Showing papers on "Non-uniform discrete Fourier transform published in 2012"

Nearly optimal sparse fourier transform

Abstract: We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k=o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = nΩ(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log (n/k) / log log n) signal samples, even if it is allowed to perform adaptive sampling.

Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis

Abstract: A systematic, unified treatment of orthogonal transform methods for signal processing, data analysis and communications, this book guides the reader from mathematical theory to problem solving in practice. It examines each transform method in depth, emphasizing the common mathematical principles and essential properties of each method in terms of signal decorrelation and energy compaction. The different forms of Fourier transform, as well as the Laplace, Z-, Walsh–Hadamard, Slant, Haar, Karhunen–Loeve and wavelet transforms, are all covered, with discussion of how each transform method can be applied to real-world experimental problems. Numerous practical examples and end-of-chapter problems, supported by online Matlab and C code and an instructor-only solutions manual, make this an ideal resource for students and practitioners alike.

Constrained least-squares spectral analysis: Application to seismic data

Abstract: An inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data. The Fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window. The method resulted in spectra that have reduced window smearing for a given window length relative to the discrete Fourier transform irrespective of window shape, and a time-frequency analysis with a combination of time and frequency resolution that is superior to the short time Fourier transform and the continuous wavelet transform. The reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window. The degree of resolution improvement relative to the short time Fourier transform increases as window length decreases. As compared with the continu...

Superfast Fourier Transform Using QTT Approximation

Abstract: We propose Fourier transform algorithms using QTT format for data-sparse approximate representation of one- and multi-dimensional vectors (m-tensors). Although the Fourier matrix itself does not have a low-rank QTT representation, it can be efficiently applied to a vector in the QTT format exploiting the multilevel structure of the Cooley-Tukey algorithm. The m-dimensional Fourier transform of an n×⋯×n vector with n=2d has \(\mathcal{O}(m d^{2} R^{3})\) complexity, where R is the maximum QTT-rank of input, output and all intermediate vectors in the procedure. For the vectors with moderate R and large n and m the proposed algorithm outperforms the \(\mathcal{O}(n^{m} \log n)\) fast Fourier transform (FFT) algorithm and has asymptotically the same log-squared complexity as the superfast quantum Fourier transform (QFT) algorithm. By numerical experiments we demonstrate the examples of problems for which the use of QTT format relaxes the grid size constrains and allows the high-resolution computations of Fourier images and convolutions in higher dimensions without the ‘curse of dimensionality’. We compare the proposed method with Sparse Fourier transform algorithms and show that our approach is competitive for signals with small number of randomly distributed frequencies and signals with limited bandwidth.

Shift-Invariant and Sampling Spaces Associated With the Fractional Fourier Transform Domain

Abstract: Shift-invariant spaces play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. A special class of the shift-invariant spaces is the class of sampling spaces in which functions are determined by their values on a discrete set of points. One of the vital tools used in the study of sampling spaces is the Zak transform. The Zak transform is also related to the Poisson summation formula and a common thread between all these notions is the Fourier transform. In this paper, we extend some of these notions to the fractional Fourier transform (FrFT) domain. First, we introduce two definitions of the discrete fractional Fourier transform and two semi-discrete fractional convolutions associated with them. We employ these definitions to derive necessary and sufficient conditions pertaining to FrFT domain, under which integer shifts of a function form an orthogonal basis or a Riesz basis for a shift-invariant space. We also introduce the fractional Zak transform and derive two different versions of the Poisson summation formula for the FrFT. These extensions are used to obtain new results concerning sampling spaces, to derive the reproducing-kernel for the spaces of fractional band-limited signals, and to obtain a new simple proof of the sampling theorem for signals in that space. Finally, we present an application of our shift-invariant signal model which is linked with the problem of fractional delay filtering.

