Showing papers on "Discrete-time Fourier transform published in 2018"
TL;DR: A fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT) is proposed, which is essentially the FFT, and is competitive with state-of-the-art algorithms.
Abstract: By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast and quasi-optimal algorithm for computing the NUDFT based on the fast Fourier transform (FFT). Our key observation is that an NUDFT and DFT matrix divided entry by entry is often well approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.
44 citations
TL;DR: An efficient and accurate numerical method for approximations of the coupled Schrodinger–Boussinesq (SBq) system based on the applications of a time-splitting Fourier spectral method and the utilizations of exponential wave integrator Fourier pseudospectral for spatial derivatives in the Boussineq-like equation.
25 citations
01 Jan 2018
TL;DR: In this chapter, fast algorithms for the computation of the DFT for d-variate nonequispaced data are described, since in a variety of applications the restriction to equispacedData is a serious drawback.
Abstract: In this chapter, we describe fast algorithms for the computation of the DFT for d-variate nonequispaced data, since in a variety of applications the restriction to equispaced data is a serious drawback. These algorithms are called nonequispaced fast Fourier transforms and abbreviated by NFFT.
21 citations
TL;DR: In this article, the Discrete Taylor-Fourier Transform (DTFT) is used to analyze power oscillating signals, and the impulse responses of low-pass differentiators, defined by a family of O-splines, are modulated.
Abstract: Power oscillating signals are analyzed with the Discrete Taylor–Fourier Transform (DTFT). This is implemented by modulating the impulse responses of its low-pass differentiators, defined by a family of O-splines. This implementation reduces its computational complexity since in practice only a small subset of filters is applied. The estimated parameters provide richer dynamic information than the traditional methods, in particular, a space representation for each dynamic component, and detection of frequency modulated events. Their estimation performance is assessed through the new proposed Total Phasor Error. To illustrate its application, and its progressive accuracy, this technique is applied to observe the synchrophasor estimates of voltages and to separate the electromechanical modes of an oscillation in a real power system. In conclusion, this multiresolution technique provides a series of increasingly precise solutions for time-frequency separation of oscillations with fluctuating frequency.
17 citations
01 Jan 2018
TL;DR: First, the real exponential is used to explain the concept of transform in reducing the multiplication operation into much simpler addition operation and the relation between the exponential and sinusoidal signals is established.
Abstract: First, the real exponential is used to explain the concept of transform in reducing the multiplication operation into much simpler addition operation Then, the complex exponential with a pure imaginary exponent is introduced and the relation between the exponential and sinusoidal signals is established It is pointed out that the DFT gives a complex exponential polynomial representation of a signal The orthogonality property of the complex exponentials is used to derive the definitions of the DFT and the IDFT The least squares error criterion of signal representation is established Several examples of computing the DFT and IDFT are presented using the matrix form of the DFT and the IDFT An image processing application concludes the chapter
13 citations
TL;DR: An improved method to detect the presence of such noncoherently sampled signals as well as an iterative algorithm to obtain accurate approximations of all the frequency components with their accompanying amplitudes and phase angles is proposed.
Abstract: Time-domain near-field scanning is gaining more and more interest within EMC engineering to analyze electromagnetic near-fields of, eg, quasi-stationar devices When using a digital oscilloscope to scan the near-fields of an electronic device, the oscilloscope measures time-domain signals that comprise in most cases a large number of frequency components For many of these components, noncoherent sampling occurs, resulting in spectral leakage when calculating the frequency spectrum of the time-domain signals with the discrete Fourier transform This paper proposes an improved method to detect the presence of such noncoherently sampled signals as well as an iterative algorithm to obtain accurate approximations of all the frequency components with their accompanying amplitudes and phase angles The algorithm excels over existing algorithms in obtaining these values especially in situations where several sinusoidal components are close to each other in the spectrum This is achieved, thanks to an iterative process of removing the influence of the multiple sinusoidal components on each other This paper contains the mathematical description of the algorithm and a numerical example evaluating the accuracy of the algorithm The algorithm has a higher accuracy than the existing approaches, eg, multipoint Interpolated Discrete Fourier Transform (IpDFTs), with only a slight increase of the computational cost
11 citations
18 Dec 2018
TL;DR: In this paper, the authors derived four interlinking formulas from four definitions of the Fourier transform pure symbolically and specified three conditions for the validity of these formulas in the tempered distributions sense.
