Non-uniform discrete Fourier transform

About: Non-uniform discrete Fourier transform is a research topic. Over the lifetime, 4067 publications have been published within this topic receiving 123952 citations.

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

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.

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.

Comments on discrete chirp-Fourier transform and its application to chirp rate estimation [with reply]

Abstract: This comment points out that the sample rate of the discrete chirp-Fourier transform (DCFT) proposed by Xia (see ibid., vol.48, p. 3122-33, 2000) is not sufficient to avoid severe "picket-fence" effect and causes some restrictions for its practical applications. By increasing the sample rate and modifying the DCFT definition, a new robust DCFT is proposed. Xia (see ibid., vol.50, no.12, p.3116, 2002) replies by first correcting an error on the analog-to-discrete parameter conversions of a chirp signal. Xia also add that when the chirp rate detection resolution is increased, it is more robust to the chirp rate error. On the other- hand, what is sacrificed by doing so is that the magnitudes of the sidelobes of the transform are increased, which may limit its capability of detecting chirps in a multicomponent or low SNR signal. Therefore, which DCFT needs to be used has to depend on the practical application.

Two-dimensional affine generalized fractional Fourier transform

Abstract: As the one-dimensional (1-D) Fourier transform can be extended into the 1-D fractional Fourier transform (FRFT), we can also generalize the two-dimensional (2-D) Fourier transform. Sahin et al. (see Appl. Opt., vol.37, no. 11, p.2130-41, 1998) have generalized the 2-D Fourier transform into the 2-D separable FRFT (which replaces each variable 1-D Fourier transform by the 1-D FRFT, respectively) and the 2-D separable canonical transform (further replaces FRFT by the canonical transform). Sahin et al., (see Appl. Opt., vol.31, no.23, p.5444-53, 1998), have also generalized it into the 2-D unseparable FRFT with four parameters. In this paper, we introduce the 1-D affine generalized fractional Fourier transform (AGFFT). It has even further extended the 2-D transforms described above. It is unseparable, and has, in total, ten degrees of freedom. We show that the 2-D AGFFT has many wonderful properties, such as the relations with the Wigner distribution, shifting-modulation operation, and the differentiation-multiplication operation. Although the 2-D AGFFT form seems very complex, in fact, the complexity of the implementation will not be more than the implementation of the 2-D separable FRFT. Besides, we also show that the 2-D AGFFT extends many of the applications for the 1-D FRFT, such as the filter design, optical system analysis, image processing, and pattern recognition and will be a very useful tool for 2-D signal processing.