Showing papers on "Non-uniform discrete Fourier transform published in 2014"

Recent Developments in the Sparse Fourier Transform: A compressed Fourier transform for big data

Abstract: The discrete Fourier transform (DFT) is a fundamental component of numerous computational techniques in signal processing and scientific computing. The most popular means of computing the DFT is the fast Fourier transform (FFT). However, with the emergence of big data problems, in which the size of the processed data sets can easily exceed terabytes, the "fast" in FFT is often no longer fast enough. In addition, in many big data applications it is hard to acquire a sufficient amount of data to compute the desired Fourier transform in the first place. The sparse Fourier transform (SFT) addresses the big data setting by computing a compressed Fourier transform using only a subset of the input data, in time smaller than the data set size. The goal of this article is to survey these recent developments, explain the basic techniques with examples and applications in big data, demonstrate tradeoffs in empirical performance of the algorithms, and discuss the connection between the SFT and other techniques for massive data analysis such as streaming algorithms and compressive sensing.

Applications of the synchrosqueezing transform in seismic time-frequency analysis

Abstract: Time-frequency representation of seismic signals provides a source of information that is usually hidden in the Fourier spectrum. The short-time Fourier transform and the wavelet transform are the principal approaches to simultaneously decompose a signal into time and frequency components. Known limitations, such as trade-offs between time and frequency resolution, may be overcome by alternative techniques that extract instantaneous modal components. Empirical mode decomposition aims to decompose a signal into components that are well separated in the time-frequency plane allowing the reconstruction of these components. On the other hand, a recently proposed method called the “synchrosqueezing transform” (SST) is an extension of the wavelet transform incorporating elements of empirical mode decomposition and frequency reassignment techniques. This new tool produces a well-defined time-frequency representation allowing the identification of instantaneous frequencies in seismic signals to highlight ...

Sparse Discrete Fractional Fourier Transform and Its Applications

Abstract: The discrete fractional Fourier transform is a powerful signal processing tool with broad applications for nonstationary signals. In this paper, we propose a sparse discrete fractional Fourier transform (SDFrFT) algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional Fourier domain. The proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise. In addition, we apply the SDFrFT to the synchronization of high dynamic direct-sequence spread-spectrum signals. Furthermore, a sparse fractional cross ambiguity function (SFrCAF) is developed, and the application of SFrCAF to a passive coherent location system is presented. The experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision.

Fast Numerical Nonlinear Fourier Transforms

Abstract: The nonlinear Fourier transform, which is also known as the forward scattering transform, decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common Fourier transform, these waves no longer have to be sinusoidal. Physically relevant waveforms are often available for the analysis instead. The details of the transform depend on the waveforms underlying the analysis, which in turn are specified through the implicit assumption that the signal is governed by a certain evolution equation. For example, water waves generated by the Korteweg-de Vries equation can be expressed in terms of cnoidal waves. Light waves in optical fiber governed by the nonlinear Schrodinger equation (NSE) are another example. Nonlinear analogs of classic problems such as spectral analysis and filtering arise in many applications, with information transmission in optical fiber, as proposed by Yousefi and Kschischang, being a very recent one. The nonlinear Fourier transform is eminently suited to address them -- at least from a theoretical point of view. Although numerical algorithms are available for computing the transform, a "fast" nonlinear Fourier transform that is similarly effective as the fast Fourier transform is for computing the common Fourier transform has not been available so far. The goal of this paper is to address this problem. Two fast numerical methods for computing the nonlinear Fourier transform with respect to the NSE are presented. The first method achieves a runtime of $O(D^2)$ floating point operations, where $D$ is the number of sample points. The second method applies only to the case where the NSE is defocusing, but it achieves an $O(D\log^2D)$ runtime. Extensions of the results to other evolution equations are discussed as well.

