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# Intro ECE 2025

GPA 3.64

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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 James McClellan in Fall. Since its upload, it has received 6 views. For similar materials see /class/233878/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

EE2025 Fall2004 Lecture 7 Fourier Series amp Spectrum 1 0Sept2004 Quiz 1 Info Quiz 1 on 13Sept04 Monday Coverage HW 1 2 and 3 Allowed one page of notes Handwritten Review Session Planned Sunday 12Sept 6pm in ECE Auditorium Check WebCT for other related announcements Old Quizzes amp Problems are linked via WebCT Word from Previous Semesters 8282004 EEVZEIZS Simgamma 1M The Rules Sinusoidal Synthesis Quizzes NO makeups given Next Quiz would count forthe one missed IF excused Excused Absence c 8282004 Must be written by an official Notify ahead of time via email onsult INFO on WebCT for more details EEVZEIZE SpringVZElUS 1M 3 Use ShortDuration Sinusoids Amp Phase Frequency amp Duration l kr l 1 Freq quotI change every RAME Then ADD several sinusoids together 8282004 EEVZEIZS SpringVZElEl 1M ANALYSIS gt SYNTHESIS Sine Synthesis SPEECH azmnm Esmz swmm m FRAME Length 10 millisec Examples Original I 9 sinusoids perframe I 4 sinusoids I 2 sinusoids Need to SMOOTH Boundaries auxmm Esmz vayvmm We a TimeVarying FREQUENCIES Diagram ASINES SYNTHESIZED SIGNAL 39 inn FhEOUEch Hz 44 E 8 E 1 s m U k g It azmnm Esmz vayvmm m 4SINES Spectrogram 44 E 8 E 1 s m U k g It auxmm Esmz vayvmm m a 9SINES Spectrogram 2SINES Spectrogram 3 E 1 5 M U E E azmnm Esmz swwm m 3 E 1 5 M U E E auxmm Esmz vayvmm m TIME SIGNALS COMPARE TIME SIGNALS ZOOM OFIIGINAL SIGNAL 2SINES SYNTHESIZED SIGNAL azmnm Esmz Svummm m 2SINES SYNTHESIZED SIGNAL auxmm Esmz vayvmm m READING ASSIGNMENTS LECTURE OBJECTIVES This Lecture Fourier Series in Ch 3 Sects 34 35 amp 36 Replaces pp 6266 in Ch 3 in DSP First Notation ak for Fourier Series Other Reading Next Lecture Sampling 8282004 EEVZEIZE SpringVZEIEIB 1M 43 ANALYSIS via Fourier Series For PERIODIC signals xtT0 xt SPECTRUM from Fourier Series ak is Complex Amplitude for kth Harmonic 8282004 EEVZEIZS SpringVZElEl 1M 14 SPECTRUM DIAGRAM Harmonic Signal Recall Complex Amplitude vs Freq 250 100 0 100 250 8282004 EEVZEIZE SpringVZElUS 1M 15 8282004 EEVZEIZS SpringVZElEl 1M 16 62 A m m 2 Sit Example Example xt 16197 3ej3m 8 8 In this case analysis i EEVZEIZS Spimgi ch EZEZEIEM just requires picking off the coef cients 15 0 1 Frequency in Hz EZEZr STRATEGY xt a ak I ANALYSIS I Get representation from the signal I Works for PERIODIC Signals I Fourier Series I Answer is an INTEGRAL over one period 1M FS Rectified Sine Wave ak nmimme a EEVZEIZS Spimgi EZEZ D34 FS Rectified Sine Wave ak SQUARE WAVE EXAMPLE e j27rT0k 1t To e j27rT0k1t T02 ak 12T0lt j27rT0gtk 1 0 12 j27rT0k1 0 1 0 g t lt T 0 1 j27rT0k 1T02 1 1 j27rT0k1T02 1 x0 1 475k 1 e 47rk1 e O 3T0 S 1 lt To 1 j7Fk 1 1 j7fk1 47rk 1e 1 47rk1 6 1 for To SCC A X 0 k odd 1 k1 k 1 k WCLD 1 1 2 k 1 t 2 O 01 02 004 t 1 k even 2 275k 1 8282004 552025 Spring2003 we 22 Fourier coefficients ak Spectrum from Fourier Series ak IS a function of k mo zit004 ZZEQS ilk k 13 Complex Amplitude for k th Harmonic ak O k 2 i2 i4 This one doesn t depend on the period TO 1 1 3 k O 7 k i1i3 2 if 39 1Hquot J 5 7 ak 0 k24 I j 12 L 239 239 i 5 1 2 5 k 0 9w 