KEEE313(03) Signals and Sysems Chang-Su Kim
Course Informaion Course homepage hp://mcl.korea.ac.kr Lecurer Chang-Su Kim Office: Engineering Bldg, Rm 508 E-mail: changsukim@korea.ac.kr Tuor 허육 (yukheo@mcl.korea.ac.kr)
Course Informaion Objecive Sudy fundamenals of signals and sysems Main opic: Fourier analysis Texbook A. V. Oppenheim and A. S. Willsky, Signals & Sysems, 2 nd ediion, Prenice Hall, 1997. Reference M. J. Robers, Signals and Sysems, McGraw Hill, 2003.
Course Informaion Prerequisie Advanced Engineering Mahemaics Manipulaion of complex numbers Assessmen Exercises 15 % Miderm Exam 30 % Final Exam 40 % Aendance 15 %
Course Schedule Firs, Chapers 1-5 of he exbook will be covered Linear ime invarian sysems Fourier analysis Then, seleced opics in Chapers 6-10 will be augh, such as Filering Sampling Modulaion Laplace Transform Z-ransform Miderm exam: 7 APR 2016 (Thursday)
Policies and Rules Exams Scope: all maerials augh Closed book A single shee of hin paper (boh sizes) Aendance Checked someimes Quizzes Pop up quizzes: no announced before # of quizzes is also varian Assignmens No lae submission is allowed
Wha Are Signals and Sysems? Signals Funcions of one or more independen variables Conain informaion abou he behavior or naure of some phenomenon Sysems Respond o paricular signals by producing oher signals or some desired behavior
Examples of Signals and Sysems (2005)
Examples of Signals and Sysems (2006)
Examples of Signals and Sysems (2007)
Examples of Signals and Sysems (2008)
Examples of Signals and Sysems (2009)
Examples of Signals and Sysems (2014)
Examples of Signals and Sysems (2016)
Audio Signals f() Funcion of ime Acousic pressure S I GN AL
Image Signals f(x,y) Funcion of spaial coordinaes (x, y) Ligh inensiy
Image Signals Color images r(x,y) g(x,y) b(x,y)
Video Signals Funcions of space and ime r(x,y,) g(x,y,) b(x,y,)
Video Signals Funcions of space and ime r(x,y,) g(x,y,) b(x,y,)
원본 Rolling shuer 왜곡 제거 안정화(warping)
Sysems Inpu signal: lef, kick, punch, righ, up, down, Oupu signal: sound and graphic daa
Sysems Image processing sysem Inpu image Oupu image
Scope of Signals and Sysems is broad Mulimedia is jus a iny porion of signal classes The concep of signals and sysems arise in a wide variey of fields Inpu signal - pressure on acceleraor pedal Oupu signal - velociy
Scope of Signals and Sysems is broad Mulimedia is jus a iny porion of signal classes The concep of signals and sysems arise in a wide variey of fields Inpu signal - ime spen on sudy Oupu signal - mark on miderm exam
Mahemaical Framework The objecive is o develop a mahemaical framework for describing signals and sysems and for analyzing hem We will deal wih signals involving a single independen variable only x() (or x[n]) For convenience, he independen variable (or n) is called ime, alhough i may no represen acual ime I can in fac represen spaial locaion (e.g. in image signal)
Coninuous-Time Signals x() vs. Discree-Time Signals x[n] CT signal DT signal x[n] is defined only for ineger values of he independen variable n n =, 2, 1, 0, 1, 2, x[n] can be obained from sampling of CT signals or some signals are inherenly discree
Examples of DT Signals
Signal Energy and Power Energy: accumulaion of squared magniudes E lim T T T N x( ) d Power: average squared magniudes 2 x( ) 2 d 2 2 E lim x[ n] x[ n] N n N n 1 2 P lim x( ) d T 2T P T T N 1 lim x[ n] N 2N 1 nn 2
Signal Energy and Power Classificaion of signals Caegory 1: Energy signal (E < and hus P = 0) e.g. Caegory 2: Power signal (E = bu P < ) e.g. Caegory 3: Remaining signals (i.e. wih infinie energy and infinie power) e.g. 0, x( ) e, x ( ) 1 x() 0 0
Transformaions of Independen Variable Three possible ime ransformaions 1. Time Shif: x( a), x[n a] Shifs he signal o he righ when a > 0, while o he lef when a < 0. 2. Time Reversal: x, x[ n] Flips he signal wih respec o he verical axis. 3. Time Scale: x(a), x[an] for a > 0. Compresses he signal lengh when a > 1, while sreching i when a < 1.
Transformaions of Independen Variable Time Reversal x(-) 1 1 2-2 -1 x() 1 Time Shif 1 x(-1) x(+1) 1-2 -1 1-3 -2-1
Transformaions of Independen Variable Time Scaling -1-1/2 Combinaions 1 x(-2) -1-1/2 x(2) 1 1-4 x(-+3) 1-3 2-2 -1-2 -1 x(/2) 1 3 4 5 6 x() 1
Transformaions of Independen Variable
Periodic Signals x() is periodic wih period T, if x() = x( + T) for all x[n] is periodic wih period N, if x[n] = x[n + N] for all n Noe ha N should be an ineger Fundamenal period (T 0 or N 0 ): The smalles posiive value of T or N for which he above equaions hold
Periodic Signals
Periodic Signals Is his periodic?
