Free Essay

Submitted By sniizzer

Words 2584

Pages 11

Words 2584

Pages 11

Jatin Kumar

Murray State University

Abstract

It has been widely recognized that waveforms are an integral part of the various universe phenomenon. Waveforms can be used to represent almost everything in the world. Therefore it is understandable that concepts related to waveforms or signals are extremely important as their applications exist in a broad variety of fields. The processes and ideas related to waveforms play a vital role in different areas of science and technology such as communications, optics, quantum mechanics, aeronautics, image processing to name a few. Even though the physical nature of signals might be completely different in various disciplines, all waveforms follow one fundamental principle; they can be represented by functions of one or more independent variables.

This paper would focus on the concept of Fourier Transform, the technique through which signals can be deconstructed and represented as sum of various elementary signals. It briefly describes Linear Time Invariant systems and their response to superimposed signals. Fourier transform has many applications in physics and Engineering. This paper would also cover some of Fourier Transform applications in telecommunication and its impact on society.

Introduction

Some of the basic signals that exist in the world and are useful in various technology fields are continuous and discrete time signals. These signals depend on a single independent variable. Generally the independent variable is considered to be time (though it is not universally true).We will focus our discussion on the signals which are considered to be a function of a single independent variable. With the use of transformation techniques, a signal can be deconstructed. The continuous and discrete signals can be represented as composite signals with the help of Fourier transform. The result of this transformation provides an output which is easy to interpret, to be changed and analyzed

To understand Fourier transform with technical details, let us start with the definitions of continuous and discrete time signals. Later we present the mathematics related to signal representations, transformation and synthesis.

Signals

A continuous signal is a signal in which the independent variable is continuous and thus the signal is defined for a continuum of values of the independent variable [1, p. 9]

A discrete time signal is only defined at discrete times, and consequently for these signals the independent variable takes on only a discrete set of values [1, p. 9]

It is also useful to define periodic functions. These functions are used to define various periodic signals

A function f(t) is said to be periodic with period T if there exists a number T > 0 such that f(t + T) = f(t) for all values of t [2, p. 4]. If such a T exists, then the smallest T for which the equation holds is called the fundamental period of the function f. Every integer multiple of the fundamental period is also a considered to be a period of the function: f(t + nT) = f(t), n= 0,±1,±2,

Continuous Time complex Exponential and Sinusoidal Signals

The continuous time complex exponential signal is of the form [1, p. 17] x(t)= 〖Ce〗^at

Where C and a are complex numbers. Depending on the values of C and a, the exponential equation can exhibit various characteristics. If we restrict the value of a to be purely imaginary, then we get signals of the form x(t)= e^(jω_o t) Important property of such type of signals is that they are periodic. Such class of signals is used to derive the sinusoidal functions. x(t)=Acos(ω_o t+ φ) ω_ois called the fundamental frequency of the signal, and φ is the phase of the signal

Sinusoidal and other periodic exponential signals are used to describe the characteristics of various real world processes.

Similarly discrete time sinusoidal signals can be defined as x[n]=Acos(ω_o n+ φ)

Linear time Invariant systems

Linear time invariant systems can be used to model many physical processes. Their response to arbitrary input signal forms the basis for development of many real world systems which includes a variety of telecommunication systems as well. Several networking and routing hardware are built on the principles of LTI systems. LTI systems provide basis for understanding the basics of how signal’s decomposition and signal processing systems works. Hence it is important to understand their definition and properties as they are at the core of various telecommunication systems.

Properties of LTI systems

Linearity of LTI systems: The input and output of the system have a linear relationship.

If an arbitrary input x1 has an output y1 and an input x2 has output y2, then the scaled summation of input(ax1 + bx2) will produce a scaled summation of output response (ay1 + by2). Here a, and b are real numbers.

