Novel fixed-point roundoff analysis of the decimation-in-time FHT

A least upper bound for the increasing factor of the magnitude of the decimation-in-time fast Hartley transform (FHT) in fixed-point arithmetic is developed and a new scaling model for the roundoff analysis in the fixed-point arithmetic computation is proposed. In this new scaling model, the input data for each computing stage of the decimation-in-time FHT only need to be divided by a constant of 2, and this can prevent overflow successfully. Hence, the novel approach would result in a higher noise-to-signal ratio for the fixed-point computation of FHT     

Design and parallel computation of regularised fast Hartley transform

The paper describes the design and parallel computation of a regularised fast Hartley transform (FHT), to be used for computation of the discrete Fourier transform (DFT) of real-valued data. For the processing of such data, the FHT has attractions over the fast Fourier transform (FFT) in terms of reduced arithmetic operation counts and reduced memory requirement, whilst its bilateral property means it may be straightforwardly applied to both forward and inverse DFTs. A drawback, however, of conventional FHT algorithms lies in the loss of regularity arising from the need for two sizes of 'butterfly' for efficient fixed-radix implementations. A generic double butterfly is therefore developed for the radix-4 FHT which overcomes the problem in an elegant fashion. The result is a recursive single-butterfly solution, referred to as the regularised FHT, which lends itself naturally to parallelisation and to mapping onto a regular computational structure for implementation with algorithmically specialised hardware.     

The fast Hartley transform

A fast algorithm has been worked out for performing the Discrete Hartley Transform (DHT) of a data sequence of N elements in a time proportional to Nlog2N. The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied. A new timing diagram (stripe diagram) is presented to illustrate the overall dependence of running time on the subroutines composing one implementation; this mode of presentation supplements the simple counting of multiplies and adds. One may view the Fast Hartley procedure as a sequence of matrix operations on the data and thus as constituting a new factorization of the DFT matrix operator; this factorization is presented. The FHT computes convolutions and power spectra distinctly faster than the FFT.     

A new split-radix FHT algorithm for length-q*2/sup m/DHTs

In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length N=q*2/sup m/, where q is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient indexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- q*2/sup m/ DHTs. Since the proposed algorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case.     

Fixed-point error analysis of radix-4 FHT algorithm with optimisedscaling schemes

The fixed-point error performance of the various fast Hartley transform (FHT) algorithms have been investigated. Scaling schemes have been proposed for each of the algorithms. However, due to their better error performance, only the decimation-in-time (DIT) FHT algorithms have been examined. The fixed-point error analysis of the radix-4 DIT algorithm is discussed first and is shown to agree closely with the simulation results. These results are then compared with the simulation results for radix-2 and split-radix algorithms. The scaling schemes are then optimised and the simulation results of the three algorithms are compared. It is concluded that the radix-4 DIT algorithm has the best error performance     

Computation of DWT via FHT-based implementation

It is well known that most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform (DWT). Although this method is efficient, noticeable computational savings have been obtained through an FFT-based implementation. The authors present a fast Hartley transform (FHT)-based implementation of the filter bank and show that noticeable overall computational savings can be obtained     

A fast Hough transform for segment detection

The authors describe a new algorithm for the fast Hough transform (FHT) that satisfactorily solves the problems other fast algorithms propose in the literature-erroneous solutions, point redundance, scaling, and detection of straight lines of different sizes-and needs less storage space. By using the information generated by the algorithm for the detection of straight lines, they manage to detect the segments of the image without appreciable computational overhead. They also discuss the performance and the parallelization of the algorithm and show its efficiency with some examples.     

Novel approach to discrete interpolation using the subsequence FHT

A novel approach to discrete interpolation of finite-duration real sequences using subsequences with the fast Hartley transform (FHT) is presented. It is shown that accuracy can be improved by decomposing the signal into an ordered set of subsequences. This development is also convenient because it permits the use of inverse fast Hartley transforms (IFHT) that are always the same size as the original FHT. This approach is also appropriate for the parallel processing of each subsequence     

New block filtered-X LMS algorithms for active noise control systems

New block formulations for an active noise control (ANC) system using only convolution machines are presented. The proposed approaches are different from conventional block least-mean-square (LMS) algorithms that use both convolution and cross-correlation machines. The block implementation is also applied to the filtering of the reference signal by the secondary-path estimate. In addition to the use of the fast Fourier transform (FFT), the fast Hartley transform (FHT) is used to develop transform-domain ANC structures for reducing computational complexity. In the proposed approach, some FFT and FHT blocks are removed to obtain an additional reduction of the computational burden resulting in the reduced-structure of FFT-based block filtered-X LMS (FBFXLMS) and FHT-based block filtered-X LMS (HBFXLMS) algorithms. The computational complexities of these new ANC structures are evaluated.     

