Showing papers on "Discrete-time Fourier transform published in 2015"

Power System Harmonics

Abstract: The French mathematician J. B. J. Fourier showed that arbitrary periodic functions could be represented by an infinite series of sinusoids of harmonically related frequencies. This chapter first defines periodic functions and orthogonal functions. A periodic function can be expanded in a Fourier series. The Fourier series of a periodic function is the sum of sinusoidal components of different frequencies. The chapter then illustrates the functions of odd or skew symmetry, even symmetry and half-wave symmetry. The odd and even symmetry has been obtained with the triangular function by shifting the origin. Fourier analysis of a continuous periodic signal in the time domain gives a series of discrete frequency components in the frequency domain. The chapter describes Dirichlet conditions and notion of power spectrum. Finally, it explains the function of convolution, which is generally carried out in the frequency domain.

Sparse Phase Retrieval from Short-Time Fourier Measurements

Abstract: We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the short-time Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements.

Direction Finding by Time-Modulated Array With Harmonic Characteristic Analysis

Abstract: A novel direction-finding method by the time-modulated array (TMA) is proposed through analyzing the harmonic characteristic of received signal, which requires only two antenna elements and a single RF channel. The signal processing of the proposed method is concise, and its calculation amount concentrates on a two-point discrete Fourier transform (DFT). Numeric simulations are provided to examine the performance of the proposed method, and a simple S band two-element TMA is constructed and tested to verify its effectiveness.

Low-Complexity Least-Squares Dynamic Synchrophasor Estimation Based on the Discrete Fourier Transform

Abstract: In this paper, the expressions for the phasor parameter estimates returned by the Taylor-based weighted least-squares (TWLS) approach, achieved using either complex-valued or real-valued variables, are derived. In particular, the TWLS phasor estimator and its derivatives are expressed as weighted sums of the discrete-time Fourier transform (DTFT) of the analyzed waveform and its derivatives. The derived expressions show that the TWLS algorithm is sensitive to lower order harmonics and interharmonics located close to the waveform frequency when few waveform cycles are analyzed. Also, the algorithm sensitivity to wideband noise is explained. The relationship between the TWLS phasor estimator and the waveform DTFT is then specifically analyzed when either a static or a second-order dynamic phasor model is assumed. Moreover, a simple and accurate procedure for evaluating the TWLS estimator of the dynamic phasor parameters is proposed. The derived expressions for the real-valued version are then approximated in order to reduce the required computational burden so as to achieve the simplified TWLS (STWLS) procedure. That procedure can be advantageously employed in real-time low-cost applications when the reference frequency used in the TWLS approach is estimated in runtime to improve estimation accuracy. Finally, computer simulations show that the phasor parameter estimates returned by the STWLS procedure when the waveform frequency is estimated by the interpolated discrete Fourier transform method comply with the M-class of performance if an appropriate number of waveform cycles is considered.

A Phase-Angle Estimation Method for Synchronization of Grid-Connected Power-Electronic Converters

Abstract: This paper proposes a positive-sequence phase-angle estimation method based on discrete Fourier transform for the synchronization of three-phase power-electronic converters under distorted and variable-frequency conditions. The proposed method is designed based on a fixed sampling rate and, thus, it can simply be employed for control applications. First, analytical analysis is presented to determine the errors associated with the phasor estimation using standard discrete Fourier transform in a variable-frequency environment. Then, a robust phase-angle estimation technique is proposed, which is based on a combination of estimated positive and negative sequences, tracked frequency, and two proposed compensation coefficients. The proposed method has one cycle transient response and is immune to harmonics, noises, voltage imbalances, and grid frequency variations. An effective approximation technique is proposed to simplify the computation of the compensation coefficients. The effectiveness of the proposed method is verified through a comprehensive set of simulations in Matlab software. Simulation results show the robust and accurate performance of the proposed method in various abnormal operating conditions.