Time invariance of LTI systems: If the output response to an input x(t) at time t is y(t), then the output response to signal x(t-T) is y(t-T). That means that the output is independent of the instant at which the input signal starts; it is only a time shifted version of output at time t. Output does not depends upon the time at which input is applied

Hence, if input to an LTI system consists of a linear combination of signals x (t) = a_1 x_1 (t) + a_2 x_2 (t)+ …

Then by superposition the response is given by y (t)= a_1 y_1 (t) + a_2 y_2 (t)+…. [1, p. 69]

Hence it is possible to represent input signals to LTI systems in terms of basic signals and use the superposition property of the system to calculate the response.

Signals can be represented in terms of impulses by decomposing them into weighted sum of shifted impulses. Figure 1 show how a discrete time signal can be represented as a sum of shifted impulse response. [1, p. 71] (a) Shows the original signal and the other five are generated by multiplying the scaled time shifted impulses with the original signal.

Figure 1

Hence an arbitrary signal can be expressed as x[n]= ∑_(k= -∞)^(+∞)▒〖x[k]δ[n-k]〗

Where δ[n-k] is the time shifted impulse function, and x[k] is the amplitude (for scaling). If we know the impulse response h[n] of the system to input impulse δ [n], we can determine the output by applying h[n] to δ [n] and then superposing the output as a linear combination of outputs to individual shifted scaled impulse x[n]. Hence the response of a system to an input x[n] can be represented by [1, p. 72] y[n]= ∑_(k= -∞)^(+∞)▒〖x[k]h[n-k]〗

This result is the convolution sum or superposition sum

Figure 2 (a) [1, p.76]shows an input signal which is non zero for t = -1,0,1

Figure 2 (b) shows impulse response of the LTI system for δ [n-1], δ [n], δ [n+1] Figure 2

Figure 3, Output of LTI system to Input Impulses[1, p.77]

Similarly for continuous time signals, continuous time LTI system response can represent output as y(t)= ∫_(-∞)^(+∞)▒〖x(τ)h(t-τ)〗

Response of continuous time LTI systems to complex periodic exponentials

When a complex exponential input is applied to an LTI system, the output is the same complex exponential with different amplitude. [1, p. 166]

(e^s )^t=〖H(s)e〗^st

Where e^st is called the eigenfunction of the system and the amplitude factor H(s) is known as the eigenvalue. Just like any arbitrary signal, the response of an LTI system to a complex exponential signal can be represented as a superposition of individual responses.

Representation of periodic signals as Fourier series

As mentioned previously a periodic signal can be represented by the equation x(t)=e^(jω_o t)

Harmonically related complex exponentials with this signal can be represented as e^(jkω_o t) , Where k = 0, +-1, +-2, +-3…, where x(t) has a fundamental period of T after which it repeats itself. This set of harmonically related signals can be represented as [1, p. 169] x(t)=∑_(k=-∞)^(+∞)▒〖a_k e^(jkω_o t) 〗

This composite signal is also periodic with fundamental period T and frequency ω_o. Terms with k as +1 and -1 have fundamental frequency and period T. Terms with k as +2 and -2 have double the fundamental frequencies and are periodic with half the fundamental period (T/2). Each such component is also called harmonics (nth harmonic in general).

This representation of a periodic signal is known as the Fourier series representation of a periodic signal.

Figure 4,

Signal x(t)=∑_(-3)^(+3)▒〖a_k e^jk2πt 〗 represented as linear combination of harmonically related sinusoidal signals[1, p.170]

If a signal can be represented with Fourier series, its coefficients are determined by the formula a_k=1/To ∫_To▒〖x(t) e^(-jkω_o t) dt〗

Coefficients are called spectral coefficients or Fourier series coefficients.

So far we have seen that Fourier series can be used to represent periodic signals. But it was also determined by Fourier that it is possible to extend this principle to represent aperiodic signals. Fourier transform pair for aperiodic signals is represented by x(t)=1/2π ∫_(-∞)^(+∞)▒〖X(ω) e^jωt dω〗

〖X(ω)〗_ =∫_(-∞)^(+∞)▒〖x(t) e^(-jωt) dt〗

Applications of Fourier Transform in Telecommunications

Modulation: We can use one signal to scale or modulate the characteristics of another signal. This can be thought of as multiplication of two signals (convolution property). This is of particular importance in communication channels. Channels typically have a frequency range over which they are best suited to transmit the signals without significant loss. For example, our atmosphere adversely affects the audible frequency range signals (10Hz to 20 KHz) whereas it allows transmission of higher frequency signals over a wide range without significant loss. Hence all the audio signals like music and speech which are transmitted through an atmosphere communication channel are modulated by high frequency carrier waves. Sinusoidal amplitude modulation is a common modulation technique used for this purpose. A high frequency sine wave acts as the information carrier signal whose frequency is in the appropriate range, and whose amplitude is modulated by various sound signals.