Even and Odd Signals x[n] is even, if x[ n] = x[n] Even funcion x[n] is odd, if x[ n] = x[n] Any signal x[n] can be divided ino even componen x e [n] and odd componen x o [n] x[n] = x e [n] + x O [n] x e [n] = (x[n] + x[ n])/2 x O [n] = (x[n] x[ n])/2 Odd funcion Similar argumens can be made for coninuous-ime signals
Even and Odd Signals Even-odd decomposiion
Exponenial and Sinusoidal Signals
Euler s Equaion Euler s formula e j cos sin cos 1 j ( e 2 1 ( e 2 j j sin j e e j Complex exponenial funcions faciliae he manipulaion of sinusoidal signals. For example, consider he sraighforward exension of differeniaion formula of exponenial funcions o complex cases. ) j )
Periodic Signals CT Signal Periodiciy condiion x() = x(+t) If T is a period of x(), hen mt is also a period, where m=1,2,3, Fundamenal period T 0 of x() is he smalles possible value of T. Exercise: Find T 0 for cos( 0 +) and sin( 0 +) DT Signal Periodiciy condiion x[n] = x[n+n] If N is a period of x[n], hen mn is also a period, where m=1,2,3, Fundamenal period N 0 of x[n] is he smalles possible value of N.
Sinusoidal Signals x() = A cos(+) or x[n] = A cos(n+) A is ampliude is radian frequency (rad/s or rad/sample) is he phase angle (rad) Noice ha alhough A cos( 0 +) A cos( 1 +), i may hold ha A cos[ 0 n+] = A cos[ 1 n+]. Do you know when?
Periodic Complex Exponenial Signals Ae j( ) j( n ) x() = or x[n] = A, and are real. z() Ae Is j periodic? Ae How abou he discree case? Is periodic? z[ n] I is periodic when /2p is a raional number Ae j n Ex 1) z[ n] e Ex 2) z[ n] e Ex 3) z[ n] e j 2p n 3 3p j n 5 2p j n 2
Review of Sinusoidal and Periodic Complex Exponenials CT case j x( ) cos( w), sin( w) or e These are periodic wih period 2p/w. Also, If w w, 1 2 cos( w ) cos( w ), sin( w ) sin( w ) and e e 1 2 1 2 j j 1 2 DT case x[ n] cos( wn), sin( wn) or e j n These are periodic only if w/2p is a raional number. Also, If w w 2 p k, 1 2 cos( w n) cos( w n), sin( w n) sin( w n) and e e 1 2 1 2 j n j n 1 2
Real Exponenial Signals x() = A e s or x[n] = A e sn A and s are real. posiive s negaive s
General Exponenial Signals x() Xe Ae e s j ( s jw) s Ae [cos( w ) j sin( w )] Real par of x() according o s ( is assumed o be 0)
Impulse and Sep Funcions
DT Uni Impulse Funcion Uni Impulse 1, n 0 [ n] 0, n 0 Shifed Uni Impulse 1, n [ nk] 0, n k k [n] 1-3 -2-1 1 2 3 [n-k] 1-1 1 k n n
DT Uni Sep Funcion Uni Sep 1, n 0 un [ ] 0, n 0 u[n] 1-3 -2-1 1 2 3 n Shifed Uni Sep u[n-k] u[ n k] 1, 0, n n k k -1 1 1 k n
Properies of DT Uni Impulse and Sep Funcions 1) [ n] u[ n] u[ n 1] 2) u[ n] [ k] [ n k] k0 3) x[ n] [ n] x[0] [ n] n k 4) x[ n] [ n n ] x[ n ] [ n n ] 0 0 0 5) x[ n] x[ k] [ n k] k
CT Uni Sep Funcion Uni Sep u() u () 1, 0 0, 0 1 Shifed Uni Sep 1, u ( ) 0, 1 u(- )
CT Uni Sep Funcion Uni sep is disconinuous a =0, so is no differeniable Approximaed uni sep 0, 0 u ( ), 0 1, 1 u () u () is coninuous and differeniable. u( ) lim u ( ) 0 du () d 1, 0 0, oherwise
CT Uni Impulse Funcion Approximaed uni impulse () 1 du (), 0 () 1/ d 0, oherwise Uni Impulse:, 0 ( ) lim ( ) 0 0, 0 b a ( ) d 1 for any a 0 and b 0. ()
CT Uni Impulse Funcion Shifed Uni Impulse (-)
Properies of CT Uni Impulse and Sep Funcions du() 1) ( ) d 2) u( ) ( ) d ( ) d 3) x( ) ( ) x(0) ( ) 0 4) x( ) ( ) x( ) ( ) 0 0 0 5) x( ) x( ) ( ) d
Comparison of DT and CT Properies Difference becomes du() [ n] u[ n] u[ n 1] ( ) differeniaion d Summaion becomes inegraion n u[ n] [ k] [ n k] u( ) ( ) d ( ) d k k0 0 Impulse funcions sample values x[ n] [ n] x[0] [ n] x( ) ( ) x(0) ( ) Shifed impulse funcions x[ n] [ n n ] x[ n ] [ n n ] x( ) ( ) x( ) ( ) 0 0 0 0 0 0 Sifing Propery: Arbirary funcions as sum x[ n] x[ k] [ n k] x( ) x( ) ( ) d or inegraion of dela funcions k
