Showing papers on "Hartley transform published in 1971"

The fast Fourier transform in a finite field

Abstract: A transform analogous to the discrete Fourier transform may be defined in a finite field, and may be calculated efficiently by the 'fast Fourier transform' algorithm. The transform may be applied to the problem of calculating convolutions of long integer sequences by means of integer arithmetic.

Computation of spectra with unequal resolution using the fast Fourier transform

Abstract: The discrete Fourier transform of a sequence, which can be computed using the fast Fourier transform algorithm, represents samples of the z transform equally spaced around the unit circle. In this letter, a technique is discussed and illustrated for transforming a sequence to a new sequence whose discrete Fourier transform is equal to samples of the z transform of the original sequence at unequally spaced angles around the unit circle.

An extension of the Hilbert transform product theorem

Abstract: A Hilbert transform multiplication theorem, well known in communicacation theory, is extended to the case when the functions involved are functions of n-dimensional real vectors.

Integral Transforms in Applied Mathematics

Abstract: An intermediate-level text on the use of integral transforms in applied mathematics and engineering. Existing works either cover the subject in more elementary form or are advanced treatises. In a very lucid style the author deals with the use of this important mathematical tool to solve ordinary and partial differential equations in problems in electrical circuits, mechanical vibration and wave motion, heat conduction, and fluid mechanics. The book is divided into five parts covering integral transform pairs, the Laplace transform, Fourier transforms, Hankel transforms, and finite Fourier transforms. A basic knowledge of complex variables and elementary differential equations is assumed. There are many exercises and examples drawn from the above fields, tables of the transform pairs needed in the text, and a glossary of terms with which the student may be unfamiliar. For the student who seeks further background on the subject, an annotated bibliography is provided.

A contribution to the theory of short-time spectral analysis with nonuniform bandwidth filters

Abstract: The mathematical representation of the short-time spectral analysis is extended from the case of uniform bandpass filters (that is, filters having the same complex envelope) to the case of nonuniform filters (that is, filters whose complex envelope depends upon their center frequency). This leads to an integral transform, formally similar to the Fourier transform, where the signal taken up to the observation time appears weighted by a function (namely, the complex envelope) depending on the frequency of analysis. Of course, for every choice of such a complex envelope (or of the equivalent set of filters), one has a corresponding integral transform to deal with. The particular case of complex envelopes as functions of the time-frequency product is studied here because of its great physical interest (it applies, for instance, to many existing "real-time audio analyzers"). The corresponding integral transform is shown to have two remarkable properties: 1) it admits an inverse integral transform; 2) it is "form invariant" under linear time scaling of the signal, and no other integral transform (that is, no other class of complex envelopes, even frequency independent) shares this property. The physical significance of such results is discussed, together with some ideas for applications and further theoretical work.

The Fourier transform of spline-function approximations to continuous data

Abstract: The transform of a spline-function approximation to continuous data is called a spline transform. In this correspondence, the spline and the discrete Fourier transforms (DFT) are compared as means for numerical computation of the Fourier integral transform. It is shown how the spline transform reduces errors introduced by the discrete transform and alleviates noise problems when the sampling rate is limited due to experimental method or hardware constraints.

New Generalized Bessel Transform and Its Relationship to the Fourier, Watson, and Kontorowich‐Lebedev Transforms

Abstract: A generalized Bessel transform is postulated, and the corresponding inversion formula is deduced. This transform is used to show the direct relationship between the Fourier, Watson, and Kontorowich‐Lebedev transforms. Thus, solutions of a certain class of boundary value problems can be expressed in terms of a single Bessel transform. Furthermore, the curvature of the boundary may range from zero to infinity.

Harmonic analysis on the poincare group. ii. the fourier transform.

Abstract: The generalized matrix elements on the Poincaré group are used to write the Fourier transform explicitly. This realizes a mapping between positive type functions on the group and generalized density matrices.

Time-varying Fourier transform

Abstract: A time-varying Fourier transform is defined along with its related power and phase spectra. A convenient recursive technique to compute this transform is also presented.

Non-linear transformations of stochastic processes associated with Fourier transforms of band-limited positive functions†

Abstract: The passage of Gaussian, processes (both stationary and non-stationary) through a device characterized by a convolution transform is studied. The output of such a device is a non-stationary stochastic process of harmonizable type. This process along with its finite Fourier transform are examined in detail. Some sampling theorems are also stated. A stochastic series representation in terms of prolate spheroidal wavefunctions is derived.

Shorter Notes: The Fourier Transform is Onto Only When the Group is Finite

Abstract: A well-known result of Henry Helson is used to prove that a locally compact abelian group is finite if and only if the Fourier transform is a surjective map. Let G be a locally compact abelian group (LCAG) with dual group G (see [3] for definitions and basic facts). It has been proved by Segal in [4] and Rajagopalan in [2] that the Fourier transform T:Ll(G)—>Co(G) is onto if and only if G is finite. Segal's proof appeals to the principal structure theorem for a LCAG and involves proving the result for groups with special properties and then for their Cartesian products, whereas Rajagopalan uses a theorem of Kakutani and Birkhoff to show that if G is extremally disconnected, then G must be discrete. In this note, we provide a short proof using a well-known result about Helson sets. Definition. A compact subset H of G (G not discrete) is a Helson set if every continuous function on H is the restriction of a Fourier transform. Lemma. Let G be nondiscrete with Haar measure m. If H is a Helson set in G, then m(H) = 0. Proof. Let/ be the characteristic function of H and a —f dm. Then if m(H)5¿0, it follows that a is a nonzero bounded Borel measure on H, and Co(G) is onto if and only if G is finite. Received by the editors March 30, 1970. A MS 1969 subject classifications. Primary 4250, 4252.