Discrete-time Fourier transform

About: Discrete-time Fourier transform is a research topic. Over the lifetime, 5072 publications have been published within this topic receiving 144643 citations. The topic is also known as: DTFT.

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.

New digital filter for the analysis of experimental data

Abstract: The extremely simple mathematical technique called ’’straightening through smoothing,’’ which is a numerical frequency filter, is generalized in order to provide a transmission function having any shape. This frequency filter requires such a small memory that it can be performed using a minicomputer or even a programmable hand held calculator and the number of channels used is not limited to a power of 2, as in the case of the fast Fourier transform. For some filtering functions the number of operations required is smaller than with the fast Fourier transform.

Discrete and continuous fourier transforms : analysis, applications and fast algorithms

Abstract: Preface Fundamentals, Analysis, and Applications Analytical and Graphical Representation of Function Contents Time and Frequency Contents of a Function The Frequency-Domain Plots as Graphical Tools Identifying the Cosine and Sine Modes Using Complex Exponential Modes Using Cosine Modes with Phase or Time Shifts Periodicity and Commensurate Frequencies Review of Results and Techniques Expressing Single Component Signals General Form of a Sinusoid in Signal Application Fourier Series: A Topic to Come Terminology Sampling and Reconstruction of Functions-Part I DFT and Band-Limited Periodic Signal Frequencies Aliased by Sampling Connection: Anti-Aliasing Filter Alternate Notations and Formulas Sampling Period and Alternate Forms of DFT Sample Size and Alternate Forms of DFT The Fourier Series Formal Expansions Time-Limited Functions Even and Odd Functions Half-Range Expansions Fourier Series Using Complex Exponential Modes Complex-Valued Functions Fourier Series in Other Variables Truncated Fourier Series and Least Squares Orthogonal Projections and Fourier Series Convergence of the Fourier Series Accounting for Aliased Frequencies in DFT DFT and Sampled Signals Deriving the DFT and IDFT Formulas Direct Conversion between Alternate Forms DFT of Concatenated Sample Sequences DFT Coefficients of a Commensurate Sum Frequency Distortion by Leakage The Effects of Zero Padding Computing DFT Defining Formulas Per Se Sampling and Reconstruction of Functions-Part II Sampling Nonperiodic Band-Limited Functions Deriving the Fourier Transform Pair The Sine and Cosine Frequency Contents Tabulating Two Sets of Fundamental Formulas Connections with Time/Frequency Restrictions Fourier Transform Properties Alternate Form of the Fourier Transform Computing the Fourier Transform Computing the Fourier Coefficients Sampling and Reconstruction of Functions-Part III Impulse Functions and Their Properties Generating the Fourier Transform Pairs Convolution and Fourier Transform Periodic Convolution and Fourier Series Convolution with the Impulse Function Impulse Train as a Generalized Function Impulse Sampling of Continuous-Time Signals Nyquist Sampling Rate Rediscovered Sampling Theorem for Band-Limited Signal Sampling of Band-Pass Signals The Fourier Transform of a Sequence Deriving the Fourier Transform of a Sequence Properties of the Fourier Transform of a Sequence Generating the Fourier Transform Pairs Duality in Connection with the Fourier Series The Fourier Transform of a Periodic Sequence The DFT Interpretation The Discrete Fourier Transform of a Windowed Sequence A Rectangular Window of Infinite Width A Rectangular Window of Appropriate Finite Width Frequency Distortion by Improper Truncation Windowing a General Nonperiodic Sequence Frequency-Domain Properties of Windows Applications of the Windowed DFT Discrete Convolution and the DFT Linear Discrete Convolution Periodic Discrete Convolution The Chirp Fourier Transform Applications of the DFT in Digital Filtering and Filters The Background Application-Oriented Terminology Revisit Gibbs Phenomenon from the Filtering Viewpoint Experimenting with Digital Filtering and Filter Design Fast Algorithms Index Mapping and Mixed-Radix FFTs Algebraic DFT versus FFT-Computed DFT The Role of Index Mapping The Recursive Equation Approach Other Forms by Alternate Index Splitting Kronecker Product Factorization and FFTs Reformulating the Two-Factor Mixed-Radix FFT From Two-Factor to Multifactor Mixed-Radix FFT Other Forms by Alternate Index Splitting Factorization Results by Alternate Expansion Unordered FFT for Scrambled Input Utilities of the Kronecker Product Factorization The Family of Prime Factor FFT Algorithms Connecting the Relevant Ideas Deriving the Two-Factor PFA Matrix Formulation of the Two-Factor PFA Matrix Formulation of the Multifactor PFA Number Theory and Index Mapping by Permutations The In-Place and In-Order PFA Efficient Implementation of the PFA On Computing the DFT of Large Prime Length Performance of FFT for Prime N Fast Algorithm I: Approximating the FFT Fast Algorithm II: Using Bluestein's FFT Bibliography Index

Random Discrete Fractional Fourier Transform

Abstract: In this letter, a new commuting matrix with random discrete Fourier transform (DFT) eigenvectors is first constructed. A random discrete fractional Fourier transform (RDFRFT) kernel matrix with random DFT eigenvectors and eigenvalues is then proposed. The RDFRFT has an important feature that the magnitude and phase of its transform output are both random. As an application example, a security-enhanced image encryption scheme based on the RDFRFT is illustrated.

Fourier transform methods for calculating action variables and semiclassical eigenvalues for coupled oscillator systems

Abstract: Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie–Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon–Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates Q1±iQ2. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three‐quarters of the dissociation energy).