Showing papers on "Non-uniform discrete Fourier transform published in 2010"

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.

Combinatorial Sublinear-Time Fourier Algorithms

Abstract: We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N≫k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of $\hat{\mathbf{A}}$, and estimate their coefficients, in polynomial(k,log N) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002; Proceedings of SPIE Wavelets XI, 2005). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008).

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.

Accurate, Guaranteed Stable, Sliding Discrete Fourier Transform [DSP Tips & Tricks]

Abstract: This article presented a novel method of computing the SDFT that we call the modulated SDFT (mSDFT). The sliding discrete Fourier transform (SDFT) is a recursive algorithm that computes a DFT on a sample-by-sample basis. The accumulated errors and potential instabilities inherent in traditional SDFT algorithms are drastically reduced in the mSDFT. We removed the twiddle factor from the feedback in a traditional SDFT resonator and thus the finite precision of its representation is no longer a problem.

Iterative algorithm of discrete Fourier transform for processing randomly sampled NMR data sets

Abstract: Spectra obtained by application of multidimensional Fourier Transformation (MFT) to sparsely sampled nD NMR signals are usually corrupted due to missing data. In the present paper this phenomenon is investigated on simulations and experiments. An effective iterative algorithm for artifact suppression for sparse on-grid NMR data sets is discussed in detail. It includes automated peak recognition based on statistical methods. The results enable one to study NMR spectra of high dynamic range of peak intensities preserving benefits of random sampling, namely the superior resolution in indirectly measured dimensions. Experimental examples include 3D 15N- and 13C-edited NOESY-HSQC spectra of human ubiquitin.

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.

Simple quasicrystals are sets of stable sampling

Abstract: This contribution is addressing an issue named in signal processing. Let be a lattice and be the dual lattice. Then the standard Shannon–Nyquist theorem says that any signal f whose Fourier transform is supported by a compact subset can be recovered from the samples if and only if the translated sets are pairwise disjoint. This sufficient condition on K is also necessary. When it is not satisfied may occur. Olevskii and Ulanovskii designed irregular sampling strategies which remedy . Then one can optimally reduce the number of measures needed to recover a signal or an image whose Fourier transform is supported by a compact set K with a given measure. The present contribution is aimed at bridging the gap between this advance on irregular sampling and the theory of quasicrystals.

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.

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.

On sampling functions and Fourier reconstruction methods

Abstract: Random sampling can lead to algorithms in which the Fourier reconstruction is almost perfect when the underlying spectrum of the signal is sparse or band-limited. Conversely, regular sampling often hampers the Fourier data recovery methods. However, 2D signals that are band-limited in one spatial dimension can be recovered by designing a regular acquisition grid that minimizes the mixing between the unknown spectrum of the well-sampled signal and aliasing artifacts. This concept can be easily extended to higher dimensions and used to define potential strategies for acquisition-guided Fourier reconstruction. The wavenumber response of various sampling operators is derived and sampling conditions for optimal Fourier reconstruction are investigated using synthetic and real data examples.

Discrete Fourier Analysis and Wavelets

Abstract: Signal and image processing is an area which continues to develop and expand. It is a subject matter which has obvious practical data applications which are underlaid by, at times, complex mathematical methods. This book is authored by two professors of Mathematics at the RoseHulman Institute of Technology and provides a good, mathematically focused, introductory text to the methods, both classical and modern, currently utilised in signal and image processing. The text aims to provide entry level material on a range of signal processing methods to those at upper-undergraduate or beginning postgraduate levels working within mathematics, engineering or other related areas. The fact that the text is aimed at a wide audience of practitioners perhaps explains why the text is less technically detailed than other books which cover a similar information base. Though, where any technical details or definitions are omitted, full references to more extensive works are provided. Therefore, whilst the book acts as a good introductory text, those readers looking for an in-depth description of the mathematics behind these techniques are advised to look elsewhere. The reader is first introduced to traditional and well-established signal processing methods, such as the discrete Fourier and discrete cosine transforms. Throughout these sections, the JPEG compression algorithm is the principal example used to promote understanding and this is an example which is thoroughly developed throughout the book. The text is then concerned with detailing other processing techniques such as convolution, filtering and windowing. More modern advancements in the field, such as filter banks and wavelet methods are then considered. Many of these ideas are clearly stated and formed in such a way that the reader is provided with at least some explanation of the derivation of them. Where clarity is lacking, the reader is helpfully directed to others works or given worked examples to help further explain concepts. One of the strengths of the text lies in the large number of illustrative examples given. Many of these examples are enhanced by pictorial demonstrations of the theory. Also, the book does provide the reader with an extensive range of exercises which are usefully supplemented with hints and some sample answers. In addition to these exercises the book also details projects for completion in MATLAB where the aim is to allow the reader to apply the mathematical theory to realistic field problems. Much of the text is given over to general introductions of mathematical theory and notation which are fundamental to the understanding of topics later introduced. A comparatively minimal portion of the text covers the rather extensive topic of wavelet methodology though a good proportion of the text is spent explaining filter banks and how they give rise to wavelets. For ease of understanding the majority of the examples on wavelet theory are based around the simplest basis (the Haar basis). In general, the book gives a rather self-contained and in-depth guide to the discrete Fourier analysis portion of the text, whilst only giving, in the main, introductory material

