Suggested Reading. Signals and Systems 4-2

Transcription

1 4 Convoluion In Lecure 3 we inroduced and defined a variey of sysem properies o which we will make frequen reference hroughou he course. Of paricular imporance are he properies of lineariy and ime invariance, boh because sysems wih hese properies represen a very broad and useful class and because wih jus hese wo properies i is possible o develop some exremely powerful ools for sysem analysis and design. A linear sysem has he propery ha he response o a linear combinaion of inpus is he same linear combinaion of he individual responses. The propery of ime invariance saes ha, in effec, he sysem is no sensiive o he ime origin. More specifically, if he inpu is shifed in ime by some amoun, hen he oupu is simply shifed by he same amoun. The imporance of lineariy derives from he basic noion ha for a linear sysem if he sysem inpus can be decomposed as a linear combinaion of some basic inpus and he sysem response is known for each of he basic inpus, hen he response can be consruced as he same linear combinaion of he responses o each of he basic inpus. Signals (or funcions) can be decomposed as a linear combinaion of basic signals in a wide variey of ways. For example, we migh consider a Taylor series expansion ha expresses a funcion in polynomial form. However, in he conex of our reamen of signals and sysems, i is paricularly imporan o choose he basic signals in he expansion so ha in some sense he response is easy o compue. For sysems ha are boh linear and ime-invarian, here are wo paricularly useful choices for hese basic signals: delayed impulses and complex exponenials. In his lecure we develop in deail he represenaion of boh coninuousime and discree-ime signals as a linear combinaion of delayed impulses and he consequences for represening linear, ime-invarian sysems. The resuling represenaion is referred o as convoluion. Laer in his series of lecures we develop in deail he decomposiion of signals as linear combinaions of complex exponenials (referred o as Fourier analysis) and he consequence of ha represenaion for linear, ime-invarian sysems. In developing convoluion in his lecure we begin wih he represenaion of discree-ime signals and linear combinaions of delayed impulses. As we discuss, since arbirary sequences can be expressed as linear combinaions of delayed impulses, he oupu for linear, ime-invarian sysems can be

2 Signals and Sysems 4-2 expressed as he same linear combinaion of he sysem response o a delayed impulse. Specifically, because of ime invariance, once he response o one impulse a any ime posiion is known, hen he response o an impulse a any oher arbirary ime posiion is also known. In developing convoluion for coninuous ime, he procedure is much he same as in discree ime alhough in he coninuous-ime case he signal is represened firs as a linear combinaion of narrow recangles (basically a saircase approximaion o he ime funcion). As he widh of hese recangles becomes infiniesimally small, hey behave like impulses. The superposiion of hese recangles o form he original ime funcion in is limiing form becomes an inegral, and he represenaion of he oupu of a linear, ime-invarian sysem as a linear combinaion of delayed impulse responses also becomes an inegral. The resuling inegral is referred o as he convoluion inegral and is similar in is properies o he convoluion sum for discree-ime signals and sysems. A number of he imporan properies of convoluion ha have inerpreaions and consequences for linear, ime-invarian sysems are developed in Lecure 5. In he curren lecure, we focus on some examples of he evaluaion of he convoluion sum and he convoluion inegral. Suggesed Reading Secion 3.0, Inroducion, pages Secion 3.1, The Represenaion of Signals in Terms of Impulses, pages Secion 3.2, Discree-Time LTI Sysems: The Convoluion Sum, pages Secion 3.3, Coninuous-Time LTI Sysems: The Convoluion Inegral, pages 88 o mid-90

3 Convoluion MARKERBOARD InAvr - causal '4s~w rkv"o' "TVNVQI;Q,,c.I, c-7t: X1I Tkiv% \ r - L,Y%3 /9 va STM recy: % clecorpose p 5 pwl invo G Lineer come'ncl4zo1. o C 0'sM basic Sina V -ha. respolase eqs o 6.~i LT I SWs ens- Co, e g Convo + I x[-i] x[0] 1 x[2] -l IOJ fr 2 x[0] x[o]8a[n] -.- e n -1 0 I2 X[1] -1 0 I 2 x[1]8[n-1] *- n x[-i]8[n+1] x[o]8[n]+x(i] 8[n -1] + x [-I]8[n+ ]+.-- +X kr =2 x[k]8[n-k] k= -c 4.1 A general discreeime signal expressed as a superposiion of weighed, delayed uni impulses I 2 X[-] x [-2]8[n +2] n

