Showing papers on "Fractional Fourier transform published in 2010"

Asymmetric cryptosystem based on phase-truncated Fourier transforms.

Abstract: We propose an asymmetric cryptosystem based on a phase-truncated Fourier transform. With phase truncation in Fourier transform, one is able to produce an asymmetric ciphertext as real-valued and stationary white noise by using two random phase keys as public keys, while a legal user can retrieve the plaintext using another two different private phase keys in the decryption process. Owing to the nonlinear operation of phase truncation, high robustness against existing attacks could be achieved. A set of simulation results shows the validity of proposed asymmetric cryptosystem.

Short-Time Fractional Fourier Transform and Its Applications

Abstract: The fractional Fourier transform (FRFT) is a potent tool to analyze the chirp signal. However, it fails in locating the fractional Fourier domain (FRFD)-frequency contents which is required in some applications. The short-time fractional Fourier transform (STFRFT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the short-time fractional Fourier domain (STFRFD). Two aspects of its performance are considered: the 2-D resolution and the STFRFD support. The time-FRFD-bandwidth product (TFBP) is defined to measure the resolvable area and the STFRFD support. The optimal STFRFT is obtained with the criteria that maximize the 2-D resolution and minimize the STFRFD support. Its inverse transform, properties and computational complexity are presented. Two applications are discussed: the estimations of the time-of-arrival (TOA) and pulsewidth (PW) of chirp signals, and the STFRFD filtering. Simulations verify the validity of the proposed algorithms.

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

Abstract: 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.

Cluster expansion and the configurational theory of alloys

Abstract: A rigorous mathematical foundation for the cluster-expansion method is presented It is shown that the cluster basis developed by Sanchez et al [Physica A 128, 334 (1984)] is a multidimensional discrete Fourier transform while the general formalism of Sanchez [Phys Rev B 48, 14013 (1993)] corresponds to a multidimensional discrete wavelet transform For functions that depend nonlinearly on the concentration, it is shown that the cluster basis corresponding to a multidimensional discrete Fourier transform does not converge, as it is usually assumed, to a finite cluster expansion or to an Ising-type model representation of the energy of formation of alloys The multidimensional wavelet transform, based on a variable basis cluster expansion, is shown to provide a satisfactory solution to the deficiencies of the discrete Fourier-transform approach Several examples aimed at illustrating the main findings and conclusions of this work are given

Image Encryption With Multiorders of Fractional Fourier Transforms

Abstract: The original information in the existing security system based on the fractional Fourier transform (FRFT) is essentially protected by only a certain order of FRFT. In this paper, we propose a novel method to encrypt an image by multiorders of FRFT. In the image encryption, the encrypted image is obtained by the summation of different orders inverse discrete FRFT of the interpolated subimages. And the original image can be perfectly recovered using the linear system constructed by the fractional Fourier domain analysis of the interpolation. The proposed method can be applied to the double or more image encryptions. Applying the transform orders of the utilized FRFT as secret keys, the proposed method is with a larger key space than the existing security systems based on the FRFT. Additionally, the encryption scheme can be realized by the fast-Fourier-transform-based algorithm and the computation burden shows a linear increase with the extension of the key space. It is verified by the experimental results that the image decryption is highly sensitive to the deviations in the transform orders.

Separation of overlapping linear frequency modulated (LFM) signals using the fractional fourier transform

Abstract: Linear frequency modulated (LFM) excitation combined with pulse compression provides an increase in SNR at the receiver. LFM signals are of longer duration than pulsed signals of the same bandwidth; consequently, in many practical situations, maintaining temporal separation between echoes is not possible. Where analysis is performed on individual LFM signals, a separation technique is required. Time windowing is unable to separate signals overlapping in time. Frequency domain filtering is unable to separate signals with overlapping spectra. This paper describes a method to separate time-overlapping LFM signals through the application of the fractional Fourier transform (FrFT), a transform operating in both time and frequency domains. A short introduction to the FrFT and its operation and calculation are presented. The proposed signal separation method is illustrated by application to a simulated ultrasound signal, created by the summation of multiple time-overlapping LFM signals and the component signals recovered with ±0.6% spectral error. The results of an experimental investigation are presented in which the proposed separation method is applied to time-overlapping LFM signals created by the transmission of a LFM signal through a stainless steel plate and water-filled pipe.

Fast linear canonical transforms

Abstract: The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande-Tukey fast Fourier transform. Results calculated with an implementation of this algorithm are presented and compared with the corresponding analytic functions.