Filtering: In many applications, it is required to change the amplitudes of frequency components of a signal or at times remove the whole frequency component itself, a process which is called filtering. Filtering is used in communication for various applications and industries. One application of filtering is in the audio systems. A filter is provided to either include the low energy (Bass) or high energy (Treble) frequency components. Filtering is also used in mobile communication to improve receiver quality even over interference limited channels.

Multiplexing: The concept of modulation can be used to transmit many signals over a channel. For example, normally microwave transmission links have bandwidth of several gigahertz which is a lot more than what is required for one voice channel. If many voice signals have overlapping frequency, their frequency content range is shifted so that they all belong to distinct frequency bands. This is achieved through amplitude modulation with a sinusoidal wave. After modulation, all the signals can be transmitted simultaneously on the same wideband channel. This concept is known as frequency division multiplexing.

Fourier Transform’s impact on society and economic value

Technology has changed the way humans see the world today. Even if we look at our recent past, we did not communicate over cell phones, communicate via video calls or fly on jet planes to different countries of the world and on space shuttles over to the moon! But we have come a long way thanks to developments in Technology. Fourier Transform has played a vital role in the development of innumerable technologies in various industries. It can be said to have had a huge impact in the way we see technology today. A lot of hardware and applications are built which utilize the principles of Fourier transform. There are many instruments in the Healthcare industry that work on the principles of FT using spectroscopy like an MRI machine. MRI machines have enabled doctors to identify various problems related to health and proper diagnosis of patients. . In the food industry, spectroscopic instruments are used to identify the adulteration of fats and edible oils

If we look at geology, FT principles are used to calculate the probability and intensity of earthquakes which can save a lot of human lives.

“Signals are ubiquitously present in manufacturing machines and systems. For example, metal removal is essential to many manufacturing processes, as seen in turning, milling, and drilling During such a process, interactions between the cutting edge of the tool and the work piece lead to removal of fragments of varying volumes, producing whereby time-varying or transient components in the vibration signals” [5]

Identification of such signals at initial stage and their removal can improve the machine performance and help reduce structural defects and machine downtime in the future. These signals are identified with the help of Fourier Transform. Hence it provides economic value to the manufacturers and machine buyers as it helps reduce manufacturing defects and costly downtime. In stock markets, various financial models for stock prices and market predictions are built using Fourier series. Looking into the future, Affine Fourier Transform (AFT-MC) can be used for transmission of data over aeronautical satellite channels. Using AFT-MC would provide higher date rate for communication when flying on airplanes. Hence we can see that Fourier Transform can be said to have played a vital role in the overall development of technology, human beings and thus the society. As we keep coming up with more innovative theories and ideas, we will find more applications of Fourier Transform and its derivatives in future.

Conclusion

We briefly described few types of signals in the paper. Then we talked about Linear Time Invariant systems and their importance in various fields of technology. We focused on the technical aspects of Fourier Transform and represented it mathematically. Later part of the paper briefly mentioned some applications of Fourier Transform in telecommunications and various other fields and also how it has had an impact on modern day technology and society.