Abstract: In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.
11 citations
TL;DR: In this paper, the amplitude and the phase parameters of a noisy real-valued sinewave by means of the two-point Interpolated Discrete-Time Fourier Transform (IpDTFT) algorithms based on the Maximum Sidelobe Decay (MSD) windows are analyzed.
11 citations
TL;DR: A novel method for parameter estimation of Newton’s rings is proposed based on concise fractional Fourier transform (CFRFT) and evaporation rate based water cycle algorithm (ER-WCA), and a pretreatment technology is further proposed to solve uneven illumination of actual Newton's rings.
10 citations
TL;DR: Experimental results show that the proposed discrete trinion Fourier transform (DTFT) is not only more robust in most standard attack situations, but also has superior time efficiency.
Abstract: A discrete trinion Fourier transform (DTFT) is proposed and computation of the DTFT is derived from the discrete Fourier transform. A block-based color image watermarking scheme is designed based on the proposed DTFT. The watermark bits are embedded by modifying the DTFT coefficients of each block by means of the quantization index modulation method. Finally, assessment of this watermarking approach is performed in terms of watermark capacity, invisibility, and robustness. A comparison with a quaternion discrete Fourier transform is also provided. Experimental results show that the proposed method is not only more robust in most standard attack situations, but also has superior time efficiency.
10 citations
TL;DR: A theory for generalized shift-invariant and sampling spaces associated with the fractional Fourier transform is developed and the numerical results validate the theoretical derivations.
03 Oct 2018
TL;DR: The impulse-invariant method approximation in the time domain Bilinear transformation Frequency transformation for digital filters Recursive versus nonrecursive design RANDOM VARIABLES, SEQUENCES, and POWER SPECTRA DENSITIES Random signals and distributions Averages Stationary processes Special random signals and probability density functions Wiener-Kintchin relations Filtering random processes Nonparametric spectra estimation
Abstract: SIGNALS AND THEIR FUNCTIONAL REPRESENTATION Some applications involving signals Fundamental representation of simple time signals Signal conditioning and manipulation Representation of signals Appendix 1: Elementary matrix algebra Appendix 2: Complex numbers Appendix 1 Problems Appendix 2 Problems LINEAR CONTINUOUS-TIME SYSTEMS Properties of systems Modeling simple continuous systems Solutions of first-order systems Evaluation of integration constants: initial conditions Block diagram representation Convolution and correlation of continuous-time signals Impulse response DISCRETE SYSTEMS Discrete systems and equations Digital simulation of analog systems Digital simulation of higher-order differential equations Convolution of discrete-time signals Appendix 1: Method of variation of parameters Appendix 2: Euler's approximation for differential equations PERIODIC CONTINUOUS SIGNALS AND THEIR SPECTRUMS Complex functions Fourier series of continuous functions Features of periodic continuous functions Linear systems with periodic inputs NONPERIODIC SIGNALS AND THEIR FOURIER TRANSFORM Direct and inverse Fourier transform Properties of Fourier transforms Some special Fourier transform pairs Effects of truncation and Gibbs' phenomenon Linear time-invariant filters Appendix SAMPLING OF CONTINUOUS SIGNALS Fundamentals of sampling The sampling theorem DISCRETE-TIME TRANSFORMS Discrete-time Fourier transform (DTFT) Summary of DTFT properties DTFT of finite time sequences Frequency response of linear time-invariant (LTI) discrete systems The discrete Fourier transform (DFT) Summary of the DFT properties Multirate digital signal processing Appendix 1: Proofs of the DTFT properties Appendix 2: Proofs of DFT properties Appendix 3: Fast Fourier transform (FFT) LAPLACE TRANSFORM One-sided Laplace transform Summary of the Laplace transform properties Systems analysis: transfer functions of LTI systems Inverse Laplace transform (ILT) Problem