6.01: Introduction to EECS I Lecture 3 February 15, 2011

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1 6.01: Iroducio o EECS I Lecure 3 February 15, : Iroducio o EECS I Sigals ad Sysems Module 1 Summary: Sofware Egieerig Focused o absracio ad modulariy i sofware egieerig. Topics: procedures, daa srucures, objecs, sae machies Lab Exercises: implemeig robo corollers as sae machies SesorIpu Brai Acio Absracio ad Modulariy: Combiaors Cascade: make ew SM by cascadig wo SM s Parallel: make ew SM by ruig wo SM s i parallel Selec: combie wo ipus o ge oe oupu Themes: PCAP February 15, 2011 Primiives Combiaio Absracio Paers 6.01: Iroducio o EECS I The iellecual hemes i 6.01 are recurrig hemes i EECS: desig of complex sysems modelig ad corollig physical sysems augmeig physical sysems wih compuaio buildig sysems ha are robus o uceraiy Iellecual hemes are developed i coex of a mobile robo. Module 2 Preview: Sigals ad Sysems Focus ex o aalysis of feedback ad corol sysems. Topics: differece equaios, sysem fucios, corollers. Lab exercises: roboic seerig sraigh ahead? seer righ seer righ seer righ sraigh ahead? seer lef seer lef Goal is o covey a disic perspecive abou egieerig. Themes: modelig complex sysems, aalyzig behaviors Aalyzig (ad Predicig) Behavior Today we will sar o develop ools o aalyze ad predic behavior. Aalyzig (ad Predicig) Behavior Make he forward velociy proporioal o he desired displaceme. Example (desig Lab 2): use soar sesors (i.e., curredisace) o move robo desireddisace from wall. desireddisace curredisace desireddisace curredisace >>> class wallfider(sm.sm):... sarsae = Noe... def genexvalues(self, sae, ip):... desireddisace = curredisace = ip.soars[3]... reur (sae,io.acio(fvel=?,rvel=0)) Fid a expressio for fvel. 1

2 6.01: Iroducio o EECS I Lecure 3 February 15, 2011 Check ourself Check ourself Which plo bes represes curredisace? desireddisace curredisace desireddisace curredisace Which expressio for fvel has he correc form? curredisace 2. curredisace-desireddisace 3. desireddisace 4. curredisace/desireddisace 5. oe of he above 5. oe of he above Example: Mass ad Sprig Example: Taks x() r 0 () y() h 1 () r 1 () h 2 () r 2 () x() y() r 0 () r 2 () mass & sprig sysem ak sysem Example: Cell Phoe Sysem Sigals ad Sysems: Widely Applicable soud ou The Sigals ad Sysems approach has broad applicaio: elecrical, mechaical, opical, acousic, biological, fiacial,... x() y() soud i r0() mass & sprig sysem soud i soud ou h1() r1() r0() r2() phoe sysem h2() r2() soud i ak sysem soud ou phoe sysem 2

3 6.01: Iroducio o EECS I Lecure 3 February 15, 2011 Sigals ad Sysems: Modular The represeaio does o deped upo he physical subsrae. Sigals ad Sysems: Hierarchical Represeaios of compoe sysems are easily combied. Example: cascade of compoe sysems soud ou soud i phoe E/M ower opic fiber ower E/M phoe soud ou soud i soud i phoe E/M FreeFoo.com. CC by-c-d. This coe is excluded from our Creaive Commos licese. For more iformaio, see hp://ocw.mi.edu/fairuse. ower opic fiber ower E/M phoe soud ou focuses o he flow of iformaio, absracs away everyhig else Composie sysem soud i phoe sysem soud ou Compoe ad composie sysems have he same form, ad are aalyzed wih same mehods. The Sigals ad Sysems Absracio Our goal is o develop represeaios for sysems ha faciliae aalysis. Coiuous ad Discree Time Ipus ad oupus of sysems ca be fucios of coiuous ime sigal i sysem sigal ou or discree ime. Examples: Does he oupu sigal overshoo? If so, how much? How log does i ake for he oupu sigal o reach is fial value? We will focus o discree-ime sysems. Differece Equaios Differece equaios are a exe way o represe discree-ime sysems. Example: y[] =x[] x[ 1] Differece equaios ca be applied o ay discree-ime sysem; hey are mahemaically precise ad compac. Differece Equaios Differece equaios are mahemaically precise ad compac. Example: y[] =x[] x[ 1] Le x[] equal he ui sample sigal δ[], { 1, if = 0; δ[] = 0, oherwise. x[] =δ[] We will use he ui sample as a primiive (buildig-block sigal) o cosruc more complex sigals. 3

