6.003: Signal Processing

Transcription

1 6.003: Signal Processing Working wih Signals Overview of Subjec Signals: Definiions, Examples, and Operaions Basis Funcions and Transforms Sepember 6, 2018

2 Welcome o Piloing a new version of focused on Signal Processing. Combines heory analysis and synhesis of signals, ime and frequency domains, convoluion and deconvoluion, filering and noise reducion, wih auhenic, real-world applicaions in music, imaging, video.

3 Role of Analysis in Science and Engineering Our common goal wih oher science / engineering endeavors is o model some aspec of he world analyze he model, and inerpre resuls o gain beer undersanding. make model Model analyze (mah, compuaion) Resul inerpre resuls World New Undersanding Classical analyses use a variey of mahs, especially calculus. We will also use compuaion which is applicable in many real-world problems ha are difficul or impossible o solve analyically srenghens ies o he real world.

4 Design as he Reverse of Analysis 1 We are ineresed no only in analyzing he behaviors of pre-exising sysems, bu also in designing new sysems. analysis Sysem Behavior design Analysis and Design are differen and complemenary aciviies. Analysis sars wih a precise saemen of he problem and proceeds o a precise saemen of a resul. convenional problem ses Design is more open-ended, wih muliple possible soluions ha differ along idiosyncraic dimensions ha were no even par of he original problem saemen. Engineering Design Problems 1 adaped from R. David Middlebrook

5 Course Mechanics Lecure: Tuesdays and Thursdays 2-3pm in Labs: Tues. and Thur. 3-5pm in 4-145, 1-150, 4-153, Homework issued Tuesdays, due following Tuesday a noon Drills: facs, definiions, and simple conceps online wih immediae feedback (no graded) inended as pracice and self-assessmen Problems inended o improve problem solving skills Exam-Type Problems: proveably correc soluions Engineering Design Problems: real-world applicaions (more open ended) Two Miderms and a Final Exam

6 Advisory Group Weekly meeings wih class represenaives help saff undersand suden perspecive learn abou eaching Tenaively mee on Wednesdays a 3pm. Ineresed?... Send o freeman@mi.edu

7 Signals Signals are funcions ha are used o convey informaion. may have 1 or 2 or 3 or even more independen variables y brighness (x, y) sound pressure () x

8 Signals Signals are funcions ha are used o convey informaion. dependen variable can be a scalar or a vecor y scalar: brighness a each poin (x, y) vecor: (red,green,blue) y a each poin (x, y) x x

9 Signals Signals are funcions ha are used o convey informaion. dependen variable can be real, imaginary, or complex-valued x() = e j2π = cos 2π +j sin 2π

10 Signals Coninuous ime (CT) versus discree ime (DT) x() x[n] n Signals from physical sysems are ofen of coninuous domain: coninuous ime measured in seconds coninuous spaial coordinaes measured in meers Compuaions usually manipulae funcions of discree domain: discree ime measured in samples discree spaial coordinaes measured in samples

11 Signals Sampling: convering CT signals o DT x() x[n] = x(nt ) 0T 2T 4T 6T 8T 10T n T = sampling inerval Imporan for compuaional manipulaion of physical daa. digial represenaions of audio signals (as in MP3) digial represenaions of images (as in JPEG)

12 Signals Reconsrucion: convering DT signals o CT zero-order hold x[n] x() n T 4T 6T 8T 10T T = sampling inerval commonly used in audio oupu devices

13 Signals Reconsrucion: convering DT signals o CT piecewise linear x[n] x() n T 4T 6T 8T 10T T = sampling inerval commonly used in rendering images

14 Signals Periodic signals consis of repeaed cycles (periods). periodic aperiodic x() = x( + T ) x() 0 T 0 x[n] = x[n + N] x[n] 0 N n 0 n

15 Signals Righ-sided signals are zero before some saring ime. Lef-sided signals are zero afer some ending ime. Signals wih finie duraion are zero excep for some range of imes. righ-sided x() lef-sided x() 0 T s T e 0 x[n] x[n] n n 0 N s N e 0