solving with Laplace transform Frequency response of LTI systems Pole location and the stability of LTI systems Feedback for linear systems Bode plots Appendix: Proofs of Laplace transform properties THE Z-TRANSFORM, DIFFERENCE EQUATIONS, AND DISCRETE SYSTEMS The z-transform Convergence of the z-transform Properties of the z-transform z-Transform pairs Inverse z-transform Transfer function Frequency response of first-order discrete systems Frequency response of higher-order digital systems z-Transform solution of first-order difference equations Higher-order difference equations Appendix: Proofs of the z-transform properties ANALOG FILTER DESIGN General aspects of filters Butterworth filter Chebyshev low-pass filter Phase characteristics Frequency transformations Analog filter design using MATLAB functions FINITE IMPULSE RESPONSE (FIR) FILTERS Properties of FIR filters FIR filters using the Fourier series approach FIR filters using windows Prescribed filter specifications using a Kaiser window MATLAB FIR filter design INFINITE IMPULSE RESPONSE (IIR) FILTERS The impulse-invariant method approximation in the time domain Bilinear transformation Frequency transformation for digital filters Recursive versus nonrecursive design RANDOM VARIABLES, SEQUENCES, AND POWER SPECTRA DENSITIES Random signals and distributions Averages Stationary processes Special random signals and probability density functions Wiener-Kintchin relations Filtering random processes Nonparametric spectra estimation LEAST SQUARE SYSTEM DESIGN, WIENER FILTER, AND THE LMS FILTER The least-squares technique The mean square error Wiener filtering examples The least mean square (LMS) algorithm Examples using the LMS algorithm APPENDIX A: MATHEMATICAL FORMULAS Trigonometric identities Orthigonality Summation of trigonometric forms Summation formulas Series expansions Logarithms Some definite integrals APPENDIX B: SUGGESTIONS AND EXPLANATIONS FOR MATLAB USE Creating a directory Help Save and load MATLAB as calculator Variable names Complex numbers Array indexing Extracting and inserting numbers in arrays Vectorization Matrices Produce a periodic function Script files Functions Subplots Figures Changing the scales of the axes of a figure Writing Greek letters Subscripts and superscripts Lines in plots INDEX Each chapter features Important Definitions and Concepts as well as Problems.
TL;DR: In this article, the authors extend the Fast Fourier Transform to a multivariate setting by considering its application to a particular model that includes losses of different types and dependency between them, and they also discuss a suitable change of measure called exponential tilting that forces the tail of the distribution to decrease at exponential rate.
Abstract: The Fast Fourier Transform provides an alternative approximate method to evaluate the distribution of aggregate losses in insurance and finance. The efficiency of this method has already been proved for univariate and bivariate insurance models; therefore, in this paper, we extend it to a multivariate setting by considering its application to a particular model that includes losses of different types and dependency between them. Since the Fourier transform method works with truncated claims distributions, it can generate aliasing errors by wrapping around the probability mass that lies at the truncation point below this point. To eliminate this problem, we also discuss a suitable change of measure called exponential tilting that forces the tail of the distribution to decrease at exponential rate. Other possible errors are also discussed. We also illustrate the method on several numerical examples.
TL;DR: The algorithm of SFT (sparse Fourier transform) is firstly used for monochromatic light spectrum reconstruction and two methods of the modern spectrum estimation, AR (Auto-Regressive) model and MUSIC (Multiple Signal Classification) are considered as high resolution spectrum estimation algorithms.
TL;DR: The proposed Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger (KGS) equation is constructed and shown to be convergent with the convergence order of in the discrete norm afterwards.