4 6.01: Iroducio o EECS I Lecure 3 February 15, 2011 Sep-By-Sep Soluios Differece equaios are coveie for sep-by-sep aalysis. Muliple Represeaios of Discree-Time Sysems Block diagrams are useful aleraive represeaios ha highligh visual/graphical paers. Fid y[] give x[] = δ[]: y[] = x[] x[ 1] Differece equaio: y[] = x[] x[ 2] =0 0 =0 y[0] = x[0] x[] =1 0 =1 y[1] = x[1] x[0] =0 1 = y[2] = x[2] x[1] =0 0 =0 y[3] = x[3] x[2] =0 0 =0... y[] = x[] x[ 1] Block diagram: x[] y[] x[] = δ[] y[] Same ipu-oupu behavior, differe sreghs/weakesses: differece equaios are mahemaically compac block diagrams illusrae sigal flow pahs Sep-By-Sep Soluios Check ourself Block diagrams are also useful for sep-by-sep aalysis. Represe y[] = x[] x[ 1] wih a block diagram: sar a res DT sysems ca be described by differece equaios ad/or block diagrams. x[] y[] Differece equaio: y[] = x[] x[ 1] 0 Block diagram: x[] y[] x[] = δ[] y[] I wha ways are hese represeaios differe? From Samples o Sigals Lumpig all of he (possibly ifiie) samples io a sigle objec he sigal simplifies is maipulaio. From Samples o Sigals Operaors maipulae sigals raher ha idividual samples. This lumpig is aalogous o represeig coordiaes i hree-space as pois represeig liss of umbers as vecors i liear algebra creaig a objec i Pyho Nodes represe whole sigals (e.g., ad ). The boxes operae o hose sigals: = shif whole sigal o righ 1 ime sep Add = sum wo sigals : muliply by Sigals are he primiives. Operaors are he meas of combiaio. 4

5 6.01: Iroducio o EECS I Lecure 3 February 15, 2011 Operaor Noaio Operaor Noaio: Check ourself Symbols ca ow compacly represe diagrams. Le R represe he righ-shif operaor: = R{} R where represes he whole ipu sigal (x[] for all ) ad represes he whole oupu sigal (y[] for all ) Represeig he differece machie Le = R. Which of he followig is/are rue: 1. y[] = x[] for all 2. y[ 1] = x[] for all 3. y[] = x[ 1] for all 4. y[ 1] = x[] for all 5. oe of he above wih R leads o he equivale represeaio = R = (1 R) Operaor Represeaio of a Cascaded Sysem Sysem operaios have simple operaor represeaios. Operaor Algebra Operaor expressios expad ad reduce like polyomials. Cascade sysems muliply operaor expressios Usig operaor oaio: 1 =(1 R) 2 =(1 R) 1 Subsiuig for 1 : 2 =(1 R)(1 R) Usig differece equaios: y 2 [] = y 1 [] y 1 [ 1] =(x[] x[ 1]) (x[ 1] x[ 2]) = x[] 2x[ 1] x[ 2] Usig operaor oaio: 2 =(1 R) 1 =(1 R)(1 R) =(1 R) 2 =(1 2R R 2 ) Operaor Approach Operaor Algebra Applies your exisig experise wih polyomials o udersad block diagrams, ad hereby udersad sysems. Operaor oaio faciliaes seeig relaios amog sysems. Equivale block diagrams (assumig boh iiially a res): Equivale operaor expressio: (1 R)(1 R)=1 2R R 2 5