A Sinusoidal Frequency Modulation Fourier Transform for Radar-Based Vehicle Vibration Estimation

Abstract: The intricate vibration of a working vehicle provides an important signature to the vehicle type. Small vibrations introduce phase modulation in radar echoes, which is referred to as micro-Doppler (m-D) phenomenon and can be modeled as sinusoidal frequency-modulated (SFM) signal. Such phase modulation induced by vibrations consists of multiple frequency components; moreover, the modulation is usually rather weak. Present parametric estimators are difficult to estimate so many parameters of every frequency component, while nonparametric approaches suffer from low precision. This paper considers the analysis of SFM signal with weak and multiple frequency components modulation on phase term. We first define the SFM signal space to bridge a gap between the SFM signal analysis and classical signal processing methods. Based on the defined signal space, a novel m-D analysis method, i.e., the sinusoidal frequency modulation Fourier transform (SFMFT), is presented. With the operations acting directly on the phase term of SFM signal, SFMFT gives the frequency spectrum of vibration traces. Unlike the existing methods, which apply a sliding short-time window to perform an instantaneous approximation, the proposed method makes use of the global data, which can provide a longer integral period gain, and consequently improves the estimation performance significantly. Simulation results indicate that the proposed method outperforms the existing methods in the spectrum accuracy, the range of estimable vibration amplitude/frequency, and the computation complexity.

The Hopping Discrete Fourier Transform [sp Tips&Tricks]

Abstract: The discrete Fourier transform (DFT) produces a Fourier representation for finite-duration data sequences. In addition to its theoretical importance, the DFT plays a key role in the implementation of a variety of digital signal-?processing algorithms. Several algorithms including the fast Fourier transform (FFT) and the Goertzel algorithm have been introduced for the fast implementation of the DFT [1], [2].

High-precision Fourier forward modeling of potential fields

Abstract: We analyzed the numerical forward methods in the Fourier domain for potential fields. Existing Fourier-domain forward methods applied the standard fast Fourier transform (FFT) algorithm to inverse transform a conjugate symmetrical spectrum into a real field. It had significant speed advantages over space-domain forward methods but suffered from problems including aliasing, imposed periodicity, and edge effect. Usually, grid expansion was needed to reduce these errors, which was equivalent to the numerical evaluation of the oscillatory Fourier integral using the trapezoidal rule with smaller steps. We tested a high-precision Fourier-domain forward method based on a combined use of shift-sampling technique and Gaussian quadrature theory. The trapezoidal rule applied by the standard FFT algorithm to evaluate the continuous Fourier transform was modified by introducing a shift parameter ξ. By choosing optimum values of ξ as Gaussian quadrature nodes, we developed a Gauss-FFT method for Fourier forward modeling of potential fields. No grid expansion was needed, the sources can be set near the boundary of the fields or even go beyond the boundary. The Gauss-FFT method converged to the space-domain solution much faster than the standard FFT method with grid expansion. Forward modeling results almost identical to space-domain ones can be obtained in less time. Numerical examples, of both simple and complex 2D and 3D source forward modeling, revealed the reliability and adaptability of the method.

Fast Discrete Fourier Transform on Generalized Sparse Grids

Abstract: In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and the cost. We construct interpolants with a computational cost complexity which is substantially lower than for the standard full grid case. The associated generalized sparse grid interpolants have the same approximation order as the standard full grid interpolants, provided that certain additional regularity assumptions on the considered functions are fulfilled. Numerical results validate our theoretical findings.

Multi-eigenvalue communication via the nonlinear Fourier transform

Abstract: Information transmission using only the discrete component of the nonlinear Fourier transform is studied and multi-eigenvalue signal sets are presented that achieve spectral efficiencies greater than 3 bits/s/Hz.

Quadratic Fourier Transforms

Abstract: In this paper we shall examine the quadratic Fourier transform which is introduced by the generalized quadratic function for one order parameter in the ordinary Fourier transform. This will be done by analyzing corresponding six subcases of the quadratic Fourier transform within a reproducing kernel Hilbert spaces framework.

A frame theoretic approach to the nonuniform fast fourier transform

Abstract: Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse process, the nonuniform fast Fourier transform (NFFT), also called convolutional gridding, is frequently employed. While various investigations have led to improvements in accuracy, efficiency, and robustness of the NFFT, not much attention has been paid to the fundamental analysis of the scheme, and in particular its convergence properties. This paper analyzes the convergence of the NFFT by casting it as a Fourier frame approximation. In so doing, we are able to design parameters for the method that satisfy conditions for numerical convergence. Our so-called frame theoretic convolutional gridding algorithm can also be applied to detect features (such as edges) from nonuniform Fourier samples of piecewise smooth functions.

Noise Analysis of Power System Frequency Estimated From Angle Difference of Discrete Fourier Transform Coefficient

Abstract: The frequency deviation of an electric power system is a reflection of its operating state. This frequency deviation can be estimated from angle variations of the discrete Fourier transform coefficient of the positive fundamental frequency. Most errors of the frequency estimate can arise from noise in the power system signal and the leakage effect of the negative fundamental frequency. This paper analyzes errors of the frequency estimates and the measured angle variations due to the noise. Their means and variances are derived theoretically with respect to signal-to-noise ratios and verified and analyzed through simulations.

