Fast Fourier Transform

Kurskod: 5DV050 Inrättad: 2008-03-31 Inrättad av: teknisk-naturvetenskapliga fakultetsnämnden Reviderad: 2012-02-29 Reviderad av: teknisk-naturvetenskapliga fakultetsnämnden Kursplan giltig från: 2012, vecka 10 Ansvarig enhet: Inst för datavetenskap SCB-ämnesrubrik: Informatik/Dataoch systemvetenskap Huvudområden och successiv fördjupning: Beräkningsteknik: Avancerad nivå, har endast kurs/er på grundnivå som förkunskapskrav (A1N) , Datavetenskap: Avancerad nivå, har endast kurs/er på grundnivå som förkunskapskrav (A1N) Betygsskala: För denna kurs ges betygen 5 Med beröm godkänd, 4 Icke utan beröm godkänd, 3 Godkänd, VG Väl godkänd, G Godkänd, U Underkänd Utbildningsnivå: Avancerad nivå

The design and analysis of parallel algorithms

Ordinary differential equations of the first order linear differential equations complex numbers and linear algebra simultaneous linear differential equations numerical methods the descriptive theory of nonlinear differential equations mechanical systems and electric circuits Fourier series Fourier integrals and Fourier transforms the Laplace transformation partial differential equations Bessel functions and Legendre polynomials applications and further properties of matrices vector analysis the calculus of variations analytic functions of a complex variable infinite series in the complex plane the theory of residues conformal mapping.

Advanced Engineering Mathematics. By C. R. Wylie. Pp. xiv, 813. 1966. (McGraw-Hill.)

The electrical conductivity of solid-state matter is a fundamental physical property and can be precisely derived from the resistance measured via the four-point probe technique excluding contributions from parasitic contact resistances. Over time, this method has become an interdisciplinary characterization tool in materials science, semiconductor industries, geology, physics, etc, and is employed for both fundamental and application-driven research. However, the correct derivation of the conductivity is a demanding task which faces several difficulties, e.g. the homogeneity of the sample or the isotropy of the phases. In addition, these sample-specific characteristics are intimately related to technical constraints such as the probe geometry and size of the sample. In particular, the latter is of importance for nanostructures which can now be probed technically on very small length scales. On the occasion of the 100th anniversary of the four-point probe technique, introduced by Frank Wenner, in this review we revisit and discuss various correction factors which are mandatory for an accurate derivation of the resistivity from the measured resistance. Among others, sample thickness, dimensionality, anisotropy, and the relative size and geometry of the sample with respect to the contact assembly are considered. We are also able to derive the correction factors for 2D anisotropic systems on circular finite areas with variable probe spacings. All these aspects are illustrated by state-of-the-art experiments carried out using a four-tip STM/SEM system. We are aware that this review article can only cover some of the most important topics. Regarding further aspects, e.g. technical realizations, the influence of inhomogeneities or different transport regimes, etc, we refer to other review articles in this field.

https://iopscience.iop.org/article/10.1088/0953-8984/27/22/223201/pdf

The 100th anniversary of the four-point probe technique: the role of probe geometries in isotropic and anisotropic systems

Two concurrent error detection (CED) schemes are proposed for N-point fast Fourier transform (FFT) networks that consists of log/sub 2/N stages with N/2 two-point butterfly modules for each stage. The method assumes that failures are confined to a single complex multiplier or adder or to one input or output set of lines. Such a fault model covers a broad class of faults. It is shown that only a small overhead ratio, O(2/log/sub 2/N) of hardware, is required for the networks to obtain fault-secure results in the first scheme. A novel data retry technique is used to locate the faulty modules. Large roundoff errors can be detected and treated in the same manner as functional errors. The retry technique can also distinguish between the roundoff errors and functional errors that are caused by some physical failures. In the second scheme, a time-redundancy method is used to achieve both error detection and location. It is sown that only negligible hardware overhead is required. However, the throughput is reduced to half that of the original system, without both error detection and location, because of the nature of time-redundancy methods. >

