Theoretical Physics Prof. Ruiz, UNC Asheville, doctorphys on YouTube Chapter R Notes. Convolution

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1 Theoreical Physics Prof Ruiz, UNC Asheville, docorphys o YouTube Chaper R Noes Covoluio R1 Review of he RC Circui The covoluio is a "difficul" cocep o grasp So we will begi his chaper wih a review of he basic RC circui, which we pla o use for our discussig covoluio The volage from boom o op o he lef side of he circui is V, which mus be he same if you go up he righ side: V VR + VC The volage across he resisor is give by Ohm's Law: VR IR This law saes ha if you icrease he volage across a resisor, you icrease he curre Thik of a simple circui wih a baery ad resisor The greaer he volage, he greaer he curre If you replace he resisor wih oe wih a greaer resisace, he you decrease he curre Ohm's Law specifically saes he resisace for a give resisor is cosa The you have a liear graph whe you plo he volage agais he curre For he capacior, he greaer he volage fashio So we wrie he volage V C, he greaer he charge q i a liear V C as proporioal o he charge q VC q The capaciace C is a cosa ha gives us a measure of how easily he capacior ca sore los of charge If he capaciace is greaer, he i ca sore more charge q, give a fixed volage capaciace: V C To make his come ou righ, we divide he charge by he V C q C Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

2 Boh VR IR V ad C q C have heir limis If you zap eiher he resisor or he capacior wih oo high a volage, you ca wase hem ad hus bur hem ou Wih hese subsiuios, our equaio V VR + VC becomes q V IR + C Imagie aachig he baery a ime where here is o charge iiially o he capacior The, iiially here is a rush of curre where VR () V IR ad V () sice q () The capacior is beig charged up Afer chargig, ie, waiig a log ime, we have o more curre VR( ) ad ( ) q( ) VC wih q( ) CV C Le's remove he baery by makig V Assume here is some iiial charge q() CV sored o he capacior e o prior chargig Now we have our sadard discharge siuaio Noe ha I C dq d, ie, he curre is he flow of charge per ui ime ierval The differeial equaio describig he discharge is dq q R + d C Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

3 We ca solve his equaio by separaig he variables q ad dq R d q C 1 1 dq q RC d q( ) 1 1 dq q RC q() 1 RC ( ) l q q q() d l q( ) l q() RC q( ) l q () RC q( ) q() e RC q( ) q() e RC Wih our prior charge of CV, we have q( ) CV e RC Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

4 Summary of he Dischargig Circui q( ) CV e RC PR1 (Pracice Problem) Solve he differeial equaio for he chargig circui ad show ha q( ) CV (1 e RC ) Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

5 R2 Square Pulse hrough Low-Pass Filer We cosider our RC circui orieed o receive a ipu wave We apply a pulse volage The capacior will begi o charge up ad he discharge If he pulse-ime is shor eough, he capacior will o fully charge Whe he pulse volage drops o zero, he capacior will discharge The oupu show is he volage across he capacior This RC circui is a low-pass filer by he way The physics ca be described by a covoluio You will udersad covoluio much beer wih his approach because you will kow everyhig abou his circui here hrough coveioal mehods We are goig o simply cas his i erms of covoluio Eve wih his said, covoluio will sill be "difficul" so we make he followig assigmes i order o cocerae o he pure mah: R 1 ad C 1 I oher words, we have a 1-ohm resisor ad a 1-Farad capacior We also ake V 1, ie, our pulse volage is 1 vol Ad you probably guessed ha we will apply he pulse for 1 secod so ha each parameer is 1 The he chargig ad dischargig equaios simplify q( ) CV (1 e RC ) becomes q( ) 1 e Chargig: Dischargig q( ) CVe RC becomes q( ) e Our applied volage is represeed by f ( ) The capacior sars chargig Bu he applied volage drops o zero afer 1 secod A he 1-secod poi he capacior has 1 e 1 charge From his mome o he discharge kicks i Noe ha he discharge is refereced o ime 1, ime greaer ha 1 secod The fucios mach a 1 Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

6 R3 Covoluio You kow ha a ispiraio of our course is Richard Feyma, a ousadig heoreical physicis Feyma always ecouraged ohers o work higs ou for hemselves i order o really udersad wha is goig o As early as a eeager, Feyma kep oebooks i which he worked ou deails for himself i his ow way Oce Feyma commeed alog hese lies i referece o experimealiss "I suddely realized why Priceo was geig resuls They were workig wih he isrume They buil he isrume; hey kew where everyhig was, hey kew how everyhig worked, I was woderful! Because hey worked wih i They did' have o si i aoher room ad push buos!" Richard Feyma Source: Surely you're jokig Mr Feyma! (Adveures of a Curious Characer, by Richard P Feyma (Auhor), Ralph Leigho (Auhor), ad Edward Huchigs (Edior), ad Alber R Hibbs (Irocio), published by W W Noro & Compay (April 17, 2997) Book availabe a wwwamazocom By 1985, he year whe his book was firs published, may so-called "Feyma" sories had amassed i he folklore of physics over he years May of hese are foud i his book ad sill more i he compaio volume Wha Do You Care Wha Oher People Thik?: Furher Adveures of a Curious Characer Followig Feyma's sress o he imporace of workig ou deails ad buildig he isrume, we have prepared he way for covoluio his way We are goig o use he example we "buil" i he previous secio - he basic RC circui covered i he irocory physics course, bu applied wih a pulse volage We sacrifice formal mahemaical derivaio (he room wih he buos) for isigh Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