Fast radix-3/9 discrete Hartley transform

An efficient algorithm for computing radix-3/9 discrete Hartley transforms (DHTs) is presented. It is shown that the radix-3/9 fast Hartley transform (FHT) algorithm reduces the number of multiplications required by a radix-3 FHT algorithm for nearly 50%. For the computation of real-valued discrete Fourier transforms (DFTs) with sequence lengths that are powers of 3, it is shown that the radix-3/9 FHT algorithm reduces the number of multiplications by 16.2% over the fastest real-valued radix-3/9 fast Fourier transform (FFT) algorithm     

快速汉克尔变换在光传输中的应用

在对各种快速汉克尔变换(FHT)算法分析的基础上,采用准离散汉克尔变换(QDHT)来研究光束在含有空间滤波器的多程激光放大系统中的传输问题。数值计算例表明,用快速汉克尔变换来模拟柱对称大型高功率激光放大系统中的光传输时,是一种非常有用的、快速的算法。     

Multidimensional vector radix FHT algorithms

In this paper, efficient multidimensional (M-D) vector radix (VR) decimation-in-frequency and decimation-in-time fast Hartley transform (FHT) algorithms are derived for computing the discrete Hartley transform (DHT) of any dimension using an appropriate index mapping and the Kronecker product. The proposed algorithms are more effective and highly suitable for hardware and software implementations compared to all existing M-D FHT algorithms that are derived for the computation of the DHT of any dimension. The butterflies of the proposed algorithms are based on simple closed-form expressions that allow easy implementations of these algorithms for any dimension. In addition, the proposed algorithms possess properties such as high regularity, simplicity and in-place computation that are highly desirable for software and hardware implementations, especially for the M-D applications. A close relationship between the M-D VR complex-valued fast Fourier transform algorithms and the proposed M-D VR FHT algorithms is established. This type of relationship is of great significance for software and hardware implementations of the algorithms, since it is shown that because of this relationship and the fact that the DHT is an alternative to the discrete Fourier transform (DFT) for real data, a single module with a little or no modification can be used to carry out the forward and inverse M-D DFTs for real- or complex-valued data and M-D DHTs. Thus, the same module (with a little or no modification) can be used to cover all domains of applications that involve the DFTs or DHTs.     

快速汉克尔变换及光束均匀化

介绍快速汉克尔变换，以及它在二元光学设计中的应用，并就光束均匀化问题进行了数值模拟，结果表明快速汉克尔变换算法是一种能用于二元光学设计的有效算法。     

A short discussion on the exact compensation of the SARrange-dependent range cell migration effect

Efficient and precise compensation of the range cell migration (RCM) effect is a key point for a fast and accurate synthetic aperture radar (SAR) data processor. In particular the range-dependent nature of the range cell migration effect complicates the compensation operation. It has been recently shown that an exact compensation of the range-dependent RCM (RDRCM) phenomenon can be carried out either by applying the chirp scaling algorithm or the chirp z-transform procedure. This paper investigates the relationship between the two methods. In particular, it is shown that the chirp z-transform based approach represents a particular implementation of the chirp scaling algorithm. A final discussion is dedicated to show how the chirp z-transform and the chirp scaling procedure can be applied within a SAR data processing algorithm     

Generalization of the Fast Hough Transform for Three-Dimensional Images

This study is devoted to the analysis of algorithms of calculating the fast Hough transform for two- and three-dimensional images. A method for calculating the fast Hough transform (FHT) for straight lines in a three-dimensional image is proposed; its space and time complexity are Θ(n⁴), where n is the characteristic linear size of the input image. The FHT algorithms for approximation in two- and three-dimensional spaces are considered, and properties of the accuracy and completeness of the corresponding sets of dyadic patterns are investigated.     

Using FHT to determine digital straight line chain codes

A novel application of the fast Hartley transform is proposed. Using the FHT, the straightness of a digital arc can be determined from the periodicity of its chain codes. Further results for the new approach in four of Lee's examples demonstrate that the new application is promising.<>     

Fast Hartley transforms for image processing

The fast Hartley transform (FHT) is used to transform two-dimensional image data. Because the Hartley transform is real-valued, it does not require complex operations. Both spectra and autocorrelations of two-dimensional ultrasound images of normal and abnormal livers were computed     

快速哈达马变换的折叠结构设计

快速哈达马变换在3G无线通信中具有广泛的应用。本文在分析快速哈达马变换算法的基础上提出了两种快速哈达马变换的折叠结构，并分别分析了这两种折叠结构的电路结构、时钟频率要求、资源消耗等因素。最后给出了这两种电路结构在Altera FPGA上实现的对比情况。     

Fast full-wave analysis of planar microstrip circuit elements instratified media

The fast Hankel transform (FHT) algorithm is implemented in the mixed-potential integral-equation (MPIE) analysis of planar microstrip circuits in stratified media. The spatial-domain Green's functions are accurately and quickly obtained by applying the FHT algorithm to the exact spectral-domain counterparts. Therefore, the entire analysis procedure has high accuracy and efficiency. A nonuniform partition scheme is used to effectively model the rapid change of current distributions around discontinuities. A generalized supplementary equation accounting for arbitrary termination conditions at both feeding and load ends is also derived. The proposed method is used to design a single-stub band-stop filter and a compensated dc block circuit. Experimental measurements are performed to validate the computation     

On fast Hartley transform algorithms

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.