The Discrete Fourier Transform, Part 6: Cross-Correlation

Abstract: This paper is part 6 in a series of papers about the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). The focus of this paper is on correlation. The correlation is performed in the time domain (slow correlation) and in the frequency domain using a Short-Time Fourier Transform (STFT). When the Fourier transform is an FFT, the correlation is said to be a “fast” correlation. The approach requires that each time segment be transformed into the frequency domain after it is windowed. Overlapping windows temporally isolate the signal by amplitude modulation with an apodizing function. The selection of overlap parameters is done on an ad-hoc basis, as is the apodizing function selection. This report is a part of project Fenestratus, from the skunk-works of DocJava, Inc. Fenestratus comes from the Latin and means, “to furnish with windows”.

Continuous and Discrete Time Signals and Systems

Abstract: 1. Introduction to signals 2. Introduction to systems Part I. Continuous Time Signals and Systems: 3. Time domain analysis of systems 4. Signal representation using Fourier series 5. Continuous-time Fourier transform 6. Laplace transform 7. Continuous-time filters 8. Case studies for CT systems Part II. Discrete Time Signals and Systems: 9. Sampling and quantization 10. Time domain analysis 11. Discrete-time Fourier series and transform 12. Discrete Fourier transform 13. Z-transform 14. Digital filters 15. FIR filter design 16. IIR filter design 17. Applications of digital signal processing Bibliography Appendices: A. Mathematical tables B. Introduction to complex numbers C. Linear constant coefficient differential equations D. Partial fraction expansion E. Introduction to MATLAB F. About the CD-ROM.

Research on the application of 4-weighted fractional Fourier transform in communication system

Abstract: The paper reveals the relationship between the weighting coefficients and weighted functions via the research of coefficients matrix and based on the original definition of 4-weighted fractional Fourier transform (4-WFRFT). The multi-parameters expression of weighting coefficients are given. Moreover, the 4-WFRFT of discrete sequences is defined by introducing DFT into it, which makes it suitable for digital communication systems. After analyzing the properties of WFRFT, a typical scheme for modulation/demodulation is proposed, which could make the statistics properties of the real and image part on both of the time and frequency domain and the phase properties have a significant variation. Such a variation could be controlled by the adjustment of transform parameters. If the WFRFT of multi-parameters is implemented, it will be more difficult to intercept and capture the modulated signals than normal.

Gabor Transform, Gabor-Wigner Transform and SPWVD as a time-frequency analysis of power quality

Abstract: The measurement algorithms applied in power quality measurement systems are based on Fourier Transformation (FT). The discrete versions of Fourier transformation - DFT (Discrete FT) and FFT (Fast Fourier Transformation) are most commonly used. That one-dimension frequency analysis is sufficient in many cases. However, to illustrate the character of the signal in a more comprehensive manner, it is crucial to represent the investigated signal on time-frequency plane. There are a lot of time-frequency representations (TFR) for presenting measured signal. The most common known are spectrogram (SPEC) and Gabor Transform (GT), which are based on direct DFT results. However, the method has relatively low time-frequency resolution. The other TFR representing Cohen class: Wigner-Ville Distribution (WVD) and its variants: Pseudo Wigner-Ville Distribution (PWVD), Smoothed Pseudo Wigner-Ville Distribution (SPWVD) and Gabor-Wigner Transform (GWT) are described in the paper. The main features of the transforms, on the basis of testing signals, were presented.