SAR-Based Vibration Estimation Using the Discrete Fractional Fourier Transform

Abstract: A vibration estimation method for synthetic aperture radar (SAR) is presented based on a novel application of the discrete fractional Fourier transform (DFRFT). Small vibrations of ground targets introduce phase modulation in the SAR returned signals. With standard preprocessing of the returned signals, followed by the application of the DFRFT, the time-varying accelerations, frequencies, and displacements associated with vibrating objects can be extracted by successively estimating the quasi-instantaneous chirp rate in the phase-modulated signal in each subaperture. The performance of the proposed method is investigated quantitatively, and the measurable vibration frequencies and displacements are determined. Simulation results show that the proposed method can successfully estimate a two-component vibration at practical signal-to-noise levels. Two airborne experiments were also conducted using the Lynx SAR system in conjunction with vibrating ground test targets. The experiments demonstrated the correct estimation of a 1-Hz vibration with an amplitude of 1.5 cm and a 5-Hz vibration with an amplitude of 1.5 mm.

Fast computing of discrete cosine and sine transforms of types vi and vii

Abstract: This disclosure presents techniques for implementing a fast algorithm for implementing odd-type DCTs and DSTs. The techniques include the computation of an odd-type transform on any real-valued sequence of data (e.g., residual values in a video coding process or a block of pixel values of an image coding process) by mapping the odd-type transform to a discrete Fourier transform (DFT). The techniques include a mapping between the real-valued data sequence to an intermediate sequence to be used as an input to a DFT. Using this intermediate sequence, an odd-type transform may be achieved by calculating a DFT of odd size. Fast algorithms for a DFT may be then be used, and as such, the odd-type transform may be calculated in a fast manner

Fourier Transforms: An Introduction for Engineers

Abstract: Preface. 1. Signals and Systems. 2. The Fourier Transform. 3. Fourier Inversion. 4. Basic Properties. 5. Generalized Transforms and Functions. 6. Convolution and Correlation. 7. Two Dimensional Fourier Analysis. 8. Memoryless Nonlinearities. A: Fourier Transform Tables. Bibliography. Index.

Fractional Fourier transform of tempered distributions and generalized pseudo-differential operator

Abstract: A brief introduction to the fractional Fourier transform and its basic properties is given. Fractional Fourier transform of tempered distributions is studied. Generalized pseudo-differential operators involving two classes of symbols and fractional Fourier transforms are investigated. An application of the fractional Fourier transform in solving a generalized heat equation is given.

Inverse Source Problems for the Helmholtz Equation and the Windowed Fourier Transform

Abstract: We consider the inverse source problem for time-harmonic acoustic or electromagnetic wave propagation in the two-dimensional free space Given the radiated far field pattern of the solution to the Helmholtz equation for a certain source term, we find that the windowed Fourier transform of this far field is related to an exponential Radon transform with purely imaginary exponent of a smoothed approximation of the source Based on this observation we set up a filtered backprojection algorithm to recover information on the unknown source term from a single far field measurement We analyze this algorithm and provide extensive numerical results that illustrate our theoretical findings As one outcome the method is shown to work better the larger the wave number Possible extensions of the reconstruction method to limited aperture data and to inverse obstacle scattering problems are briefly sketched

Fractional S transform — Part 1: Theory

Abstract: The S transform, which is a time-frequency representation known for its local spectral phase properties in signal processing, uniquely combines elements of wavelet transforms and the short-time Fourier transform (STFT). The fractional Fourier transform is a tool for non-stationary signal analysis. In this paper, we define the concept of the fractional S transform (FRST) of a signal, based on the idea of the fractional Fourier transform (FRFT) and S transform (ST), extend the S transform to the time-fractional frequency domain from the time-frequency domain to obtain the inverse transform, and study the FRST mathematical properties. The FRST, which has the advantages of FRFT and ST, can enhance the ST flexibility to process signals. Compared to the S transform, the FRST can effectively improve the signal time-frequency resolution capacity. Simulation results show that the proposed method is effective.

