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# Intro ECE 2025

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This 0 page Class Notes was uploaded by Cassidy Effertz on Monday November 2, 2015. The Class Notes belongs to ECE 2025 at Georgia Institute of Technology - Main Campus taught by Christopher Rozell in Fall. Since its upload, it has received 38 views. For similar materials see /class/233880/ece-2025-georgia-institute-of-technology-main-campus in ELECTRICAL AND COMPUTER ENGINEERING at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

ECE2025 Fall 2010 LECTURE 2 Complex Numbers School of Electrical amp Computer Engineering Georgia Institute of Technology August 27 2010 8302010 ECE2025 rel2mm JMLVEHJ TSQUARE Info Check the Forums amp Announcements for msgs MAKE YOUR OWN POSTINGS PDF Files on Web Lectures are being posted 4 per page Lab 1 has been posted Get PDF file of Lab 1 from Web Hard copy of InstructorVerification Sheet HW 1 was posted as PDF HW 1 due next week at recitation HW 2 will be posted soon 8302010 ECE2025 Fallzm JML39BHJ INFORMATION MATLAB issues ask your TA for help LABS start NEXT week Monday Attend correctassigned section in Klaus2440 Login for ECE Lab computers Georgia Tech password Select your Windows Domain to be AD RECITATIONS Bring Calculatorto Recitation next week Practice Complex arithmetic Attend your assigned time ACTIVE participation 5 POINTS the crucial 5 POINTS can make a difference 8302010 ECE2025 rel2mm JMLVEHJ Homework Info HWs will be posted on SunMon Covered in Rec during the following Week Due the week after that 7 days later Format info on t square Cover page for Homework Prob Set 1 due in RECITATION next week At the beginning of class Solutions will be posted to tsquare after the last Recitation on Thursday afternoon 8302010 ECE2025 Fallzm JML BHJ Homework Formatting Honor Code Include a cover page with 39 Written BXChange NO Name E t h NO Lab section ie L05 L20 etc 9 romo 9X0 angev 39 ReCita on Props name Spoken Discussion OK after that 39 DOWHIOad example quot40m 115 quotare Work out and write the solutions yourself Write on ONE side only Use Engineer s paper or plain white paper 39 Brtgm Linei tlf YOILJtQ fe 0r rtehceiVi Wd rittentr ee ronicmaeria o romo ersu ens i sa STAPLE STAPLE STAPLE W Intelligent Tutoring System READING ASSIGNMENTS 0 Login with GT Login httpitsvipgatechedu This Lecture 0 1500 problems organized by chapter in SPFirst textbook Optional Chapter 2 Sects 23 to 25 Appendix A Complex Numbers 39 quot Appendix B MATLAB Next Lecture finiSh Chap 2 Section 26 to end 0 Each chapter opened for limited time 0 Full participation would be 10 questionschapter 0 Replace lowest HW score with ITS score 0 ITS score 75participation 25correct mmum Rang mm mm 7 wan2mm EEE2E125 Faiizmu imam LECTURE OBJECTIVES COMPLEX NUMBERS Understand complex numbers and how to do the arithmetic Lead to introduction of an ABSTRACTION Complex Numbers represent Sinusoids Complex Exponential Signal next lecture Coming Monday 8302010 ECEZUZE FallQmU JMrrEHJ 9 To solve 22 1 Z j Math and Physics use 2 i Complex number 2 x jy 8302011 ECEZDZS Fallrzm JMtrEHJ 10 PLOT COMPLEX NUMBERS COMPLEX Addition VECTOR Addition E y Real part if 7 1 En 2 15 Imaginary part Re5JO 5 13 39 5 j0 r Re2j5 2 Im2j5 5 3 j3 4 3 8302010 ECEZUZE FallVZEIlEI JME EHJ 11 z3 21 22 4 j32j5 42j 356j2 y Z 2 2 15 DISPLACED m VERSION of z X 5 1 lt z 3 6 12 E ReZ ziZRezi ReaAxis x i i Z14j3 Im ZziZImzi I 8302010 ECEZUZE FallVZEHEI JME EHJ 12 Coordinate Systems RectangularCartesian y Notation for a point in a 20 space 3 z xy 23 I Polar z r 6m tan l 3 polar Bl3 l7 l Eczmza rmmu Maw 13 POLARH RECTANGULAR Relate xy to r6 INeed a notation for POLAR FORM wan2mm Eczzuz rmmu imam lo Dual Representations Z 352 Zj5 zxjyrej9 Real part 4 lmwmary AXlS 333J e5n4 3xEe737r4 ElmZulu Eczmza rmmu JMvBHJ 15 Euler s FORMULA Complex Exponential 