Abstract: In this paper, we focus on constructing and analyzing a new Fourier pseudo-spectral conservative scheme for the Klein-Gordon-Schrodinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can be cast into a canonical finite-dimensional Hamiltonian form. Then, an energy-preserving and charge-preserving scheme is constructed by using the symmetric discrete gradient method. Based on the discrete conservation laws and the equivalence of the semi-norm between the Fourier pseudo-spectral method and the finite difference method, the pseudo-spectral solution of the proposed scheme is proved to be bounded in the discrete $L_{\infty}$ norm. The proposed scheme is shown to be convergent with the convergence order of $\mathcal{O}(J^{-r} + \tau^2)$ in the discrete $L_2$ norm afterwards, where J is the number of nodes and τ is the time step size. Numerical experime...
TL;DR: An explicit formula is given for the element of the inverse where is a band Toeplitz matrix with left bandwidth s and right bandwidth r, whose elements are the discrete Fourier transform of where f is the symbol of .
Abstract: In this paper, we give an explicit formula for the element of the inverse where is a band Toeplitz matrix with left bandwidth s and right bandwidth r. The formula involves determinants, , whose elements are the discrete Fourier transform of where f is the symbol of .
01 Jan 2018
TL;DR: The discrete-time Fourier series (DTFS) expansion of a periodic sequence is obtained and it is shown that the DTFS represents the DFT of a finite length sequence.
Abstract: The DTFT of a discrete-time signal is a continuous function of the frequency (\( \omega \)), and hence, the relation between \( X\left( {\text{e}^{{j}\omega } } \right) \) and \( x(n) \) is not a computationally convenient representation. However, it is possible to develop an alternative frequency representation called the discrete Fourier transform (DFT) for finite duration sequences. The DFT is a discrete-time sequence with equal spacing in frequency. We first obtain the discrete-time Fourier series (DTFS) expansion of a periodic sequence. Next, we define the DFT of a finite length sequence and consider its properties in detail. We also show that the DTFS represents the DFT of a finite length sequence. Further, evaluation of linear convolution using the DFT is discussed. Finally, some fast Fourier transform (FFT) algorithms for efficient computation of DFT are described.
01 Jan 2018
TL;DR: The DTFT of a discrete periodic signal and the relation between the DFT and the DTFT spectra are established in this paper, and the properties of DTFT are derived and examples are given.
Abstract: The DTFT version of the Fourier analysis is presented. The DTFT primarily represents aperiodic discrete signals by a continuous periodic spectrum. The DTFT definition is derived starting from the DFT definition. Examples of determining the DTFT of signals are given. The DTFT of a discrete periodic signal and the relation between the DFT and the DTFT spectra are established. The properties of the DTFT are derived and examples are given. Typical applications of the DTFT are presented. The numerical approximation of the DTFT by the DFT concludes the chapter.
TL;DR: In this article, a lecture demonstration of convolution and correlation between two spatial signals using the Fourier transform tool is presented, where both simulation and optical experiments are possible using a variety of object transparencies.
Abstract: Convolution operation is indispensable in studying analog optical and digital signal processing. Equally important is the correlation operation. The time domain community often teaches convolution and correlation only with one dimensional time signals. That does not clearly demonstrate the effect of convolution and correlation between two signals. Instead if we consider two dimensional spatial signals, the convolution and correlation operations can be very clearly explained. In this paper, we propose a lecture demonstration of convolution and correlation between two spatial signals using the Fourier transform tool. Both simulation and optical experiments are possible using a variety of object transparencies. The demonstration experiments help to clearly explain the similarity and the difference between convolution and correlation operations. This method of teaching using simulation and hands-on experiments can stimulate the curiosity of the students. The feedback of the students, in my class teaching, has been quite encouraging.
Patent•
20 Dec 2018TL;DR: In this paper, a sliding windowed infinite Fourier transform (SWIFT) algorithm for digital signal processing is described. And the SWIFT algorithm is stable and allows for improved computational efficiency, improved frequency resolution, improved sampling, reduced memory requirements, and reduced spectral leakage.
