Efficient algorithm for 2-D arithmetic Fourier transform

This article presents an efficient algorithm for the two-dimensional (2-D) arithmetic Fourier transform (AFT) based on the Mobius inversion formula of odd number series. It requires fewer multiplications and has less complexity over previous algorithms. In addition, a technique is proposed to carry out the on-axis Fourier coefficients. A parallel VLSI architecture is developed for the new algorithm     

A generalized Mobius transform and arithmetic Fourier transforms

A general approach to arithmetic Fourier transforms (AFT) is developed. The implementation is based on the concept of killer polynomials and the solution of an arithmetic deconvolution problem pertaining to a generalized Mobius transform. This results in an extension of the Bruns (1903) procedure, valid for all prime numbers, and in an AFT that extracts directly the sine coefficients from the Fourier series     

On factor graphs and the Fourier transform

We introduce the concept of convolutional factor graphs, which represent convolutional factorizations of multivariate functions, just as conventional (multiplicative) factor graphs represent multiplicative factorizations. Convolutional and multiplicative factor graphs arise as natural Fourier transform duals. In coding theory applications, algebraic duality of group codes is essentially an instance of Fourier transform duality. Convolutional factor graphs arise when a code is represented as a sum of subcodes, just as conventional multiplicative factor graphs arise when a code is represented as an intersection of supercodes. With auxiliary variables, convolutional factor graphs give rise to "syndrome realizations" of codes, just as multiplicative factor graphs with auxiliary variables give rise to "state realizations." We introduce normal and co-normal extensions of a multivariate function, which essentially allow a given function to be represented with either a multiplicative or a convolutional factorization, as is convenient. We use these function extensions to derive a number of duality relationships among the corresponding factor graphs, and use these relationships to obtain the duality properties of Forney graphs as a special case.     

有互感和电源的电容耦合电路的量子涨落

在[Optics Letters 28(2003)680]一文中引入了复分数富里哀变换,它不同于通常的两维实分数富里哀变换.在本文中我们从Wigner分布的复形的转动的观点以及从在梯度折射率介质中光的传播的观点分别阐明复分数富里哀变换的定义.新建立的量子力学纠缠态表象把复分数富里哀变换和Wigner分布联系起来.     

Time-varying Fourier transform

A time-varying Fourier transform is defined along with its related power and phase spectra. A convenient recursive technique to compute this transform is also presented.     

A methodology for designing,modifying, and implementing Fourier transform algorithms on various architectures

Fourier transform algorithms are described using tensor (Kronecker) products and an associated class of permutations. Algebraic properties of tensor products and the related permutations are used to derive variants of the Cooley-Tukey fast Fourier transform algorithm. These algorithms can be implemented by translating tensor products and permutations to programming constructs. An implementation can be matched to a specific computer architecture by selecting the appropriate variant. This methodology is carried out for the Cray X-MP and the AT&T DSP32.This work was performed at the Center for Large Scale Computation, Suite 400, 25 West 43rd Street, New York, New York 10036, USA, and was supported by a grant from DARPA/ACMP.     

Square-wave Fourier transform

Unless inconveniently high sampling rates are used, the fast-Fourier-transform (f.f.t.) algorithm, being a trapezoidal integration rule, gives errors in the tails of spectra. A transform with a square-wave kernel, which can be evaluated accurately in a simple manner, is proposed to complement the f.f.t. An example involving measured data is included.     

Computation of the discrete cosine transform using the iterative arithmetic fourier transform

A method of computing the discrete cosine transform using the iterative arithmetic Fourier transform is presented. It overcomes the difficulty of dense, Farey-fraction sampling in the time domain or space domain when using the original arithmetic Fourier transform (AFT) algorithm. Also, dense transform-domain samples are obtained without any interpolation or zero-padding. These dense samples can be advantageously used for accurate parameter estimation or determination of a few principal components. The inverse discrete cosine transform can be efficiently computed from these dense, Farey-fraction transform-domain samples using the original AFT. The resulting structure is suitable for VLSI implementation.     