A Robust Sparse Fourier Transform in the Continuous Setting

Abstract: In recent years, a number of works have studied methods for computing the Fourier transform in sub linear time if the output is sparse. Most of these have focused on the discrete setting, even though in many applications the input signal is continuous and naive discretization significantly worsens the sparsity level. We present an algorithm for robustly computing sparse Fourier transforms in the continuous setting. Let x(t) = x*(t) + g(t), where x* has a k-sparse Fourier transform and g is an arbitrary noise term. Given sample access to x(t) for some duration T, we show how to find a k-Fourier-sparse reconstruction x'(t) with [frac{1}{T}int0T abs{x(t) - x(t)}2 mathrm{d} t lesssim frac{1}{T}int0T abs{g(t)}2 mathrm{d}t. The sample complexity is linear in k and logarithmic in the signal-to-noise ratio and the frequency resolution. Previous results with similar sample complexities could not tolerate an infinitesimal amount of i.i.d. Gaussian noise, and even algorithms with higher sample complexities increased the noise by a polynomial factor. We also give new results for how precisely the individual frequencies of x* can be recovered.

Fast Inverse Nonlinear Fourier Transform For Generating Multi-Solitons In Optical Fiber

Abstract: The achievable data rates of current fiber-optic wavelength-division-multiplexing (WDM) systems are limited by nonlinear interactions between different subchannels. Recently, it was thus proposed to replace the conventional Fourier transform in WDM systems with an appropriately defined nonlinear Fourier transform (NFT). The computational complexity of NFTs is a topic of current research. In this paper, a fast inverse NFT algorithm for the important special case of multi-solitonic signals is presented. The algorithm requires only $\mathcal{O}(D\log^{2}D)$ floating point operations to compute $D$ samples of a multi-soliton. To the best of our knowledge, this is the first algorithm for this problem with $\log^{2}$-linear complexity. The paper also includes a many samples analysis of the generated nonlinear Fourier spectra.

High-Speed Image Registration Algorithm with Subpixel Accuracy

Abstract: A new, fast and computationally efficient lateral subpixel shift registration algorithm is presented. It is limited to register images that differ by small subpixel shifts otherwise its performance degrades. This algorithm significantly improves the performance of the single-step discrete Fourier transform approach proposed by Guizar-Sicairos and can be applied efficiently on large dimension images. It reduces the dimension of Fourier transform of the cross correlation matrix and reduces the discrete Fourier transform (DFT) matrix multiplications to speed up the registration process. Simulations show that our algorithm reduces computation time and memory requirements without sacricing the accuracy associated with the usual FFT approach accuracy.

Recovery of piecewise smooth images from few fourier samples

Abstract: We introduce a Prony-like method to recover a continuous domain 2-D piecewise smooth image from few of its Fourier samples. Assuming the discontinuity set of the image is localized to the zero level-set of a trigonometric polynomial, we show the Fourier transform coefficients of partial derivatives of the signal satisfy an annihilation relation. We present necessary and sufficient conditions for unique recovery of piecewise constant images using the above annihilation relation. We pose the recovery of the Fourier coefficients of the signal from the measurements as a convex matrix completion algorithm, which relies on the lifting of the Fourier data to a structured low-rank matrix; this approach jointly estimates the signal and the annihilating filter. Finally, we demonstrate our algorithm on the recovery of MRI phantoms from few low-resolution Fourier samples.

Multichannel Random Discrete Fractional Fourier Transform

Abstract: We propose a multichannel random discrete fractional Fourier transform (MRFrFT) with random weighting coefficients and partial transform kernel functions First, the weighting coefficients of each channel are randomized Then, the kernel functions, selected based on a choice scheme, are randomized using a group of random phase-only masks (RPOMs) The proposed MRFrFT can be carried out both electronically and optically, and its main features and properties have been given Numerical simulation about one-dimensional signal demonstrates that the MRFrFT has an important feature that the magnitude and phase of its output are both random Moreover, the MRFrFT of two-dimensional image can be viewed as a security enhanced image encryption scheme due to the large key space and the sensitivity to the private keys

High-Precision, Permanently Stable, Modulated Hopping Discrete Fourier Transform

Abstract: A new modulated hopping Discrete Fourier Transform (mHDFT) algorithm which is characterized by its merits of high accuracy and constant stability is presented. The proposed algorithm, which is based on the circular frequency shift property of DFT, directly moves the k-th DFT bin to the position of k = 0, and computes the DFT by incorporating the successive DFT outputs with arbitrary time hop L. Compared to previous works, since the pole of mHDFT precisely settles on the unit circle in the Z-plane, the accumulated errors and potential instabilities, which are caused by the quantization of the twiddle factor, are always eliminated without increasing much computational effort. The numerical simulation results verify the effectiveness and superiority of the proposed algorithm.