A Space-Vector Discrete Fourier Transform for Unbalanced and Distorted Three-Phase Signals

Abstract: In this paper, a space-vector discrete-time Fourier transform is proposed for fast and precise detection of the fundamental-frequency and harmonic positive- and negative-sequence vector components of three-phase input signals. The discrete Fourier transform is applied to the three-phase signals represented by Clarke's αβ vector. It is shown that the complex numbers output from the Fourier transform are the instantaneous values of the positive- and negative-sequence harmonic component vectors of the input three-phase signals. The method allows the computation of any desired positive- or negative-sequence fundamental-frequency or harmonic vector component of the input signal. A recursive algorithm for low-effort online implementation is also presented. The detection performance for variable-frequency and interharmonic input signals is discussed. The proposed and other usual method performances are compared through simulations and experiments.

Multichannel Sampling and Reconstruction of Bandlimited Signals in Fractional Fourier Domain

Abstract: The classical multichannel sampling theorem for common bandlimited signals has been extended differently to fractional bandlimited signals associated with the fractional Fourier transform (FRFT). However, the implementation of those existing extensions is inefficient because of the effect of spectral leakage and hardware complexity. The purpose of this letter is to introduce a practical multichannel sampling theorem for fractional bandlimited signals. The theorem which is constructed by the ordinary convolution in the time domain can reduce the effect of spectral leakage and is easy to implement. The classical multichannel sampling theorem and the well-known sampling theorem for the FRFT are shown to be special cases of it. Some potential applications of this theorem are also presented. The validity of the theoretical derivations is demonstrated via simulations.

The approach to carrier scheme convergence based on 4-weighted fractional fourier transform

Abstract: We propose a new variant of a block transmission based on 4-Weighted Fractional Fourier Transform (4-WFRFT), which contains a compatible process of single-carrier (SC) and multi-carrier (MC) modulation. The 4-WFRFT system has an even and symmetrical bit energy distribution on the timefrequency plane that it could achieve a better performance than or equal to those of SC and MC systems in selective fading channels. The approach of 4-WFRFT may facilitate unification of the two competitive carrier schemes, and improve the distortion resistance capability of communication system.

A new integral transform and associated distributions

Abstract: In this paper, we generalize the concepts of a new integral transform, namely the Sumudu transform, to distributions and study some of their properties. Further, we also apply this transform to solve one-dimensional wave equation having a singularity at the initial conditions.

Uniqueness results for the phase retrieval problem of fractional fourier transforms of variable order

Abstract: In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if u and v are such that fractional Fourier transforms of orderhave same modulus jFuj= jFvjfor some setof �'s, then v is equal to u up to a constant phase factor. The setdepends on some extra assumptions either on u or on both u and v. Cases considered here are u, v of compact support, pulse trains, Hermite functions or linear combinations of translates and dilates of Gaussians. In this last case, the setmay even be reduced to a single point (i.e. one fractional Fourier transform may suffice for uniqueness in the problem).

Double-image encryption based on the affine transform and the gyrator transform

Abstract: We propose a kind of double-image-encryption algorithm by using the affine transform in the gyrator transform domain. Two original images are converted into the real part and the imaginary part of a complex function by employing the affine transform. And then the complex function is encoded and transformed into the gyrator domain. The affine transform, the encoding and the gyrator transform are performed twice in this encryption method. The parameters in the affine transform and the gyrator transform are regarded as the key for the encryption algorithm. Some numerical simulations have validated the feasibility of the proposed image encryption scheme.

Fractional Fourier Transform, Wigner Distribution, and Filter Design for Stationary and Nonstationary Random Processes

Abstract: In this paper, we derive the relationship among the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the stationary and nonstationary random processes. We find many interesting properties. For example, if we perform the FRFT for a stationary process, although the result is no longer stationary, the amplitude of the autocorrelation function is still independent of time. We also find that the LCT of a white noise is still a white one. For the FRFT of a stationary process, the ambiguity function (AF) is a tilted line and the Wigner distribution function (WDF) is invariant along a certain direction. We also define the “fractional stationary random process” and find that a nonstationary random process can be expressed by a summation of fractional stationary random processes. In addition, after performing the filter designed in the FRFT domain for a white noise, we can use the segment length of the ω -axis on the WDF plane to estimate the power of the noise and use the area circled by cutoff lines to estimate its energy. Thus, in communication, to reduce the effect of the white noise, the “area” of the WDF of the transmitted signal should be as small as possible.