16 Signals The minima and maxima of bounded signals are finie. bounded unbounded x() x() 0 0 x[n] x[n] 0 n 0 n

17 Signals Even signals are symmeric abou ime zero. Odd signals are anisymmeric abou ime zero. even x() = x( ) odd x() = x( ) 0 x[n] = x[ n] x[n] = x[ n] 0 n 0 n

18 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

19 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

20 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

21 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

22 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

23 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

24 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

25 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? f 1 () = f(2) f 2 () = f() f 3 () = f(2) f 4 () = 1 3 f()

26 Check Yourself Compuer generaed speech (by Rober Donovan) f() Lisen o he following four manipulaed signals: f 1 (), f 2 (), f 3 (), f 4 (). How many of he following relaions are rue? 2 f 1 () = f(2) f 2 () = f() X f 3 () = f(2) X f 4 () = 1 3 f()

27 y y y y Check Yourself f(x, y) x How many images mach he expressions beneah hem? f 1 (x, y)=f(2x, y)? x f 2 (x, y)=f(2x 250, y)? x f 3 (x, y)=f( x 250, y)? x

28 y y y y Check Yourself f(x, y) x f 1 (x, y) = f(2x, y)? x x x f 2 (x, y) = f(2x 250, y)? f 3 (x, y) = f( x 250, y)? x = 0 f 1 (0, y) = f(0, y) x = 250 f 1 (250, y) = f(500, y) X x = 0 f 2 (0, y) = f( 250, y) x = 250 f 2 (250, y) = f(250, y) x = 0 f 3 (0, y) = f( 250, y) X x = 250 f 3 (250, y) = f( 500, y) X

29 y y y y Check Yourself f(x, y) x How many images mach he expressions beneah hem? f 1 (x, y)=f(2x, y)? x f 2 (x, y)=f(2x 250, y)? x f 3 (x, y)=f( x 250, y)? x

30 Musical Sounds as Signals Signals are funcions ha are used o convey informaion. Example: a musical sound can be represened as a funcion of ime. sound pressure Alhough his ime funcion is a complee descripion of he sound, i does no expose many of he imporan properies of he sound.

31 Musical Sounds as Signals Even hough hese sounds have he same pich, hey sound differen. piano cello bassoon oboe horn alosax violin bassoon sec I s no clear how he differences relae o properies of he signals. (from hp://heremin.music.uiowa.edu)

32 Musical Signals as Sums of Sinusoids One way o characerize differences beween hese signals is express each as a sum of sinusoids. ( f() = ck cos kω o + d k sin kω o ) k=0 cos 0 2π ωo sin 0 2π ωo cos ωo 2π ωo sin ωo 2π ωo cos 2ωo. 2π ωo sin 2ωo. 2π ωo Since hese sounds are (nearly) periodic, he frequencies of he dominan sinusoids are ineger muliples of a fundamenal frequency ω o.

33 Harmonic Srucure The weighs of he componens (c k and d k ) describe he harmonic srucure of he signal. ( f() = ck cos kω o + d k sin kω o ) k=0 c 2 k +d2 k ωo 2ωo 3ωo 4ωo 5ωo 6ωo ω harmonic # DC fundamenal second harmonic hird harmonic fourh harmonic fifh harmonic sixh harmonic

34 Harmonic Srucure Harmonic srucure plays an imporan role in generaing he characerisic sounds of musical insrumens. piano piano k bassoon bassoon k violin violin k (from hp://heremin.music.uiowa.edu)

35 Two Views of he Same Signal The harmonic expansion provides an alernaive view of he signal. ( f() = ck cos kω o + d k sin kω o ) k=0 We can view he musical signal as a funcion of ime f(), or as a sum of harmonics wih ampliudes [c 0, d 0, c 1, d 1,...]. Boh views are useful. For example, he peak sound pressure is more easily seen in f(), while consonance is more easily analyzed by comparing harmonics. We call such an alernaive view of a signal a ransform.

