The discrete rotational Fourier transform

We define a discrete version of the angular Fourier transform and present the properties of the transform that show it to be a rotation in time-frequency space, this new transform is a generalization of the DFT. Efficient algorithms for its computation can then be based on the FFT and the eigenstructure of the DFT     

The discrete fractional Fourier transform

We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform     

Application of the discrete Fourier transform method to thindielectric structures [EM scattering]

The discrete Fourier transform method is a numerical technique designed to take advantage of the convolution structure that often appears in the differential-integral equations arising from electromagnetic scattering. It used the fast Fourier transform (FFT) to compute both the derivatives and the convolution integrals. As a consequence, this method is easy to program, uses less computer memory than comparable methods, yields accurate predictions, and in general, offers a better rate of convergence. This technique, which is particularly suited for solving problems where the scatterer is made of a dielectric material and has a shape which can be approximated accurately by a rectangular grid, is applied to thin dielectric slabs with both electric and magnetic properties. It is shown that by choosing the conductivity sufficiently large, a thin dielectric slab behaves like a metallic plate. On the other hand, with a suitable choice of conductivity, a particular thin dielectric slab will act like a resistive plate     

Application of the discrete Fourier transform method tononorthogonal dihedral reflectors

The discrete Fourier transform method (DFTM) is used to calculate the radar cross section (RCS) or orthogonal or nonorthogonal dihedral reflectors. In order to accurately model the mutual interactions between the two plates, a four-point bivariate Lagrange interpolation is incorporated into the DFTM. The resulting algorithm is simple and efficient. Results obtained with it compare favorably with measurements and predictions computed using the method of moments     

Source coding of the discrete Fourier transform

Distortion-rate theory is used to derive absolute performance bounds and encoding guidelines for direct fixed-rate minimum mean-square error data compression of the discrete Fourier transform (DFT) of a stationary real or circularly complex sequence. Both real-part-imaginary-part and magnitude-phase-angle encoding are treated. General source coding theorems are proved in order to justify using the optimal test channel transition probability distribution for allocating the information rate among the DFT coefficients and for calculating arbitrary performance measures on actual optimal codes. This technique has yielded a theoretical measure of the relative importance of phase angle over the magnitude in magnitude-phase-angle data compression. The result is that the phase angle must be encoded with 0.954 nats, or 1.37 bits, more rate than the magnitude for rates exceeding 3.0 nats per complex element. This result and the optimal error bounds are compared to empirical results for efficient quantization schemes.     

Vectorized mixed radix discrete Fourier transform algorithms

A number of previous attempts at the vectorization of the fast Fourier transform (FFT) algorithm have fallen somewhat short of achieving the full potential speed of vector processors. The algorithm formulation and implementation described here not only achieves full vector utilization but successfully copes with the problems of hierarchical storage. In the present paper, these techniques are described and extended to the general mixed radix algorithms, prime factor algorithm (PFA), the multidimensional discrete Fourier transform (DFT), the rectangular transform convolution algorithms, and the Winograd fast Fourier transform algorithm. Some of the methods were used in the Engineering Scientific Subroutine Library for the IBM 3090 Vector Facility. Using this approach, very good and consistent performance was obtained over a very wide range of transform lengths.     

Binary windows for the discrete Fourier transform

An efficient structure is suggested for the frequency-domain windowing of discrete Fourier transforms. In this scheme, multiplications are replaced by shifts in the position of the binary point. Three new window functions are described which can be realized by the suggested structure.     

基于矩的无乘法离散傅立叶变换

提出了一种无乘法实现离散傅立叶变换(DFT)的新算法:通过模运算和泰勒展开,把DFT的计算转化为离散矩和常系数乘积的形式;然后,通过在二进制系统中进行比特运算和移位运算,把浮点乘积转化为定点的整数加法.离散矩可由全加法实现,因此新算法只涉及整数加法和移位运算.此外,为该算法设计出脉动阵列VLSI结构,并和现有结构进行了对比分析.分析结果表明新结构不涉及乘法运算,节约了硬件资源,加快了运算速度.该方法也可以推广到其他离散变换的计算.     

The discrete fractional Fourier transform based on the DFT matrix

We introduce a new discrete fractional Fourier transform (DFrFT) based on only the DFT matrix and its powers. Eigenvectors of the DFT matrix are obtained in a simple-yet-elegant and straightforward manner. We show that this DFrFT definition based on the eigentransforms of the DFT matrix mimics the properties of continuous fractional Fourier transform (FrFT) by approximating the samples of the continuous FrFT. By appropriately combining existing commuting matrices we obtain a new commuting matrix which performs better. We show the validity of the proposed algorithms by computer simulations comparing DFrFT points and continuous FrFT samples for various signals.     