References Alan V. Oppenheim, Signals and systems, Prentice Hal International, Inc., London, August 6th 1996 Prof. Brad Osgood, Fourier Transform and its applications, Lecture Notes, Stanford University http://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf Lisheng Fan; Fukawa, K.; Suzuki, H., "MAP Receiver with Spatial Filters for Suppressing Cochannel Interference in MIMO-OFDM Mobile Communications," Vehicular Technology Conference, 2008. VTC 2008-Fall. IEEE 68th , vol., no., pp.1,5, 21-24 Sept. 2008 doi: 10.1109/VETECF.2008.188, Accessed on 10/30/2013 http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4657020&isnumber=4656832 Weinstein, S.; Ebert, P., "Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform," Communication Technology, IEEE Transactions on , vol.19, no.5, pp.628,634, October 1971 doi: 10.1109/TCOM.1971.1090705, , Accessed on 11/05/2013 http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1090705&isnumber=23757 Schey JA, Introduction to manufacturing processes. 3rd edn, McGraw-Hill

Science/Engineering/Math, New York, 1999, Page 5 N. Vlachos, “Applications of Fourier transform-infrared spectroscopy to edible oils”

Food Technology Department, Faculty of Food Technology and Nutrition, Technological Educational Institute of Athens, Ag. Spyridonos Str., 12210 Egaleo, Athens, Greece, July 2006, Accessed on 12/01/2013 http://www.sciencedirect.com/science/article/pii/S000326700601035X

Free Essay

...ENTS 699R: Lecture 1d support ENTS 699R Lecture 1d support: Fourier Transform Tables Alejandra Mercado June, 2013 1 Transform Pairs The following is a table of basic transform pairs that can be used as building blocks to derive more complicated transform pairs: Time domain function, with dummy variable t 1 2 3 4 5 6 7 F F F Frequency domain function, with dummy variable f δ(t) ⇐⇒ 1 1 ⇐⇒ δ(f ) δ(t − t0 ) ⇐⇒ e−j2πf t0 sin (2πf0 t + φ) ⇐⇒ F F j −jφ δ(f 2 [e 1 −jφ δ(f 2 [e + f0 ) − ejφ δ(f − f0 )] cos (2πf0 t + φ) ⇐⇒ + f0 ) + ejφ δ(f − f0 )] 1 |t| ≤ T F t 2 rect ( T ) = ⇐⇒ T sinc(f T ) = T sin(πf T ) πf T 0 o.w. sinc(βt) ⇐⇒ F 1 β f · rect ( β ) = 1 β ·1 |f | ≤ o.w. β 2 0 Page 1 ENTS 699R: Lecture 1d support 2 Properties For the table of Fourier Transform properties, assume that we already know that: g(t) ⇐⇒ G(f ) h(t) ⇐⇒ H(f ) and that α, β, T, φ, f0 , t0 are all arbitrary constants. Time domain function, with dummy variable t A B C D E F F G H I F F F F F Frequency domain function, with dummy variable f Property name time/frequency reversal duality time shift frequency shift linearity g(−t) ⇐⇒ G(−f ) G(t) ⇐⇒ g(−f ) g(t − t0 ) ⇐⇒ g(t)ej2πf0 t F F F e−j2πf t0 G(f ) ⇐⇒ G(f − f0 ) 1 2 j 2 αg(t) + βh(t) ⇐⇒ αG(f ) + βH(f ) g(t) cos(2πf0 t) ⇐⇒ g(t) sin(2πf0 t) ⇐⇒ F F F (G(f − f0 ) + G(f + f0 )) modulation (G(f + f0 ) − G(f − f0 )) modulation multipl. in time domain convolution in time domain time scaling g(t) ×......

Words: 332 - Pages: 2

Free Essay

...PE, etc. Bio-composite materials that can be decomposed naturally will be a solution to solve these problems. The aim of this study was to utilize the waste shredded coconut after the coconut milk was extracted, and used as reinforcing agent bio-composite material with matrix of Polylacticacid(PLA) that can decompose naturally. During this time, grated coconut is simply discarded without further exploited. Alkalization chemical treatment with 5% NaOH 1 hour to Grated Coconut Milk Residue (GCMR) was done to improve compatibility with PLA matrix. Bio-composite material was made with a fraction of 0%, 15%, 30% w/w using compression molding technique. The chemical structure of GCMR before and after chemical treatment was observed with Fourier Transform Infrared. The mechanical property of bio-composite material was observed with tensile test. Whereas the morphological characteristics of GCMR and fracture on the surface of bio-composite observed using Field Emission Scanning Electron Microscope. Then, thermal properties of bio-composite observed with Simultaneous Thermal Analysis. Result showed the addition of grated coconut fibers improve the mechanical properties, thermal stability, speed recrystallization PLA and PLA interfacial bonding between the filler GCMR. Thus, this material has the potential to reduce the environmental crisis caused by non-biodegradable material and non-renewable. Kata Kunci—Instruksi; Poly(latic acid)(PLA), Reinforcing Agent, Grated Coconut Residue,......

