Showing papers on "Hartley transform published in 2009"

Making sense of the Legendre transform

Abstract: The Legendre transform is a powerful tool in theoretical physics and plays an important role in classical mechanics, statistical mechanics, and thermodynamics. In typical undergraduate and graduate courses the motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform. We then discuss examples to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations.

A Convolution and Product Theorem for the Linear Canonical Transform

Abstract: The linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, however, the convolution theorems don't have the elegance and simplicity comparable to that of the Fourier transform (FT), which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. The purpose of this letter is to introduce a new convolution structure for the LCT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters. Some of well-known results about the convolution theorem in FT domain, fractional Fourier transform (FRFT) domain are shown to be special cases of our achieved results.

Analog and Digital Signals and Systems

Abstract: This book presents a systematic, comprehensive treatment of analog, discrete signal analysis and synthesis and an introduction to analog communication theory. The material is divided into five parts. The first part (Chapters 1-3) is mathematically oriented and deals with continuous-time (analog) signals. The second part (Chapters 4-5) is again mathematically based and deals with Fourier transforms, Hartley transforms, Laplace and Hilbert transforms. The third part (Chapters 6-7) presents basic system analysis, approximations, and analog filter circuits using op-amps. The fourth part (Chapters 8-9) deals with discrete signals, fast algorithms and digital filters. The fifth part deals with an introduction to analog communication systems.

Comparative study of Hilbert–Huang transform, Fourier transform and wavelet transform in pavement profile analysis

Abstract: This study employs the Hilbert–Huang transform (HHT), the wavelet transform and the Fourier transform to analyse the road surface profiles of three pavement profiles. The wavelet and Fourier transforms have been the traditional spectral analysis methods, but they are predicated on a priori selection of basis functions that are either of infinite length or have fixed finite widths. The central idea of HHT is the empirical mode decomposition, which decomposes a signal into basis functions called the intrinsic mode functions (IMFs). The Hilbert transform can then be applied to the IMFs to generate an energy–time–frequency spectrum called the Hilbert spectrum. The strength of HHT is the ability to process non-stationary and non-linear data. Unlike the Fourier transform, which transforms information from the time domain into the frequency domain, the HHT does not lose temporal information after transformation, i.e. energy–frequency information is maintained in the time domain. This paper attempts to reveal the...

Fractional Fourier transform of Lorentz-Gauss beams

Abstract: Lorentz-Gauss beams are introduced to describe certain laser sources that produce highly divergent beams. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz-Gauss beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz-Gauss beam passing through an FRFT system has been derived. By using the derived expression, the properties of a Lorentz-Gauss beam in the FRFT plane are graphically illustrated with numerical examples.

Brief Notes in Advanced DSP: Fourier Analysis with MATLAB

Abstract: Discrete Fourier Transform Properties of the discrete Fourier transform Fourier transform splitting Fast Fourier transform Codes for the paired FFT Paired and Haar transforms Integer Fourier Transform Reversible integer Fourier transform Lifting schemes for DFT One-point integer transform DFT in vector form Roots of the unit Codes for the block DFT General elliptic Fourier transforms Cosine Transform Partitioning the DCT Paired algorithm for the N-point DCT Codes for the paired transform Reversible integer DCT Method of nonlinear equations Canonical representation of the integer DCT Hadamard Transform The Walsh and Hadamard transform Mixed Hadamard transformation Generalized bit and transformations T-decomposition of Hadamard matrices Mixed Fourier transformations Mixed transformations: continuous case Paired Transform-Based Decomposition Decomposition of 1D signals 2D paired representation Fourier Transform and Multiresolution Fourier transform Representation by frequency-time wavelets Time-frequency correlation analysis Givens-Haar transformations References Index

Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

Abstract: 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.

Hilbert Transform and Its Engineering Applications

Abstract: The Hilbert transform is useful in calculating instantaneous attributes of a time series, especially the envelope amplitude and instantaneous frequency. The instantaneous envelope is the amplitude of the complex Hilbert transform; the instantaneous frequency is the time rate of change of the instantaneous phase angle. These properties can be applied to identify dynamic characteristics of a linear as well as a nonlinear system. However, the conventional discrete Hilbert transform, which is based on fast Fourier transform and inverse Fourier transform, has shown the lack of accuracy for time-derivative calculations. In this paper, we first introduce the Hilbert transform and its applications to the nonlinear system parameter identification. Then we address the practical issues in applying the Hilbert transform to engineering applications. To increase the accuracy of the envelope detection, we propose a Hilbert transform technique based on local-maxima interpolation. Analyses and simulations are carried out to demonstrate the advantages of the proposed technique. Finally, we employ the proposed local-maxima-interpolation technique in identifying the nonlinear dynamic characteristics of industrial examples.