7w 5n I I 9 175 75 25 0 25 75 125 175 225 8282004 EE2025 Spring2003 We 23 8282004 EE2025 Spring2003 We 24 Fourier Series Synthesis HOW do you APPROXIMATE xt T 0 ak T Jxte2 T0ktdt 0 0 Use FINITE number of coefficients 8282004 EE2025 Spring2003 We 25 Fourier Series Synthesis 00 611 at a3 75 a i 5 7 9 I I l i I I I I 13f0 7f0 3f0 0 f0 3f0 5f0 7f0 9f0 13f0 fHZ Spectrum Plot To 2 Period N Number of Coef cients ak kfo versus f A A Fourier Synthesis 0k Approximate the Signal xN t Fourier Analysis Extract Sinusoids V V xt gt 7i N ak I fAfwfanIyldr f0 i HZ XNU Z aka227knt quot N 0 T0 1 L 0 mt r 1 t m 0 n n 2 m 0 n m 2 Synthesis 1 st amp 3rd Harmonics ya l cos27 25t cos2 z75gt quot 2 7 37 A l 2 L J Jr j J J 39 L L L g 37 5 9 7T 97TI 7HI 5HI i 39 i I I I 175 75 25 0 25 75 125 175 225 1 V V V 03 06 04 02 0 A A A Lv v Lv V Lv v 8282004 034 4302 0 umgti sec 004 006 003 27 Synthesis up to 7th Harmonic 1 2 2 5 y t cos50m sgt1n150m 2 s1 n2 507 l srn350m 2 7 37539 57 7739 l 2 L 1 7T 7T j J J j j J J 371 371 57139 77f 97 9H1 7 1 5711 i 7 i l l 1 175 75 25 O 25 75 125 175 225 1 VAVAVA AVAVAVA AVAVAVA oi i 06 04 x 0 A A A A A A A A A LU V V V LU V V VI LV V V VI 8282 039J4 002 0 002 004 006 008 28 1 time in sec EE2025 Fall2004 Lecture 6 Fourier Series Analysis 3Sept2004 General Info Help Sessions M6 T 6 and W6 Of ce Hours Visit any Prof or TA Bulletin Board OFFICIAL ANNOUNCEMENTS Quiz 1 on 13Sept Coverage HW1 2 and 3 Old Quizzes amp Problems are linked via WebCT Lab 3 bring headphones Lots to read Concept Maps Lab 2 report due at beginning of lab Ask your grading TA about hisher format Prob Set 3 due NEXT Week Prob Set 4 will be clue during week of 13Sept 8312004 EEVZEIZS SpringVZEIEIA 1M The Rules INSTANTANEOUS FREQ of the Chirp I Quizzes NO makeups given Next Quiz would count forthe one missed IF excused Excused Absence Must be written by an officialquot Notify ahead of time via email Consult INFO on WebCT for more details Late Labs are 10 points per day No late Homework 8312004 E32025 SpringVZElUA 1M Chirp Signals have Quadratic phase Freq will change LINEARLY vs time xt Acosar2 Bt p gtirar2 8rp 8312004 E32025 SpringVZElUA rm CHIRP SPECTROG RAM i iEEIEI aw CHiRP CENTERED at aan Hz Question Create a Chirp Chirp should start at 200 Hz and end at 3200 Hz and last for 15 sec xt Acosat2 it p 3 a aimmm aimmm EEmz Swinva m READING ASSIGNMENTS LECTURE OBJECTIVES This Lecture Fourier Series in Ch 3 Sects 34 35 amp 36 Replaces pp 6266 in Ch 3 in DSP First Notation ak for Fourier Series Work with the Fourier Series Integral Other Reading ANALYSIS via Fourier Series Next Lecture More Fourier Series 39 Formsignals XtTo Xt Later spectrum from the Fourier Series aimmm EEmz Swinva m aimmm EEmz Swinva m HISTORY Jean Baptiste Joseph Fourier 1807 thesis memoir I On the Propagation of Heat in Solid Bodies Heat Napoleonic era Joseph Fourier lived from 1768 to 1830 Fourier studied the mathematical theory of heat conduction He established the partial differential equation governing heat i fusion and solved It by using Infinite series of trigonometric functions wquot i i i i unininin Findoutmoreat EGWUUA Eszuzs Spmgi m lME 9 8312004 hnpllwwwrmstory mcsstrandrews acuklhistorylMatnematicianleourierhtml SPECTRUM DIAGRAM Harmonic Signal Recall Complex Amplitude vs Freq M 831 gum EEVZEIZS springrz m Mo 12 851mm sazuzs SPnngrz m lMI Fourier Series Synthesis