Words: 265 - Pages: 2

Free Essay

...2007-2008 JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY, HYDERABAD B.TECH. ELECTRONICS AND COMMUNICATION ENGINEERING I YEAR COURSE STRUCTURE |Code |Subject |T |P/D |C | | |English |2+1 |- |4 | | |Mathematics - I |3+1 |- |6 | | |Mathematical Methods |3+1 |- |6 | | |Applied Physics |2+1 |- |4 | | |C Programming and Data Structures |3+1 |- |6 | | |Network Analysis |2+1 |- |4 | | |Electronic Devices and Circuits |3+1 |- |6 | | |Engineering Drawing |- |3 |4 | | |Computer Programming Lab. |- |3 |4 | | |IT Workshop |- |3 |4 | | |Electronic Devices and Circuits Lab |- |3...

Words: 26947 - Pages: 108

Free Essay

...Lovely Professional University, Punjab Course Code MTH251 Course Category Course Title FUNCTION OF COMPLEX VARIABLE AND TRANSFORM Courses with Numerical focus Course Planner 16423::Harsimran Kaur Lectures 3.0 Tutorials Practicals Credits 2.0 0.0 4.0 TextBooks Sr No T-1 Title Advanced Engineering Mathematics Reference Books Sr No R-1 R-2 Other Reading Sr No OR-1 Journals articles as Compulsary reading (specific articles, complete reference) Journals atricles as compulsory readings (specific articles, Complete reference) , Title Higher Engineering Mathematics Advanced Modern Engineering Mathematics Author Grewal, B. S. Glyn James Edition 40th 3rd Year 2007 2011 Publisher Name Khanna Publishers Pearson Author Jain R. K. and Iyenger S. R. K. Edition 3rd Year 2007 Publisher Name Narosa Relevant Websites Sr No RW-1 RW-2 (Web address) (only if relevant to the course) www2.latech.edu/~schroder/comp_var_videos.htm freescienceonline.blogspot.com/2010_04_01_archive.html Salient Features Topic videos available Complex Analysis Reference Material Available LTP week distribution: (LTP Weeks) Weeks before MTE Weeks After MTE Spill Over 7 6 2 Detailed Plan For Lectures Week Number Lecture Number Broad Topic(Sub Topic) Chapters/Sections of Text/reference books Other Readings, Lecture Description Relevant Websites, Audio Visual Aids, software and Virtual Labs Introduction Functions of a Complex Variable Learning Outcomes Pedagogical Tool Demonstration/ Case......

Words: 3054 - Pages: 13

Free Essay

...student version of MATLAB 7.x available under general software in the UCCS bookstore. Other specific programming tools will be discussed in class. 1.) 2.) 3.) 4.) 5.) 6.) Graded homework worth 20%. Quizzes worth 15% total Laboratory assignments worth 20% total. Mid-term exam worth 15%. Final MATLAB project worth 10%. Final exam worth 20%. Topics Text 1.1–1.4 2.1–2.9 3.1–3.9 4.1–4.6 5.1–5.9 6.1–6.9 7.1–7.10 8.1–8.12 9.1–9.10 10.1–10.6 11.1–11.11 12.1–12.4 Weeks 0.5 1.0 1.0 1.0 1.5 1.5 (exam) 1.0 2.0 1.5? 0.5? 1.5? 1.5 (project) 1. Course Overview and Introduction 2. Sinusoids 3. Spectrum Representation 4. Sampling and Aliasing 5. FIR filters 6. Frequency response of FIR filters 7. z-Transforms 8. IIR Filters 9. Continuous-Time Signals and Systems 10. Frequency Response 11. Continuous-Time Fourier Transform 12. Filtering, Modulation, and Sampling Note: that topics 9–12 will most likely only be overviewed at the end of the semester....