7 You have already me your firs covoluio The fucio a he righ below is a covoluio of a square pulse f ( ) wih he dischargig fucio g( ) e of he capacior We will show you wha we mea by his ow Sar wih q( ) where 1 Le's play wih his soluio q( ) (1 e ) e 1 ( 1) ( 1) q( ) e e q( ) e ( e 1) q e e e 1 ( ) ( ) Wha abou for 1 Le's check 1 1 u u q( ) e e e e u u q ( ) e e e e e ( e 1) 1 e Sice f ( u ) 1 for u 1 ad elsewhere, we ca wrie for all : u q( ) e f ( u) e Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

8 Our equaio u q( ) e f ( u) e ca also be wrie as ( ) ( ) ( u) q f u e This is your covoluio So wha do we mea mahemaically whe we sae ha he fucio q( ) is a covoluio of a square pulse f ( ) wih he dischargig fucio g( ) e? We simply mea his "covolued" iegral or more formally he followig ( ) ( ) ( u) q f u e q( ) f ( u) g( u) We covolue f wih g Or we ake he covoluio of f ad g The oaio for covoluio is give below f ( )* g( ) f ( u) g( u) Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

9 R4 Covoluio is Commuaive We demosrae i his secio, usig our example, ha he covoluio operaor is commuaive Our wo fucios for : f ( ) 1 for 1 ad g( ) e The covoluio f ( )* g( ) f ( u) g( u) i he las secio gave us ( u) u ( )* ( ) ( ) ( ) f g f u e e f u e Wha abou? g( )* f ( ) g( u) f ( u) How do you shif he pulse fucio? I is jus 1 for he ierval or 1 secod This is bes hadled by a chage of variable Defie a ew iegraio variable z u The dz ad as u goes from o, we have z goig from o Our covoluio u wih he ew variable is g( )* f ( ) e f ( u) z z g( )* f ( ) e f ( z)( dz) e f ( z) dz I he las sep he mius was used o flip he order of he iegraio limis Bu his is iegraio variable z g( )* f ( ) e e f ( z) dz u f ( )* g( ) e f ( u) e sice we ca choose ay leer for he Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

10 Therefore, covoluio is commuaive f ( )* g( ) g( )* f ( ) f ( )* g( ) f ( u) g( u) g( )* f ( ) g( u) f ( u) R5 The Laplace Trasform ad Covoluio We reur o our circui problem o see how Laplace rasforms are relaed o covoluios We reur o he RC circui q V IR + C We ake he Laplace rasform of boh sides Wih V f ( ), R 1, ad C 1, dq f ( ) I + q, ie, q f ( ) d + dq L{ } + L{ q} L{ f ( )} d We eed he Laplace rasform of our box fucio s L{ f ( )} f ( ) e d Bu we will o eed o acually calculae i sice we are aimig for a more geeral relaioship Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

11 dq L{ } + L{ q} L{ f ( )} d sq( s) q() + Q( s) F( s) There is o iiial charge, herefore q () sq( s) + Q( s) F( s) Q( s)( s + 1) F( s) 1 Q( s) F( s) s + 1 Bu we kow he aswer is, where g( ) e q( ) f ( )* g( ) f ( u) g( u) Bu WAIT! From ou ables, a L{ e } 1 s a Therefore, 1 L{ g( )} s + 1 Check his ou Q( s) F( s) G( s) The Laplace rasform of a covoluio is equal o he proc of he Laplace rasforms L{ f ( )* g( )} F( s) G( s) L{ f ( )* g( )} L{ f ( )} L{ g( )} Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

12 f ( )* g( ) f ( u) g( u) F( s) G( s ) R6 Covoluio from Power Series A( x) a Take wo power series: x ad o B( x) b x o The capial leers are i a world similar o Laplace-rasform space whe compared o heir respecive lile leers The lile leers refer o our world Muliply he big oes i rasform space The we mus have some sor of covoluio for he lile "guys" Noe ha we chose a differe summaio idex for each If we did o, we would oly ge he diagoal erms whe k l k k k l A( x) B( x) a x b x l l A( x) B( x) k l a b x k l k + l Image Grid from wwwhelpigwihmahcom k A( x) B( x) a b x Le k + l We will sum k from o ad he from o ifiiy o do he job k k Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese

13 A( x) B( x) akb k x c x C( x) k The ew lile "guys" are relaed o old as follows akb k k c This is he covoluio - he discree versio Le's move o he coiuous case c a b k sice k 1 k k k Chage dela o d, rip off idexes (promoig o coiuous variables), ad chage he summaio sig io a "sake" Now replace wih ad k wih u c( ) a( k) b( k) dk c( ) a( u) b( u) Oh, wha he heck, replace "a" wih "f" ad "b" wih "g" ad you have f ( )* g( ) f ( u) g( u) THE CONVOLUTION! THE END Michael J Ruiz, Creaive Commos Aribuio-NoCommercial-ShareAlike 3 Upored Licese