Abstract: Systems and methods for digital signal processing using a sliding windowed infinite Fourier transform (“SWIFT”) algorithm are described. A discrete-time Fourier transform (“DTFT”) of an input signal is computed over an infinite-length temporal window that is slid from one sample in the input signal to the next. The DTFT with the temporal window at a given sample point is effectively calculated by phase shifting and decaying the DTFT calculated when the temporal window was positioned at the previous sample point and adding the current sample to the result. The SWIFT algorithms are stable and allow for improved computational efficiency, improved frequency resolution, improved sampling, reduced memory requirements, and reduced spectral leakage.
01 Jan 2018
TL;DR: In this paper, the importance of the z-transform in the analysis of LTI systems is established, and the solution of state-space equations of discrete-time LTI system is presented.
Abstract: The DTFT may not exist for all sequences due to the convergence condition, whereas the z-transform exists for many sequences for which the DTFT does not exist. Also, the z-transform allows simple algebraic manipulations. As such, the z-transform has become a powerful tool in the analysis and design of digital systems. This chapter introduces the z-transform, its properties, the inverse z-transform, and methods for finding it. Also, in this chapter, the importance of the z-transform in the analysis of LTI systems is established. Further, one-sided z-transform and the solution of state-space equations of discrete-time LTI systems are presented. Finally, transformations between continuous-time systems and discrete-time systems are discussed.
Patent•
01 Feb 2018
TL;DR: In this paper, the received signal is split into a plurality of processing sub-channels, each corresponding to one or more hop frequencies, and within each sub-channel: i) subtracting any sub-carrier frequency from the receiving signal; ii) filtering the signal from (i) using a chip-matched filter; iii) selecting a sub-set of samples from the filtered signal; iv) correlating the sampled signal from step (iii) with a known reference signal to produce at least one correlator output.
Abstract: A method for the reception of a frequency hopped direct sequence spread spectrum signal comprises acquiring the signal by splitting the received signal into a plurality of processing sub-channels, each corresponding to one or more hop frequencies; and within each sub-channel: i) subtracting any sub-carrier frequency from the received signal; ii) filtering the signal from (i) using a chip-matched filter; iii) selecting a sub-set of samples from the filtered signal; iv) correlating the sampled signal from step (iii) with a known reference signal to produce at least one correlator output. The output(s) from each sub-channel are provided to an input of a corresponding one or more common discrete time Fourier transforms (DTFT), and an output therefrom having a peak above a predetermined threshold is selected for further processing in the receiver. Once acquired, the method provides for a less computationally expensive way of tracking the signal. The method, and system implementing the method, may be used for demodulating appropriately modulated signals, and has particular application in satellite navigation receivers.
01 Jan 2018
TL;DR: In this chapter, the importance of the z-transform in the analysis of LTI systems is established and methods for finding it are introduced.
Abstract: The DTFT may not exist for all sequences due to the convergence condition, whereas the z-transform exists for many sequences for which the DTFT does not exist. Also, the z-transform allows simple algebraic manipulations. As such, the z-transform has become a powerful tool in the analysis and design of digital systems. This chapter introduces the z-transform, its properties, the inverse z-transform, and methods for finding it. Also, in this chapter, the importance of the z-transform in the analysis of LTI systems is established.
01 Jan 2018
TL;DR: Fast computation of the 1-D and 2-D linear convolution and correlation operations by using the DFT is presented and the convolution of long sequences using the overlap–save method along with the D FT is explained.
Abstract: Fast computation of the 1-D and 2-D linear convolution and correlation operations by using the DFT is presented. Implementing the convolution of long sequences using the overlap–save method along with the DFT is explained. Some applications, using the DFT to implement the convolution and correlation operations, in image processing and digital communications applications are presented in detail.
Patent•
16 Jul 2018
TL;DR: In this paper, a standard for designing an efficient filter in a QAM-FBMC system, a filter suitable for a multi-path fading channel environment, and a design method for the same is presented.
