6.003: Signal Processing Lecture 1 September 6, 2018

Transcription

1 63: Sigal Processig Workig wih Overview of Subjec : Defiiios, Eamples, ad Operaios Basis Fucios ad Trasforms Welcome o 63 Piloig a ew versio of 63 focused o Sigal Processig Combies heory aalysis ad syhesis of sigals, ime ad frequecy domais, covoluio ad decovoluio, filerig ad oise reducio, wih auheic, real-world applicaios i music, imagig, video Sepember 6, 28 Role of Aalysis i Sciece ad Egieerig Our commo goal wih oher sciece / egieerig edeavors is o model some aspec of he world aalyze he model, ad ierpre resuls o gai beer udersadig make model Model World aalyze (mah, compuaio) Resul ierpre resuls New Udersadig Classical aalyses use a variey of mahs, especially calculus We will also use compuaio which is applicable i may real-world problems ha are difficul or impossible o solve aalyically sreghes ies o he real world Desig as he Reverse of Aalysis We are ieresed o oly i aalyzig he behaviors of pre-eisig sysems, bu also i desigig ew sysems Sysem aalysis desig Behavior Aalysis ad Desig are differe ad complemeary aciviies Aalysis sars wih a precise saeme of he problem ad proceeds o a precise saeme of a resul coveioal problem ses Desig is more ope-eded, wih muliple possible soluios ha differ alog idiosycraic dimesios ha were o eve par of he origial problem saeme Egieerig Desig Problems adaped from R David Middlebrook Course Mechaics Lecure: Tuesdays ad Thursdays 2-3pm i -9 Labs: Tues ad Thur 3-5pm i 4-45, -5, 4-53, -9 Homework issued Tuesdays, due followig Tuesday a oo Drills: facs, defiiios, ad simple coceps olie wih immediae feedback (o graded) ieded as pracice ad self-assessme Problems ieded o improve problem solvig skills Eam-Type Problems: proveably correc soluios Egieerig Desig Problems: real-world applicaios (more ope eded) Advisory Group Weekly meeigs wih class represeaives help saff udersad sude perspecive lear abou eachig Teaively mee o Wedesdays a 3pm Ieresed? Sed o freema@miedu Two Miderms ad a Fial Eam

2 are fucios ha are used o covey iformaio may have or 2 or 3 or eve more idepede variables are fucios ha are used o covey iformaio depede variable ca be a scalar or a vecor y brighess (, y) y scalar: brighess a each poi (, y) vecor: (red,gree,blue) y a each poi (, y) soud pressure () are fucios ha are used o covey iformaio depede variable ca be real, imagiary, or comple-valued () = e j = cos +j si Coiuous ime (CT) versus discree ime (DT) () [] from physical sysems are ofe of coiuous domai: coiuous ime measured i secods coiuous spaial coordiaes measured i meers Compuaios usually maipulae fucios of discree domai: discree ime measured i samples discree spaial coordiaes measured i samples Samplig: coverig CT sigals o DT Recosrucio: coverig DT sigals o CT zero-order hold () [] = (T ) [] () T 2T 4T 6T 8T T T = samplig ierval Impora for compuaioal maipulaio of physical daa digial represeaios of audio sigals (as i MP3) digial represeaios of images (as i JPEG) T 4T 6T 8T T T = samplig ierval commoly used i audio oupu devices 2

3 Recosrucio: coverig DT sigals o CT piecewise liear Periodic sigals cosis of repeaed cycles (periods) periodic aperiodic [] () () = ( + T ) () T 4T 6T 8T T T [] = [ + N] [] T = samplig ierval N commoly used i rederig images Righ-sided sigals are zero before some sarig ime Lef-sided sigals are zero afer some edig ime wih fiie duraio are zero ecep for some rage of imes righ-sided () lef-sided () The miima ad maima of bouded sigals are fiie bouded ubouded () () T s T e [] [] [] [] N s N e Eve sigals are symmeric abou ime zero Odd sigals are aisymmeric abou ime zero Check Yourself Compuer geeraed speech (by Rober Doova) f() eve () = ( ) odd () = ( ) Lise o he followig four maipulaed sigals: f (), f 2 (), f 3 (), f 4 () How may of he followig relaios are rue? [] = [ ] [] = [ ] f () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 3 f() 3

