Order N log2 N WHT/DHT algorithm

An efficient algorithm is proposed which computes the coefficients of the higher order discrete Hartley transform (DHT) directly from the coefficients of lower-order DHTs. With this new development, the two-stage Walsh-Hadamard transform/discrete Hartley transform (WHT/DHT) is comparable to the existing fast algorithms. The same approach can also be used for the computation of DCT coefficients     

Novel FPGA implementations of Walsh-Hadamard transforms for signalprocessing

The paper describes two approaches suitable for a field-programmable gate-array (FPGA) implementation of fast Walsh-Hadamard transforms. These transforms are important in many signal-processing applications including speech compression, filtering and coding. Two novel architectures for the fast Hadamard transforms using both a systolic architecture and distributed arithmetic techniques are presented. The first approach uses the Baugh-Wooley multiplication algorithm for a systolic architecture implementation. The second approach is based on both a distributed arithmetic ROM and accumulator structure, and a sparse matrix-factorisation technique. Implementations of the algorithms on a Xilinx FPGA board are described. The distributed arithmetic approach exhibits better performances when compared with the systolic architecture approach     

On fast M-sequence transforms (Corresp.)

The equivalence ofM-sequence matrices to Walsh-Hadamard matrices can be exploited to take advantage of the latter's fast transform algorithm. The nature of the obvious equivalence, its relation to theM-sequence and its implementation as address modifications realized by linear feedback shift-registers are discussed.     

哈德码变换域等均值等方差最近邻矢量量化码字搜索算法

本文提出一种基于哈德码变换的等均值等方差最近邻(HTEENNS)快速矢量量化码字搜索算法.在编码前,该算法预先计算每个码字的哈德码变换,然后根据各码字哈德码变换的第一维系数大小的升序排列对码字进行排序.在编码过程中,首先计算输入矢量的哈德码变换和方差,然后选取与输入矢量哈德码变换的第一维系数最近的码字作为初始匹配码字,然后利用两条有效的删除准则在该码字附近进行上下搜索与输入矢量最近的码字.测试结果表明,本文算法比等均值最近邻搜索算法(ENNS)、等均值等方差最近邻搜索(EENNS)算法和哈德码变换域部分失真搜索算法等算法有效得多.     

One-dimensional Hadamard naturalness-preserving transformreconstruction of signals degraded by nonstationary noise processes

The problem of reconstructing a time-varying one-dimensional signal from segments of its Hadamard naturalness preserving transform (NPT) is considered. Closed-form reconstruction formulas are derived for specific formats of available Hadamard NPT segments. Two fast reconstruction schemes, which in a single iteration provide the same solution as the general iterative reconstruction algorithm, are introduced for these formats. Simulation results of the algorithms when applied to first-order Gauss-Markov and image signals are good. However, the reconstruction of speech signals is poor due to the poor estimates derived from the available segments of the transformed signal. The reconstruction algorithm is proposed for interframe two-layer packet transmission of image signals over an Orwell-ring-based network. Good pictorial results are obtained at up to 50% packet loss rate     

A General Description of Linear Time-Frequency Transforms and Formulation of a Fast, Invertible Transform That Samples the Continuous S-Transform Spectrum Nonredundantly

Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.     

Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

A new transform known as conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) is presented in this paper. The transform matrix of this transform possesses sequency ordering and the spectrum obtained by the CS-SCHT is conjugate symmetric. Some of its important properties are discussed and analyzed. Sequency defined in the CS-SCHT is interpreted as compared to frequency in the discrete Fourier transform. The exponential form of the CS-SCHT is derived, and the proof of the dyadic shift invariant property of the CS-SCHT is also given. The fast and efficient algorithm to compute the CS-SCHT is developed using the sparse matrix factorization method and its computational load is examined as compared to that of the SCHT. The applications of the CS-SCHT in spectrum estimation and image compression are discussed. The simulation results reveal that the CS-SCHT is promising to be employed in such applications.     