Chapter 2 The Fourier Transform in Clifford Analysis

Abstract: Publisher Summary This chapter focuses on the Fourier transform in Clifford analysis. This chapter includes an introductory section on Clifford analysis, and each section starts with an introductory situation. This chapter presents the new Clifford–Fourier transform is given in terms of an operator exponential, or alternatively, by a series representation. Particular attention is directed to the two-dimensional (2D) case since then the Clifford–Fourier kernel can be written in a closed form. This chapter also discusses the fractional Fourier transform wherein, it is shown that the traditional and the Clifford analysis approach coincide. This chapter develops the theory for the Clifford–Hermite and Clifford–Gabor filters for early vision. This chapter faced with the following situation: In dimension greater than two, we have a first Clifford–Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension, two both transforms coincide.

Signals, Systems, Transforms, and Digital Signal Processing with MATLAB

Abstract: Signals, Systems, Transforms, and Digital Signal Processing with MATLAB has as its principal objective simplification without compromise of rigor. Graphics, called by the author, "the language of scientists and engineers", physical interpretation of subtle mathematical concepts, and a gradual transition from basic to more advanced topics are meant to be among the important contributions of this book. After illustrating the analysis of a function through a step-by-step addition of harmonics, the book deals with Fourier and Laplace transforms. It then covers discrete time signals and systems, the z transform, continuous- and discrete-time filters, active and passive filters, lattice filters, and continuous- and discrete-time state space models. The author goes on to discuss the Fourier transform of sequences, the discrete Fourier transform, and the fast Fourier transform, followed by Fourier-, Laplace, and z-related transforms, including WalshHadamard, generalized Walsh, Hilbert, discrete cosine, Hartley, Hankel, Mellin, fractional Fourier, and wavelet. He also surveys the architecture and design of digital signal processors, computer architecture, logic design of sequential circuits, and random signals. He concludes with simplifying and demystifying the vital subject of distribution theory. Drawing on much of the authors own research work, this book expands the domains of existence of the most important transforms and thus opens the door to a new world of applications using novel, powerful mathematical tools.

Fourier Transform Methods in Finance

Abstract: In recent years, Fourier transform methods have emerged as one of the major methodologies for the evaluation of derivative contracts, largely due to the need to strike a balance between the extension of existing pricing models beyond the traditional Black-Scholes setting and a need to evaluate prices consistently with the market quotes.Fourier Transform Methods in Finance is a practical and accessible guide to pricing financial instruments using Fourier transform. Written by an experienced team of practitioners and academics, it covers Fourier pricing methods; the dynamics of asset prices; non stationary market dynamics; arbitrage free pricing; generalized functions and the Fourier transform method.Readers will learn how to: compute the Hilbert transform of the pricing kernel under a Fast Fourier Transform (FFT) technique characterise the price dynamics on a market in terms of the characteristic function, allowing for both diffusive processes and jumps apply the concept of characteristic function to non-stationary processes, in particular in the presence of stochastic volatility and more generally time change techniques perform a change of measure on the characteristic function in order to make the price process a martingale recover a general representation of the pricing kernel of the economy in terms of Hilbert transform using the theory of generalised functions apply the pricing formula to the most famous pricing models, with stochastic volatility and jumps. Junior and senior practitioners alike will benefit from this quick reference guide to state of the art models and market calibration techniques. Not only will it enable them to write an algorithm for option pricing using the most advanced models, calibrate a pricing model on options data, and extract the implied probability distribution in market data, they will also understand the most advanced models and techniques and discover how these techniques have been adjusted for applications in finance.