4 y Check Yourself How may images mach he epressios beeah hem? 25y 25 25y 25 y Musical Souds as f(, y) are fucios ha are used o covey iformaio Eample: a musical soud ca be represeed as a fucio of ime soud pressure Alhough his ime fucio is a complee descripio of he soud, i does o epose may of he impora properies of he soud f (, y)=f(2, y)? f 2 (, y)=f(2 25, y)? f 3 (, y)=f( 25, y)? Musical Souds as Eve hough hese souds have he same pich, hey soud differe piao oboe cello hor violi bassoo alosa bassoo 262 sec I s o clear how he differeces relae o properies of he sigals (from hp://heremimusicuiowaedu) Musical as Sums of Siusoids Oe way o characerize differeces bewee hese sigals is epress each as a sum of siusoids f() = cos cos cos 2 ( ck cos kω o + d k si kω o ) k= si si si 2 Sice hese souds are (early) periodic, he frequecies of he domia siusoids are ieger muliples of a fudameal frequecy ω o Harmoic Srucure The weighs of he compoes (c k ad d k ) describe he harmoic srucure of he sigal ( f() = ck cos kω o + d k si kω o ) k= c 2 k +d2 k harmoic # DC fudameal secod harmoic hird harmoic fourh harmoic fifh harmoic sih harmoic ω Harmoic Srucure Harmoic srucure plays a impora role i geeraig he characerisic souds of musical isrumes piao bassoo violi (from hp://heremimusicuiowaedu) piao bassoo violi k k k 4

5 Two Views of he Same Sigal The harmoic epasio provides a aleraive view of he sigal ( f() = ck cos kω o + d k si kω o ) k= We ca view he musical sigal as a fucio of ime f(), or as a sum of harmoics wih ampliudes [c, d, c, d, ] Boh views are useful For eample, he peak soud pressure is more easily see i f(), while cosoace is more easily aalyzed by comparig harmoics We call such a aleraive view of a sigal a rasform Trasforms There are may kids of rasforms ad epasios based o differe ypes of basis fucios Eamples: epasios based o derivaives of fucios Maclauri epasio: f() = f() + f () + f () 2 + f () 3 +! 2! 3! The fucio of ime f() is represeed by is derivaives a = Taylor epasio abou = a: f() = f(a) + f (a) ( a) + f (a) ( a) 2 + f (a) ( a) 3 +! 2! 3! The fucio of ime f() is represeed by is derivaives a = a Series Represeaios of Maclauri series f() = f()! + f ()! + f () 2 2! + f () 3 3! + Basis fucios: /! 2 /2! 4 /4! Noice ha eve powers of are eve fucios of ime, ad odd powers of are odd fucios of ime The epasio for > implicily deermies he fucio for < /! 3 /3! 5 /5! Series Represeaios of To model a sigal ha sars a a paricular ime (say = ), we eed a differe se of basis fucios f() = f( + )u () + f ( + )u () + f ( + )u 2 () + f ( + )u 3 () + Basis fucios: u() = u () u 2 () = 2 2! u() u 4 () = 4 4! u() u () =! u() u 3 () = 3 3! u() u 5 () = 5 5! u() The firs of hese fucios is he ui sep u() = u () Subseque fucios are iegrals of heir predecessors u + () = u (τ)dτ Series Represeaios of Series Represeaios of This se of fucios [u (), u (), u 2 (), ] is closed uder iegraio, ie, if f() ca be epressed as a sum of hese fucios, he he iegral of f() ca also Eample: f() = cos()u() = u() 2 2! u() + 4 4! 6 8 u() u() + 6! 8! u() + = u () u 2 () + u 4 () u 6 () + u 8 () + g() = f(τ)dτ = si()u() = u() 3 3! u() + 5 5! = u () u 3 () + u 5 () u 7 () + u 9 () 7 9 u() u() + 7! Sice f() ca be wrie as a sum of fucios from he se, i follows ha g() = f(τ)dτ ca also f() is represeed by he sequece of coefficies [,,,,,,,, ] g() is represeed by he sequece of coefficies [,,,,,,,, ] Same se [u (), u (), u 2 (), ] is o closed uder differeiaio Is here a meaigful way o defie he ime derivaive of a sep? Le u () represe he followig sigal u () 9! u() + ad u () represe is derivaive: he u () lim u () = u() ad lim u () = a impulse 5

6 Impulse Fucio Le δ() represe a impulse, which is defied by wo properies: δ() = for all, ad δ()d = area uder impulse = Focus o Fourier Represeaios (Siusoidal Bases) Harmoic srucure deermies cosoace ad dissoace ocave (D+D ) fifh (D+A) D+E δ() These defiiios oly make sese as limis While moivaed by a square pulse (u ()), ay fucio wih ui area ad fiie duraio also works ime(periods of "D") D' A E Wih all of is area i he shores possible ierval of ime δ() is complemeary o he broades sigal () = cosa D D harmoics D Focus o Fourier Represeaios (Siusoidal Bases) Siusoidal decomposiios are useful hroughou physics Diffusio equaio: f 2 f 2 eamples: hea rasfer (origial applicaio), chemical raspor Wave equaio: 2 f 2 2 f 2 eamples: ligh, radio waves, -rays, acousics, fluid dyamics Labs Focus o Applicaios The broad applicabiliy of Fourier mehods provides may ieresig applicaios Eamples iclude music eg, music figerpriig ala Shazam image processig eg, MRI video eg moio magificaio Reaso: f(, ) f(, ) where f(, ) = e j(ω k) = cos(ω k) + j si(ω k) Today s Lab: Souds as Geerae audio sigals ad ivesigae heir properies Break io secios Fid your secio umber o he 63 websie: hp://miedu/63 uder Week, Lab B 6