Slant Haar transform

The slant Haar transform (SHT) is defined and related to ,the slant Walsh-Hadamard transform (SWHT). A fast algorithm for the SHT is presented and its computational complexity computed. In most applications, the SHT is faster and performs as well as the SWHT.     

A fast and accurate Fourier algorithm for iterative parallel-beamtomography

We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires O(N²logN) floating-point operations per iteration to reconstruct an N×N image from P view angles, as compared to O(N ²P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary     

Relationship between two algorithms for discrete cosine transform

Two contributions are made to the implementation of fast discrete cosine transform algorithms. The first uses Hadamard ordering to improve the regularity of the Lee fast cosine transform (FCT) algorithm for the discrete cosine transform (DCT). The second derives a close relationship between the Lee FCT and the recursive algorithm for the DCT.<>     

Convergence analysis of alias-free subband adaptive filters based on a frequency domain technique

Convergence analysis of alias-free subband adaptive filters (SADFs) is presented based on a frequency-domain technique where instead of analyzing the adaptive algorithms in the time-domain, the averaging method and the ordinary differential equation (ODE) method are applied to the frequency-domain expressions of the adaptive algorithms converted by the discrete Fourier transform. As an alias-free SADF algorithm, the SADF proposed by Pradhan and Reddy is known. In this paper, this technique is first applied to the Pradhan's SADF. The stability of the Pradhan's SADF is verified in the frequency domain, and a simple formula to evaluate the mean square error (MSE) of the algorithm is theoretically derived. By using a slight modification, the technique can be applied to the two-band delayless subband adaptive filter (DLSADF) with the Hadamard transform. The stability condition and the MSE of the DLSADF with the Hadamard transform are also obtained. Simulation results of both algorithms show the validity of the theoretical results.     

Fast computation of discrete Hartley transform via Walsh?Hadamard transform

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.     

Fast Block Center Weighted Hadamard Transform

Motivated by the Hadamard transforms and center weighted Hadamard transforms, a new class of block center weighted Hadamard transforms (BCWHT) are proposed, which weights the region of midspatial frequencies of the signal more than the Hadamard transform. Based on the Kronecker product, direct sum operations, the identity matrix and recursive relations, the proposed one and 2-D fast BCWHTs algorithms through sparse matrix factorization are simply obtained.     

VOIR: a volumetric image reconstruction algorithm based on Fouriertechniques for inversion of the 3-D Radon transform

A novel volumetric image reconstruction algorithm known as VOIR is presented for inversion of the 3-D Radon transform or its radial derivative. The algorithm is a direct implementation of the projection slice theorem for plane integrals. It generalizes one of the most successful methods in 2-D Fourier image reconstruction involving concentric-square rasters to 3-D; in VOIR, the spectral data, which is calculated by fast Fourier techniques, lie on concentric cubes and are interpolated by a bilinear method on the sides of these concentric cubes. The algorithm has great computational advantages over filtered-backprojection algorithms; for images of side dimension N, the numerical complexity of VOIR is O(N(3) log N) instead of O(N (4)) for backprojection techniques. An evaluation of the image processing performance is reported by comparison of reconstructed images from simulated cone-beam scans of a contrast and resolution test object. The image processing performance is also characterized by an analysis of the edge response from the reconstructed images.     

A Generalized Reverse Block Jacket Transform

Jacket matrices motivated by the center weight Hadamard matrices have played important roles in signal processing, communication, image compression, cryptography, etc. In this paper we propose a notation called block Jacket matrix which substitutes elements of the matrix into common matrices or even block matrices. Employing the well-known Pauli matrices which are very important in many subjects, block Jacket matrices with any size are investigated in detail, and some recursive relations for fast construction of the block Jacket matrices are obtained. Based on the general recursive relations, several special block Jacket matrices are constructed. To decompose high order block Jacket matrices, a fast decomposition algorithm for the factorable block Jacket matrices is suggested. After that some properties of the block Jacket matrices are investigated. Finally, several remarks are presented. These remarks are associated with comparisons between the Clifford algebra and the block Jacket matrices, generations of orthogonal and quasi-orthogonal sequences, and relations of the block Jacket matrices to the orthogonal transforms for signal processing. Since the Pauli matrices are actually infinitesimal generators of $SU(2)$ group, the proposed construction and decomposition algorithms for the block Jacket matrices are available in the signal processing, communication, quantum signal processing and information theory.      

