Subspaces of FM~mlet transform

The subspaces of FMmlet transform are investigated. It is shown that some of the existing transforms like the Fourier transform, short-time Fourier transform, Gabor transform, wavelet transform, chirplet transform, the mean of signal, and the FM-1let transform, and the butterfly subspace are all special cases of FMmlet transform. Therefore the FMmlet transform is more flexible for delineating both the linear and nonlinear time-varying structures of a signal.     

A new matrix approach to real FFTs and convolutions of length 2<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

A new matrix, scaled odd tail, SOT, is introduced. This new matrix is used to derive real and complex FFT algorithms for lengths n = 2 ^k . A compromise is reached between Fourier transform and polynomial transform methods for computing the action of cyclic convolutions. Both of these methods lead to arithmetic operation counts that are better than previously published results. A minor improvement is also demonstrated that enables us to compute the actions of Fermat prime order FFTs in fewer additions than previously available algorithms.     

Research progress on discretization of fractional Fourier transform

As the fractional Fourier transform has attracted a considerable amount of attention in the area of optics and signal processing, the discretization of the fractional Fourier transform becomes vital for the application of the fractional Fourier transform. Since the discretization of the fractional Fourier transform cannot be obtained by directly sampling in time domain and the fractional Fourier domain, the discretization of the fractional Fourier transform has been investigated recently. A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented in this paper. The discretizations include sampling in the fractional Fourier domain, discrete-time fractional Fourier transform, fractional Fourier series, discrete fractional Fourier transform （including 3 main types： linear combination-type; sampling-type; and eigen decomposition-type）, and other discrete fractional signal transform. It is hoped to offer a doorstep for the readers who are interested in the fractional Fourier transform.     

The fractional Fourier transform: theory, implementation and error analysis

The fractional Fourier transform is a time–frequency distribution and an extension of the classical Fourier transform. There are several known applications of the fractional Fourier transform in the areas of signal processing, especially in signal restoration and noise removal. This paper provides an introduction to the fractional Fourier transform and its applications. These applications demand the implementation of the discrete fractional Fourier transform on a digital signal processor (DSP). The details of the implementation of the discrete fractional Fourier transform on ADSP-2192 are provided. The effect of finite register length on implementation of discrete fractional Fourier transform matrix is discussed in some detail. This is followed by the details of the implementation and a theoretical model for the fixed-point errors involved in the implementation of this algorithm. It is hoped that this implementation and fixed-point error analysis will lead to a better understanding of the issues involved in finite register length implementation of the discrete fractional Fourier transform and will help the signal processing community make better use of the transform.     

Comments on “Generalised finite Radon transform for N × N images”

In Kingston and Svalbe [1], a generalized finite Radon transform (FRT) that applied to square arrays of arbitrary size N × N was defined and the Fourier slice theorem was established for the FRT. Kingston and Svalbe asserted that “the original definition by Matúš and Flusser was restricted to apply only to square arrays of prime size,” and “Hsung, Lun and Siu developed an FRT that also applied to dyadic square arrays,” and “Kingston further extended this to define an FRT that applies to prime-adic arrays”. It should be said that the presented generalized FRT together with the above FRT definitions repeated the known concept of tensor representation, or tensor transform of images of size N × N which was published earlier by Artyom Grigoryan in 1984-1991 in the USSR. The above mentioned “Fourier slice theorem” repeated the known tensor transform-based algorithm of 2-D DFT [5-11], which was developed for any order N₁ × N₂ of the transformation, including the cases of N × N, when N = 2^r, (r > 1), and N = L^r, (r ≥ 1), where L is an odd prime. The problem of “over-representation” of the two-dimensional discrete Fourier transform in tensor representation was also solved by means of the paired representation in Grigoryan [6-9].     

Applying the Fourier–modified Mellin transform (FMMT) to Doppler-distorted waveforms

The magnitude spectrum of a time domain signal has the property of delay-invariance. Similar to the delay-invariance property of the Fourier transform, the Mellin transform has the property of scale-invariance. By combining these two transforms together one can form the Fourier–Mellin transform that yields a signal representation which is independent of both delay and scale change. Due to the undesired low-pass property of the Mellin transform (MT), the modified Mellin transform (MMT) which is also scale-invariant is applied in our approach. Therefore the Fourier–modified Mellin transform (FMMT) of the original signal and the Doppler-distorted signal will be identical. This signal representation is useful in signal detection and target recognition. Several examples dealing with different waveforms have been simulated to illustrate the applicability of this approach. The performance of the Fourier–modified Mellin transform under different levels of noise in the signal are also illustrated in this paper.     