Image fusion based on image decomposition using self-fractional Fourier functions

Abstract: Image fusion has been receiving increasing attention in the research community in a wide spectrum of applications. Several algorithms in spatial and frequency domains have been developed for this purpose. In this paper we propose a novel algorithm which involves the use of fractional Fourier domains which are intermediate between spatial and frequency domains. The proposed image fusion scheme is based on decomposition of source images (or its transformed version) into self-fractional Fourier functions. The decomposed images are then fused by maximum absolute value selection rule. The selected images are combined and inverse transformation is taken to obtain the final fused image. The proposed decomposition scheme and the use of some transformation before the decomposition step offer additional degrees of freedom in the image fusion scheme. Simulation results of the proposed scheme for different transformation of the source images for two different sets of images are also presented. It is observed through the simulation results that the use of taking the transformation before the decomposition step improves the quality of fused image. In particular the results of using the fractional Fourier transform and discrete cosine transform before the decomposition step are encouraging.

Image and video processing using discrete fractional transforms

Abstract: The mathematical transforms such as Fourier transform, wavelet transform and fractional Fourier transform have long been influential mathematical tools in information processing. These transforms process signal from time to frequency domain or in joint time–frequency domain. In this paper, with the aim to review a concise and self-reliant course, the discrete fractional transforms have been comprehensively and systematically treated from the signal processing point of view. Beginning from the definitions of fractional transforms, discrete fractional Fourier transforms, discrete fractional Cosine transforms and discrete fractional Hartley transforms, the paper discusses their applications in image and video compression and encryption. The significant features of discrete fractional transforms benefit from their extra degree of freedom that is provided by fractional orders. Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractional cosine transform is better in encryption of image and video. Mean square error and peak signal-to-noise ratio with optimum fractional order are considered quality check parameters in image and video.

Two-dimensional discrete fourier transform (2d-dft) based codebook for elevation beamforming

Abstract: The present disclosure relates to systems and methods for a two-dimensional discrete Fourier transform based codebook for elevation beamforming. A two-dimensional discrete Fourier transform based codebook is determined for elevation beamforming. The codebook supports single stream codewords and multistream codewords. The two-dimensional discrete Fourier transform based codebook is generated by stacking the columns of the matrix product of two discrete Fourier transform codebook matrices. The codebook size may be flexibly designed based on required beam resolution in azimuth and elevation. A best codebook index is selected from the generated two-dimensional discrete Fourier transform based codebook. The selected codebook index is provided in a channel state information report. The channel state information report is transmitted to a base station.

Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform

Abstract: A movie encryption scheme is proposed using a discrete multiple-parameter fractional Fourier transform and theta modulation. After being modulated by sinusoidal amplitude grating, each frame of the movie is transformed by a filtering procedure and then multiplexed into a complex signal. The complex signal is multiplied by a pixel scrambling operation and random phase mask, and then encrypted by a discrete multiple-parameter fractional Fourier transform. The movie can be retrieved by using the correct keys, such as a random phase mask, a pixel scrambling operation, the parameters in a discrete multiple-parameter fractional Fourier transform and a time sequence. Numerical simulations have been performed to demonstrate the validity and the security of the proposed method.

The SVFT-Based Control

Abstract: In this paper, a controller based on the space vector Fourier transform (SVFT) is proposed. The vector controller is implemented in the stationary αβ reference frame and can be designed to track unbalanced and distorted three-phase reference signals containing specific positive-sequence, negative-sequence, and harmonic components. In order to evaluate the proposed controller closed-loop response, three-phase unbalanced and distorted reference currents are imposed in a grid-connected voltage source converter. The ability of the controller to follow the references with zero steady-state error is proved through frequency response plots and stability analysis. Experimental results are obtained to confirm the properties of the proposed scheme.

