Fast algorithms for the discrete Fourier preprocessing transforms

It is pointed out that the two-stage Fourier algorithms are useful in a variety of applications in signal/image processing and recognition. The first stage, known as the DFPT (discrete Fourier preprocessing transform), has the potential of very fast and low-cost implementation, say, in a VLSI chip, as well as high-quality performance in applications without the second stage. The DFT (discrete Fourier transform), the RDFT (real DFT), or any other DFPT can be obtained if the second stage is completed. Since the second stage consists of independent blocks of circular correlations, they can be computed in fast sequential or parallel architectures. There are two groups of fast algorithms for the computation of DFPTs. The first group involves the representation of a DFPT in terms of skew-circular correlations (SCCs), which are then computed by fast SCC algorithms or parallel architectures such as semisystolic arrays. The second group involves the fast computation of a DFPT, including the DFT and the RDFT, by the implementationally simplest DFPT, such as class 2, case V DFPT     

Fast biased polynomial transforms for 2D prime length discrete Fourier transforms

The fast biased polynomial transform (FBPT) is defined directly over a ZN ? 1 ring instead of the conventional cyclotomic polynomial rings. For N prime, these FBPTs can be used for the efficient evaluation of the 2D prime length DFT.     

Fast algorithm for calculation of both Walsh-Hadamard and Fourier transforms (FWFTs)

In the letter a fast and efficient algorithm is presented for calculating both DFT and the WHT. This is achieved through the factorisation of the intermediate transform T into a product of sparse matrices. The algorithm can be implemented using a single butterfly structure, and is amenable for both software and hardware implementations.<>     

Discrete Fourier and Hadamard transforms

Elementary properties of discrete Fourier transforms (d.f.t.) have appeared in recent literature. Corresponding properties of BIFORE (Hadamard) transforms are summarised and compared with those of the d.f.t.     

Fourier, Laplace and Hilbert transforms

For a simple evaluation of the Hilbert transform, iterated Fourier- or Laplace-transform schemes have been suggested. In this letter, an alternative iterated transform scheme, more general than the one previously reported, has been put forward.     

Multiplicity of fractional Fourier transforms and theirrelationships

The multiplicity of the fractional Fourier transform (FRT), which is intrinsic in any fractional operator, has been claimed by several authors, but never systematically developed. The paper starts with a general FRT definition, based on eigenfunctions and eigenvalues of the ordinary Fourier transform, which allows us to generate all possible definitions. The multiplicity is due to different choices of both the eigenfunction and the eigenvalue classes. A main result, obtained by a generalized form of the sampling theorem, gives explicit relationships between the different FRTs     

Closed-form discrete fractional and affine Fourier transforms

The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT     

Fourier transforms in propagation and scattering problems

Three-dimensional Fourier transforms are applied to electromagnetic wave propagation problems. After obtaining functions in transform space, inversion and contour integration yield a plane wave integral representation.     

Composite spectrogram using multiple Fourier transforms

The authors propose a time?frequency (T?F) analysis method that uses a time- and frequencydependent resolution to represent a signal. The method is based on the idea of splitting the T?F plane into equal-TF-area Heisenberg boxes in some optimal way that closely matches spectral events. Compared with existing methods based on orthogonal decompositions, by lifting the orthogonality constraint, extra freedom is gained in the way the T?F plane can be partitioned, which enables time and frequency adaptation at the same time. A best tiling selection algorithm of quadratic complexity is derived using dynamic programming to find the optimal frame from a family. Experiments show the advantage of this more flexible representation.     

复数阶分数傅里叶变换的光学解释

基于分数傅里叶变换的级联性,将变换级次由实数扩展到复数,获得了复数阶分数傅里叶变换的定义,推证了分数傅里叶变换的帕赛瓦尔定理,分析了其与吸收介质内部光传过程的等效性,诠释了复数阶分数傅里叶变换的物理机制,计算机模拟实验验证了结论的可靠性。     

Fractional Fourier transforms of hypercomplex signals

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given.     

Doppler processing using Walsh and hard-limited Fourier transforms

The use of Walsh and hard-limited Fourier transforms to synthesize banks of contiguous Doppler filters is examined for the case of a 32-point transform. It is found that the former is generally inferior to the latter from the point of view of number and size of spurious sidelobes.     

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     

互补分形图像的光学傅里叶变换

从理论上推导了SC分形图像的衍射强度表达式,并用光学方法进行了SC分形互补屏的傅里叶变换,在频谱面上除中心处,分形互补屏给出了相同的衍射图样,而且也具有对称性和自相似嵌套结构。实验结果与理论上推导出的表达式相吻合。同时,验证了巴俾涅原理。     

Hypercomplex Fourier transforms of color images.

Fourier transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed.     

Multidimensional chirp algorithms for computing Fourier transforms

Continuous versions of the multidimensional chirp algorithms compute the function G(y)=F(My), where F(y) is the Fourier transform of a function f(x) of a vector variable x and M is an invertible matrix. Discrete versions of the algorithms compute values of F over the lattice L(2)=ML(1 ) from values of f over a lattice L(1), where L(2) need not contain the lattice reciprocal to L(1). If M is symmetric, the algorithms are multidimensional versions of the Bluestein chirp algorithm, which employs two pointwise multiplication operations (PMOs) and one convolution operation (CO). The discrete version may be efficiently implemented using fast algorithms to compute the convolutions. If M is not symmetric, three modifications are required. First, the Fourier transform is factored as the product of two Fresnel transforms. Second, the matrix M is factored as M=AB, where A and B are symmetric matrices. Third, the Fresnel transforms are modified by the matrices A and B and each modified transform is factored into a product of two PMOs and one CO.     

椭圆高斯光束的分数傅里叶变换

利用分数傅里叶变换与Wigner分布函数旋转等效的性质 ,推导出了椭圆高斯光束在分数傅里叶变换平面上的光强分布和束宽的解析公式。研究了椭圆高斯光束光强和束宽随分数傅里叶变换阶数变化的规律 ,并给出了数值计算结果     

Fast Hartley transforms for image processing

The fast Hartley transform (FHT) is used to transform two-dimensional image data. Because the Hartley transform is real-valued, it does not require complex operations. Both spectra and autocorrelations of two-dimensional ultrasound images of normal and abnormal livers were computed