Integer fast Fourier transform

A concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier transform is introduced. Unlike the fixed-point fast Fourier transform (FxpFFT), the new transform has the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2^N-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the transform, is presented and confirmed by a simulation. The transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power     

微型傅里叶变换光谱仪

采用硅微机械加工技术制造的高分辨率光谱仪不久将从实验室进入商品应用市场。一些基于微电子机械系统(MEMS)技术的仪器，例如由瑞士Neuchatel's Institute of Microtechnology大学开发研究的片状光栅干涉仪将会以低成本批量生产。     

`Instant? Fourier transform

The fast Fourier transform andd the fast Walsh transform are too slow for some real-time applications. For binary data, an `instant? Fourier transform is based on harmonic analysis in a space of 2n-tuples of 0s and 1s. Simple, modular logic finishes transforming 2n real-time serial binary data one clock pulse after the last datum arrives.     

Beamforming using the fractional Fourier transform

We present a new method of beamforming using the fractional Fourier transform (FrFT). This method encompasses the conventional minimum mean-squared error (MMSE) beamforming in the frequency domain or spatial domain as special cases. It is especially useful for applications involving chirp signals such as signal enhancement problems with accelerating sinusoidal sources where the Doppler effect generates chirp signals and a frequency shift and active radar problems where chirp signals are transmitted. Numerical examples demonstrate the potential advantage of the proposed method over the ordinary frequency or spatial domain beamforming for a moving source scenario.     

The fast Fourier transform

The fast Fourier transform (FFT), a computer algorithm that computes the discrete Fourier transform much faster than other algorithms, is explained. Examples and detailed procedures are provided to assist the reader in learning how to use the algorithm. The savings in computer time can be huge; for example, an N = 210-point transform can be computed with the FFT 100 times faster than with the use of a direct approach.     

Signal duration and the Fourier transform

Using the variance measure of duration, it is shown that the duration of a signal is composed of two terms. The first term is the duration of the zero-phase equivalent signal, and the second term is the variance of the phase derivative of the Fourier transformed signal.     

Interpolation using the fast Fourier transform

The fast Fourier transform (FFT) algorithm has had widespread influence in many areas of computation since its "rediscovery" by Cooley and Tukey [1]. An efficient and accurate method for interpolation of functions based on the FFT is presented. As an application, the generation of the characteristic polynomial in the "generalized eigenvalue problem" [2] is considered.     

Precise expression relating the Fourier transform of a continuoussignal to the fast Fourier transform of signal samples

The fast Fourier transform (FFT) signature of a finite duration, constant frequency, time signal displays sidebands which are a sampling artifact. An analytical expression is derived which precisely predicts artifact behavior. Using this expression, a precise spectral estimate (PSE) is derived. It is demonstrated that this method outperforms the FFT with and without Hamming weighting both in estimating the magnitudes and phases of the spectral components of a time series. Furthermore, PSE is capable of resolving frequency to a fraction of an FFT frequency resolution cell     

On the use of windows for harmonic analysis with the discrete Fourier transform

This paper makes available a concise review of data windows and their affect on the detection of harmonic signals in the presence of broad-band noise, and in the presence of nearby strong harmonic interference. We also call attention to a number of common errors in the application of windows when used with the fast Fourier transform. This paper includes a comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared. Finally, an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude.     

What is the fast Fourier transform?

The fast Fourier transform is a computational tool which facilitates signal analysis such as power spectrum analysis and filter simulation by means of digital computers. It is a method for efficiently computing the discrete Fourier transform of a series of data samples (referred to as a time series). In this paper, the discrete Fourier transform of a time series is defined, some of its properties are discussed, the associated fast method (fast Fourier transform) for computing this transform is derived, and some of the computational aspects of the method are presented. Examples are included to demonstrate the concepts involved.     

Speech spectrograms using the fast Fourier transform

An important aid in the analysis and display of speech is the sound spectrogram, which represents a time-frequency?intensity display of the short-time spectrum.1-3 With many modern speech facilities centering around small or medium-size computers, it is often useful to generate spectrograms digitally, online. The fast Fourier transform algorithm provides a mechanism for implementing this efficiently.