滑动离散傅里叶算法输出稳定性研究

受数字系统有限字长的影响,滑动离散傅里叶变换(滑动DFT)算法的频率单元存在输出不稳定的缺点。利用改进Goertzel算法的递归单元对滑动DFT算法的频率单元改造后,不仅可以直接计算起始频谱值,而且滑动DFT算法可以每隔N个输出值就对频率单元清零,并能提供准确的新谱值,保证了滑动DFT算法的频率单元可以长时间连续不断的处理输入数据,而不会出现输出不稳定现象。这种方法在连续地、实时地进行时频谱分析中具有重要的意义。     

A new inequality in discrete Fourier theory

Discrete Fourier theory has been applied successfully in digital communication theory. In this correspondence, we prove a new inequality linking the number of nonzero components of a complex valued function defined on a finite Abelian group to the number of nonzero components of its Fourier transform. We characterize the functions achieving equality. Finally, we compare this inequality applied to Boolean functions to the inequality arising from the minimal distance property of Reed-Muller codes.     

离散傅里叶变换性质的可视化教学设计

离散傅里叶变换是“数字信号处理”的重要教学内容,然其定义及主要性质涉及大量数学公式推导表征,较为抽象枯燥,不利于学生直观深入理解;且课程内容中对理论知识的具体应用场景涉及较少,难以激发学生的学习兴趣。针对此问题,提出了将数字图像频谱分析作为离散傅里叶变换性质教学的辅助手段,对线性、循环移位、对称性、对偶性等主要性质,分别进行直观形象的可视化展示。课程实践结果表明,该教学设计能有效提高学生的学习兴趣和知识掌握度,取得了较好的教学效果。     

Derivation of Fourier transform of discrete series from z-transform

The relationship between z-transform and Fourier transform of a discrete series is presented. The Fourier transform of a discrete series whose z-transform has poles on the unit circle can be derived using these results.<>     

加窗离散傅里叶变换性能分析和比对

离散傅里叶变换（DFT）是数字信号处理的常用方法,但通过加窗对信号进行时域截短会使频谱分析出现误差。不同窗函数对信号频谱产生的影响不同,对5种常用窗函数的特性进行了比较和适用性分析,通过仿真实验证明了针对不同信号特点和处理要求,应选择合适的窗函数,以减小频谱泄漏和栅栏效应对信号分析的影响。     

单透镜成象的分数傅里里叶变换分析

利用描述光波傍轴传输的光学分数傅里叶变换基本关系式,对单透镜系统进行分析,结果表明,单透镜系统可实现任意阶分数傅里叶变换,Lohmann第一类光学分数傅里叶变换装置是其特例,单透镜成像过程是进行阶次为π的分数傅里叶变换,进一步表明光学分数傅里叶变换的物理实质是光波的菲涅耳衍射。     

提高窗口傅立叶变换精度的方法

由于CCD采集的变形条纹图是空域有限的,在对其作窗口傅立叶变换处理时,高斯窗在图像边缘的不完整会产生测量误差。针对这种情况,提出了采用Gerchberg迭代法对条纹进行外插延拓,扩展窗口傅立叶变换处理的有效区域范围,进而改善窗口傅立叶变换的测量精度。     

一种基于DSTFT解调2FSK信号的新方法

现有解调２ＦＳＫ信号的方法,大多采用时域解调的方式。本文提出了一种利用离散短时傅里叶变换（ＤＳＴＦＴ）在时频域解调２ＦＳＫ信号的新方法,仿真表明效果良好。     

Calculating Chebyshev shading coefficients via the discrete Fourier transform

A previous technique for deriving Chebyshev shading coefficients using a cosine series is rewritten in the form of an inverse discrete Fourier transform (DFT) thus allowing one to take advantage of standard DFT algorithms. The reduced accuracy required for intermediate calculations is retained. Additionally, the fast Fourier transform can be used giving computational savings.     

Fast computation of the discrete Fourier transform of real data

Fast algorithms for the computation of the discrete Fourier transform (DFT) of real signals are important since the signals in practical situations are mostly real. The more efficient algorithms for real data are those that are derived from the algorithms for complex data. So far, all such algorithms use a real array to store the data. However, as the data values are real and their transform values are mostly complex, two possible data structures can be used for these algorithms: real or complex. DFT algorithms for real data that use a complex array for storing both the real data and their transform values are derived from the Cooley-Tukey radix-2 algorithm for complex data. This approach reduces the number of bit-reversal and array-index updating operations, eliminates independent data-swapping operations, and yields a computational structure that is almost as regular as that of the algorithms for complex data. Detailed derivations of the proposed algorithms for the computation of both the DFT of real data and the inverse DFT of the transform of real data, as well as their computational complexities, are presented. A C-language program of one of the proposed algorithms is given, illustrating the use of all the features of the new approach in software implementation. Comparison results are included to show that the proposed algorithms are faster and simpler than the real-valued split-radix and other algorithms     

基2DFT的最快速算法

本文提出一种FFT新算法,其计算量不大于现有的各种基2DFT算法.然后,与Winograd小DFT(4,8,16点)结合使用,得出一种计算DFT的最快速算法.