Cosine Transform∗ Gilbert Strang† Abstract. Each discrete cosine transform (DCT) uses N real basis vectors whose components are π cosines. In the DCT-4‚ for example‚ the jth component of vk is cos(j + 1 )(k + 1 ) N . These 2 2 basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the intensities along a row of pixels‚ its cosine series ck vk has the coefficients ck = (x‚ vk )/N . They are quickly computed from a Fast Fourier Transform. But a
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ELE374 Fourier Analysis and Synthesis of Waveforms. By Anthony Njuguna EE08U122 – 080947424 Anthony Njuguna ee08u122 Abstract Many applications in communication and systems are concerned with propagation of signals through networks. The resultant output signal is dependent on the properties of both the input signal and the processes acting on the signal. This is a laboratory Report will be focusing on using Fourier series to analyze waveforms and the synthesis of waveforms. The report highlights
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[pic] Fourier Series: Basic Results [pic] Recall that the mathematical expression [pic] is called a Fourier series. Since this expression deals with convergence‚ we start by defining a similar expression when the sum is finite. Definition. A Fourier polynomial is an expression of the form [pic] which may rewritten as [pic] The constants a0‚ ai and bi‚ [pic]‚ are called the coefficients of Fn(x). The Fourier polynomials are [pic]-periodic functions. Using the trigonometric
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Introduction to Laplace Transforms for Engineers C.T.J. Dodson‚ School of Mathematics‚ Manchester University 1 What are Laplace Transforms‚ and Why? This is much easier to state than to motivate! We state the definition in two ways‚ first in words to explain it intuitively‚ then in symbols so that we can calculate transforms. Definition 1 Given f‚ a function of time‚ with value f (t) at time t‚ the Laplace transform of f is ˜ denoted f and it gives an average value of f taken over all
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The Hilbert transform Mathias Johansson Master Thesis Mathematics/Applied Mathematics Supervisor: BÄrje Nilsson‚ VÄxjÄ University. o a o Examiner: BÄrje Nilsson‚ VÄxjÄ University. o a o Abstract The information about the Hilbert transform is often scattered in books about signal processing. Their authors frequently use mathematical formulas without explaining them thoroughly to the reader. The purpose of this report is to make a more stringent presentation of the Hilbert transform but still
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Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis q Table of contents q Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk Table of contents 1. 2. 3. 4. 5. 6. 7. Theory Exercises Answers Integrals Useful trig results Alternative notation Tips on using solutions Full worked solutions Section 1: Theory 3 1. Theory q A graph of periodic function f (x) that has period L exhibits the same pattern every L units along the
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The Discrete Cosine Transform (DCT): Theory and Application 1 Syed Ali Khayam Department of Electrical & Computer Engineering Michigan State University March 10th 2003 1 This document is intended to be tutorial in nature. No prior knowledge of image processing concepts is assumed. Interested readers should follow the references for advanced material on DCT. ECE 802 – 602: Information Theory and Coding Seminar 1 – The Discrete Cosine Transform: Theory and Application 1. Introduction
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Discrete wavelet transform 2 Others Other forms of discrete wavelet transform include the non- or undecimated wavelet transform (where downsampling is omitted)‚ the Newland transform (where an orthonormal basis of wavelets is formed from appropriately constructed top-hat filters in frequency space). Wavelet packet transforms are also related to the discrete wavelet transform. Complex wavelet transform is another form. Properties The Haar DWT illustrates the desirable properties of wavelets
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action alone causes transformation. Transformation is not just change. Change is often reversible and external. But transformation is lasting and happens from within. A caterpillar transforms into a butterfly. A child transforms into a youth and an adult. A country transforms into a nation. A nation transforms from a developing nation to a developed nation. We have transformed as a country to attain political freedom and have come a long way since then. We have the potential to lead the world
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Laplace Transforms Gilles Cazelais May 2006 Contents 1 Problems 1.1 Laplace Transforms . . . . . . 1.2 Inverse Laplace Transforms . 1.3 Initial Value Problems . . . . 1.4 Step Functions and Impulses 1.5 Convolution . . . . . . . . . . 2 Solutions 2.1 Laplace Transforms . . . . . . 2.2 Inverse Laplace Transforms . 2.3 Initial Value Problems . . . . 2.4 Step Functions and Impulses 2.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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