Fast computation of transform coefficients for a subadjacent blockfor a transform family

The authors examine several different kinds of subadjacent blocks, and consider fast computation for a transform family including the Walsh-Hadamard transform and the Rh transform as special cases. The approach proposed here provides a direct frequency-frequency procedure. Results indicate that, for this transform family, a reduction of the number of multiplications and additions is achieved by a factor of two-thirds. An example for the Rh transform shows that further reduction of arithmetic operations is also possible. The results of this method are even better for the Walsh-Hadamard transform. The new algorithms can reduce the number of additions from the level O(N log ₂ N) to the level O(N), as compared to the traditional method     

Filter design and comparison for two fast CWT algorithms

The continuous wavelet transform (CWT) is a powerful technique for signal analysis. Direct CWT computation by FFT requires O(N log₂ N) operations per scale, where N is the data length. The a trous algorithm and the Shensa (1992) algorithm are two fast methods to compute CWT recursively that require only O(N) operations per scale. Both of them can be described by the multiresolution analysis (MRA) structure but with different MRA filters. This paper proposes methods to design the MRA filters of the two algorithms to improve their accuracy on CWT computation. We begin with the formulation of the CWT computation error using the MRA structure. The MRA filters of the two algorithms are then designed to minimize the error. In either algorithm, both the lowpass and bandpass MRA filters can be optimized. The a trous algorithm has closed-form solutions for the two filters. The Shensa algorithm, on the other hand, has an analytic solution for the bandpass filter only. Finding the optimum lowpass filter requires a multidimensional numerical search. Simulation studies show that by using the proposed optimum filters, the Shensa algorithm, in general, outperforms the a trous algorithm     

Hardware structure for Walsh-Hadamard transforms

An efficient hardware structure is reported for the Walsh-Hadamard transform. Based on the radix-2 algorithm, the structure can be implemented with shift registers and controllable adder/subtracters for high processing throughput. Such an implementation is particularly suited to applications that require real-time operations and interface directly with sequential input data     

2D grid architectures for the DFT and the 2D DFT

New algorithms for the DFT and the 2-dimensional DFT are presented. The DFT and the 2-dimensional DFT matrices can be expressed as the Kronecker product of DFT matrices of smaller dimension. These algorithms are synthesized by combining the efficient factorization of the Kronecker product of matrices with the highly hardware efficient recursive implementation of the smaller DFT matrices, to yield these algorithms. The architectures of the processors implementing these algorithms consist of 2-dimensional grid of processing elements, have temporal and spatial locality of connections. For computing the DFT of sizeN or for the 2D DFT of sizeN=N ₁ byN ₁, these algorithms require 2N multipliers and adders, take approximately computational steps for computing a transform vector, and take approximately computation steps between the computation of two successive transform vectors.     

改进的算术傅立叶变换（AFT）算法

算术傅立叶变换（AFT）是一种非常重要的傅立叶分析技术。AFT的乘法量少（仅为O（N））,算法结构简单,非常适合VLSI设计,具有广泛的应用。但AFT的加法量很大,为O（N∧2）,因此减少AFT的加法运算是很重要的工作。本文通过分析AFT的采样特点,给出了奇函数和偶函数的AFT的改进算法。然后在此基础上给出了一般函数的AFT的改进算法。改进算法比原算法的加法运算量降低了一半,因此计算速度快了一倍。本文改进的偶函数和奇函数的AFT算法还分别可以用来计算离散余弦变换（DCT）和离散正弦变换（DST）。