1 1 Discree-Time Covoluio Noes 03 largely plagiarized by %khc Le s begi our discussio of covoluio i discree-ime, sice life is somewha easier i ha domai. We sar wih a sigal x[] ha will be he ipu io our LTI sysem H. Firs, we break x[] io he sum of appropriaely scaled ad shifed impulses, as i Figure 1(a), wih he impulse fucio defied i discree-ime as i Figure 1(b). I Figure 1(a), x[] x[0] δ [] x[1] δ [ 1] x[] δ [ ] = + + (a) ay sigal ca be decomposed io he sum of scaled ad shifed impulses δ [] h[] (b) a impulse ad he correspodig impulse respose x[0] δ [] x[0] h[] x[1] δ [ 1] x[1] h[ 1] x[] δ [ ] x[] h[ ] (c) scalig ad shifig he impulse produces a scaled ad shifed impulse respose x[] y[] (d) superposiio of he hree sigals o he lef from (c) gives x[]; likewise, superposiio of he hree sigals o he righ gives y[]; so if x[] is ipu io our sysem wih impulse respose h[], he correspodig oupu is y[] Figure 1: Discree-ime covoluio. wehave decomposed x[] iohesumof x[0][], x[1][, 1], ad x[][, ]. I geeral, ay x[] ca bebroke up io he sum of x[k][, k],wherex[k] is he appropriae scalig for a impulse [, k] ha is ceered a = k.
EE10: Sigals ad Sysems; v5.0.0 I oher words, we have: x[] = 1X k= x[k][, k] Nex, we are give h[], he respose of he sysem o a impulse ceered a = 0. I oher words, if [] is he ipu io his sysem, he oupu is h[]. Uilizig he homogeeiy propery of a liear sysem, if x[0][] is he ipu, he oupu is x[0]h[]. If x[1][, 1] is he ipu, he oupu is x[0]h[, 1] by boh he homogeeiy ad ime-ivaria properies of H. A similar argume ca be made for x[][, ] as ipu producig x[]h[, ] as oupu. This is illusraed i Figure 1(c). Fially, he superposiio propery of a liear sysem permis us o add up he oupus o form he composie oupu o he x[] from Figure 1(a). The fial oupu y[] is show i Figure 1(d). This fial oupu is jus: Superposiio Iegral y[] = 1X k= x[k]h[, k] Assume we have a liear sysem which performs he operaio H o is ipu x(), givig he oupu y(). I symbols, y() = H[x()] = H[ = = x( )(, )d ] Z 1 H[x( )(, )]d Z 1 x( )H[(, )]d where we have used he sifig iegral o subsiue for x() i he secod lie, he addiiviy propery of lieariy [x 1 () +x ()! y 1 ()+y ()] i he hird, ad he homogeeiy propery of lieariy [x()! y()] i he fourh. Wha is H[(, )]? Well, firs we d beer ask wha (, ) is; i s a impulse fucio ceered a =. H[(, )] is he he respose of he sysem H a ime o a impulse ceered a ime. Le s deoe H[(, )] by g(; ). Noe ha g is a fucio of wo variables. I s a fucio of because he respose of he sysem is goig o be a fucio of ime. I s a fucio of, because he sysem may be ime varyig, so i eed o kow where he impulse is ceered. If he sysem is ime varyig, he respose of he sysem o a impulse ceered a = 1 will i geeral be differe from he respose of he sysem o a impulse ceered a =. Ayway, he oupu of he sysem y() o ay ipu x() becomes y() = x( )g(; )d Bu if he sysem is ime-ivaria, we ca furher massage g(; ) io somehig more palaable. Cosider a ime ivaria sysem, which respods o a impulse ceered a = a as i Figure (b); his sysem respose is deoed g(; a). Now, because he sysem is ime ivaria, if i shif he impulse by, he sysem respose should also shif by, as i Figure (d). This ew sysem respose is deoed g(, ;a, ). Why he secod argume shifs is easy o see; he secod argume ells us where he impulse is ceered, ad sice he ceer has bee shifed by, he ew ceer is a,. Big deal. Wha abou he firs argume? The bes way o see his is o oe wha happes o he poi (p; q); igesshifedby also. Now le s make a =. I oher words, we shif he impulse o he origi by a uis ad he sysem respose ges shifed, by a oal of a uis, as i Figure (f). So he oupu of he sysem y() o ay ipu x() becomes y() = x( )g(, ;0)d
EE10: Sigals ad Sysems; v5.0.0 3 δ ( a) g(,a) q (a) a (b) b p δ ( (a )) g(,a ) q (c) a (d) b p δ () g( a,0) q (e) (f) b a p a Figure : (a) impulse ceered a = a, (b) respose of sysem o impulse ceered a = a, (c) impulse ceered a = a,, (d) respose of sysem o impulse ceered a = a,, (e) impulse ceered a = 0, (f) respose of sysem o impulse ceered a = 0. We assume ha he sysem is ime ivaria, so ha (c) ad (d) are simply (a) ad (b) shifed. For he paricular choice of a =, we obai (e) ad (f) from (c) ad (d). Noe wha happes o he poi (p; q) i (b), (d), ad (f). We ca he defie a ew impulse respose h(, )=g(, ;0) ad fially we have he superposiio iegral as see i lecure: y() = x( )h(, )d We he oe ha he superposiio iegral coveiely correspods o he covoluio operaio, as previously see i mah classes. So, o summarize: y() = x() h() = x( )h(, )d Bu why boher? Well, if you have he impulse respose h() of ay LTI sysem, ad i give you ay ipu x(), you ca ell me wha y() is by doig a covoluio of x ad h. Tha meas you do have o keep o goig dow o he lab ad puig ipus io your LTI sysem o figure ou wha your oupu is, sice you ca calculae i. Keep i mid ha he covoluio iegral wih h(, ) oly works for liear ime ivaria sysems. Of course, i real life TM, may sysems are oliear. bu wih appropriae liearizaio echiques, you ca approximae hem by LTI sysems i he regios of ieres ad aalyze hem usig echiques developed i his course. 