Performance of methods based on the fractional fourier transform for the detection of linear frequency modulated signals

Abstract: Analysis of detectors of linear frequency modulated (LFM) signals based on the fractional Fourier transform (FrFT) is presented. This allows one to conduct a fair comparison of the performance of these methods with those based on the Fourier transform (FT). In order to facilitate this analysis, expressions for the distribution of the coefficients of the FrFT are presented. Analytic approximations for the FT of the LFM signals are also developed. It is shown that the FrFT methods achieve superior performance if the sweep rate is sufficiently fast or the data length is sufficiently large.

Efficient phase retrieval of sparse signals

Abstract: We consider the problem of one dimensional (1D) phase retrieval, namely, recovery of a 1D signal from the magnitude of its Fourier transform. This problem is ill-posed since the Fourier phase information is lost. Therefore, prior information on the signal is needed in order to recover it. In this work we consider the case in which the prior information on the signal is that it is sparse, i.e., it consists of a small number of nonzero elements. We propose a fast local search method for recovering a sparse 1D signal from measurements of its Fourier transform magnitude. Our algorithm does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images. Simulation results indicate that the proposed algorithm is fast and more accurate than existing techniques.

Spectral Analysis of Sampled Signals in the Linear Canonical Transform Domain

Abstract: The spectral analysis of uniform or nonuniform sampling signal is one of the hot topics in digital signal processing community. Theories and applications of uniformly and nonuniformly sampled one-dimensional or two-dimensional signals in the traditional Fourier domain have been well studied. But so far, none of the research papers focusing on the spectral analysis of sampled signals in the linear canonical transform domain have been published. In this paper, we investigate the spectrum of sampled signals in the linear canonical transform domain. Firstly, based on the properties of the spectrum of uniformly sampled signals, the uniform sampling theorem of two dimensional signals has been derived. Secondly, the general spectral representation of periodic nonuniformly sampled one and two dimensional signals has been obtained. Thirdly, detailed analysis of periodic nonuniformly sampled chirp signals in the linear canonical transform domain has been performed.

On robust phase retrieval for sparse signals

Abstract: Recovering signals from their Fourier transform magnitudes is a classical problem referred to as phase retrieval and has been around for decades. In general, the Fourier transform magnitudes do not carry enough information to uniquely identify the signal and therefore additional prior information is required. In this paper, we shall assume that the underlying signal is sparse, which is true in many applications such as X-ray crystallography, astronomical imaging, etc. Recently, several techniques involving semidefinite relaxations have been proposed for this problem, however very little analysis has been performed. The phase retrieval problem can be decomposed into two tasks — (i) identifying the support of the sparse signal from the Fourier transform magnitudes, and (ii) recovering the signal using the support information. In earlier work [13], we developed algorithms for (i) which provably recovered the support for sparsities upto O(n1/3−∊). Simulations suggest that support recovery is possible upto sparsity O(n1/2−∊). In this paper, we focus on (ii) and propose an algorithm based on semidefinite relaxation, which provably recovers the signal from its Fourier transform magnitude and support knowledge with high probability if the support size is O(n1/2−∊).

The Discrete Yang-Fourier Transforms in Fractal Space

Abstract: The Yang-Fourier transform (YFT) in fractal space is a generation of Fourier transform based on the local fractional calculus. The discrete Yang-Fourier transform (DYFT) is a specific kind of the approximation of discrete transform, used in Yang- Fourier transform in fractal space. This paper points out new standard forms of discrete Yang-Fourier transforms (DYFT) of fractal signals, and both properties and theorems are investigated in detail. Keywords -Fractal, Signal, Discrete, Yang-Fourier transforms

A new DFT-based frequency estimator for single-tone complex sinusoidal signals

Abstract: Frequency estimation for single-tone complex sinusoidal signals under additive white Gaussian noise is a classical and fundamental problem in many applications, such as communications, radar, sonar and power systems. In this paper, we propose a new algorithm by interpolating discrete Fourier transform (DFT) samples. Different from other existing interpolation methods for frequency estimation, our algorithm is based on a much simpler expression and has mathematically tractable bias expression in closed form, which can potentially assist future bias correction. Simulations confirm that our proposed algorithm outperforms all existing alternatives in the literature with comparable complexity.

