6.003: Signal Processing Lecture 2a September 11, 2018

Transcription

1 6003: Sigal Processig Buildig Block Sigals represeig a sigal as a sum of simpler sigals represeig a sample as a sum of samples from simpler sigals Remiders From Las Time Piloig a ew versio of 6003 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 Auheic coecios o real-world applicaios are key o developig a deep ad useful udersadig of he coe Sepember, 208 Imporace of Compuaio Our goal is o develop heories ha help solve real-world problems make model Model aalyze (mah, compuaio) Resul ierpre resuls Desig as he Reverse of Aalysis We are ieresed o oly i aalyzig he behaviors of pre-exisig sysems, bu also i desigig ew sysems Sysem aalysis Behavior World New Udersadig desig 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 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 Weekly Aciviies Lecure: Tuesdays ad Thursdays 2-3pm i -90 Labs: Tues ad Thur 3-5pm i 4-45, -50, 4-53, -90 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 Exam-Type Problems: proveably correc soluios Egieerig Desig Problems: real-world applicaios (more ope eded) Advisory Group Weekly meeigs wih class represeaives sar omorrow: help saff udersad sude perspecive lear abou eachig We will mee o Wedesdays a 3pm Ieresed? Sed o freema@miedu Piazza simple aswers o simple quesios Office Hours beer for deeper more cocepual quesios Thursdays 7pm-9pm Sudays am-5pm Meeigs wih ay saff member by appoime

2 Souds as Sigals Sigals are fucios ha are used o covey iformaio Example: a musical soud ca be represeed as a fucio of ime Expressig Sigals as Sums of Simpler Sigals Expressig a sigal as a sum of siusoids a aleraive view ( f() = ck cos kω o + d k si kω o ) k=0 soud pressure Alhough his ime fucio is a complee descripio of he soud, i does o expose may of he impora properies of he soud cos 0 cos cos 2 Boh views are useful For example, si 0 si si 2 he peak soud pressure is more easily see i f(), while cosoace is easier o see from he frequecy compoes Expressig Sigals as Sums of Simpler Sigals As a simpler example, cosider expasios based o derivaives Maclauri expasio: f() = f(0) + f (0) + f (0) 2 + f (0) 3 +! 2! The fucio of ime f() is represeed by is derivaives a = 0 Taylor expasio abou = a: f() = f(a) + f (a) ( a) + f (a) ( a) 2 + f (a) ( a) 3 +! 2! The fucio of ime f() is represeed by is derivaives a = a Here, he basis fucios are polyomials, ad he coefficies are proporioal o successive derivaives a = 0 or = a Series Represeaios of Sigals Maclauri series f() = f(0) 0! + f (0)! + f (0) 2 2! + f (0) 3 + Basis fucios: /0! 2 /2! 4 /4! Noice ha eve powers of are eve fucios of ime, ad odd powers of are odd fucios of ime The expasio for > 0 implicily deermies he fucio for < 0 /! 3 / 5 /5! Righ-Sided Basis Fucios To model a sigal ha sars a a paricular ime (say = 0), we ca use basis fucios ha sar a = 0 f() = f(0)u 0 () + f (0)u () + f (0)u 2 () + f (0)u 3 () + Basis fucios: u() = u 0 () u 2 () = 2 2! u() u 4 () = 4 4! u() u () =! u() u 3 () = 3 u() u 5 () = 5 5! u() The firs of hese fucios is he ui sep u() = u 0 () Subseque fucios are iegrals of heir predecessors u + () = u (τ)dτ Visualizig Series Maclauri expasio of he cosie fucio ( > 0) f() = cos() = ( ) 2 (2)! f() = + 0 0! 2 2! +4 4! 6 6! +8 8! 0 0! +2 2! f() 2