Can you represen hese funcions using sep funcions? x() y() a b c 1-1 1 z() 2 w() -1 1-2 1-1 1 2
Can you represen hese funcions using sep funcions? -1 1 x[n] 1 N n y[n] 1-3 -2-1 1 2 3 4 5 n
Basic Sysem Properies
Wha is a Sysem? Sysem is a black box ha akes an inpu signal and convers i o an oupu signal. DT Sysem: y[n] = H[x[n]] x[n] H y[n] CT Sysem: y() = H(x()) x() H y()
Inerconnecion of Sysems Series (or cascade) connecion: y() = H 2 ( H 1 ( x() ) ) x() H 1 H 2 y() e.g. a radio receiver followed by an amplifier Parallel connecion: y() = H 2 ( x() ) + H 1 ( x() ) x() H 1 H 2 + y() e.g. Carrying ou a eam projec
Inerconnecion of Sysems Feedback connecion: y() = H 1 ( x()+h 2 ( y() ) ) x() + H 1 y() H 2 e.g. cruise conrol Various combinaions of connecions are also possible
Memoryless Sysems vs. Sysems wih Memory Memoryless Sysems: The oupu y() a any insance depends only on he inpu value a he curren ime, i.e. y() is a funcion of x() Sysems wih Memory: The oupu y() a any insance depends on he inpu values a pas and/or fuure ime insances as well as he curren ime insance Examples: A resisor: y() = Rx() A capacior: y = 1 C න x τ dτ A uni delayer: y[n] = x[n 1] An accumulaor: n y n = k= x[k]
Causaliy Causaliy: A sysem is causal if he oupu a any ime insance depends only on he inpu values a he curren and/or pas ime insances. Examples: y[n] = x[n] x[n 1] y() = x( + 1) Is a memoryless sysem causal? Causal propery is imporan for real-ime processing. Bu in some applicaions, such as image processing, daa is ofen processed in a non-causal way. image processing
Applicaions of Lowpass Filering Preprocessing before machine recogniion Removal of small gaps
Applicaions of Lowpass Filering Cosmeic processing of phoos
Inveribiliy Inveribiliy: A sysem is inverible if disinc inpus resul in disinc oupus. If a sysem is inverible, hen here exiss an inverse sysem which convers he oupu of he original sysem o he original inpu. x() Sysem y() Inverse Sysem w() = x() Examples: y( ) w( ) 4x( ) 1 4 y( ) y[ n] x[ k] n k w[ n] y[ n] y[ n 1] y( ) x( ) d dy() w () d
Sabiliy Sabiliy: A sysem is sable if a bounded inpu yields a bounded oupu (BIBO). In oher words, if x < k 1 hen y() < k 2. Examples: y( ) x( ) d 0 y[ n] 100x[ n]
Lineariy A sysem is linear if i saisfies wo properies. Addiiviy: Homogeneiy: x = x 1 () + x 2 () y() = y 1 () + y 2 () x = c x 1 () y() = c y 1 (), for any consan c The wo properies can be combined ino a single propery. lineariy: x = ax 1 () + bx 2 () y() = ay 1 () + by 2 () Examples y( ) x 2 ( ) y[ n] nx[ n] y( ) 2 x( ) 3
Time-Invariance A sysem is ime-invarian if a delay (or a ime-shif) in he inpu signal causes he same amoun of delay in he oupu. x = x 1 ( 0 ) y() = y 1 ( 0 ) Examples: y[ n] nx[ n] y( ) x(2) y( ) sin x( )
Superposiion in LTI Sysems For an LTI sysem: Given response y() of he sysem o an inpu signal x(), i is possible o figure ou response of he sysem o any signal x 1 () ha can be obained by scaling or ime-shifing he inpu signal x(), because x 1 = a 0 x 0 + a 1 x 1 + a 2 x 2 + y 1 = a 0 y 0 + a 1 y 1 + a 2 y 2 + Very useful propery since i becomes possible o solve a wider range of problems. This propery will be basis for many oher echniques ha we will cover hroughou he res of he course.
Superposiion in LTI Sysems Exercise: Given response y() of an LTI sysem o he inpu signal x(), find he response of ha sysem o anoher inpu signal x 1 () shown below. x() 2 y() 1 1-1 1 x 1 () 2-1 1 3