4 Signals and Sysems 4.2 The convoluion sum for linear, imeinvarian discree-ime sysems expressing he sysem oupu as a weighed sum of delayed uni impulse responses. jj )x[ x[2] -1 0 I 2 X 10{ x[0]8[n] -I X x[1]8[n+1] n m- x [0] h [n] 0 0 x [ 1] h [n-i] x[ 2 91T ] x[-2]8[n+2] -1 0 i 2 n 0 0 x [-I] h [n+1. x [-2] h [n.2] x[n] =E k = -o x[k] S[n-k] 4.3 One inerpreaion of he convoluion sum for an LTI sysem. Each individual sequence value can be viewed as riggering a response; all he responses are added o form he oal oupu. Linear Sysem: If Time-invarian: +0n y [n] =E x[k] hk [n] k = [n - k] - h [n] hk [n] = h + [n-k] o0 LTI: y[n] x[k] h[n - k]i =E Convoluion Sum

5 Convoluion 4-5 x(-2a)&a(+2a)a x(-2a) -2A -A -A Approximaion of a coninuous-ime signal as a linear combinaion of weighed, delayed, recangular pulses. [The ampliude of he fourh graph has been correced o read x(o).] I X (0) X (0) BA A 0 A - xa) x (A) 86 (-A) A 2A x() X() x() x(o) 6A() A + x(a) 6 A( + x(- A) 6A( + A)A+... x(k A) 6,( - k A) A = lim 1 x(k A) S( - k A) A A+O k=-oc +W f x(,r) 6( -,r) d-r As he recangular pulses in Transparency 4.4 become increasingly narrow, he represenaion approaches an inegral, ofen referred o as he sifing inegral.

6 Signals and Sysems 4-6 x() = Eim ( x(k A) 'L+0 k=-o 56( - ka) A 4.6 Derivaion of he convoluion inegral represenaion for coninuous-ime LTI sysems. Linear Sysem: +o y() = 0 x(ka) +O k=- o +00 =f xt) ht() dr hk() A If Time-Invarian: hkj ) = ho( - ka) LTI: h,() = he ( - r) +01 v() f x(r) h(-7) dr- 1-0 Convoluion Inegral x() 0 i 4.7 Inerpreaion of he convoluion inegral as a superposiion of he responses from each of he recangular pulses in he represenaion of he inpu. x (0) x(a) x (0) h M x(ka) oa AA y() ka 0 x() y() 0 oa

7 Convoluion 4-7 Convoluion Sum: +0o x[n] =E x[k] k= -0ok y [n] = x [k] k= -o00 S[n-k] h[n-k] =x[n] * h[n] 4.8 Comparison of he convoluion sum for discree-ime LTI sysems and he convoluion inegral for coninuous-ime LTI sysems. Convoluion Inegral: x() y() +00 =f x(-) 6(-r) dr +fd = X(r) h(-,r) dr- = x() -* h() y [n] Z x[k]h [n-k] x [n]= u [n] h[n]=an u[n] 0 n x [n] h [n] 4.9 Evaluaion of he convoluion sum for an inpu ha is a uni sep and a sysem impulse response ha is a decaying exponenial for n > 0. x [k] k h [n-k] n k

8 Signals and Sysems y()f x(r)h(-r)dr x() u () 4.10 Evaluaion of he convoluion inegral for an inpu ha is a uni sep and a sysem impulse response ha is a decaying exponenial for > 0. h ( )=e~43 u ) O x () h () 0 r x (r) h (-r) T MARKERBOARD 4.2

9 Convoluion MARKERBOARD 4.3 v4egva.z: ). Te Lk (- -C jt C k L-T)aT 1 U -e 3 (0 o- eoverkp ~3e ee ov0 C- L h*o,<0o E J r)

10 MIT OpenCourseWare hp://ocw.mi.edu Resource: Signals and Sysems Professor Alan V. Oppenheim The following may no correspond o a paricular course on MIT OpenCourseWare, bu has been provided by he auhor as an individual learning resource. For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.