Harmonic Signal 3 Freqs X Akejgoquot 0312004 E52025 SpringZOiM NC 13 a H 3 50 l l I 1 l 3 3I 50 30 10 0 10 7 in Hz Sum of Cosine Waves with Harmonic Frequencies i i i i i i i 13312004 Time 1 sec SYNTHESIS vs ANALYSIS STRATEGY xt a ak I SYNTHESIS I ANALYSIS I Easy I Hard I Given wkAkltk create I Given Xt extract Xt mew44 I How many I Synthesis can be I Need algorithm for HARD computer I Synthesize Speech so that it sounds good 8312004 E52025 Spr1n972004 m 15 I ANALYSIS I Get representation from the signal I Works for PERIODIC Signals I Fourier Series I Answer is an INTEGRAL over one period 8312004 E52025 SprmngOOlI m 16 INTEGRAL Property of expj ORTHOGONALITY of expj INTEGRATE over ONE PERIOD T0 T J e j27rT0mzdt ITO e j27rT0mz 0 0 127rm 0 2 To e J39Z 39m j27rm T0 Ie j392 T0mldt 0 0 8312004 EEVZEIZE Spymum 1M 47 PRODUCT of expj and exp j T 1 To 0 k 7quot f Jej27139T0Zte j27rT0ktdt 0 0 k E 8312004 EEVZEIZS Spymum 1M 18 SQUARE WAVE EXAMPLE F8 for a SQUARE WAVE ak 1 osm o xt 0 TOStltTO for TO 2004 sec xt 139 l 702 61 02 0304 t 8312004 E32025 Spyquotaim 1M 19 T 1 0 ak my WW dz k at 0 To 0 02 1 1e 27r04ktdt e j2 O4kt ak 04 04 27rk04 0 ej k 1 1 1k j27zk j27zk 8312004 E32025 Spyquotaim 1M 20 0 General Information EE2025 Fall2004 Bulletin Board OFFICIAL ANNOUNCEMENTS Old Quizzes amp Problems are linked via WebCT Lecture 5 Quiz 1 on 13Sept04 Monday I All d f t H d 39tt Periodic Signals iIarmonlcs CaTchJTatjgepiffnitZdn es an W en amp TimeVarying Smusouds Review Session planned for Sunday 12Sept 6pm 30Aug04 HW 3 due NEXT WEEK In Reeltatlon Solutions will be posted Thurs evening Concept Map for Chapter 3 Lab Info i Prepare for Online PrePostLab Questions Practice version is available Take advantage of Help Sessions Lab 2 Report Turn in at beginning your lab time Writeup lab report on Beamforming Discuss lab report standards with your TA Miscellaneous ERRORS ALWAYS Check Bulletin Board Complete INSTRUCTOR VERIFICATION in Lab showing frequency delermlned m n by variations wnn tlme do i representation of are symmetric about navrng linear 5 o multiplying signal by 932004 gems spyquotgaunt in 3 932004 EEVZEIZS SpringVZEIEIA ill READING ASSIGNMENTS Problem Solving Skills This Lecture Math Formula Plotamp Sketches 39 Chapter 3 Sections 32 and 33 Sum of Cosines Stversust Chapter 3 Sections 37 and 38 39 Amp Freq Phase 5 E 39 Spectrum I Next Lecture Recorded Signals MATLAS Speech Numerical 39 FOUI39IeI39 Serles Music Computation sections 34 35 and 36 No simple formula Plotting list of numbers LECTURE OBJECTIVES SPECTRUM DIAGRAM Signals with HARMONIC Frequencies Add Sinusoids with fk kf0 N 39 j7Z393 j7Z393 xl A0 ZAk cos2 Jk 4ejr2 7e 7e 4ej7r2 kzl I L5 7 f FREQUENCY can change vs TIM I Chirps a I 250 100 0 100 250 Introduce Spectrogram Visualization specgramm x0 10 14 COS2 1 00y 3 plotspec m 8 75 932004 EEVZEIZS SpringVZElUA 1M 7 Recall Complex Amplitude vs Freq A 932004 Eszuzs SpringVZEIEIA 1M 8 SPECTRUM for PERIODIC Nearly Periodic in the Vowel Region Period is Approximately T 00065 sec Speech BAT 02 39 04 08 L V V 028 285 029 0295 time sec 932004 EE72025 SpringZOOA lMc PERIODIC SIGNALS Repeat every T secs Definition xt xt T xt 003231 Example Speech can be quasiperiodic 932004 E52025 Sprln92004 lMc Period of Complex Exponential xt 6 Hm xt m 36 1 gtaT27rk 11 Harmonic Signal Spectrum N xt A0 ZAk cos27rk or pk k1 N W X0 2 Xke w