39 Real part is cosine sine Imaginary part is sine W Magnitude is one rejg rcos6 jrsin6 wan2mm EEE2E125 rmmu JMvBHJ we Something Interesting Frequently Encountered Values The derivative of a sinusoidal function is also smuso39dal icosg sin 0 ising cos 0 d0 d0 The derivative of an exponential is also 39 d exponential eb6 beb baconstant Check 2 3 s1n6 cos6 z cos6js1n6 d6 J 39 6 If 2ng 26 3 d6 d6 je6 jcos6jsin6 sin6jcos6 830ZUlU ECEZCIZS Fallrzm JMcrEHJ i7 Common Values 11j0ej2quot 1 1 jO WM j Z j e jej72 ej37I2 392 ZZ01ZOeJ M foranyz g 4 lijz eJ i1j 8302Eil0 ECEZEIZE FallVZEIlEI JMrrEHJ 18 Complex Exponential Has Advantages More Examples of Advantage Ease in derivation oftrig identities Ex eSiwz 61616162 cos 01 j sin 01cos 02 j sin 02 cos 01 cos 02 j2 sin 01 sin 02 jcos 01 sin 02 sin 01 cos 02 cos 01 cos 02 sin 01 sin 02 jcos 01 sin 02 sin 01 cos 02 cos6 1 62 Reej39913992 cos 61 cos 2 sin 61 sin 62 sin6162Imej399192 cos 61 sin 62 sin 61 cos 62 8302010 EcEznzs FallVZEHEI JMcrEHJ 19 Recall ReZzl ZRezl Imz 21 21mg 2 lt3 Igze dgie 61 j 62 91 11 sin0 jcos0 because icosB isin8 isinB cos 8 d8 8 Re 66619 JReegd6 Icos d sin 6 1m 66619 J1me d6 Isin 6d6 cos6 Also check 8302010 ECEZEIZS FallVZEHU JMLVEKHJ 2n Other Important quotFactsquot MULTIPLICATION In real analysis M 1 In complex analysis 1 ej392n7r1m ejZnIrm Id ejamlwm ej Z e for arbitrary integern m 1 Z 9 J ejO zl jz3j 3 j27z3 1 JE 3 M 6 3T 1 He 1 1 3 ej4 3J e 3 12 8302010 ECEZEIZS FallZmU JML39EHJ 2 l CARTESIAN use polynomial algebra Zl X Zz x1jy1gtltx2 132 x1x2 y1y2 jx1y2 x2y1 M easier because you can leverage the properties of exponentials 161 162 J3961 92 ZIXZZ 1 16 X726 irze Multiply the magnitudes and add the angles 8302010 znna JH McClellan1 va Sclraler 22 DIVISION Complex Multiply VECTOR ROTATION CARTESIAN use complex conjugate to convert to multiplication x1jy1 Z12 Z12 Zz x2jy2 222 Izzl2 POLAR simplerto subtract exponents 13961 Z1 ie Zie gfgz 13962 7392 z2 rZe 8302010 ZEIEIS JH McClellan amp WW Scharer 23 Multiplicationdivision scales and rotates vectors y 2122r1r28j9162 lt1 S 5 9 zzzrzel 2 E z1r1eIe1 Real Axis 8302010 ZUEICLJH McClellan aRW Scharer 24 ECE2025 Fall 2010 LECTURE 1 Intro to Signal Processing amp Sinusoids School of Electrical amp Computer Engineering Georgia Institute of Technology August 23 2010 Introduction to Signal Processing What is a signal A signal is a characterization of information represented as a function of time or space a sequence of symbols or simply a hand gesture For our purposes we are most interested in signals that are function oftime examples sound waves speech music and the like visual images pictures video neural activities radio waves and others For processing these functions are turned into sequences of numbers SHEIND EERDZ mmm imam 2 Audio and Video Applications Image Processing Ema2mm Ecauz mm Maw Image sharpening and other enhancements SHEIND Ecauz Medical Diagnostics and Treatment Medical Imaging Mmophane r 39t u n J Eat with ochlear implant 8182010 ECEZEIZE Femu N am 5 8182010 ECE2EI25 FaiZCHEI JME39BHJ Neural Signals Communications Neural spikes 8182010 ECE2EI25 FaHrZEIEI JMoEHJ 7 8182010 ECEZEIZS FaH 2mm JMcrEHJ 8 Financial Markets Main Idea m w Lots of different application areas many in EE for historical reasons Want to decompose complicated signals into simpler components to make them W Easier to analyze and understand Easier to process to produce desired effects A Use mathematical abstractions to let us develop tools useful for ANY signal Ellamm EcEznzS mm Maw SHEEMU Ecauz mmm imam COURSE INFORMATION Homework Info LABS HWs will be posted on SunMon Covered in Rec during the Week I Room 2440 In Klaus BU de Due the week afterthat 7 clays later I MATLAB based computer projects HW Set 1 already posted on t square Start from 2quotd week Format info on tsquare Cover page for Homework MATLAB Help in the evenings Tentative Times