is similar. It is shown that the product of the transforms of two sequences is congruent to the transform of their circular convolution. Therefore, a method of computing circular convolutions without quantization error and with only very few multiplications is revealed.

Logical convolution and discrete Walsh and Fourier power spectra

Abstract: The Walsh power spectrum of a sequence of random samples is defined as the Walsh transform of the logical autocorrelation function of the random sequence. The "logical" autocorrelation function is defined in a similar form as the "arithmetic" autocorrelation function. The Fourier power spectrum, which is defined as the Fourier transform of the arithmetic autocorrelation function, can be obtained from the Walsh power spectrum by a linear transformation. The recursive relations between the logical and arithmetic auto-correlation functions are derived in this paper. For a given process with computed or modeled autocorrelation function the Fourier and Walsh power spectra are computed by using the fast Fourier and Walsh transforms, respectively. Examples are given from the speech and imagery data.

A Recurrence Technique for Expanding a Function in Spherical Harmonics

Abstract: A recurrence technique is described that enables the use of proven efficient Fourier transform techniques to be applied to the expansion of a given function in terms of spherical harmonics.

Real-time Fourier transform with a surface-wave convolver

Abstract: The generation of Fourier transforms of electronic signals in real time with an acoustic-surface-wave convolver is demonstrated. Two matched chirps propagating through each other behave as a narrowband filter in wavevector space, with the filter centre moving at a rate proportional to the acoustic velocity and the chirp rate.

Accurate calculation of Fourier transform of two-center Slater orbital products†

Abstract: A method is presented which leads to accurate Fourier transform values of any 1s−1s Slater-type orbital overlap distribution. The numerical merits are discussed and illustrated by some examples.

A fast Fourier transform algorithm for symmetric real-valued series

Abstract: A new algorithm is presented for calculating the real discrete Fourier transform of a real-valued input series with even symmetry. The algorithm is based on the fast Fourier transform algorithm for arbitrary real-valued input series (FTRVI) [1], [2]. By eliminating all unnecessary steps and storage locations, and by rearranging the intermediate results and the operation sequence, it is possible to reduce the computation time and the required core storage by a factor of 2 as compared to the case of arbitrary real input or by a factor of 4 as compared to the general fast Fourier transform for complex inputs.

Spectral errors resulting from random sampling-position errors in fourier transform spectroscopy.

Abstract: In the design and operation of a two-beam interferometer in Fourier spectroscopy, it is important to known how much the lack of precision in the setting of the optical path differences will affect the measured spectra. This problem has been considered by Surh, and by Sakai, who has given a relation for the stan­ dard deviation in the size of the \"ghost\" lines due to the random error in the sampling of the interferogram of a monochromatic spectral line. We will reformulate and extend the results to apply to a more general spectrum. Consider an interferogram P(x) which is sampled at optical path differences x = jΔx giving values Pj for N, conveniently an even integer, values of the integer j in the range — N/2 ≤ j < N/2. The discrete Fourier transform pair

Interpolation using the fast Fourier transform

Abstract: The fast Fourier transform (FFT) algorithm has had widespread influence in many areas of computation since its "rediscovery" by Cooley and Tukey [1] An efficient and accurate method for interpolation of functions based on the FFT is presented As an application, the generation of the characteristic polynomial in the "generalized eigenvalue problem" [2] is considered

Anharmonic frequency analysis

Abstract: A new numerical method of frequency analysis is described, designed mainly to search for discrete frequencies in a time series. An integral transform is applied twice to the data for different reference times. A complex amplitude within a selected narrow frequency band is obtained for each transform. The frequency is then determined from the phase change of the complex amplitude over the difference of the two reference times. Very high precision is obtained, which is demonstrated in two examples. 1. Introduction. We consider a real time function X(t) and assume that it can be described either by a set of discrete frequencies and their complex amplitudes (at a given reference time) or by a continuous complex amplitude density function over some frequency range or a combination of both. The term frequency analysis is used here for a process which determines the frequencies, amplitudes and phases of the spectral components of the time function. There are three classical methods available which perform frequency analysis under specific conditions: the Fourier series, the Fourier integral and Prony's method. Each is restricted in its application, depending on the properties of the time function. Since all three methods are well known and described in standard textbooks, we will only repeat some of their essential properties. (a) Fourier series. If X(t) is periodic with a period T, it can be described by a constant term and a finite or infinite set of harmonic frequencies, where the basic frequency is equal to the reciprocal of the period T and all other frequencies are integer multiples of the basic frequency. The complex amplitude is obtained by the well-known integral transform of X(t) over the interval T. The integration extends only over the (finite) time interval T, since X(t) is periodic, and no additional in- formation is introduced into the process, if the integral transform were to be ex- tended over several periods. On the other hand, when a time function X(t) is given only for a time interval T and this integral transform is applied, it is automatically assumed that the time function is periodic with the period T, and no additional information is obtained by an extrapolation. Therefore, for example, the Fourier series has no real application in prediction problems. (b) Fourier integral. Here, a frequency continuum is provided for the analysis. The integral transform now determines the amplitude density as a function of fre- quency. The time function is aperiodic and must not contain discrete frequencies with finite amplitudes (equivalent to infinite amplitude density), otherwise the trans- form is not convergent. The integration has to be performed over the infinite time range from - a to + a, unless the time function is zero outside a certain time

A note on the multidimensional sampling theorem

Abstract: The multidimensional sampling theorem is extended to allow nonuniform, but periodic, sampling with explicit and simple reconstruction formulas. The extension is applicable to hexagonal and other nonrectangular lattices; such sampling schemes can be as efficient as rectangular sampling.