3 Two Views The sigal cos ca be represeed as a fucio of ime f() or as a sequece of coefficies F [k] f() = f(0) 0! + f (0)! + f (0) 2 2! + f (0) 3 + = f(0) u 0 () + f (0) u () + f (0) u 2 () + f (0) u 3 () + f(0) = f (0) = 0 f (0) = f (0) = 0 f (4) (0) = f(0) (5) = 0 f(0) (6) = F [k] = [, 0,, 0,, 0,, 0, ] Series Represeaios of Sigals This se of fucios [u 0 (), u (), u 2 (), ] is closed uder iegraio, ie, if f() ca be expressed as a sum of hese fucios, he he iegral of f() ca also Example: f() = cos()u() = u() 2 2! u() + 4 4! 6 8 u() u() + 6! 8! u() + = u 0 () u 2 () + u 4 () u 6 () + u 8 () + g() = f(τ)dτ = si()u() = u() 3 u() + 5 5! = u () u 3 () + u 5 () u 7 () + u 9 () 7 9 u() u() + 7! 9! u() + 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 [, 0,, 0,, 0,, 0, ] g() is represeed by he sequece of coefficies [0,, 0,, 0,, 0,, ] Visualizig Series Maclauri Expasio of he Sie Fucio ( > 0) f() = si() = ( ) 2+ (2 + )! Visualizig Series Maclauri Expasio of he Sie Fucio ( > 0) f() = si() = ( ) 2+ (2 + )! x() = +! ! 7 7! +9 9!! +3 x() A Aleraive View The sigal si ca be represeed as a fucio of ime g() or as a sequece of coefficies G[k] g() = g(0) 0! + g (0)! + g (0) 2 2! + g (0) 3 + = g(0) u 0 () + g (0) u () + g (0) u 2 () + g (0) u 3 () + Summary: Series Represeaios of Sigals Whe we expad a fucio as a Maclauri series, we are developig a aleraive represeaio of ha fucio Aleraive represeaios ca lead o ew approaches/operaios: eg, iegraio i ime shifig Maclauri coefficies Aleraive represeaios ca eve ew isighs (see oday s lab) g(0) = = 0 g (0) = f(0) = g (0) = f (0) = 0 g (0) = f (0) = g (4) (0) = f (0) = 0 g(0) (5) = f(0) (4) = g(0) (6) = f(0) (5) = 0 { 0 if k = 0 G[k] = F [k ] oherwise 3

4 Pulses as Buildig Block Sigals We ed o hik abou he ime represeaio of a D sigal as he represeaio of a sigal Bu he ime represeaio ca jus as easily be regarded as a expasio usig a differe se of basis fucios This is especially clear for discree-ime (DT) sigals Pulses as Buildig Block Sigals The ui-sample sigal δ[] is he simples o-rivial DT sigal I has a sigle o-zero sample {, if = 0; δ[] = 0, oherwise δ[] 0 This sigal is useful for cosrucig more complex sigals Pulses as Buildig Block Sigals A arbirary sigal x[] ca be wrie as a weighed sum of dela fucios x[] = x[k]δ[ k] k= Here he basis fucios are shifed versios of he dela fucio δ[ + 2] δ[ + ] δ[] δ[ ] δ[ 2] δ[ 3] The weighs are he sample values: [, x[ 2], x[ ], x[0], x[], x[2], ] Ui-Impulse Sigal The ui-impulse fucio is represeed by a arrow wih he umber, which represes is area or weigh δ() I has wo seemigly coradicory properies: i is ozero oly a = 0, ad is defiie iegral (, ) is oe! Boh of hese properies follow from hikig abou δ() as a limi: δ() = lim ɛ 0 p ɛ () 2ɛ ɛ p ɛ () ɛ ui area Ui-Impulse Sigal The ui-impulse sigal acs as a pulse wih ui area bu zero widh Ui-Impulse Sigal The impulse idea is aalogous o he oio of a poi mass δ() = lim ɛ 0 p ɛ () 2ɛ ɛ p ɛ () ɛ ui area If he mass of a poi were really a posiive umber, he he desiy would be ifiie p /2 () p /4 () p /8 ()

