Showing papers on "Hartley transform published in 1987"

Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint

Abstract: Previously it was shown that one can reconstruct an object from the modulus of its Fourier transform (solve the phase-retrieval problem) by using the iterative Fourier-transform algorithm if one has a nonnegativity constraint and a loose support constraint on the object. In this paper it is shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint. Sufficiently strong support constraints include certain special shapes and separated supports. Reconstruction results are shown, including the effect of tapered edges on the object’s support.

The DFT : an owner's manual for the discrete Fourier transform

Abstract: Preface 1. Introduction. A Bit of History An Application Problems 2. The Discrete Fourier Transform (DFT). Introduction DFT Approximation to the Fourier Transform The DFT-IDFT pair DFT Approximations to Fourier Series Coefficients The DFT from Trigonometric Approximation Transforming a Spike Train Limiting Forms of the DFT-IDFT Pair Problems 3. Properties of the DFT. Alternate Forms for the DFT Basic Properties of the DFT Other Properties of the DFT A Few Practical Considerations Analytical DFTs Problems 4. Symmetric DFTs. Introduction Real sequences and the Real DFT (RDFT) Even Sequences and the Discrete Cosine Transform (DST) Odd Sequences and the Discrete Sine Transform (DST) Computing Symmetric DFTs Notes Problems 5. Multi-dimensional DFTs. Introduction Two-dimensional DFTs Geometry of Two-Dimensional Modes Computing Multi-Dimensional DFTs Symmetric DFTs in Two Dimensions Problems 6. Errors in the DFT. Introduction Periodic, Band-limited Input Periodic, Non-band-limited Input Replication and the Poisson Summation Formula Input with Compact Support General Band-Limited Functions General Input Errors in the Inverse DFT DFT Interpolation - Mean Square Error Notes and References Problems 7. A Few Applications of the DFT. Difference Equations - Boundary Value Problems Digital Filtering of Signals FK Migration of Seismic Data Image Reconstruction from Projections Problems 8. Related Transforms. Introduction The Laplace Transform The z- Transform The Chebyshev Transform Orthogonal Polynomial Transforms The Discrete Hartley Transform (DHT) Problems 9. Quadrature and the DFT. Introduction The DFT and the Trapezoid Rule Higher Order Quadrature Rules Problems 10. The Fast Fourier Transform (FFT). Introduction Splitting Methods Index Expansions (One ---> Multi-dimensional) Matrix Factorizations Prime Factor and Convolution Methods FFT Performance Notes Problems Glossary of (Frequently and Consistently Used) Notations References.

The Fast Hartley Transform Algorithm

Abstract: The fast Hartley transform (FHT) is similar to the Cooley-Tukey fast Fourier transform (FFT) but performs much faster because it requires only real arithmetic computations compared to the complex arithmetic computations required by the FFT. Through use of the FHT, discrete cosine transforms (DCT) and discrete Fourier transforms (DFT) can be obtained. The recursive nature of the FHT algorithm derived in this paper enables us to generate the next higher order FHT from two identical lower order FHT's. In practice, this recursive relationship offers flexibility in programming different sizes of transforms, while the orderly structure of its signal flow-graphs indicates an ease of implementation in VLSI.

Improved Fourier and Hartley transform algorithms: Application to cyclic convolution of real data

Abstract: This paper highlights the possible tradeoffs between arithmetic and structural complexity when computing cyclic convolution of real data in the transform domain. Both Fourier and Hartley-based schemes are first explained in their usual form and then improved, either from the structural point of view or in the number of operations involved. Namely, we first present an algorithm for the in-place computation of the discrete Fourier transform on real data: a decimation-in-time split-radix algorithm, more compact than the previously published one. Second, we present a new fast Hartley transform algorithm with a reduced number of operations. A more regular convolution scheme based on FFT's is also proposed. Finally, we show that Hartley transforms belong to a larger class of algorithms characterized by their "generalized" convolution property.

Optical correlator performance of binary phase-only filters using Fourier and Hartley transforms

Abstract: Theoretical studies of the performance capabilities of binary phase-only filters (BPOFs), constructed using both Fourier and Hartley transforms, are presented. A thorough analysis of the Fourier BPOF is given. We show that, although BPOFs constructed using Fourier transforms perform well in optical correlator systems, they are also subject to additional noise sources and have the possibility of generating large false correlation signals. We then present an analysis of BPOFs constructed using the Hartley transform. We show that BPOFs made using the Hartley transform provide superior false correlation rejection and more uniformly sized correlation signals for heavily multiplexed BPOFs, compared with those made using the Fourier transform. We also present a technique for constructing Hartley BPOFs. Therefore, although it is well known that the quality of the correlation signal depends on the object, this work demonstrates that the quality of the correlation signal can also depend on the technique used in the synthesis of the BPOF.

Fast computation of the discrete cosine transform and the discrete Hartley transform

Abstract: A new factorization of the discrete Hartley transform (DHT) is presented. It is used to derive new algorithms for the DHT and the discrete cosine transform (DCT) with reduced number of multiplications.

Selective Fourier transform localization.