Meta-modelling and graph grammars for multi-paradigm modelling in AToM<Superscript>3</Superscript>

This paper presents the combined use of meta-modelling and graph grammars for the generation of visual modelling tools for simulation formalisms. In meta-modelling, formalisms are described at a meta-level. This information is used by a meta-model processor to generate modelling tools for the described formalisms. We combine meta-modelling with graph grammars to extend the model manipulation capabilities of the generated modelling tools: edit, simulate, transform into another formalism, optimize and generate code. We store all (meta-)models as graphs, and thus, express model manipulations as graph grammars.We present the design and implementation of these concepts in AToM³ (A_To_ol for M_ulti-formalism, M_eta-M_odelling). AToM³ supports modelling of complex systems using different formalisms, all meta-modelled in their own right. Models in different formalisms may be transformed into a single common formalism for further processing. These transformations are specified by graph grammars. Mosterman and Vangheluwe [18] introduced the term multi-paradigm modelling to denote the combination of multiple formalisms, multiple abstraction levels, and meta-modelling. As an example of multi-paradigm modelling we present a meta-model for the Object-Oriented Continuous Simulation Language OOCSMP, in which we combine ideas from UML class diagrams (to express the OOCSMP model structure), Causal Block Diagrams (CBDs), and Statecharts (to specify the methods of the OOCSMP classes). A graph grammar is able to generate OOCSMP code, and then a compiler for this language (C-OOL) generates Java applets for the simulation execution.     

Fast, prime factor, discrete Fourier transform algorithms over GF(2) for 8 ? m ? 10

In this paper it is shown that Winograd’s algorithm for computing convolutions and a fast, prime factor, discrete Fourier transform (DFT) algorithm can be modified to compute Fourier-like transforms of long sequences of 2^m − 1 points over GF(2^m), for 8 ? m ? 10. These new transform techniques can be used to decode Reed-Solomon (RS) codes of block length 2^m − 1. The complexity of this new transform algorithm is reduced substantially from more conventional methods. A computer simulation verifies these new results.     

遗传算法在周期信号傅立叶变换中的应用

为了克服周期信号进行傅立叶变换时各次谐波幅度值需要进行复杂的理论计算问题,提出一种利用遗传算法进行周期信号傅立叶变换的方法,介绍了周期信号傅立叶变换和遗传算法的基本原理;给出了使用遗传算法对周期信号进行分解适应度函数的实现方法和确定各次谐波幅值计算的方法。并提供使用遗传算法对周期信号进行分解的具体步骤。仿真实验结果表明:该方法能够满足周期信号傅立叶变换的要求,与周期信号傅立叶变换理论计算方法相比,其突出优点是算法简单,易于实现。     

The fast Fourier transform for general order

An elementary and transparent representation of the fast Fourier transform is given. Instead of using the usual and highly algebraic approach it is shown how a Fourier transform of the ordern=p·m can be reduced top Fourier transforms of orderm by performing essentiallym Fourier transforms of orderp on the data. The resulting process is discussed in more detail forn=3 ^q andn=5 ^q . The problem of retrieval of the wanted coefficients from the final data is solved by a simple argument. The generalization for an ordern equal to a product of powers of prime numbers is immediate.     

Fractional dual tree complex wavelet transform and its application to biometric security during communication and transmission

In this paper, the dual tree complex wavelet transform, which is an important tool and recent advancement in signal and image processing, has been generalized by coalescing dual tree complex wavelet transform and fractional Fourier transform. The new transform, i.e. the fractional dual tree complex wavelet transform (FrDT-CWT) inherits the excellent mathematical properties of dual tree complex wavelet transform and fractional Fourier transform. Possible applications of the proposed transform are in biometrics, image compression, image transmission, transient signal processing etc. In this paper, biometric is chosen as the primary application and hence a new technique is proposed for securing biometrics during communication and transmission over insecure channel.     

Fourier变换和小波变换一般形式 —线调频小波变换和多普勒小波变换

Fourier变换、窗口Fourier变换与小波变换在许多领域得到广泛的应用。该文回顾了Fourier变换和小波变换的发展;介绍了两种新的处理非平稳信号的方法,即线调频小波变换和多普勒小波变换;分析了线调频小波变换是短时Fourier变换和小波变换的时频分析的统一时频表示形式,Fourier变换、小波变换以及线调频小波变换都是多普勒小波变换的特殊情况。线调频小波变换和多普勒小波变换比Fourier变换和小波变换更具灵活性,为图像、信号处理提供了新的方法和工具。     

Signal-adaptive discrete evolutionary transform as a sparse time–frequency representation

Discrete evolutionary transform (DET) has usually been applied to signals in a blind-way without using any parameters to characterize the signal. For this reason, it is not optimal and needs improvement by using some information about the signal. In this paper, we propose an improvement for the discrete evolutionary transform in order to obtain a sparse representation and redefine the generalized time-bandwidth product optimal short-time Fourier transform as a special case of it. In case of linear FM-type signals, the optimized kernel function of the transform is determined according to signal parameters including the instantaneous frequency. The performance of the adaptive-DET is illustrated on three distinct cases. In case of multi-component LFM signals, when the concentration of the proposed distribution is compared to the ordinary sinusoidal-DET, the improvement is computed as 28% in terms of the ratio of norms. Furthermore we define a new and a general class of distribution functions named as the short-time generalized discrete Fourier transform (ST-GDFT) which is a larger set of signal representations including the adaptive-DET.     