Discrete Fourier transform based pattern classifiers

Abstract: A technique for pattern classification using the Fourier tra nsform combined with the nearest neighbor classifier is proposed. The multidimensional fast Fourier transform ( FFT) is applied to the patterns in the data base. Then the magnitudes of the Fourier coefficients are sorted in desc ending order and the first P coefficients with largest magnitudes are selected, where P is a design parameter. These coefficients are then used in fur ther processing rather than the original patterns. When a noisy pattern is presente d for classification, the pattern’s P Fourier coefficients with largest magnitude are extracted. The coefficients are a rranged in a vector in the descending order of their magnitudes. The obtained vector is referred to as the signat ure vector of the corresponding pattern. Then the distance between the signature vector of the pattern to be cl assified and the signature vectors of the patterns in the data base are computed and the pattern to be classified is matc hed with a pattern in the data base whose signature vector is closest to the signature vector of the pattern bein g classified.

Synchronization of Sampling-Based Measuring Systesm

Abstract: The amplitude and the phase of a measured single-tone signal can accurately be calculated from the discrete Fourier transform (DFT) components in case of coherent sampling. When the synchronization is not perfect, spectral leakage appears and errors are introduced in the determination of the signal parameters. In this paper, a new method to synchronize the sampling frequency with the frequency of the measured signal is presented. The method is based on the information contained in the fundamental and the sideband components of the DFT of the measured signal. A single measurement gives the synchronization error and permits the correction of the sampling frequency required to become coherent. Moreover, the error on the amplitude measurement relative to a quasi-coherent sampling is also discussed and quantitatively modeled. The results of the model as well as the synchronization process are experimentally verified and compared with the interpolated DFT algorithm.

The S-transform using a new window to improve frequency and time resolutions

Abstract: The S-transform presents arbitrary time series as localized invertible time–frequency spectra. This transformation improves the short-time Fourier transform and the wavelet transform by merging the multiresolution and frequency-dependent analysis properties of wavelet transform with the absolute phase retaining of Fourier transform. The generalized S-transform utilizes a combination of a Fourier transform kernel and a scalable-sliding window. The common S-transform applies a Gaussian window to provide appropriate time and frequency resolution and minimizes the product of these resolutions. However, the Gaussian S-transform is unable to obtain uniform time and frequency resolution for all frequency components. In this paper, a novel window based on the $$t$$ student distribution is proposed for the S-transform to achieve a more uniform resolution. Simulation results show that the S-transform with the proposed window provides in comparison with the Gaussian window a more uniform resolution for the entire time and frequency range. The result is suitable for applications such as spectrum sensing.

A novel method of filtration by the discrete heap transforms

Abstract: In this paper, we describe the method of filtering the frequency components of the signals and images, by using the discrete signal-induced heap transforms (DsiHT), which are composed by elementary rotations or Givens transformations. The transforms are fast, because of a simple form of decomposition of their matrices, and they can be applied for signals of any length. Fast algorithms of calculation of the direct and inverse heap transforms do not depend on the length of the processed signals. Due to construction of the heap transform, if the input signal contains an additive component which is similar to the generator, this component is eliminated in the transform of this signal, while preserving the remaining components of the signal. The energy of this component is preserved in the first point, only. In particular case, when such component is the wave of a given frequency, this wave is eliminated in the heap transform. Different examples of the filtration over signals and images by the DsiHT are described and compared with the known method of the Fourier transform.

Robust Causality Check for Sampled Scattering Parameters via a Filtered Fourier Transform

Abstract: We introduce a robust numerical technique to verify the causality of sampled scattering parameters given on a finite bandwidth. The method is based on a filtered Fourier transform and includes a rigorous estimation of the errors caused by missing out-of-band samples. Compared to existing techniques, the method is simpler to implement and provides a useful insight on the time-domain characteristics of the violation. Through an applicative example, we shows its usefulness to improve the accuracy of macromodeling techniques used to convert sampled scattering parameters into transient simulation models.

A new improved Synchrosqueezing Transform based on adaptive short time fourier transform

Abstract: The vibration signals during the run-up and run-down periods of rotating machinery are nonstationary. The normal types of time frequency (TF) analysis algorithms have been widely used in the nonstationary condition. But those methods show imperfection for drastically changing vibration signal. Synchrosqueezing Transform (SST) is an adaptive and invertible transform. However, the original SST method is not suitable for vibration signals with abruptly instantanous frequency (IF). In this paper, SST based on adaptive short time Fourier transform(ASTFT) is proposed to improve the time resolution of drastically changing vibration signal. Compared with Short Time Fourier Transform (STFT), SST and Generalized Synchrosqueezing Transform (GST) based on STFT, SST based on ASTFT is of good performance with high computing speed. Experiments were conducted by using SQI motor simulator to validate the proposed method. The vibration signals of a motor with fault were collected during the run-down process and analysed by the proposed method.The results showed that the fault feature frequency of the motor can be accurately captured, which suggests that the proposed approach is effective for analyzing the actute variation signals.