Research and Comparison of Time-frequency Techniques for Nonstationary Signals

Abstract: Most of signals in engineering are nonstationary and time-varying. The Fourier transform as a traditional approach can only provide the feature information in frequency domain. The time-frequency techniques may give a comprehensive description of signals in time-frequency planes. Based on some typical nonstationary signals, five time-frequency analysis methods, i.e., the short-time Fourier transform (STFT), wavelet transform (WT), Wigner-Ville distribution (WVD), pseudo-WVD (PWVD) and the Hilbert- Huang transform (HHT), were performed and compared in this paper. The characteristics of each method were obtained and discussed. Compared with the other methods, the HHT with a high time-frequency resolution can clearly describe the rules of the frequency compositions changing with time, is a good approach for feature extraction in nonstationary signal processing.

Optical OFDM based on the fractional Fourier transform

Abstract: We describe a innovative OFDM scheme based on orthogonal chirped subcarriers, that corresponds to the fractional Fourier transform (FrFT) of the input signal. The FrFT can be electronically implemented with a complexity equivalent to the conventional fast Fourier transform (FFT); on the other hand, the planar device that implements the FrFT in the optical domain is similar to the passive arrayed waveguide grating (AWG) device that performs the FFT. We analyze the spectral efficiency, the peak-to-average power ratio (PAPR) and the frequency offset sensitivity of a FrFT-based optical OFDM system, and make an accurate comparison with the standard FFT-based implementation.

Methods of Applied Fourier Analysis

Abstract: Periodic functions hardy spaces prediction theory discrete systems and control theory harmonic analysis in Euclidean space distributions functions with restricted transforms phase space wavelet analysis the discrete Fourier transform the Hermite functions.

Enhancing the Resolution of the Spectrogram Based on a Simple Adaptation Procedure

Abstract: This work is concerned with improving the quality of signal localization for the short-time Fourier transform by properly adjusting the size of its analysis window over time. The adaptation procedure involves the estimation of an area in the time-frequency plane which is more compact than the support of the fixed-window spectrogram. Then, at each time instant, the optimal window is selected such that the signal energy is maximized within the identified area. The proposed method achieves its objectives, and can compare favorably with alternative time-adaptive spectrograms as well as with advanced quadratic representations.

A stable and accurate butterfly sparse fourier transform

Abstract: Recently, the butterfly approximation scheme was proposed for computing Fourier transforms with sparse and smooth sampling in the frequency and spatial domains. We present a rigorous error analysis which shows how the local expansion degree depends on the target accuracy and the nonharmonic bandwidth. Moreover, we show that the original scheme becomes numerically unstable if a large local expansion degree is used. This problem is removed by representing all approximations in a Lagrange-type basis instead of the previously used monomial basis. All theoretical results are illustrated by numerical experiments.

A New Formulation of the Fast Fractional Fourier Transform

Abstract: By using a spectral approach, we derive a Gaussian-like quadrature of the continuous fractional Fourier transform. The quadrature is obtained from a bilinear form of eigenvectors of the matrix associated to the recurrence equation of the Hermite polynomials. These eigenvectors are discrete approximations of the Hermite functions, which are eigenfunctions of the fractional Fourier transform operator. This new discrete transform is unitary and has a group structure. By using some asymptotic formulas, we rewrite the quadrature in terms of the fast Fourier transform (FFT), yielding a fast discretization of the fractional Fourier transform and its inverse in closed form. We extend the range of the fractional Fourier transform by considering arbitrary complex values inside the unit circle and not only at the boundary. We find that this fast quadrature evaluated at $z=i$ becomes a more accurate version of the FFT and can be used for nonperiodic functions.