Robust color texture features based on ranklets and discrete Fourier transform

Abstract: We present a set of multiscale, multidirectional, rotation-invariant features for color texture characterization. The proposed model is based on the ranklet transform, a technique relying on the calculation of the relative rank of the intensity level of neighboring pixels. Color and texture are merged into a compact descriptor by computing the ranklet transform of each color channel separately and of couples of color channels jointly. Robustness against rotation is based on the use of circularly symmetric neighborhoods together with the discrete Fourier transform. Experimental results demonstrate that the approach shows good robustness and accuracy.

Reduction of computational complexity of Hough transforms using a convolution approach

Abstract: The Hough transform finds wide application in machine and robot vision. The family of Hough transforms allows a variety of geometric objects to be located and described in an image. However, the classical Hough transform is computationally complex when targeting complex objects. This renders the Hough transform unsuitable for many real-time applications. We present a new algorithm for calculating the circle Hough transform by recasting it as a convolution. This new approach allows the transform to be calculated using the Fast Fourier transform, yielding an algorithm with lower computational complexity than the traditional approach. Although the convolution approach is applicable to the same range of different targets as the traditional Hough transform, we limit ourselves to a consideration of the circle Hough transform in this treatment.

Fragile watermarking using finite field trigonometrical transforms

Abstract: Fragile digital watermarking has been applied for authentication and alteration detection in images. Utilizing the cosine and Hartley transforms over finite fields, a new transform domain fragile watermarking scheme is introduced. A watermark is embedded into a host image via a blockwise application of two-dimensional finite field cosine or Hartley transforms. Additionally, the considered finite field transforms are adjusted to be number theoretic transforms, appropriate for error-free calculation. The employed technique can provide invisible fragile watermarking for authentication systems with tamper location capability. It is shown that the choice of the finite field characteristic is pivotal to obtain perceptually invisible watermarked images. It is also shown that the generated watermarked images can be used as publicly available signature data for authentication purposes.

Orthogonal expansion of ground motion and PDEM-based seismic response analysis of nonlinear structures

Abstract: This paper introduces an orthogonal expansion method for general stochastic processes. In the method, a normalized orthogonal function of time variable t is first introduced to carry out the decomposition of a stochastic process and then a correlated matrix decomposition technique, which transforms a correlated random vector into a vector of standard uncorrelated random variables, is used to complete a double orthogonal decomposition of the stochastic processes. Considering the relationship between the Hartley transform and Fourier transform of a real-valued function, it is suggested that the first orthogonal expansion in the above process is carried out using the Hartley basis function instead of the trigonometric basis function in practical applications. The seismic ground motion is investigated using the above method. In order to capture the main probabilistic characteristics of the seismic ground motion, it is proposed to directly carry out the orthogonal expansion of the seismic displacements. The case study shows that the proposed method is feasible to represent the seismic ground motion with only a few random variables. In the second part of the paper, the probability density evolution method (PDEM) is employed to study the stochastic response of nonlinear structures subjected to earthquake excitations. In the PDEM, a completely uncoupled one-dimensional partial differential equation, the generalized density evolution equation, plays a central role in governing the stochastic seismic responses of the nonlinear structure. The solution to this equation will yield the instantaneous probability density function of the responses. Computational algorithms to solve the probability density evolution equation are described. An example, which deals with a nonlinear frame structure subjected to stochastic ground motions, is illustrated to validate the above approach.

High-order seislet transform and its application of random noise attenuation

Abstract: The seislet transform is a wavelet-like transform that analyzes seismic data by following variable slopes of seismic events across different scales.It generalizes the discrete wavelet transform(DWT)in the sense that DWT in the lateral direction is simply the seislet transform with zero slopes.An earlier work used low-order versions of DWT to construct the seislet transform.In this work,we extend this approach to a higher order,using the Cohen-Daubechies -Feauveau 9/7 biorthogonal wavelet transform(the basis for the JPEG2000 compression scheme)as a template.Using synthetic and field-data examples,we demonstrate that the new transform can provide a better compression rate for seismic events than the Fourier transform,DWT,or the low-order seislet transform.Therefore,the high-order seislet transform can be more suitable for data processing tasks such as data regularization and noise attenuation.