6 6.01: Iroducio o EECS I Lecure 3 February 15, 2011 Operaor Algebra Operaor oaio prescribes operaios o sigals, o samples: e.g., sar wih, subrac 2 imes a righ-shifed versio of, ad add a double-righ-shifed versio of! : R : Operaor Algebra Expressios ivolvig R obey may familiar laws of algebra, e.g., commuaiviy. R(1 R) = (1 R)R This is easily proved by he defiiio of R, ad i implies ha cascaded sysems commue (assumig iiial res) R 2 : is equivale o y = 2R R 2 : Operaor Algebra Muliplicaio disribues over addiio. Operaor Algebra The associaive propery similarly holds for operaor expressios. Equivale sysems Equivale sysems 2 2 Equivale operaor expressio: R(1 R)=R R 2 Equivale operaor expressio: ( ) ( ) (1 R)R (2 R)=(1 R) R(2 R) Check ourself Explici ad Implici Rules How may of he followig sysems are equivale? Recipes versus cosrais. 2 2 =(1 R) 4 Recipe: oupu sigal equals differece bewee ipu sigal ad righ-shifed ipu sigal. = R (1 R) = 4 Cosrais: fid he sigal such ha he differece bewee ad R is. Bu how? 6

7 6.01: Iroducio o EECS I Lecure 3 February 15, 2011 Example: Accumulaor Try sep-by-sep aalysis: i always works. Sar a res. x[] y[] Fid y[] give x[] = δ[]: y[] = x[] y[ 1] y[0] = x[0] y[] =1 0=1 y[1] = x[1] y[0] =0 1=1 y[2] = x[2] y[1] =0 1=1... x[] = δ[] y[] Example: Accumulaor The respose of he accumulaor sysem could also be geeraed by a sysem wih ifiiely may pahs from ipu o oupu, each wih oe ui of delay more ha he previous Persise respose o a rasie ipu! =(1 R R 2 R 3 ) Example: Accumulaor Example: Accumulaor These sysems are equivale i he sese ha if each is iiially a res, hey will produce ideical oupus from he same ipu. (1 R) 1 = 1? 2 =(1 R R 2 R 3 ) 2 Proof: Assume 2 = 1 : The sysem fucioal for he accumulaor is he reciprocal of a polyomial i R. (1 R) = 2 =(1 R R 2 R 3 ) 2 =(1 R R 2 R 3 ) 1 =(1 R R 2 R 3 )(1 R) 1 = ((1 R R 2 R 3 ) (R R 2 R 3 )) 1 = 1 I follows ha 2 = 1. The produc (1 R) (1 R R 2 R 3 ) equals 1. Therefore he erms (1 R) ad (1 R R 2 R 3 ) are reciprocals. Thus we ca wrie = 1 1 R =1 R R2 R 3 R 4 Example: Accumulaor The reciprocal of 1 R ca also be evaluaed usig syheic divisio. 1 R R 2 R 3 1 R 1 1 R R R R 2 R 2 R 2 R 3 R 3 R 3 R 4 Check ourself A sysem is described by he followig operaor expressio: = 1 12R. Deermie he oupu of he sysem whe he ipu is a ui sample. Therefore 1 =1 R R 2 R 3 R 4 1 R 7

8 6.01: Iroducio o EECS I Lecure 3 February 15, 2011 Liear Differece Equaios wih Cosa Coefficies Ay sysem composed of adders, gais, ad delays ca be represeed by a differece equaio. Check ourself Deermie he differece equaio ha relaes x[ ] ad y[ ]. y[] a 1 y[ 1] a 2 y[ 2] a 3 y[ 3] = b 0 x[] b 1 x[ 1] b 2 x[ 2] b 3 x[ 3] x[] y[] Such a sysem ca also be represeed by a operaor expressio. (1 a 1 R a 2 R 2 a 3 R 3 ) =(b 0 b 1 R b 2 R 2 b 3 R 3 ) We will see ha his correspodece provides isigh io behavior. This correspodece also reduces algebraic edium. 1. y[] = x[ 1] y[ 1] 2. y[] = x[ 1] y[ 2] 3. y[] = x[ 1] y[ 1] y[ 2] 4. y[] = x[ 1] y[ 1] y[ 2] 5. oe of he above Sigals ad Sysems Muliple represeaios of discree-ime sysems. Differece equaios: mahemaically compac. y[] = x[] x[ 1] Block diagrams: illusrae sigal flow pahs. x[] y[] Operaor represeaios: aalyze sysems as polyomials. =(1 R) Labs: represeig sigals i pyho corollig robos ad aalyzig heir behaviors. 8

9 MIT OpeCourseWare hp://ocw.mi.edu 6.01SC Iroducio o Elecrical Egieerig ad Compuer Sciece Sprig 2011 For iformaio abou ciig hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.