Showing papers on "Hartley transform published in 1979"

A Sinusoidal Family of Unitary Transforms

Abstract: A new family of unitary transforms is introduced. It is shown that the well-known discrete Fourier, cosine, sine, and the Karhunen-Loeve (KL) (for first-order stationary Markov processes) transforms are members of this family. All the member transforms of this family are sinusoidal sequences that are asymptotically equivalent. For finite-length data, these transforms provide different approximations to the KL transform of the said data. From the theory of these transforms some well-known facts about orthogonal transforms are easily explained and some widely misunderstood concepts are brought to light. For example, the near-optimal behavior of the even discrete cosine transform to the KL transform of first-order Markov processes is explained and, at the same time, it is shown that this transform is not always such a good (or near-optimal) approximation to the above-mentioned KL transform. It is also shown that each member of the sinusoidal family is the KL transform of a unique, first-order, non-stationary (in general), Markov process. Asymptotic equivalence and other interesting properties of these transforms can be studied by analyzing the underlying Markov processes.

Is computing with the finite Fourier transform pure or applied mathematics

Abstract: Let me begin with my view of a bit of history. Before the Second World War mathematics in the United States was a servant of the needs of others and mathematicians taught service courses. Indeed, while A. Weil was teaching at an Eastern university it would be only a slight exaggeration to say that he was forbidden from presenting proofs in class and was called on the carpet by a dean for breaking this structure. In the years after the War, mathematics became a subject in its own right. Proofs became acceptable, as the creation of the "new math" proved to the world. Mathematicians were in demand, were men in their own right and no one's servants. However, this growth period had a very unfortunate side affect. While mathematics was becoming a subject in its own right, many of its practitioners wanted to rid themselves of their former servant image. They had felt denigrated by the service role; so they denigrated service mathematics. Unfortunately, they lumped together service mathematics and applied mathematics. And so during this growth period of mathematics, there sprang up a distinction between pure and applied mathematics. During these years, the applied mathematicians felt the pure mathematicians looked down on them, and so the communications between the pure and applied mathematicians virtually dried up. In this paper we willl show that there is really not much difference between pure and applied mathematics. Indeed, we will cite instances of pure and applied mathematicians doing the same or analogous mathematics, but because of the lack of communication neither knew of the others' work. With these broad generalities stated, let me try to explain how I came to the writing of this paper. This may perhaps serve as an example of how the gap between pure and applied mathematicians can be bridged. I became interested in the study of the finite Fourier transform because I needed to know the eigenvalues of the finite Fourier transform. This arose in the study of the multiplicity of the regular representation of a solvmanifold. This problem was solved and the solution can be found in [8, p. 95]. Tolimieri, and Tolimieri and I, took up this problem in [18] and [3] and related the eigenvalue problem of the finite Fourier transform to a certain algebra of theta functions as discussed in Chapter I of this paper. I felt that

Gegenbauer Transforms via the Radon Transform

Abstract: Use is made of the Radon transform on even dimensional spaces and Gegenbauer functions of the second kind to obtain a general Gegenbauer transform pair. In the two-dimensional limit the pair reduces to a Chebyshev transform pair.

Coherent Optical Implementation Of Generalized Two-Dimensional Transforms

Abstract: A coherent optical method capable of performing arbitrary two-dimensional linear transformations has recently been studied, in which transform coefficients are given by two-dimensional inner products of the input image and a set of basis functions. Since the inner product of two functions is equal to the value of their correlation when there is zero shift between the functions, it is possible to use an optical correlator to solve for the coefficients of the transform. By using random phase masks in the input and the filter planes of the correlator, we have been able to pack many coefficients close together in the output plane, and thus take better advantage of the space-bandwidth product of the optical system. Both the input random phase mask and the spatial filter are computer-generated holographic elements, created by a computer-controlled laser beam scanner. The system can be "pro-grammed" to perform arbitrary two-dimensional linear transformations. For demonstration, the set of two-dimensional Walsh functions was chosen as a transform basis. When the resolution of the Walsh functions was limited to 128 x 128, up to 256 transform coefficients were obtained in parallel. The signal-to-noise and accuracy of the transform coefficients were compared to the theory.

A winograd-Fourier transform algorithm for real-valued data

Abstract: A modified version of the Winograd-Fourier transform algorithm is presented for use in transforming real vectors. The new algorithm uses real arithmetic and real storage of intermediate results throughout while retaining the economy of Winograd's basic method. The derivation of the transform is explained and some programming techniques are discussed and illustrated.

Discrete Fourier transform via Walsh transform

Abstract: The purpose of this correspondence is to point out a number of significant references, in the area of Walsh-Fourier transform conversion, that were missed in a recent paper [1].

D.F.T. computation by fast polynomial transform algorithms

Abstract: A new method is introduced for the fast computation of multidimensional discrete Fourier transforms (d.f.t.). We show that some multidimensional d.f.t.s are mapped efficiently into one-dimensional d.f.t.s by using a single polynomial transform and some auxiliary calculations. Since polynomial transforms can be computed without multiplications, this approach reduces significantly the number of operations over the conventional fast Fourier transform (f.f.t.) and is therefore attractive for image-processing applications.

The tempered Fourier transform

Abstract: A constant‐Q digital spectral analysis scheme is described which exploits the ’’perfect fifth’’ symmetry of the 12‐tone musical scale.

New polynomial transform algorithms for fast DFT computation

Abstract: Polynomial transforms defined in rings of polynomials, have been introduced recently and shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper, we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFTs and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA).

Matrix multipliers for calculating orthogonal transforms

Abstract: The use of MOS transistors and weighted capacitors in a device to calculate the transform of a set of signal samples in 300ns will be discussed. Experimental results for a 16-point Hadamard and an 8-point complex Fourier transform will be given.

The Modification of Simple Images by Fourier Transform Manipulation

Abstract: Knowledge of how an image is changed by a given modification of its Fourier transform is important when attempting to find solutions to problems in image processing. Several types of filters were used to modify the Fourier transforms of an edge and a bartarget. The images were reconstructed from the modified transforms to determine the change produced by a given modification. Various regions of the Fourier transform determine specific characteristics of the image. The amplitude of the low frequencies controls the form and average level of the image, while the amplitude of the high frequencies effects sharp transitions and noise.

Parallelism in the computation of the FFT and the WFTA

Abstract: Arithmetic concurrencies, such as those found in special-purpose fast Fourier transform (FFT) hard-ware, are surveyed and categorized. Similar structures are then derived for the Winograd Fourier transform algorithm (WFTA). Relative time-efficiency plots are obtained for the 1024-point radix-4 FFT and the 1008-point WFTA as a function of the number of real arithmetic operations executable in parallel. This comparison shows that the relative time efficiency of the two algorithms in sequential computations generally carries over to cases where arithmetic parallelism is exploited.