Abstract: The present invention provides a standard for designing an efficient filter in a QAM-FBMC system, a filter suitable for a multi-path fading channel environment, and a design method for the same. According to the present invention, the design method for a filter considering regionality of a time area in a filter bank-based multi-carrier system includes: a step of applying the filter applied with a movement of frequencies by the filter in each of multiple sub-carriers as a pulse forming filter, matching each filter by reception filtering, and defining a whole pulse shape; a step of defining self-SIR of QAM-FBMC transmission signal on a normalized value by calculating total interference power in the multiple sub-carriers; a step of determining power distribution density (PSD) of the QAM-FBMC transmission signal by discrete time Fourier transform (DTFT) of the filter and determining relation of a fall-off rate of the DTFT of the filter and a filter coefficient by differentiability of continuity of the filter; and a step of performing regionalization in a frequency area by the fall-off rate and a tap of the filter and performing the regionalization in the time area using normalized time and energy.
01 Jan 2018
TL;DR: The Fourier transform has arguably had the most significant impact over the course of the 20th century as discussed by the authors, and it is the most widely used numerical method in the world today.
Abstract: Of all the numerical methods we have seen so far, the Fourier transform has arguably had the most significant impact over the course of the 20th century.
Patent•
18 Dec 2018
TL;DR: In this article, the authors proposed a fast discrete-time Fourier transform method for parallel iterative processing in which the correlation of the time-domain signal sequence of the batch before and after, can avoid a large number of unnecessary redundant calculations, and remarkably improves the real-time performance of the DFT.
Abstract: The invention belongs to the technical field of signal and information processing and relates to a fast discrete-time Fourier transform method for parallel iterative processing In the discrete-time Fourier transform of the time-domain signal sequence intercepted by the sliding window mode, the invention fully utilizes the correlation of the time-domain signal sequence of the batch before and after, can avoid a large number of unnecessary redundant calculations, and remarkably improves the real-time performance of the discrete-time Fourier transform Compared with the common fast Fourier transform method, the speed increase multiple of the fast discrete-time Fourier transform method of the parallel iterative processing of the present invention becomes larger with the increase of the data length, so that the discrete-time Fourier transform can be more efficiently completed, and further the real-time requirement of the signal processing can be more satisfied
01 Aug 2018
Patent•
26 Jan 2018
TL;DR: In this paper, a multifrequency interpolation iteration frequency estimation method based on all phase spectrum analysis, and an estimator is presented, which comprises the following steps that: 1) carrying out all phase FFT (Fast Fourier Transform) spectrum analysis processing on an input signal, and searching a peak value spectrum position; 2) calculating the all phase DTFT (Discrete Time Fourier transform) amplitudes of two-side frequency points of the peak value spectrums, carrying out interpolation iterations, and obtaining a frequency estimation result; 3) judging whether the relative difference
Abstract: The invention discloses a multifrequency interpolation iteration frequency estimation method based on all phase spectrum analysis, and an estimator. The method comprises the following steps that: 1) carrying out all phase FFT (Fast Fourier Transform) spectrum analysis processing on an input signal, and searching a peak value spectrum position; 2) calculating the all phase DTFT (Discrete Time Fourier Transform) amplitudes of two-side frequency points of the peak value spectrums, carrying out interpolation iteration, and obtaining a frequency estimation result; 3) judging whether the relative difference of two all phase DTFT spectrum values meets an iteration termination condition or not; and 4) if the relative difference of two all phase DTFT spectrum values does not meet the iteration termination condition, calculating frequency offset, regulating the positions of the peak value spectrums, and repeating the steps 2)-3); and if the relative difference of two all phase DTFT spectrum values meets the iteration termination condition, finishing iteration, and outputting a frequency estimation result. The estimator comprises that a simulated input signal is sampled, a sampled digital signal is segmented, each section of data and filter coefficients are stored into an external RAM (Random Access Memory), a DSP (Digital Signal Processor) carries out all phase FFT, all phase DTFT and interpolation iteration processing on the above input data, frequency is estimated, and finally, a frequency value is displayed in virtue of an output driver and a display module thereof.