Multimodal circular filtering using Fourier series

Abstract: Recursive filtering with multimodal likelihoods and transition densities on periodic manifolds is, despite the compact domain, still an open problem. We propose a novel filter for the circular case that performs well compared to other state-of-the-art filters adopted from linear domains. The filter uses a limited number of Fourier coefficients of the square root of the density. This representation is preserved throughout filter and prediction steps and allows obtaining a valid density at any point in time. Additionally, analytic formulae for calculating Fourier coefficients of the square root of some common circular densities are provided. In our evaluation, we show that this new filter performs well in both unimodal and multimodal scenarios while requiring only a reasonable number of coefficients.

Discrete Fresnel Transform and Its Circular Convolution

Abstract: Discrete trigonometric transformations, such as the discrete Fourier and cosine/sine transforms, are important in a variety of applications due to their useful properties. For example, one well-known property is the convolution theorem for Fourier transform. In this letter, we derive a discrete Fresnel transform (DFnT) from the infinitely periodic optical gratings, as a linear trigonometric transform. Compared to the previous formulations of DFnT, the DFnT in this letter has no degeneracy, which hinders its mathematic applications, due to destructive interferences. The circular convolution property of the DFnT is studied for the first time. It is proved that the DFnT of a circular convolution of two sequences equals either one circularly convolving with the DFnT of the other. As circular convolution is a fundamental process in discrete systems, the DFnT not only gives the coefficients of the Talbot image, but can also be useful for optical and digital signal processing and numerical evaluation of the Fresnel transform.

A new single-phase PLL based on discrete fourier transform

Abstract: A new single-phase synchronous reference frame phase-locked loop (SRF-PLL) is proposed in this paper. The key point to implement single-phase SRF-PLL is how to generate the orthogonal signal accurately, even under polluted grids with harmonics and DC components. In this paper, the generation of the orthogonal signal is achieved by the discrete Fourier transform (DFT) filter. Compared with conventional SRF-PLLs, the DFT-based PLL (DFT-PLL) proposed offers better harmonics immunity and DC component rejection capacity. Besides, the proposed method incorporates grid frequency variations by adjusting the sampling period of DFT according to the estimated frequency. Also, digital implementation of DFT-PLL is simple and straightforward with low computational burden. These advantages make the proposed DFT-PLL especially suitable for harmonically distorted and frequency-varying applications. Experimental results and comparisons with two another widely used SRF-PLLs are given to validate the effectiveness and advantages of the proposed method.

Convolution and Product Theorem for the Special Affine Fourier Transform.

Abstract: The Special Affine Fourier Transform or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Unlike the Fourier transform, the SAFT does not work well with the standard convolution operation. Recently, Q. Xiang and K. Y. Qin introduced a new convolution operation that is more suitable for the SAFT and by which the SAFT of the convolution of two functions is the product of their SAFTs and a phase factor. However, their convolution structure does not work well with the inverse transform in sofar as the inverse transform of the product of two functions is not equal to the convolution of the transforms. In this article we introduce a new convolution operation that works well with both the SAFT and its inverse leading to an analogue of the convolution and product formulas for the Fourier transform. Furthermore, we introduce a second convolution operation that leads to the elimination of the phase factor in the convolution formula obtained by Q. Xiang and K. Y. Qin.

Explicit Hermite-type eigenvectors of the discrete Fourier transform

Abstract: The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions---the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical computations. In this paper we provide a solution to these problems. First, we construct an explicit basis of (nonorthogonal) eigenvectors of the discrete Fourier transform, thus extending the results of [F. N. Kong, IEEE Trans. Circuits Syst. II. Express Briefs, 55 (2008), pp. 56--60]. Applying the Gram--Schmidt orthogonalization procedure we obtain an orthogonal eigenbasis of the discrete Fourier transform. We prove that the first eight eigenvectors converge to the corresponding Hermite functions, and we conjecture that this convergence result remains true for all eigenvectors.

