Showing papers on "Discrete Hartley transform published in 1987"
TL;DR: A new implementation of the real-valued split-radix FFT is presented, an algorithm that uses fewer operations than any otherreal-valued power-of-2-length FFT.
Abstract: This tutorial paper describes the methods for constructing fast algorithms for the computation of the discrete Fourier transform (DFT) of a real-valued series. The application of these ideas to all the major fast Fourier transform (FFT) algorithms is discussed, and the various algorithms are compared. We present a new implementation of the real-valued split-radix FFT, an algorithm that uses fewer operations than any other real-valued power-of-2-length FFT. We also compare the performance of inherently real-valued transform algorithms such as the fast Hartley transform (FHT) and the fast cosine transform (FCT) to real-valued FFT algorithms for the computation of power spectra and cyclic convolutions. Comparisons of these techniques reveal that the alternative techniques always require more additions than a method based on a real-valued FFT algorithm and result in computer code of equal or greater length and complexity.
489 citations
TL;DR: It is shown that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves.
Abstract: This paper describes the discrete Radon transform (DRT) and the exact inversion algorithm for it. Similar to the discrete Fourier transform (DFT), the DRT is defined for periodic vector-sequences and studied as a transform in its own right. Casting the forward transform as a matrix-vector multiplication, the key observation is that the matrix-although very large-has a block-circulant structure. This observation allows construction of fast direct and inverse transforms. Moreover, we show that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves. In fact, we describe not a single transform, but a class of transforms, representatives of which correspond in one way or another to discrete versions of the RT and its generalizations. An interesting observation is that the exact inversion algorithm cannot be obtained directly from Radon's inversion formula. Given the fact that the RT has no nontrivial one-dimensional analog, exact invertibility makes the DRT a useful tool geared specifically for multidimensional digital signal processing. Exact invertibility of the DRT, flexibility in its definition, and fast computational algorithm affect present applications and open possibilities for new ones. Some of these applications are discussed in the paper.
426 citations
Book•
01 Jan 1987
TL;DR: This book discusses the Discrete Fourier Transform (DFT) and a few applications of the DFT, as well as some of the techniques used in real sequences and the Real DFT.
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.
354 citations
TL;DR: Through use of the fast Hartley transform, discrete cosine transforms (DCT) and discrete Fourier transforms (DFT) can be obtained and 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 F HT's.
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.
175 citations
TL;DR: 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 and a new fast Hartley transform algorithm with a reduced number of operations.
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.
131 citations
TL;DR: This work demonstrates that the quality of the correlation signal can also depend on the technique used in the synthesis of the BPOF, and that BPOFs made using the Hartley transform provide superior false correlation rejection and more uniformly sized correlation signals for heavily multiplexed BPOs.
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.
95 citations
TL;DR: A new factorization of the discrete Hartley transform is presented and it is used to derive new algorithms for the DHT and the discrete cosine transform with reduced number of multiplications.
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.
88 citations
TL;DR: A single-mode star network, made from polarization-preserving components, can perform the spatial discrete Fourier transform of coherent light patterns presented at the inputs.
Abstract: A single-mode star network, made from polarization-preserving components, can perform the spatial discrete Fourier transform of coherent light patterns presented at the inputs. This can be accomplished with passive components, such as 2 x 2 couplers, and propagation delays. The Hadamard transform can be performed similarly.
83 citations
TL;DR: A new and simple algorithm for computing a discrete Hankel transform, which does not rely on the fast Fourier transform, can provide a major improvement in speed and accuracy over previously described methods.
71 citations
TL;DR: 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.
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.
59 citations
TL;DR: It is shown that the lower bound for the computation of the multidimensional transform is O(n2 log2 n) and an optimal architecture based on arrays of processors computing one-dimensional Fourier transforms and a rotation network or rotation array is proposed.
Abstract: It is often desirable in modern signal processing applications to perform two-dimensional or three-dimensional Fourier transforms. Until the advent of VLSI it was not possible to think about one chip implementation of such processes. In this paper several methods for implementing the multidimensional Fourier transform together with the VLSI computational model are reviewed and discussed. We show that the lower bound for the computation of the multidimensional transform is O(n2 log2 n). Existing nonoptimal architectures suitable for implementing the 2-D transform, the RAM array transposer, mesh connected systolic array, and the linear systolic matrix vector multiplier are discussed for area time tradeoff. For achieving a higher degree of concurrency we suggest the use of rotators for permutation of data. With ``hybrid designs'' comprised of a rotator and one-dimensional arrays which compute the one-dimensional Fourier transform we propose two methods for implementation of multidimensional Fourier transform. One design uses the perfect shuffle for rotations and achieves an AT2 p of O(n2 log2 n· log2 N). An optimal architecture for calculation of multidimensional Fourier transform is proposed in this paper. It is based on arrays of processors computing one-dimensional Fourier transforms and a rotation network or rotation array. This architecture realizes the AT2 p lower bound for the multidimensional FT processing.
01 Feb 1987
TL;DR: A three-dimensional (3-D) Discrete Fourier Transform (DFT) algorithm for real data using the one-dimensional Fast Hartley Transform (FHT) is introduced that is simpler and retains the speed advantage that is characteristic of the Hartley approach.
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.
IBM1
TL;DR: The algorithm formulation and implementation described here not only achieves full vector utilization but successfully copes with the problems of hierarchical storage.