Fault-tolerant FFT networks

Signals, Systems, Transforms, and Digital Signal Processing with MATLAB has as its principal objective simplification without compromise of rigor. Graphics, called by the author, "the language of scientists and engineers", physical interpretation of subtle mathematical concepts, and a gradual transition from basic to more advanced topics are meant to be among the important contributions of this book. After illustrating the analysis of a function through a step-by-step addition of harmonics, the book deals with Fourier and Laplace transforms. It then covers discrete time signals and systems, the z transform, continuous- and discrete-time filters, active and passive filters, lattice filters, and continuous- and discrete-time state space models. The author goes on to discuss the Fourier transform of sequences, the discrete Fourier transform, and the fast Fourier transform, followed by Fourier-, Laplace, and z-related transforms, including WalshHadamard, generalized Walsh, Hilbert, discrete cosine, Hartley, Hankel, Mellin, fractional Fourier, and wavelet. He also surveys the architecture and design of digital signal processors, computer architecture, logic design of sequential circuits, and random signals. He concludes with simplifying and demystifying the vital subject of distribution theory. Drawing on much of the authors own research work, this book expands the domains of existence of the most important transforms and thus opens the door to a new world of applications using novel, powerful mathematical tools.

Signals, Systems, Transforms, and Digital Signal Processing with MATLAB

In this paper, a hybrid technique using the discrete cosine transform (DCT) and the discrete wavelet transform (DWT) is presented. We show evaluation and comparative results for DCT, DWT and hybrid DWT-DCT compression techniques. Using the Power Signal to Noise Ratio (PSNR) as a measure of quality, we show that DWT with a two-threshold method named “improved-DWT” provides a better quality of image compared to DCT and to DWT with a one-threshold method. Finally, we show that the combination of the two techniques, named improved-DWT-DCT compression technique, showing that it yields a better performance than DCT-based JPEG in terms of PSNR.

A hybrid image compression technique based on DWT and DCT transforms

The organization and functional design of a parallel radix-4 fast Fourier transform (FFT) computer for real-time signal processing of wide-band signals is introduced.

A Parallel Radix-4 Fast Fourier Transform Computer

The author proposes a generalisation of the theory of generalised functions, also known as the theory of distributions, by extending the theory to include generalised functions of a complex variable, both in the complex plane associated with continuous-time functions and that with discrete-time functions. The generalisation provides, among others, mathematical justifications of the properties of recently introduced generalised Dirac-delta impulses, using the principles of distribution theory. Properties of generalised functions of a complex variable are explored both in the Laplace domain associated with continuous-time functions and the z domain associated with discrete-time functions. Shifting of distributions, scaling, derivation, convolution with distributions and convolution with ordinary functions are evaluated in Laplace and z domains. Three-dimensional generalisations of sequences leading to generalised impulses, and of test functions in Laplace and z domains are presented. New expanded Laplace and z transforms are obtained using the proposed generalisation.

Complex-variable distribution theory for Laplace and z transforms

A generalisation of the Dirac-delta impulse and its derivatives as two generalised distributions, namely, the xi and zeta impulses, and their derivatives, defined on the complex s-plane and z-plane of continuous-time and discrete-time functions, respectively, is proposed. The generalised impulses extend the existence of Laplace and z transforms to a large class of infinite duration two-sided functions, which hitherto had no transform or had only a Fourier transform in the form of distributions. The proposed generalised impulses are shown to bridge the gap between the theory of generalised functions and both the unilateral and bilateral Laplace and z transforms. The generalised impulses extend the existence of Laplace and z transforms to include both functions that have a Fourier transform as a distribution as well as exponentially rising infinite duration two-sided functions that have no Fourier transform. It is shown that a modulation theory can now be added to the properties of bilateral transforms. No such theorem has hitherto existed for these transforms. The proposed generalised impulses and the resulting extended Laplace and z transforms are shown to lead to new complex-plane operations, such as spatial convolution, and to simplify operations such as ordinary convolution, sampling and the solution of differential and difference equations. Bilateral Laplace and z transforms may receive greater attention now that these transforms can be applied to a new, large and basic class of functions, such as two-sided infinite duration exponentials and rising trigonometric and hyperbolic functions.

Michael J. Corinthios

Papers

Signals, Systems, Transforms, and Digital Signal Processing with MATLAB

A hybrid image compression technique based on DWT and DCT transforms

A Parallel Radix-4 Fast Fourier Transform Computer

Complex-variable distribution theory for Laplace and z transforms

Generalisation of the Dirac-delta impulse extending Laplace and z transform domains