Calculation of the vertical gradient of the gravity field using the fourier transform

Abstract: Fourier transform techniques have been used to calculate the theoretical filter (amplitude) response function of Nth order vertical derivative continuation operation. The amplitude response functions of the vertical gradient and its continuation follow from the same. These response functions are subsequently used to calculate the weighting coefficients suitable for two dimensional equispaced data. A shortening operator has been incorporated to limit the extent of the operator. For comparative study, some of the developed coefficient sets and the one presented in this paper are analysed in the frequency domain and their merits and demerits are discussed.

Fast convolution for recursive digital filters

Abstract: A physically motivated procedure is presented for evaluating the response of recursive digital filters via the fast Fourier transform (FFT). The key idea of the method is to employ a short string of artificial impulses as an alternative representation of the initial conditions implicit in processing a long excitation in segments. The method is computationally more efficient or of wider applicability than existing FFT techniques.

On the localization of rectangular partial sums for multiple Fourier series

Abstract: The question of the localization for rectangular partial sums of the multiple Fourier series for functions of Sobolev spaces is settled.

An algorithm for the on-line computation of fourier spectra

Abstract: An on-line method to compute the spectra associated with the Fourier transform M of a data sequence is developed, whereM and N are finite positive integers. This method provides a simple means of generating time-frequency-amplitude plots of Fourier power and phase spectra. Such plots may be used to display Fourier spectra for pedagogical purposes, and in the general area of the classification of transient waveforms whose durations are unknown. An illustrative example is included.

Summability and Fourier analysis.

Abstract: Let s = {sn} be an infinite sequence of complex numbers, that is, a continuous function on the discrete additive semigroup of natural numbers N. The sequence s has a continuous extension s to βN, the Stone-Cech compactification of N (s takes the value if s is unbounded). Throughout the paper, the symbol βZ denotes the Stone-Cech compactification of the space Z, and the continuous extension of a function / defined on Z to βZ will be denoted by f; for a description of the Stone-Cech compactification we refer the reader to [2, pp. 82-93], We impose the norm

Automatic equalization and discrete Fourier transform techniques (Ph.D. Thesis abstr.)

Abstract: The primary objective of this thesis is the study and analysis of techniques that use discrete Fourier transforms (DFT) in conjunction with fast Fourier transform (FFT) algorithms for automatic equalization of synchronous data transmission. W e show that the problem of minimizing mean-square intersymbol interference can be analyzed in the discrete f requency domain as an optimization problem with constraints. Various solutions to this problem are studied including Rosen’s gradient projection method, Lagrange multipliers, and direct substitution. W e prove that the rate of convergence toward the opt imum parameter setting is faster for the gradient projection scheme, for channels usually d iscussed in the literature, than for the corresponding t ime-domain technique. W e then develop an alternative gradient projection method that provides savings proport ional to N/log, N in the number of required computat ions, where N is the number of discrete f requency parameters. Finally, we devise a scheme for finding an approximate solution in one iteration using gradient projection. This solution was found to be almost exact for the particular cases we simulated, even in the presence of noise. The speed of convergence for all the schemes is dependent on the overall channel characteristics and each method has advantages in certain special situations. W e show that all the aforement ioned methods converge in the mean in the presence of noise. A var iance bound indicates that the var iance about this setting is finite and can be made as small as desired by reducing the gradient step size and hence the speed of convergence. Our second objective is to use the DFT and FFT algorithms to make time domain equalization computationally more efficient. W e show that the number of computat ions needed to set a time domain equalizer can be made proport ional to M log, M instead of MZ, where M is the number of adjustable parameters, by use of FFT algorithms. Savings always ensue for sufficiently large M and grow rapidly thereafter. The breakeven point is approximately M = 16. W e also derive tight and easily obtainable bounds on the eigenvalues of the t ime-domain iteration matrix in terms of DFT coefficients. This allows us to increase the speed of convergence. Leonard P. W inkler, “Opt imum and adapt ive detector arrays,” Ph.D., Dep. Elec. Eng., Polytech. Inst. Brooklyn, Brooklyn, N.Y., June 1971. Adviser: Mischa Schwartz.

Convolution and module theory of linear systems

Abstract: This paper shows that both discrete and continuous time-invariant, linear, dynamic systems can be cast in a module framework The work here is based on convolution theory and employs Mikuainski's convolution quotients, which are more powerful and appropriate for this purpose than the conventional Laplace and Z transform techniques.