Abstract: A number of previous attempts at the vectorization of the fast Fourier transform (FFT) algorithm have fallen somewhat short of achieving the full potential speed of vector processors. The algorithm formulation and implementation described here not only achieves full vector utilization but successfully copes with the problems of hierarchical storage. In the present paper, these techniques are described and extended to the general mixed radix algorithms, prime factor algorithm (PFA), the multidimensional discrete Fourier transform (DFT), the rectangular transform convolution algorithms, and the Winograd fast Fourier transform algorithm. Some of the methods were used in the Engineering Scientific Subroutine Library for the IBM 3090 Vector Facility. Using this approach, very good and consistent performance was obtained over a very wide range of transform lengths.
TL;DR: In this paper, it was shown that the two-dimensional Hartley transform is mathematically equivalent to the Fourier transform, but is real valued; amplitude alone fully represents everything.
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.
01 Feb 1987
TL;DR: The computational effort required for multidimensional Hartley transforming was estimated in this article, where the computational effort was also shown to be exponential in the number of transversal transformations.
Abstract: The computational effort required for multidimensional Hartley transforming is estimated.
TL;DR: The design and architecture of an array processor for the 2-dimensional discrete cosine transform, DCT is discussed, aimed for used in high speed applications, e.g., transform coding of TV images in real time.
01 Apr 1987
TL;DR: The computational method uses the sprit-radix algorithm which requires the least number of operations compared with other Hartley algorithms and is compared with those using the fast Fourier 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.
01 Aug 1987
TL;DR: The cas-cas transform as mentioned in this paper is a real-to-real transform for convolutional arrays and power spectra, which can be used to compute 2D power spectrum.
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.
01 Apr 1987
TL;DR: A new approach is developed for the fast computation and VLSI implementation of all discrete trigonometric transforms in the least number of operations and pipelining stages in terms of the fast algorithm (FRFT) for the real discrete Fourier transform.
Abstract: A new approach is developed for the fast computation and VLSI implementation of all discrete trigonometric transforms in the least number of operations and pipelining stages. This is achieved in terms of the fast algorithm (FRFT) for the real discrete Fourier transform. FRFT is based upon Givens' plane rotation as the basic unit of computation in contrast to FFT's which are based upon the complex butterfly.
06 Apr 1987
TL;DR: An algorithm for evaluating the Discrete Fourier Transform at particular output frequency is derived using a technique called summation by parts (SBP), which is shown to reduce the number of multiplications and the numbers of bits per multiplicative coefficient needed to implement the DFT.
Abstract: An algorithm for evaluating the Discrete Fourier Transform (DFT) at particular output frequency is derived using a technique called summation by parts (SBP). This technique is shown to reduce the number of multiplications and the number of bits per multiplicative coefficient needed to implement the DFT. For many transform lengths, only two one-bit multiplications or simple memory shifts are needed to implement the DFT. When the DFT length is prime, a SBP algorithm designed for a fixed output frequency index can be used to evaluate the DFT at any other non-zero output frequency index simply by appropriately changing the order of the input sequence.
TL;DR: A variety of efficient DHT algorithms including Bracewell's original decimation-in-time radix-2 algorithm are summarized and statistical models for fixed- and floating-point arithmetic roundoff errors are used as the basis for a theoretical study of roundoff noise characteristics of several discrete Hartley transform algorithms.
Abstract: The principal objective of this paper is the study of the arithmetic roundoff error characteristics of several discrete Hartley transform (DHT) algorithms. We first summarize a variety of efficient DHT algorithms including Bracewell's original decimation-in-time radix-2 algorithm. Statistical models for fixed- and floating-point arithmetic roundoff errors are then used as the basis for a theoretical study of roundoff noise characteristics of a number of the DHT algorithms. The results of a detailed experimental study of roundoff noise are compared to the theoretical predictions. In fixed-point implementation of the decimation-in-time and frequency radix-2 algorithms, it is found that the noise-to-signal ratio increases approximately 1.1 bits per stage. For the floating-point implementation, the number of bits of rms noise-to-signal ratio for all the algorithms increase as \sqrt{\log_{2}N} , so that doubling the number of points produced a mild increase in the output noise.
TL;DR: This tutorial is an outgrowth of a course in signal processing given by Julius O. Smith at Stanford University in the fall of 1984 and provides an elementary mathematical introduction to spectrum analysis.
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.
TL;DR: In this paper, a fast algorithm for computing the discrete Hartley transform (DHT) via the Walsh-Hadamard transform (WHT) is proposed, which is carried out on an interframe basis in (N × N) data blocks, where N is an integer power of two.
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.
TL;DR: This correspondence presents details of a new implementation of the prime factor FFT algorithm (PFA) for computing the discrete Fourier transform (DFT) that saves about 40 percent of the execution time of the conventional one.
Abstract: This correspondence presents details of a new implementation of the prime factor FFT algorithm (PFA) for computing the discrete Fourier transform (DFT). This implementation applies a program generation technique to the PFA algorithm and saves about 40 percent of the execution time of the conventional one.
01 Jan 1987
TL;DR: The error-performance of radix-2 decimation-in-time and decimation -in-frequency form of the fast Hartley transform algorithm has been studied and the expressions obtained are similar to those obtained in the case of FFT for the corresponding cases.
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.
01 Jul 1987
TL;DR: In this article, the equivalence between pre-and post-permutation algorithms for the fast Hartley transform (FHT) has been discussed, and improvements are made to two recently published FHT programs.
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.
Proceedings Article•
01 Jan 1987