This week Klaus 2440 Wed amp Staple staple staple Thurs at6pm Prob Set 1 due in RECITATION next week RECITATIONS always in VL361 39 Atlne beginning l Class Solutions will be posted to t square I EMPHASIS on PrOblem SOlVlng alterthe last Recitation on Thursday a ernoon BMBZEHEI ems mm Maw mum Ecauz mm mm Computer Ownership I Hardware I All students own computers I Software I MATLAB is part of the required suite I ECE2025 Lab I Bring your own laptop if possible Bl BZD l D ECE2EI25 FallVZEI lEI iMeEHJ REMINDERS I TSquare Login I GT username amp password I ECE Computer Account I All ECE Students have an account I www ace help gatech t edufaq index A html I HW 1 posted get the PDF from t square I Due at recitation next week EMSZEN El ECEDZ FalLZEIlEI Memo tsquaregatechedu Why is this class hard INTRODUCTION To SIGNAL PROCESSING quot 9 yua AMDUMEMEHB lriLrudLlclJuri in Signal Prunessmg REED 5 ECE2025 Introduction to Signal Processing Assignments Fall 2010 Gradebuuk EmailArchwe Quick Links Site in site Stals gt Lecture Schedule 3 Notes 52m N gt Homework Pmblems gt Lab Projects gt Quizzes and Exams gt Course Guidelines gt M gt MATLAB Movies gt SP First CDROM gt PruressurTA Office Huuis EMS2mm I It is an introductory course topics have strong sequential dependency I Many people don t I Manage their time well so they get behind making the learning of new concepts harder and harder I Use the course resources recitation professors and TAs all with office hours I Read the book EMS2m u ECEDZ FalLZEIlEI iMeEHJ READING ASSIGNMENTS COURSE OBJECTIVE This Lecture Chapter 2 pp 917 Appendix A Complex Numbers Appendix B MATLAB Chapter 1 Introduction 8182010 ECEZEIZE FallZEI1D lMcrEHJ 17 Students will be able to Understand mathematical descriptions of signal processing algorithms and express those algorithms as computer implementations MATLAB codes What are your objectives What do you expect to learn in this class 8182010 ECEZEIZE Faiizmn lMcrEHJ 18 WHY USE DSP LECTURE OBJECTIVES DSP Digital Signal Processing Digital means computational Computational algorithms programming are flexible amp easy to implement Applications provide a physical context Mathematical abstractions lead to generalization and systematic design of new processing techniques 8182010 ECEZEIZS Fall 2010 lMcrEHJ 19 Write general formula for a sinusoidal waveform or signal as function of time From the formula plot the sinusoid versus time What s a signal in general It s a function of time or space or both We are interested in its mathematical representation 8182010 ECEZEIZS Fa112E1E lMcrEHJ 20 TU NING FORK EXAMPLE TUNING FORK A440 Waveform CDROM demo A is at 440 Hertz Hz E Waveform is a SINUSOIDAL SIGNAL Computer plot looks like a sine wave This should be the mathematical formula SHE20m ECEZEIZS FallVZEIlD iMoBHJ Zl AMPLlTUDE ElEZEHEI AMPLlTUDE ZOOM TIME lrnillisec In on TWO PERIODS arl ll i l Gilli glut Mill 1 w TIME millisec 4 SPEECH EXAMPLES Speech Signal BAT E W l l time EMSZulu ECEZDZS FallVZEIlH iMoEHJ v WNNMEE Nearly Periodic in Vowel Region Period is Approximately T 00065 sec Speech BAT BlEZEHEI 0285 029 tlme sec ECEDZS FallVZEIlEI iMcrEHJ 0295 SPEECH SIGNAL SINES and COSINES More complicated than a single sinusoid Theory tells us Vowel sounds are essentially periodic and can be reasona Iy approxima ed by a sum of several sinusoids where are these sinusoids their frequencies Use FOURIER Spectral ANALYSIS Break signal into its sinusoidal components the FREQUENCY SPECTRUM Synthetic AH Always use the COSINE FORM Sine is a special case sinat cosat Xixn i scans mm mm m SINUSOIDAL SIGNAL EXAMPLE of SINUSOID A cosa t go 39 AMPLITUDE 39 PERIOD in sec Magnitude I 39 FREQU ENCY Radianssec I PHASE ec Hertz cycless a27z39f Xlxn w scans mm mm 2 Given the Formula Make a plot Sinusoidal Waveloml 2 a Time secs xiimnin scans mm mm H

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