Efficient Iterative Estimation of the Parameters of a Damped Complex Exponential in Noise

Abstract: The estimation of the frequency and decay factor of a single decaying exponential in noise is a problem of prime importance. A popular estimation scheme uses the computationally efficient implementation of the Discrete Fourier transform, the FFT, to obtain a coarse estimate which is then improved by a fine estimation stage. Such estimators, however, show a performance that degrades and departs from the Cramer–Rao Lower Bound (CRLB) as the number of samples increases. To overcome this problem, we propose an iterative, exponentially windowed algorithm. We derive the new estimator’s theoretical performance and study its behavior under different decay rates of the window. We show that the estimator has excellent performance that tracks the CRLB with increasing number of samples if the window decay rate is appropriately set.

Fast and accurate simulation of the turbulent phase screen using fast Fourier transform

Abstract: This work describes an accurate method for simulating turbulent phase screens. The phase screen is divided into a fast Fourier transform (FFT)-based screen and a tilt screen. The simulation of the FFT-based screen is different from that of the standard method. In the simulation, the discrete power spectrum of the turbulence is obtained from the discrete Fourier transform of the phase autocorrelation matrix, not from the theoretical power spectrum. This method avoids the drawbacks of the undersampling of the low frequency and high frequency components which occurs in the standard FFT-based method. The maximum error in the phase structure function can be reduced to <0.13% , and the additional execution time increases by only several percents. This method is only suitable for square screens.

Numerical Approximation of Probability Mass Functions via the Inverse Discrete Fourier Transform

Abstract: First passage distributions of semi-Markov processes are of interest in fields such as reliability, survival analysis, and many others. Finding or computing first passage distributions is, in general, quite challenging. We take the approach of using characteristic functions (or Fourier transforms) and inverting them to numerically calculate the first passage distribution. Numerical inversion of characteristic functions can be unstable for a general probability measure. However, we show they can be quickly and accurately calculated using the inverse discrete Fourier transform for lattice distributions. Using the fast Fourier transform algorithm these computations can be extremely fast. In addition to the speed of this approach, we are able to prove a few useful bounds for the numerical inversion error of the characteristic functions. These error bounds rely on the existence of a first or second moment of the distribution, or on an eventual monotonicity condition. We demonstrate these techniques with two examples.

Non-Uniform FFT and Its Applications in Particle Simulations

Abstract: Ewald summation method, based on Non-Uniform FFTs (ENUF) to compute the electrostatic interactions and forces, is implemented in two different particle simulation schemes to model molecular and soft matter, in classical all-atom Molecular Dynamics and in Dissipative Particle Dynamics for coarse-grained particles. The method combines the traditional Ewald method with a non-uniform fast Fourier transform library (NFFT), making it highly efficient. It scales linearly with the number of particles as () log  NN , while being both robust and accurate. It conserves both energy and the momentum to float point accuracy. As demonstrated here, it is straightforward to implement the method in existing computer simulation codes to treat the electrostatic interactions either between point-charges or charge distributions. It should be an attractive alternative to mesh-based Ewald methods.

Robust Digital Image Reconstruction via the Discrete Fourier Slice Theorem

Abstract: The discrete Fourier slice theorem is an important tool for signal processing, especially in the context of the exact reconstruction of an image from its projected views. This paper presents a digital reconstruction algorithm to recover a two dimensional (2-D) image from sets of discrete one dimensional (1-D) projected views. The proposed algorithm has the same computational complexity as the 2-D fast Fourier transform and remains robust to the addition of significant levels of noise. A mapping of discrete projections is constructed to allow aperiodic projections to be converted to projections that assume periodic image boundary conditions. Each remapped projection forms a 1-D slice of the 2-D Discrete Fourier Transform (DFT) that requires no interpolation. The discrete projection angles are selected so that the set of remapped 1-D slices exactly tile the 2-D DFT space. This permits direct and mathematically exact reconstruction of the image via the inverse DFT. The reconstructions are artefact free, except for projection inconsistencies that arise from any additive and remapped noise. We also present methods to generate compact sets of rational projection angles that exactly tile the 2-D DFT space. The improvement in noise suppression that comes with the reconstruction of larger sized images needs to be balanced against the corresponding increase in computation time.