Words: 276 - Pages: 2

Free Essay

...of the Fourier transform and the frequency domain, and how they apply to image enhancement. Background Introduction to the Fourier Transform and the Frequency Domain DFT Smoothing Frequency-Domain Filters Sharpening Frequency-Domain Filters 4.1 Background • Any function that periodically repeats itself can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient (Fourier series). • Even functions that are not periodic (but whose area under the curve is finite) can be expressed as the integral of sines and/or cosines multiplied by a weighting function (Fourier transform). • The advent of digital computation and the “discovery” of fast Fourier Transform (FFT) algorithm in the late 1950s revolutionized the field of signal processing, and allowed for the first time practical processing and meaningful interpretation of a host of signals of exceptional human and industrial importance. • The frequency domain refers to the plane of the two dimensional discrete Fourier transform of an image. • The purpose of the Fourier transform is to represent a signal as a linear combination of sinusoidal signals of various frequencies. = Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient – a Fourier series 4.2 Introduction to the Fourier Transform and the Frequency Domain • The one-dimensional Fourier......

Words: 3417 - Pages: 14

Free Essay

...Digital Image Processing: PIKS Inside, Third Edition. William K. Pratt Copyright © 2001 John Wiley & Sons, Inc. ISBNs: 0-471-37407-5 (Hardback); 0-471-22132-5 (Electronic) DIGITAL IMAGE PROCESSING DIGITAL IMAGE PROCESSING PIKS Inside Third Edition WILLIAM K. PRATT PixelSoft, Inc. Los Altos, California A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York • Chichester • Weinheim • Brisbane • Singapore • Toronto Designations used by companies to distinguish their products are often claimed as trademarks. In all instances where John Wiley & Sons, Inc., is aware of a claim, the product names appear in initial capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. Copyright 2001 by John Wiley and Sons, Inc., New York. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic or mechanical, including uploading, downloading, printing, decompiling, recording or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM. This publication is......

Words: 173795 - Pages: 696

Free Essay

...Information Hiding – Steganography Steganography Types and Techniques Abstract— Information hiding has been one of the most crucial element of information technology in recent years. Unlike Cryptography, Steganography does not only keep the content of information secret, its also keeps the existence of the information secret. This is achieved by hiding information behind another information. This paper gives an overview of Steganography and its techniques, types, and also its advantages and disadvantages. Keywords-component; steganography; information hiding; security; confidentiality; techniques. INTRODUCTION Steganography help hides the fact that communication is taking place by hiding some information behind another information thereby making the communication invisible. The origin of the word “steganography” is from the Greek words “steganos” and “graphia” which is interpreted as “covered” and “writing” respectively thereby defining steganography as “covered writing”. Steganography and cryptography are both part of information hiding but neither alone is without flaws. The goal of steganography is defeated once the hidden message is found or noticed even while still in the original message (carrier). For safely transmission of hidden messages or information in steganography, multimedia files like audio, video and images are mostly used has the carrier or cover source. METHODOLOGY The Methodology used for this paper is based on the...

Words: 1519 - Pages: 7

Free Essay

...Abstract. We propose, analyze, and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also the isotropic forms of TV discretizations. The per-iteration computational complexity of the algorithm is three fast Fourier transforms. We establish strong convergence properties for the algorithm including ﬁnite convergence for some variables and relatively fast exponential (or q-linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the lagged diﬀusivity algorithm for TV-based deblurring. Some extensions of our algorithm are also discussed. Key words. half-quadratic, image deblurring, isotropic total variation, fast Fourier transform AMS subject classiﬁcations. 68U10, 65J22, 65K10, 65T50, 90C25 DOI. 10.1137/080724265 1. Introduction. In this paper, we propose a fast algorithm for reconstructing images from blurry and noisy observations. For simplicity, we assume that the underlying images have square domains, but all discussions can be equally applied to rectangle domains. Let 2 2 2 u0 ∈ Rn be an original n×n......