ngemw k1 E52025 Sprln92004 NC 12 Define FUNDAMENTAL FREQ Harmonic Signal 3 Freqs N xl A0 ZAk cos2 gok k1 5 a fk kfo 00 Z 39fo 50 30 10 o 10 30 50 70 Hz f0 fundamental Frequency T 0 fundamental Period 932004 E32025 Spyquot92004 N 13 932004 E32025 Spy4004 N 14 POP QUIZ FUNDAMENTAL IRRATIONAL SPECTRUM Here s another spectrum 10 7ej7r3 7e j7r3 397r2 397r2 4e 1 4e I L 250 100 0 100 250 932004 E52025 SpurgVZEIEIA m 5 932004 E52025 SpurgVZEIEIA m 6 Harmonic Signal 3 Freqs NONHarmonic Signal Sum of Cosine Waves with Nonharmonic Frequencies Sum of Cosme Waves wnh Harmonic Frequencles 0 02 014 0i6 08 12 114 16 18 2 39 Timemec 39 Timesec 932004 E92025 SprinngElEllt 1M 17 932004 E92025 SprinngElEllt 1M TimeVarying FREQUENCY ANALYSIS FREQUENCIES Diagram Now a much HARDER problem Given a recording of a song have the computer write the music n Can a machine extract frequencies I Figure 318 Sheet musxc notation is a timcifrequency diagram Yes If we COMPUTE the spectrum for xt During short intervals Time is the horizontal axis 932004 E32025 SprinngElElzl in 19 932004 E92025 SprinngElElli in SIMPLE TEST SIGNAL Cmajor SCALE stepped frequencies Frequency is constant for each note Frequencies of C Major Scale 700 600 500 400 A Frequency Hz 03 D C 0 200 400 600 800 1000 1 200 1400 1 600 Time msec 200 932004 21 Rrated ADULTS ONLY I SPECTROGRAM Tool I MATLAB function iS specgramm I SPFirst has plotspec m amp spectgr m ANALYSIS program Takes xt as input amp Produces spectrum values Xk Breaks xt into SHORT TIME SEGMENTS I Then uses the FFT Fast Fourier Transform 932004 EE2025 Spring2004 No 22 SPECTROGRAM EXAMPLE FREQUENCY Hz Two Constant Frequencies Beats M BEAT SIGNAL FREQS 672 Hz and 648 Hz 800 700 BEATS F0 660 Hz Fm 12 Hz O O O 01 O O b O O 012 014 016 018 02 TIME sec 00 O O CENTER FREQ 660 Hz I llllnnlll A39I39IRIIquot EDEn IVIUIJUI HIII U rnEu N O O h I N L O O O O 005 01 015 02 TIME sec AM Radio Signal I Same as BEAT Notes 6 127mm e j27z660t ej27r12t e j27r12t 392 2 392 2 392 4 392 4 41 6 7567 le J 7567 le 7r6 8te J 7r6 8t 932004 EE2025 Spring2004 ch 24 SPECTRUM of AM Beat 4 complex exponentials in AM 1 1 i i 4l l4 4l l4 1672 648 o 648 672 932004 552025 Spring2004 We 25 STEPPED FREQUENCIES Cmajor SCALE successive sinusoids Frequency is constant for each note Frequencies of CMujor Scale 700 600 500 Frequency Hz 400 300 200 0 200 400 100 800 1000 1200 1400 1600 932004 Time msec SPECTROGRAM of CScale n Sinusoids ONLY II I I I a I I I a a l l V A O E d v 700 600 500 400 Frequency Hz 300 200 932004 200 0 200 400 600 800 1000 1200 1400 27 Time msec Spectrogram of LAB SONG Beethovens FIFTH Robby GRIFFIN FREQUENCY Hz 932004 TimeVarying Frequency New Signal Linear FM Frequency can change vs time Continuously not stepped I FREQUENCY MODULATION FM CHIRP SIGNALS Linear Frequency Modulation LFM 932004 EEzuzs swingzum 1M 29 Called Chirp Signals LFM I Quadratic phase Freq will change LINEARLY vs time Example of Frequency Modulation FM Define instantaneous frequency 932004 EEznzs ail2004 1M 30 INSTANTANEOUS FREQ INSTANTANEOUS FREQ of the Chirp Definition x0 mesa0 3 allt 120 For Sinusoid xt Acos27zf0t 1 Mr 2 27zf0t 0 2wiltrgtwltr2zrfo A 932004 EEVZEIZS Spur372mm 1M 3 Chirp Signals have Quadratic phase Freq will change LINEARLY vs time xt Acosat2 Bt p gt Mr 2 052 8tp 932004 EEznzs ail2004 1M 32

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