Generalized Sampling Expansion for Bandlimited Signals Associated With the Fractional Fourier Transform

Abstract: The aim of the generalized sampling expansion (GSE) is the reconstruction of an unknown continuously defined function f(t), from the samples of the responses of M linear time invariant (LTI) systems, each sampled by the 1/M th Nyquist rate. In this letter, we investigate the GSE in the fractional Fourier transform (FRFT) domain. Firstly, the GSE for fractional bandlimited signals with FRFT is proposed based on new linear fractional systems, which is the generalization of classical generalized Papoulis sampling expansion. Then, by designing fractional Fourier filters, we obtain reconstruction method for sampling from the signal and its derivative based on the derived GSE and the property of FRFT. Last, the potential application of the GSE is presented to show the advantage of the theory.

Uniform Discrete Curvelet Transform

Abstract: An implementation of the discrete curvelet transform is proposed in this work. The transform is based on and has the same order of complexity as the Fast Fourier Transform (FFT). The discrete curvelet functions are defined by a parameterized family of smooth windowed functions that satisfies two conditions: i) 2π periodic; ii) their squares form a partition of unity. The transform is named the uniform discrete curvelet transform (UDCT) because the centers of the curvelet functions at each resolution are positioned on a uniform lattice. The forward and inverse transform form a tight and self-dual frame, in the sense that they are the exact transpose of each other. Generalization to M dimensional version of the UDCT is also presented. The novel discrete transform has several advantages over existing transforms, such as lower redundancy ratio, hierarchical data structure and ease of implementation.

On some applications of a new integral transform

Abstract: In this work a new integral transform, namely Sumudu transform was applied to solve linear ordinary differential equation with and without constant coefficients having convolution terms. In particular we apply Sumudu transform technique to solve Spring-Mass systems, Population Growth and financial problem. Mathematics Subject Classification: Primary 35G15, 44A85; Secondary 44A35

ICI Analysis for FRFT-OFDM Systems to Frequency Offset in Time-Frequency Selective Fading Channels

Abstract: This letter presents the effects of frequency offset to orthogonal frequency division multiplexing systems based on fractional Fourier transform (FRFT-OFDM) in time-frequency selective fading channels. FRFT-OFDM systems generalize the OFDM systems based on discrete Fourier transform by the deployment of FRFT. The expressions of signal-to-interference ratio (SIR) due to ICI are derived in different fading channels. In a flat channel, the performances of both systems are the same. In a frequency selective channel, the FRFT-OFDM systems have superior SIR performances by choosing the optimal fractional factor, when Doppler spread is comparable to the inverse of the symbol duration or carrier offset exists in the system.

Chaos based multiple image encryption using multiple canonical transforms

Abstract: We propose a new method for multiple image encryption using linear canonical transforms and chaotic maps. Three linear canonical transforms and three chaotic maps are used in the proposed technique. The three linear canonical transforms that have been used are the fractional Fourier transform, the extended fractional Fourier transform and the Fresnel transform. The three chaotic maps that have been used are the tent map, the Kaplan–Yorke map and the Ikeda map. These chaotic maps are used to generate the random phase masks and these random phase masks are known as chaotic random phase masks. The mean square error and the signal to noise ratio have been calculated. Robustness of the proposed technique to blind decryption has been evaluated. Optical implementation of the technique has been proposed. Experimental and simulations results are presented to verify the validity of the proposed technique.

Sampling and Reconstruction of Sparse Signals in Fractional Fourier Domain

Abstract: Sampling theory for continuous time signals which have a bandlimited representation in fractional Fourier transform (FrFT) domain-a transformation which generalizes the conventional Fourier transform-has blossomed in the recent past. The mechanistic principles behind Shannon's sampling theorem for fractional bandlimited (or fractional Fourier bandlimited) signals are the same as for the Fourier domain case i.e. sampling (and reconstruction) in FrFT domain can be seen as an orthogonal projection of a signal onto a subspace of fractional bandlimited signals. As neat as this extension of Shannon's framework is, it inherits the same fundamental limitation that is prevalent in the Fourier regime-what happens if the signals have singularities in the time domain (or the signal has a nonbandlimited spectrum)? In this paper, we propose a uniform sampling and reconstruction scheme for a class of signals which are nonbandlimited in FrFT sense. Specifically, we assume that samples of a smoothed version of a periodic stream of Diracs (which is sparse in time-domain) are accessible. In its parametric form, this signal has a finite number of degrees of freedom per unit time. Based on the representation of this signal in FrFT domain, we derive conditions under which exact recovery of parameters of the signal is possible. Knowledge of these parameters leads to exact reconstruction of the original signal.

A novel transform for image compression

Abstract: In this paper, we propose an orthogonal multiplication-free transform of order that is an integral power of two by an appropriate extension of the well-known fourthorder integer discrete cosine transform. Moreover, we develop an efficient algorithm for its fast computation. It is shown that the computational and structural complexities of the algorithm are similar to that of the Hadamard transform. By applying the proposed transform to image compression, we show that it outperforms the existing transforms having complexities similar to that of the proposed one.