Words: 12310 - Pages: 50

Free Essay

...Task 1: Task one was intended to build off of ideas implemented in the previous lab. More specifically formulating a V-I curve for the resistor circuit displayed in the prelab. Graph 1 shows the resulting V-I curve. Graph 1: V-I curve for the potentiometer. Like the previous lab, the signal was generated by a function generator and then sent through a potentiometer followed by a precision resistor in series. Figure 1 shows the LabView Block diagram that was used to record voltages of the two resistors. Figure 1: LabView Block diagram used to record voltages across the precision resistor and potentiometer. Because the resistance of the precision resistor was known to be 10Ω the recorded voltage was used to calculate the current using Ohm’s Law shown in equation 1. V=I*R Equation 1: Ohm’s law. Used to calculate the current. Task 2: Task two was designed to demonstrate how different sampling rates and different frequencies affect data and how it appears. In this section, the function generator was set to 5 Vpp with a frequency of 1000Hz and plugged into the compact DAQ module. Next the experiment was performed using five sampling rates (500, 1000, 2000, 3752, & 20000 ) with samples of 10, 40, 50, 200, and 2000. The LabView block diagram shown in figure 2 was responsible for collecting the mean and standard deviation of the amplitude and frequency. ...

Words: 868 - Pages: 4

Free Essay

...present. * * Microsoft Excel carries many qualitative tools used to analyze data. One tool is the Random Number Generation analysis tool. This tool fills a range with random numbers that a drawn from one of several different dispersals. For example you can use this tool in business to show the difference of a coin flip. The next tool is Rank and Percentile. This tool is used to analyze a set of data in an arrange set. The next tool is the Moving Average analysis tool. This tool projects values in the forecast period which is based on the average value. For example, you are able to create algebraic equations using this tool. The final example I chose is Fourier Analysis tool. This tool solves problems of linear systems and also analyzes periodic data by using the FFT (Fast Fourier Transform) method which actually transforms the data. These tools greatly assist users with their business decisions by allowing companies to create data that can assist with payroll, expense reports, employee time management, customer information and can help complete...

Words: 729 - Pages: 3

Free Essay

...% % % dt = 1/100; % sampling rate % et = 4; % end of the interval % t = 0:dt:et; % sampling range % y = 3*sin(4*2*pi*t) + 5*sin(2*2*pi*t); % % Y = fft(y); % compute Fourier transform % n = size(y,2); % 2nd half are complex conjugates % amp_spec = abs(Y)/n; % % % figure; % subplot(3,1,1); % first of two plots % plot(t,y); grid on % plot with grid % axis([0 et -8 8]); % adjust scaling % % subplot(3,1,2); % second of two plots % freq = (0:size(amp_spec,2)-(1/(n*dt)))/(n*dt); % abscissa viewing window % plot(freq,amp_spec); grid on % % subplot(3,1,3); % second of two plots % freq1 = ((-size(amp_spec,2)+1)/2:(size(amp_spec,2)-1)/2)/(n*dt); % abscissa viewing window % FTy = fftshift(amp_spec) % plot(freq1,FTy); grid on % clear all close all clc x=[] for i = -5:1/1000:5 if i > 0.5 | i < -0.5 x = [x 0]; else x = [x 1]; end end figure;plot(-5:1/1000:5,x) dt = 1/1000; Xf1 = fft(x); n= length(Xf1); Xf = abs(Xf1)/n; Freq = (0:size(Xf,2)-1)/(n*dt) figure; plot(Freq ,Xf) freq1 = ((-size(Xf,2)+1)/2:(size(Xf,2)-1)/2)/(n*dt) Xf = fftshift(Xf) figure; plot(freq1,Xf) f=[-2:0.01:2] H1 = 1./sqrt(1 + (f/1).^2); figure;plot(f,H1); H = 1./sqrt(1 + (freq1/100).^2); REP_mod = H.* Xf; figure;plot(freq1,abs(REP_mod)); Xf1 = fftshift(Xf1) REP = H.*Xf1; REP = ifftshift(REP); IXF = ifft(REP); %IXF = abs(IXF); figure;...