Representation of the Fourier Transform by Fourier Series

The analysis of the mathematical structure of the integral Fourier transform shows that the transform can be split and represented by certain sets of frequencies as coefficients of Fourier series of periodic functions in the interval . In this paper we describe such periodic functions for the one- and two-dimensional Fourier transforms. The approximation of the inverse Fourier transform by periodic functions is described. The application of the new representation is considered for the discrete Fourier transform, when the transform is split into a set of short and separable 1-D transforms, and the discrete signal is represented as a set of short signals. Properties of such representation, which is called the paired representation, are considered and the basis paired functions are described. An effective application of new forms of representation of a two-dimensional image by splitting-signals is described for image enhancement. It is shown that by processing only one splitting-signal, one can achieve an enhancement that may exceed results of traditional methods of image enhancement.     

基于深度卷积神经网络的变换域通信网络抗干扰优化算法

为了有效抑制变换域通信网络干扰信号,改善信噪比,研究了基于深度卷积神经网络的变换域通信网络抗干扰优化算法。应用傅里叶变换方法将信号从时域转换到频域,并以傅里叶变换通信信号获得的参数为依据构建干扰信号模型;嵌入干扰信号模型以形成接收信号,然后对接收信号进行处理并存储在干扰数据库中,利用深度卷积神经网络完成干扰信号的特征学习与干扰估计,并根据干扰估计结果,在接收信号中去除干扰信号,完成变换域通信网络抗干扰优化。实验结果表明：该算法可有效完成变换域通信网络抗干扰优化,优化后通信信号的信噪比改善性能与误码性能均较佳,输出的通信信号几乎无干扰信号存在。     

一种电力故障录波数据压缩算法的研究

电力系统故障录波数据是分析电网故障的主要依据,录波数据压缩有益于减小数据存储容量和提高数据传输效率。针对电力故障录波数据的格式及构成特点,提出了一种基于傅里叶变换和小波包变换的数据压缩新算法。采用离散傅里叶变换对录波模拟量通道的B时段数据进行压缩和重构,根据重构误差判断该通道是否为故障通道;对故障通道的暂态扰动时段采用小波包变换进行压缩,对正常通道及故障通道的其他时段采用傅里叶变换进行压缩。大量录波文件的压缩结果和工程实际应用表明,所提算法可以同时获得很高的压缩率和压缩精度,具有广阔的应用前景。     

小波变换在脉象信号消噪处理中的应用

首先简单介绍了小波变换消噪的原理,然后分别利用小波变换和傅里叶变换,对临床采集的脉象信号进行了消噪处理.结果表明,利用小波消噪能够有效地保留脉象信号中的尖峰和突变部分,最大限度地反映了原信号本身的特性,消噪效果较好.     

Research progress of the fractional Fourier transform in signal processing

While solving a heat conduction problem in 1807, a French scientist Jean Baptiste Jo-seph Fourier, suggested the usage of the Fourier theorem. Thereafter, the Fourier trans-form (FT) has been applied widely in many scientific disciplines, and has played i…     

分数阶小波变换

小波变换是对信号时域-频域(Fourier域)的多分辨率分析,也可看作是一种Fourier域伸缩带通滤波.分数阶Fourier变换是对传统Fourier变换的推广,对信号分析处理有更大的灵活性,为了将多分辨率分析理论推广到时域-广义频域(分数阶Fourier域),提出了一种分数阶小波变换,分析了分数阶小波变换在广义频域伸缩带通滤波特性,分析信号时的时域-广义频域平面的多分辨率分析网格划分.分数阶小波变换是传统小波变换的推广,在对原小波变换核作一定改动后增加了小波变换对信号处理的灵活性.可以看到,将分数阶小波变换的变换角度取为π/2,便得到与传统小波变换多分辨率分析理论完全一致的结果.理论分析和计算机仿真表明了所提理论的正确性和有效性.     

An Improved Sliding‐Mode Repetitive Learning Control Scheme Using Wavelet Transform

This paper presents an enhanced sliding‐mode repetitive learning control (SMRLC) scheme using the wavelet transform. Distinct from previous wavelet transform‐based repetitive control schemes, the proposed SMRLC learns from a switching signal that is equivalent to the compensation error of the SMRLC, thereby speeding up the learning process. The wavelet analysis is employed to decompose the switching signal, capture its low‐frequency components effectively, and synthesize a de‐noised, high‐scale signal for the learning process. Experimental study on a directly driven Hoekens straight‐line mechanism is conducted, and the proposed wavelet transform‐based SMRLC is experimentally compared with a Fourier series‐based SMRLC. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society