An in-place truncated fourier transform and applications to polynomial multiplication

Abstract: The truncated Fourier transform (TFT) was introduced by van der Hoeven in 2004 as a means of smoothing the "jumps" in running time of the ordinary FFT algorithm that occur at power-of-two input sizes. However, the TFT still introduces these jumps in memory usage. We describe in-place variants of the forward and inverse TFT algorithms, achieving time complexity O(n log n) with only O(1) auxiliary space. As an application, we extend the second author's results on space-restricted FFT-based polynomial multiplication to polynomials of arbitrary degree.

The discrete multiple-parameter fractional Fourier transform

Abstract: As a generalization of the Fourier transform (FT), the fractional Fourier transform (FRFT) has many applications in the areas of optics, signal processing, information security, etc. Therefore, the efficient discrete computational method is the vital fundament for the application of the fractional Fourier transform. The multiple-parameter fractional Fourier transform (MPFRFT) is a generalized fractional Fourier transform, which not only includes FRFT as special cases, but also provides a unified framework for the study of FRFT. In this paper, we present in detail the discretization method of the MPFRFT and define the discrete multiple-parameter fractional Fourier transform (DMPFRFT). Then, we utilize the tensor product to define two-dimensional multiple-parameter fractional Fourier transform (2D-MPFRFT) and the corresponding two-dimensional discrete multiple-parameter fractional Fourier transform (2D-DMPFRFT). Finally, as an application, a novel image encryption method based on 2D-DMPFRFT is proposed. Numerical simulations are performed to demonstrate that the proposed method is reliable and more robust to blind decryption than several existing methods.

Fractional fourier transform

Abstract: A brief introduction to the fractional Fourier transform and its properties is given. Its relation to phase-space representations (time- or space-frequency representations) and the concept of fractional Fourier domains are discussed. An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.

A new LFM-signal detector based on fractional Fourier transform

Abstract: A new LFM-signal detector formulated by the integration of the 4th-power modulus of the fractional Fourier transform is proposed. It has similar performance to the modulus square detector of Radon-ambiguity transform because of the equivalence relationship between them. But the new detector has much lower computational complexity in the case that the number of the searching angles is far less than the length of the signal. Moreover, it is proved that the new detector can be generalized to the integration of the nth-power (2 ≶ n) modulus of the fractional Fourier transform via mathematical derivation. Computer simulation results have confirmed the effectiveness of the proposed detector in LFM-signal detection.

Structure analysis and surface simulation of woven fabrics using fast Fourier transform techniques

Abstract: Fourier transform techniques are particularly suitable for modelling woven fabrics with periodic structures. These techniques allow us to analyse the periodicity and the directionality of repeated ...

A high accuracy conformal method for evaluating the discontinuous fourier transform

Abstract: A highly accurate, fast algorithm is proposed to evaluate the flnite Fourier transform of both continuous and discontinues functions. As the discretization is conformal to the function discontinuities, this method is called the conformal Fourier transform (CFT) method. It is applied to computational electromagnetics to calculate the Fourier transform of induced electric current densities in a volume integral equation. The spectral discrimination in the CFT method can be arbitrary and the spectral range can be as large as needed. As no discretization for the Fourier exponential kernel is needed, the CFT method is not restricted by the Nyquist sampling theorem, thus avoiding the aliasing distortions that exist in other traditional methods. The accuracy of the CFT method is greatly improved since the method is based on high order interpolation and the closed-form Fourier transforms for polynomials partly reduce the error due to discretization. Assuming Ns and N are the numbers

FPGA implementation of discrete fractional Fourier transform

Abstract: Since decades, fractional Fourier transform has taken a considerable attention for various applications in signal and image processing domain. On the evolution of fractional Fourier transform and its discrete form, the real time computation of discrete fractional Fourier transform is essential in those applications. On this context, we have proposed new hardware architecture for implementing a Discrete Fractional Fourier Transform (DFrFT) which requires hardware complexity of O(4N), where N is transform order. This proposed architecture has been simulated and synthesized using verilogHDL, targeting a FPGA device (XLV5LX110T). The simulation results are very close to the results obtained by using MATLAB. The result shows that, this architecture can be operated on a maximum frequency of 217MHz.