Tensor representation of color images and fast 2D quaternion discrete Fourier transform

Abstract: In this paper, a general, efficient, split algorithm to compute the two-dimensional quaternion discrete Fourier transform (2-D QDFT), by using the special partitioning in the frequency domain, is introduced. The partition determines an effective transformation, or color image representation in the form of 1-D quaternion signals which allow for splitting the N × M-point 2-D QDFT into a set of 1-D QDFTs. Comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given. In particular, a proposed method of calculating the 2r × 2r -point 2-D QDFT uses 18N2 less multiplications than the well-known column-row method and method of calculation based on the symplectic decomposition. The proposed algorithm is simple to apply and design, which makes it very practical in color image processing in the frequency domain.

Fast Fourier transform-based Retinex and alpha-rooting color image enhancement

Abstract: Efficiency in terms of both accuracy and speed is highly important in any system, especially when it comes to image processing. The purpose of this paper is to improve an existing implementation of multi-scale retinex (MSR) by utilizing the fast Fourier transforms (FFT) within the illumination estimation step of the algorithm to improve the speed at which Gaussian blurring filters were applied to the original input image. In addition, alpha-rooting can be used as a separate technique to achieve a sharper image in order to fuse its results with those of the retinex algorithm for the sake of achieving the best image possible as shown by the values of the considered color image enhancement measure (EMEC).

Low Complexity FFT-Based Frequency Offset Estimation for M-QAM Coherent Optical Systems

Abstract: Frequency offset estimation algorithms for M-ary quadrature amplitude modulation based on the maximization of the periodogram of fourth-power samples have been investigated. In this letter, a low complexity estimation algorithm using interpolated discrete Fourier transform approach is introduced to replace the gradient descent (GD) or the chirp $z$ transform (CZT) methods. The numerical simulations and complexity analysis demonstrate that the proposed estimator has much lower computational complexity than that of GD and CZT without performance degradation.

Fast Sparse Period Estimation

Abstract: The problem of estimating the period of a point process from observations that are both sparse and noisy is considered. By sparse it is meant that only a potentially small unknown subset of the process is observed. By noisy it is meant that the subset that is observed, is observed with error, or noise. Existing accurate algorithms for estimating the period require O(N) operations where N is the number of observations. By quantizing the observations we produce an estimator that requires only O(N log N) operations by use of the chirp z-transform or the fast Fourier transform. The quantization has the adverse effect of decreasing the accuracy of the estimator. This is investigated by Monte-Carlo simulation. The simulations indicate that significant computational savings are possible with negligible loss in statistical accuracy.

Discrete Fourier Transform

Abstract: The Fourier transform is an integral of the product of a signal (waveform) to be analyzed and a complex exponential function with an arbitrary frequency (see Eq. ( 2.37)). In theoretical discussions, it is possible to deal with continuous functions. However, since a continuous function requires an infinite number of points to describe, it is not suitable for a numerical analysis using a digital computer.

On the Duality of Discrete and Periodic Functions

Abstract: Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Periodization Theorem on generalized functions. We define discretization and periodization on generalized functions and show that the Fourier transform of periodic functions are discrete functions and, vice versa, the Fourier transform of discrete functions are periodic functions.

Recovering signals from the Short-Time Fourier Transform magnitude

Abstract: The problem of recovering signals from the Short-Time Fourier Transform (STFT) magnitude is of paramount importance in many areas of engineering and physics. This problem has received a lot of attention over the last few decades, but not much is known about conditions under which the STFT magnitude is a unique signal representation. Also, the recovery techniques proposed by researchers are mostly heuristic in nature. In this work, we first show that almost all signals can be uniquely identified by their STFT magnitude under mild conditions. Then, we consider a semidefinite relaxation-based algorithm and provide the first theoretical guarantees for the same. Numerical simulations complement our theoretical analysis and provide many directions for future work.