Abstract: We have introduced the selective Fourier transform technique for spectral localization. This technique allows the acquisition of a high-resolution spectrum from a selectable location with control over the shape and size of the spatial response function. The shape and size of the spatial response are defined during data acquisition and the location is selectable through processing after the data acquisition is complete. The technique uses pulsed-field-gradient phase encoding to define the spatial coordinates. In this paper the theoretical basis of the selective Fourier transform technique is developed and experimental results are presented, including comparisons of spectral localization using either the selective Fourier transform method or conventional multidimensional Fourier transform chemical-shift imaging. © Academic Press, Inc.

An improved digit-reversal permutation algorithm for the fast Fourier and Hartley transforms

Abstract: All radix-B fast Fourier transforms (FFT) or fast Hartley transforms (FHT) performed "in-place" require at some point that the sequence elements he permuted such that, indexing the elements 0 to N - 1, the element with index i is swapped with the element whose index is j. The permutation is called digit-reversing, because if i is represented as a string of digits, base B, then j is that index whose representation is the same string of digits written in reverse order. N is a power of B and B \geq 2 . An elegant algorithm has been found that Performs this "perfect shuffle" more efficiently and, according to timing experiments, runs about eight times faster than the fastest other algorithm known to the author. The algorithm is of order O(N) and led, for example, to a saving of 7 percent in the total (radix-2) FFT running time for N = 1024.

A three-dimensional DFT algorithm using the fast Hartley transform

Abstract: A three-dimensional (3-D) Discrete Fourier Transform (DFT) algorithm for real data using the one-dimensional Fast Hartley Transform (FHT) is introduced. It requires the same number of one-dimensional transforms as a direct FFT approach but is simpler and retains the speed advantage that is characteristic of the Hartley approach. The method utilizes a decomposition of the cas function kernel of the Hartley transform to obtain a temporary transform, which is then corrected by some additions to yield the 3-D DFT. A Fortran subroutine is available on request.

A new approach to recursive Fourier transform

Abstract: In the recursive Fourier transform, the data window can be chosen such that the number of computations required to update the transform at each frequency upon reception of a new data sample is independent of the transform block length.

Optical phase obtained by analogue Hartley transformation

Abstract: When the two-dimensional Fourier transformation is performed with a lens the optical amplitude and phase in the output plane represent the complex transform. It can be shown that the two-dimensional Hartley transform is mathematically equivalent to the Fourier transform, but is real valued; amplitude alone fully represents everything. This is significant because ordinary optical detectors do not respond to phase. Here we describe the construction of an optical system in the form of a modified Michelson interferometer which physically demonstrates that it is possible to produce the Hartley transform of a plane luminous object. It is thus possible to encode in the form of amplitude the half of the information in a diffraction pattern that normally is carried in the form of phase.

Multidimensional Hartley transforms

Abstract: The computational effort required for multidimensional Hartley transforming is estimated.

Fast interpolation algorithm using fast Hartley transform

Abstract: The use of fast Hartley transform for fast discrete interpolation is considered. The computational method uses the sprit-radix algorithm which requires the least number of operations compared with other Hartley algorithms. Results from this method are compared with those using the fast Fourier transform.

A separable Hartley-like transform in two or more dimensions

Abstract: This letter introduces a discrete, separable, real-to-real transform, called the cas-cas transform. Theorems for the two-dimensional (2-D) case are presented, and the cas-cas transform is compared to the Hartley transform as an alternative way to convolve 2-D arrays and compute 2-D power spectra.

Spectrum Analysis Tutorial, Part 1: The Discrete Fourier Transform

Abstract: This tutorial is an outgrowth of a course in signal processing given by Julius O. Smith at Stanford University in the fall of 1984 (see Smith 1981, as well). It provides an elementary mathematical introduction to spectrum analysis. This is the first of two parts. In part one, the discrete Fourier transform is introduced and analyzed in depth. In part two, some fundamental spectrum analysis theorems and applications are discussed. The only mathematical background assumed is high school trigonometry, algebra, and geometry. No calculus is required. Familiarity with summation formulae, complex numbers, and vectors is helpful, although not essential.

Fast computation of discrete Hartley transform via Walsh-Hadamard transform

Abstract: A new fast algorithm is proposed to compute the discrete Hartley transform (DHT) via the Walsh-Hadamard transform (WHT). The processing is carried out on an interframe basis in (N × N) data blocks, where N is an integer power of two. The WHT coefficients are obtained directly, and then used to obtain the DHT coefficients. This is achieved by a transform matrix, the H-transform matrix, which is ortho-normal and has a block-diagonal structure. A complete derivation of the block-diagonal structure for the H-transform matrix is given.