The Infinite-Line Inverse Scattering Transform

Abstract: Publisher Summary This chapter focuses on the inverse scattering transform (IST) solution of the infinite-line Korteweg–deVries (KdV) equation and on the numerical analysis of this problem It provides insight about the classical problem of nonlinear, shallow-water wave motion and its IST solution for infinite-line boundary conditions where the concept of the soliton has its roots The chapter also provides tools for the wavenumber or frequency domain analysis of nonlinear experimental data and of computer-generated nonlinear wave motion; both of these problems are assumed to be governed approximately by, but to evolve at an order somewhat greater than, the Korteweg–de Vries equation Fourier transform solution to the linearized KdV equation is reviewed These results provide a basis for discussing the scattering transform solution to the KdV equation on the infinite interval The relationship between the linear Fourier transform and the scattering transform is discussed, where particular emphasis is placed on those results that are necessary for the development of the numerical algorithm for the direct scattering transform (DST) and for physically interpreting nonlinear wave motion governed by the KdV equation Some of the important assumptions leading to discrete Fourier methods are also discussed

DHT domain digital watermarking with low loss in image informations

Abstract: The paper proposes a digital image watermarking scheme that selects regions for data embedding based on information measure. Two valued kernels of Hadamard transformation cause smaller image information change during embedding compared to other transform domains such as DCT (discrete cosine transform), DFT (discrete Fourier transform), Fourier–Mellin and wavelet-based embedding. Moreover, the usage of Hadamard transform as signal decomposition tool offers advantages in terms of simpler implementation, low computation cost and high resiliency at low quality compression considering both JPEG and JPEG 2000 framework. Compression resiliency is further improved using adaptive negative modulation. These facts are validated comparing the performance with some other existing watermarking schemes as well as DCT domain implementation of the proposed scheme.

Investigating advantages and disadvantages of the analysis of a geometrical surface structure with the use of fourier and wavelet transform

Abstract: Nowadays a geometrical surface structure is usually e valuated with the use of Fourier transform. This type of transform allows for accurate analysis of harmonic components of surface profiles. Due to its funda mentals, Fourier transform is particularly efficient when eval uating periodic signals. Wavelets are the small waves that are oscillatory and limited in the range. Wavelets ar e special type of sets of basis functions that are useful in the description of function spaces. They are particularly useful for the description of non-continuous and irregular functions that appear most often as responses of real physical systems. Bases of wavelet functions are usually well located in the frequency and in the time domain. In the case of periodic signals, the Fourier transform is still extremely useful. It allows to obtain accurate inform ation on the analyzed surface. Wavelet analysis does not provide as accurate information about the measured surface as the Fourier tra nsform, but it is a useful tool for detection of irregularities of the profile. Therefore , wavelet analysis is the better way to detect scratches or cracks that sometimes occur on the surface. The pape r presents the fundamentals of both types of transform. It presents also the comparison of an evaluation of the roundness profile by Fourier and wavelet transforms.

Advanced Hough Transform Using A Multilayer Fractional Fourier Method

Abstract: The Hough transform (HT) is a commonly used technique for the identification of straight lines in an image. The Hough transform can be equivalently computed using the Radon transform (RT), by performing line detection in the frequency domain through use of central-slice theorem. In this research, an advanced Radon transform is developed using a multilayer fractional Fourier transform, a Cartesian-to-polar mapping, and 1-D inverse Fourier transforms, followed by peak detection in the sinogram. The multilayer fractional Fourier transform achieves a more accurate sampling in the frequency domain, and requires no zero padding at the stage of Cartesian-to-polar coordinate mapping. Our experiments were conducted on mix-shape images, noisy images, mixed-thickness lines and a large data set consisting of 751 000 handwritten Chinese characters. The experimental results have shown that our proposed method outperforms all known representative line detection methods based on the standard Hough transform or the Fourier transform.

Fractional Fourier transform in temporal ghost imaging with classical light

Abstract: We investigate temporal, second-order classical ghost imaging with long, incoherent, scalar plane-wave pulses. We prove that in rather general conditions, the intensity correlation function at the output of the setup is given by the fractional Fourier transform of the temporal object. In special cases, the correlation function is shown to reduce to the ordinary Fourier transform and the temporal image of the object. Effects influencing the visibility and the resolution are considered. This work extends certain known results on spatial ghost imaging into the time domain and could find applications in temporal tomography of pulses.