5 Ui-Impulse Sigal Impulses are useful for hikig abou samplig ad recosrucio Check Yourself Assume ha we sample a CT sigal x() oce every T secods Samplig is sraighforward: x[] = x(t ) Recosrucio is rickier How ca we recosruc a coiuousime sigal from is samples? Exac recosrucio is o possible because samplig discards iformaio x() 0T 2T 4T 6T 8T 0T x[] = x(t ) However, we ca approximae he CT sigal wih a sum of small icremes (Riema sum), ad he represe each piece wih a appropriaely weighed impulse y() = x[]w δ( T ) We will look a samplig more closely i he ex lecure ad use he samples o make a sigal y() o approximae x(): y() = x[]w δ( T ) Wha should be he weigh W of he impulses so ha he recosrucio coverges o x() as T 0 Represeig a Sample as a Sum Whe we represe sigals as sums of sigals, we are implicily represeig samples as sums of samples Example Deermie he sum of a ifiie geomeric series C = a = + a + a 2 + a 3 + Represeig a Sample as a Sum Whe we represe sigals as sums of sigals, we are implicily represeig samples as sums of samples Example Deermie he sum of a fiie geomeric series N D = a = + a + a 2 + a a N Focus o Siusoidal Basis Fucios We ca simplify he oaio (ad resulig work) by usig complex expoeials o represe rigoomeric fucios: e jk = cos kω o +j si kω o 0 (Euler s Equaio) 4π Now a sigle complex umber capures boh cos ad si depedece, ( x() = ck cos kω o + d k si kω o ) = a k e jk k=0 k= bu we eed o iclude boh posiive ad egaive values of k sice cos kω o = 2 ejk + 2 e jk ad si kω o = 2j ejk 2j e jk Noe o Euler s Equaio Euler s equaio exeds he expoeial fucio (wih real domai) o a complex fucio of complex domai e θ = + θ + θ2 2! + θ3 + θ4 4! + θ5 5! + θ6 6! + θ7 7! + e jθ = + jθ + j2 θ 2 2! + j3 θ 3 + j4 θ 4 4! = + jθ θ2 2! jθ3 = ( θ2 2! + θ4 4! θ6 6! + e jθ = cos θ + j si θ + θ4 4! + jθ5 5! ) + j5 θ 5 5! + j6 θ 6 6! + j7 θ 7 7! + θ6 6! jθ7 + 7! ) + j (θ θ3 + θ5 5! θ7 7! + Richard Feyma called Euler s equaio he mos remarkable formula i mahemaics 5

6 Noe o Euler s Equaio Euler s equaio ca be ierpreed as a relaio bewee polar ad caresia coordiaes of a ui vecor a agle θ Im θ cos θ e jθ = cos θ + j si θ si θ If θ = ω he he agle θ icreases a a cosa rae (i radias/secod or cycles/sec) The real par is he cos ω ad he imagiary par is si ω Re Rules of Trigoomery Evaluaig eve simple expressios si a + cos a rigoomery si(a + b) = si(a) cos(b) + cos(a) si(b) si(a b) = si(a) cos(b) cos(a) si(b) cos(a + b) = cos(a) cos(b) si(a) si(b) cos(a b) = cos(a) cos(b) + si(a) si(b) a(a + b) = (a(a) + a(b))/( a(a) a(b)) a(a b) = (a(a) a(b))/( + a(a) a(b)) si A + si B = 2 si(a + B)/2 cos(a B)/2 si A si B = 2 cos(a + B)/2 si(a B)/2 cos A + cos B = 2 cos(a + B)/2 cos(a B)/2 cos A cos B = 2 si(a + B)/2 si(a B)/2 si(a + b) + si(a b) = 2 si(a) cos(b) si(a + b) si(a b) = 2 cos(a) si(b) cos(a + b) + cos(a b) = 2 cos(a) cos(b) cos(a + b) cos(a b) = 2 si(a) si(b) 2 cos A cos B = cos(a B) + cos(a + B) 2 si A si B = cos(a B) cos(a + B) 2 si A cos B = si(a + B) + si(a B) 2 cos A si B = si(a + B) si(a B) Rules for Combiig Complex Numbers Maipulaig complex expressios is easier e jθ = cos θ + j si θ (a + jb) + (c + jd) = (a + c) + j(b + d) (a + jb) (c + jd) = (ac bd) + j(ad + bc) bu sill akes pracice Check Yourself Complex umbers How may of he followig are rue? A cos θ+j si θ = cos θ j si θ B (cos θ + j si θ) = cos θ + j si θ C 2 + j2 + e jπ/4 = 2 + j2 + e jπ 4 D Im(j j ) > Re(j j ) E a 2 + a 3 = a