36 Transforms There are many kinds of ransforms and expansions based on differen ypes of basis funcions. Examples: expansions based on derivaives of funcions. Maclaurin expansion: f() = f(0) + f (0) + f (0) 2 + f (0) 3 + 1! 2! 3! The funcion of ime f() is represened by is derivaives a = 0. Taylor expansion abou = a: f() = f(a) + f (a) ( a) + f (a) ( a) 2 + f (a) ( a) 3 + 1! 2! 3! The funcion of ime f() is represened by is derivaives a = a.

37 Series Represenaions of Signals Maclaurin series f() = f(0) 1 0! + f (0) 1! + f (0) 2 2! + f (0) 3 3! + Basis funcions: 1/0! /1! /2! 3 /3! /4! 5 /5! Noice ha even powers of are even funcions of ime, and odd powers of are odd funcions of ime. The expansion for > 0 implicily deermines he funcion for < 0.. 1

38 Series Represenaions of Signals To model a signal ha sars a a paricular ime (say = 0), we need a differen se of basis funcions. f() = f(0 + )u 0 () + f (0 + )u 1 () + f (0 + )u 2 () + f (0 + )u 3 () + Basis funcions: u() = u 0 () u 1 () = 1! u() 1 u 2 () = 2 2! u() 1 1 u 3 () = 3 3! u() 1 1 u 4 () = 4 4! u() 1 1 u 5 () = 5 5! u() The firs of hese funcions is he uni sep u() = u 0 (). Subsequen funcions are inegrals of heir predecessors u n+1 () = u n(τ)dτ.

39 Series Represenaions of Signals This se of funcions [u 0 (), u 1 (), u 2 (),...] is closed under inegraion, i.e., if f() can be expressed as a sum of hese funcions, hen he inegral of f() can also. Example: f() = cos()u() = u() 2 2! u() + 4 4! 6 8 u() u() + 6! 8! u() + = 1u 0 () 1u 2 () + 1u 4 () 1u 6 () + 1u 8 () +... g() = f(τ)dτ = sin()u() = u() 3 3! u() + 5 5! = 1u 1 () 1u 3 () + 1u 5 () 1u 7 () + 1u 9 (). 7 9 u() u() + 7! 9! u() Since f() can be wrien as a sum of funcions from he se, i follows ha g() = f(τ)dτ can also. f() is represened by he sequence of coefficiens [1, 0, 1, 0, 1, 0, 1, 0,...] g() is represened by he sequence of coefficiens [0, 1, 0, 1, 0, 1, 0, 1,...]

40 Series Represenaions of Signals Same se [u 0 (), u 1 (), u 2 (),...] is no closed under differeniaion. Is here a meaningful way o define he ime derivaive of a sep? Le u () represen he following signal. u () 1 0 and u () represen is derivaive: u () 1 0 hen lim 0 u () = u() and lim 0 u () = an impulse

41 Impulse Funcion Le δ() represen an impulse, which is defined by wo properies: δ() = 0 for all 0, and δ()d = area under impulse = 1. δ() 1 0 These definiions only make sense as limis. While moivaed by a square pulse (u ()), any funcion wih uni area and finie duraion also works. Wih all of is area in he shores possible inerval of ime δ() is complemenary o he broades signal x() = consan.

42 Focus on Fourier Represenaions (Sinusoidal Bases) Harmonic srucure deermines consonance and dissonance. ocave (D+D ) fifh (D+A) D+E ime(periods of "D") D' A E D D harmonics D

43 Focus on Fourier Represenaions (Sinusoidal Bases) Sinusoidal decomposiions are useful hroughou physics. Diffusion equaion: f 2 f x 2 examples: hea ransfer (original applicaion), chemical ranspor Wave equaion: 2 f 2 2 f x 2 examples: ligh, radio waves, x-rays, acousics, fluid dynamics Reason: n f(x, ) n n f(x, ) x n where f(x, ) = e j(ω kx) = cos(ω kx) + j sin(ω kx)

44 Labs Focus on Applicaions The broad applicabiliy of Fourier mehods provides many ineresing applicaions. Examples include music e.g., music fingerprining ala Shazam image processing e.g., MRI video e.g. moion magnificaion

45 Today s Lab: Sounds as Signals Generae audio signals and invesigae heir properies. Break ino secions. Find your secion number on he websie: hp://mi.edu/6.003 under Week 1, Lab B.