Words: 266 - Pages: 2

Free Essay

...Jordan University of Science and Technology Faculty of Engineering Department of Mechanical Engineering Course: Graduation project Project Title: Experimental Modal Analysis Name: Hamzeh Ahmad Alqaisi I.D.: 20080025119 Instructor: Dr.Yousef Najjar Supervisor: Dr.Naem Alkhader Due date: 14/11/2012 EXPERIMENTAL MODAL ANALASYS ABSTRACT Experimental modal analysis has grown steadily in popularity since the advent of the digital FFT spectrum analyzer in the early 1970’s. Today, impact testing (or bump testing) has become widespread as a fast and economical means of finding the modes of vibration of a machine or structure. Contents TITLE PAGE NO. Nomenclature………………………………………………………………………………………………………………………………… CHAPTER 1: Introduction………………………………………………………………………………………………………………………………….. CHAPTER 2: Experiment setup……………………………………………………………………………………………………………………….. FRF Calculations……………………………………………………………………………………………………………………………. CHAPTER 3: Results………………………………………………………………………………………………………………………………………….. Conclusion…………………………………………………………………………………………………………………………………….. References…………………………………………………………………………………………………………………………………… Appendices Computer program………………………………………………………………………………………………………………………. CHAPTER 1: Introduction: Modes are used as a simple and efficient means...

Words: 2216 - Pages: 9

Free Essay

...TABLE OF CONTENTS PAGE NO. CHAPTER 1- INTRODUCTION 1.1 INTRODUCTION 6 1.2 WHAT IS OFDMA? 9 CHAPTER 2 - SUBCARRIER ALLOCATION SCHEMES 2.1 STATIC SUBCARRIER ALLOCATION 12 2.2 DYNAMIC SUBCARRIER ALLOCATION 14 2.2.1 THE PROPOSED DSA SCHEME 15 CHAPTER 3- FLOW CHART AND ALGORITHM 4.1 FLOW CHART 17 4.1.1 MAIN FLOWCHART 17 4.1.2 WHEN 2 USERS FALL IN RANGE NUMBERED 3,5,8,14 20 4.1.3 WHEN 3 USERS FALL IN RANGE NUMBERED 3,5,8,14 22 4.2 ALGORITHM 24 CHAPTER 5-BLOCK DIAGRAMS 5.1 CONTROL FRAME TRANSCEIVER 28 5.2 DATA TRANSMISSION 29 5.3 DATA RECEPTION 31 CHAPTER 6- IMPLEMENTATION AND RESULTS 6.1 MATLAB 34 6.1.1 INSTRUCTIONS USED IN CODE 35 6.2 CONTROL FRAME TRANSMISSION AND RECEPTION 39 6.3 DATA......

Words: 7933 - Pages: 32

Premium Essay

...------------------------------------------------- Contents * SECTION :2 * SECTION :3 * SECTION :4 * SECTION :5 ------------------------------------------------- SECTION :2 Multiply the following polynomials in z by using the fft algorithm a=1+2*z^{-1}+ 4*z^{-2}+7*z^{-3}+12*z^{-4}+25*z^{-5} b=1-3*z^{-1}+ 7*z^{-2}+15*z^{-3}-12*z^{-4}+13*z^{-5} Transform polynomials equation to vector ------------------------------------------------- a = [1 2 4 7 12 25]; ------------------------------------------------- b = [1 -3 7 15 -12 13]; ------------------------------------------------- % FFT function in Matlab using Circular Convolution, instead avoid the ------------------------------------------------- % affect of circular convolution, we add zero paddles at end of the vector ------------------------------------------------- ------------------------------------------------- a0 = [a zeros(1, 5)]; ------------------------------------------------- b0 = [b zeros(1 , 5)]; ------------------------------------------------- ------------------------------------------------- % Convolution in time domain equivalent the mutiply in frequency domain. ------------------------------------------------- % First equation doing convolution operation of two polynomials equation......

Words: 1316 - Pages: 6