3 Useful Covoluio Properies Covoluio is associaive, commuaive, ad disribuive over addiio. Exercises Prove he above saeme. Use he defiiio of he superposiio iegral. Covoluio is o disribuive over muliplicaio. Exercise Cosider u() u() =u() [1u()] = [u() 1][u() u()]. Covolvig x() wih:
EE10: Sigals ad Sysems; v5.0.0 () gives ẋ(). () gives x(). The ideiy uder covoluio is he ui impulse. (, 0 ) gives x(, 0 ). R u() gives x()d. Exercises Prove hese. Of he hree, he firs is he mos difficul, ad he secod he easies. Time Ivariace, Causaliy, ad BIBO Sabiliy Revisied Now ha we have he covoluio operaio, we ca recas he es for ime ivariace i a ew ligh. If x()! y(), we look o see if x() (, 0 )! y() (, 0 ). The impulse respose h() of a give sysem gives he respose of ha sysem o a impulse ceered a = 0. If he sysem is a res ad is causal, he impulse respose should o begi o chage from zero uil i sees he impulse a = 0. So a alerae way of provig causaliy is o deermie if h() =0for<0. The superposiio iegral also gives us aoher way of lookig a BIBO sabiliy. If h() is o absolulely R 1 R iegrable (ha is, if jh()jd does o coverge), he here is o way ha y() = 1 x( )h(, )d is goig R 1 x( )x(, )d liear? Time ivaria? Causal? Memoryless? o coverge. Exercise Is y() = 5 A Example Of course, heory is ice, bu if you ca apply i, i is o erribly useful. This example is worked usig a cookbook approach o covoluio. There are more isighful ways of doig his example. Oe mehod is hied a i he exercise a he ed of his secio. Ayway, he cookbook approach for LTI sysems oly: 1. Se up x( ).. Se up h(, ) by flippig. 3. Deermie regios of iegraio by draggig h(, ).. Do he iegraio of he produc of x( ) ad h(, ). The choice of which fucio is x ad ad which is h is arbirary. Sice covoluio is commuaive, you ll probably wa o choose he simpler oe (such as he oe symmeric abou = 0) o flip ad shif. Too bad i have doe ha i my example (oe also ha i have used isead of i made creaig he figure slighly easier). To geerae x( ), use i place of. To geerae h(, ) is slighly harder. Firs, creae h( ) from h() by replacig wih. Themake h(, ) by flippig h( ) abou = 0. Fially, realize h(, ) by relabelig he pois o he axis; = a becomes = + a. Noe ha by icreasig, h(, ) moves o he righ. A his poi i ime, reread he previous paragraph ad ry o figure ou why you relabel he poi = a as = + a (hi: draw h(0,) ad h(1,), ad he geeralize). Try o o cofuse wih. The differece bewee ad is he idex io h. Deermiig he differe regios of iegraio is he mos difficul sep. We drag h(, ) over x( ) by chagig he value of (o ). We he look for values of where: boh h(, ) ad x( ) are ozero (if oe of hem were zero, heir produc would be zero, ad he iegral would be rivial). he limis o he iegral will chage.
EE10: Sigals ad Sysems; v5.0.0 5 makig x() x() flippig h() o make h( ) h() = : draggig h( ) o x() o deermie he differe regios of iegraio =: x() h(0 ) = 1: =3: h(1 ) =0: =: 1 h( ) =1: Figure 3: A example of he flip ad drag covoluio echique. I my example, h(, ) ad x( ) are boh ozero oly bewee =, ad =. We have he followig regios of iegraio: 8 0 if <, x( )h(, )d if, <<0 For, <<0: y() = >< >: Z R ț R R,, x( )h(, )d if 0 << x( )h(, )d if << 0 if > For 0 <<: Z 0 ( + )d +, For <<: ( + )d = + 8j ț, Z 0 = + 8, (,8) = ( + ) (, + )d = [ + 8 ]j 0, +[, + 8 ]j 0 =,(, ), 8(, ), + 8 =, + 8 + 8 =,(,, ) Z (, + )d =, + 8j,, = (, ), 8(, ) +8 = (, )
EE10: Sigals ad Sysems; v5.0.0 6 Does his make sese? Well, he produc of x ad h icreases from 0 a =,oamaximumvaluea = 1ad he falls back o zero a =. Noe ha here is o discoiuiy a =,, = 0, =, or =. I geeral, if your x ad h are piecewise coiuous, y should also be coiuous. Exercise Redo his example. The ry i for he same x(), bu wih h() = for oly <<, zero elsewhere. Exercise Redo his example. However, cosider he pulse as he sum of wo seps, ad he riagle as he sum of hree ramps. Do oly oe covoluio of a sep ad a ramp, ad he use lieariy ad ime ivariace o cosruc he whole soluio. 6 A Look Ahead More covoluio fu follows. If you are ucomforable wih how o do i, ry workig some of he examples i he exbook or askig ay of us for help.