Development of a New Transform:MRT

Abstract: Fourier transform methods are employed heavily in digital signal processing. Discrete Fourier Transform (DFT) is among the most commonly used digital signal transforms, The exponential kernel of the DFT has the properties of symmetry and periodicity. Fast Fourier Transform (FFT) methods for fast DFT computation exploit these kernel properties in different ways. In this thesis, an approach of grouping data on the basis of the corresponding phase of the exponential kernel of the DFT is exploited to introduce a new digital signal transform, named the M-dimensional Real Transform (MRT), for l-D and 2-D signals. The new transform is developed using numbertheoretic principles as regards its specific features. A few properties of the transform are explored, and an inverse transform presented. A fundamental assumption is that the size of the input signal be even. The transform computation involves only real additions. The MRT is an integer-to-integer transform. There are two kinds of redundancy, complete redundancy & derived redundancy, in MRT. Redundancy is analyzed and removed to arrive at a more compact version called the Unique MRT (UMRT). l-D UMRT is a non-expansive transform for all signal sizes, while the 2-D UMRT is non-expansive for signal sizes that are powers of 2. The 2-D UMRT is applied in image processing applications like image compression and orientation analysis. The MRT & UMRT, being general transforms, will find potential applications in various fields of signal and image processing.

The Vector-Valued Big q-Jacobi Transform

Abstract: Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L-2-space of functions on R with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform F with two different big q-Jacobi functions as a kernel, and we construct the inverse of F.

Fractional Fourier transform of Lorentz beams

Abstract: This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent flelds. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the deflnition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.

Sinusoidal Polynomial Parameter Estimation Using the Distribution Derivative

Abstract: In this paper, we present a method to estimate the parameters of a generalized sinusoidal model. A generalized sinusoid x is defined as a polynomial in the log domain, with complex coefficients alphai : x(t)=exp(Sigmai alphai t i), where i=0...Q. The method is based on the distribution derivative of the signal and operates in the transform domain. The method is very general and can use any linear transform such as the Fourier transform or the wavelet transform, or even combinations of linear transforms. Examples with the Fourier transform are given. The Fourier-based estimation methods are evaluated using synthetic signals and have performance very close to the theoretical bound.

Properties, digital implementation, applications, and self image phenomena of the gyrator transform

Abstract: The gyrator transform was introduced in recent years and useful for image processing. As the fractional Fourier transform, the gyrator transform can also be viewed as an extension of the Fourier analysis. In this paper, we discuss the digital implementation algorithms of the gyrator transform. We also discuss the eigenfunctions and the self-imaging phenomena of the gyrator transform. Many properties, such as the differential and dilation preservation properties, are also discussed. We also discuss the possible applications of the gyrator transform in filter design and signal sampling.

Beurling's theorems and inversion formulas for certain index transforms

Abstract: The familiar Beurling theorem (an uncertainty principle), which is known for the Fourier transform pairs, has recently been proved by the author for the Kontorovich-Lebedev transform. In this paper analogs of the Beurling theorem are established for certain index transforms with respect to a parameter of the modified Bessel functions. In particular, we treat the generalized Lebedev-Skalskaya transforms, the Lebedev type transforms involving products of the Macdonald functions of different arguments and an index transform with the Nicholson kernel function. We also find inversion formulas for the Lebedev-Skalskaya operators of an arbitrary index and the Nicholson kernel transform.

Image processing using spatial transform

Abstract: This paper deals with image processing using spatial (geometric) transforms such as translation, rotation, and scaling, shearing, and projective transform. These transforms can be used for image correction. Translation, rotation, and scaling transform are also called affine transform which is a subset of projective transform. All the methods are given experimental results implemented in Matlab.

Chapter 6 – Feature Generation I: Data Transformation and Dimensionality Reduction

Abstract: Publisher Summary This chapter discusses the feature generation stage using data transformations and dimensionality reduction. Feature generation is important in any pattern recognition task. Given a set of measurements, the goal is to discover compact and informative representations of the obtained data. The basic approach followed in this chapter is to transform a given set of measurements to a new set of features. If the transform is suitably chosen, transform domain features can exhibit high information packing properties compared with the original input samples. The chapter reviews Karhunen–Loeve transform and the singular value decomposition as dimensionality reduction techniques. The independent component analysis, nonnegative matrix factorization, and nonlinear dimensionality reduction techniques are presented. Then the discrete Fourier transform, discrete cosine transform, discrete sine transform, Hadamard, and Haar transforms are defined. The rest of the chapter focuses on the discrete time wavelet transform. The chapter also shows that the Fourier transform is just one of the tools from a palette of possible transforms.