Spectrum Analysis Tutorial, Part 2: Properties and Applications of the Discrete Fourier Transform

Abstract: In part one of this tutorial (Jaffe 1987), we introduced the discrete Fourier transform (DFT). To review, the DFT takes a waveform as input and produces as output the spectrum of that waveform. One way to understand this process is to consider the samples of the waveform as a vector and to see the DFT as the projection of this vector onto a set of complex sinusoidal basis vectors. In this manner, the DFT produces a sequence of spectral components equally spaced in frequency, with a length equal to that of the original waveform. Each element of the spectrum is a coefficient of the projection given by the inner product of the waveform with one of the basis sinusoids. This coefficient can be represented in polar coordinates to give the amplitude and phase of the corresponding sinusoid. The equation for the DFT is:

Alternative to split-radix Hartley transform

Abstract: Radix-4 transforms, which have a speed advantage but have been restricted to data lengths which are powers of 4, can also be used for odd powers of 2 also by splitting the data sequence into two interleaved parts, applying the radix-4 algorithm to each in turn, and combining the results.

Solving for non-ideal conditions in two-dimensional Fourier transform magnetic resonance imaging using a generalised inverse transform

Abstract: Magnetic resonance imaging uses real and complex forms of the Fourier transform and to a lesser degree the Radon transform and their appropriate inverses. Under ideal conditions, these solutions are fast and optimal in the sense of signal-to-noise ratio (S/N). In practice, though, the phase of the signal may not be ideal so that the effective forward transform is no longer a Fourier transform. Using the inverse Fourier transform would then result in an image with artifacts. In this paper the generalised matrix inverse problem is addressed for such non-ideal conditions. Solutions are obtained for the following non-ideal circumstances: non-uniform sampling; imaging in the presence of motion; deconvolution of T2 effects; resolution enhancement; one-sided data reconstruction. The method is applicable to other deviant models as well. The goal is to maintain some specified property of the image, such as resolution, with minimal production of artifacts. A concomitant loss in S/N is inevitable for such trade-offs, but is often not a serious problem compared with the artifacts themselves.

Fixed-point error analysis of fast Hartley transform

Abstract: Fast Hartley transform (FHT) has been proposed recently by Bracewell. This is closely related to the fast Fourier transform (FFT) However, it has two advantages over the FFT, namely, the forward and inverse transforms are the same; and the Hartley transformed outputs are real-valued, rather than complex data, Hence, the speed of computation can be increased by 50% for performing fast convolution or correlation. These properties have led to investigations to use the Hartley transform for time-efficient discrete Fourier analysis of real signals. In this paper, the error-performance of radix-2 decimation-in-time and decimation-in-frequency form of the fast Hartley transform algorithm has been studied. The analysis assumes fixed-point sign magnitude arithmetic. The analysis is carried out for decimation-in-time and decimation-in-frequency form of the fast Hartley transform algorithms, assuming all the errors to be uncorrelated. Then, the analysis is carried out, assuming the truncation errors to be correlated, in the case of decimation-in-frequency form of FHT. The predicted results are compared with computer simulation studies and those obtained in the case of fast Fourier transform. It has been observed that the expressions obtained in the analysis are similar to those obtained in the case of FFT for the corresponding cases.

On fast Hartley transform algorithms

Abstract: This letter discusses the equivalence between the pre- and post-permutation algorithms for the fast Hartley transform (FHT). Some improvements are made to two recently published FHT programs.

Further symmetries of in-place FFTs

Abstract: It has long been known that an in-place version of the Fast Fourier Transform (FFT) exists for real sequences of data. More recently, in-place FFTs have been devised for real sequences with even, odd, or quarter wave symmetries. All of these symmetric FFTs take the input sequence in scrambled (bit-reversed) order and produce the transform sequence in natural order. For many applications, this is the opposite of what is needed, i.e., one would like to provide the input sequence in natural order. In this paper, an in-place version of the FFT is presented which takes a real sequence in natural order and produces the transform in scrambled order. The algorithm requires half of the operations and storage of the complex algorithms. Analogous in-place algorithms are also given for naturally ordered even, odd, quarter wave even and quarter wave odd sequences.

Reconstruction of complex-valued propagating wave fields using the Hilbert–Hankel transform

Abstract: In this paper the theory of the unilateral inverse Fourier transform and the unilateral Hankel transform is developed. The consistency between each transform and its bilateral version leads to an approximate real-part sufficiency condition for complex-valued one-dimensional even signals and two-dimensional circularly symmetric signals. The two-dimensional result is used in a reconstruction algorithm that is applied to synthetic -and experimental underwater acoustic fields.

Real-valued algorithms for the FFT

Abstract: Since 1965, when Cooley and Tukey published their famous paper on the radix-2 fast Fourier transform, much effort has gone into developing even more efficient algorithms. Most algorithms, however, do not directly handle real-valued data very well, and them exist several ways to solve that problem. This paper derives a new algorithm; the decimation-in-time real-valued split-radix FFT, which can transform any length N = 2Msequence but uses less operations than any other known real-valued FFF, which is the fastest Cooley-Tukey real-valued transform in use. Instead of breaking the transform down equally as in traditional algorithms, the even and odd indexed parts are broken down differently in the split-radix algorithm. This gives a significant savings in both additions and multiplications over any fixed radix Cooley-Tukey FFT. The paper compares the split-radix transform with several of the already existing methods such as the Hartley transform, the prime factor, Winograd, Cooley-Tukey etc, and shows in which cases a specific algorithm is faster than the rest.