Convoluion Lecure #6 C.3 8
Deiniion When we compue he ollowing inegral or τ and τ we say ha he we are convoluing wih g d his says: ae τ, lip i convolve in ime -τ, hen displace i in ime by seconds -τ, and muliply i by τ. Finally, inegrae he produc over all τ.
Properies o Convoluion Firs some shorhand: Commuaive: Associaive: Disribuive: d ] [ ] [ 3 3 ] [ ] [ ] [ 3 3
A Graphical Example o How o Perorm a Convoluion =A- τ/ A B hτ C h d h-τ B h-τ B τ τ - τ -- τ We need o loo a 5 cases: <, < <, 3 < <, 4 < < 3, & 5 > 3 For case and case 5, C= since here is no overlap. B h-τ, < A h-τ, >3 A B -- -- τ 3 τ 3
Graphical Convoluion Example Coninued Case : < < Case 3: < < Case 4: < < 3 AB/ 3 h-τ, < < B A -- h-τ, < < 3 A B -- 3 h-τ, < < B A -- C τ C τ τ AB d AB AB / AB d C AB[ AB / AB AB d ] 4 AB
Malab Code clear all; endpulse=; s=.; endpoin=; n=-endpoin:s:endpoin; nn=-endpoin*:s:endpoin*; pulser=n>=&n<=endpulse; pulse=*pulser; ripulse=n>=&n<=; ri=-n.*ripulse; subplo,, plon,ri,'r',n,pulse,'b'; ile'signals'; xlabel'ime Seconds'; axis[- min[minri minpulse].*max[maxri maxpulse]]; c=convpulse,ri*s; subplo,, plonn,c; ile'convoluion'; xlabel'shi Seconds'; axis[- minc.*maxc];.5-3 4 5 6 7 8 9 ime Seconds Convoluion.4. Signals - 3 4 5 6 7 8 9 Shi Seconds 5
Inegraion o Convoluion Inegral Case : C AB d AB AB { } AB { } AB { } AB { } AB{ } Case 3: From he inegraion rom Case C AB d AB AB{ } AB{ } AB Case 4: 3 From he inegraion rom Case C AB d AB AB { } AB { } AB { } AB { } AB{ } 6
7 Convoluion and Sysems For an LI sysem, le s deine h as he sysem response o a uni impulse source, δ. hen he ollowing mus be rue: superposiion scalar ime invariance h x x h x x h h y x
Convoluion and Sysems Coninued Consruc x as he sum o uni impulse slices. he irs expression represens he slices o he source oaled as approaches ininiy. Le Side Equals lim x approaches x d x x x h x x h d y Consruc y as he sum o slices o he response due o an uni impulse uncion: xδh-δ. he inegral on he righ is he convoluion o x and h. Righ Side Equals lim x h approaches x h d his resul is very imporan since i says ha i one nows he impulse response o a sysem hen he oupu response or any given inpu source can be ound by convolving he inpu wih he impulse response. 8
Impulse Response and Causaliy y o does no depend on x i ha occur a imes aer o, i > o. y h x d h x d h x d o o o o For his inegral h x d, is posiive,, and or x x we have i o i o o o o For his inegral h x d, is negaive,, and or x x we have i o i o hereore or y o be causal his h x d mus be zero. For his o happen h,. o o o o 9
Calculaing he Uni Impulse Response, h R Vs i L Le s irs loo a mehods:. Narrow Pulse approximaion. Diereniaing u Beer Mehods o Come 3
Narrow Pulse Approximaion ε = widh o pulse ε heigh o pulse As ε, he narrow pulse = /ε [u-u-ε] δ So hen le s irs loo a he response o u: di Vs u i R L d i = Vs/R -e -R/L u Now we consruc he narrow pulse response: i = Vs/R /ε {-e -R/L u - e -R/L-ε u-ε} 3
Narrow Pulse Approximaion Coninued i = Vs/R/ε [-e -R/L u -e -R/L-ε u-ε] OR, or < i= Vs/R/ε [-e -R/L ] or < ε Vs/R/ε [-e -R/L ] -e -R/L-ε ], or > ε OR, or < i= Vs/R/ε [-e -R/L ], or < ε Vs/R/ε [e R/Lε e -R/L ], or > ε 3
aylor Series Approximaion or e x ax ax e! or x, e ax ax! 3 ax ax ax...!! 3! we can drop he higher order erms : ax! ax 33
Narrow Pulse Approximaion Coninued Applying he approximaion or e x, Vs= he uni impulse uncion and α = R/L, or < i= Vs/R/ε [-e -R/L ]= Vs/R/εR/L=Vs/Rα/ε, or < ε Vs/R/ε [e +R/Lε e -R/L ]= Vs/R/εαεe -α, or > ε, or < i= /Rα/ε=α/R, or <ε /R αe -α, or > ε h lim i e R u 34
Diereniaing he Uni Sep uncion he response due o a Uni Sep uncion is i = Vs/R -e -α u and since du, hen d h di d h [ e u e ] R [ e e u ] R e R u 35
Convoluion or Discree Sysems For an LI sysem, le s deine h[n] as he sysem response o a uni impulse source, δ[n]. δ[n]=, n= and or n We have: x[n]= Σ x[m]δ[n-m] y[n]=σ x[m] h[n-m] In addiion he same convoluion properies hold: Commuaive Associaive Disribuive [ n] [ n] [ n] [ n] [ n] { [ n] 3[ n]} { [ n] [ n]} 3[ n] [ n] { [ n] 3[ n]} { [ n] [ n]} { [ n] 3[ ]} 36
Sabiliy o Sysems Coninuous Sysems I a sysem is sable, i.e., Bounded Inpu, Bounded Oupu BIBO, he ollowing mus be rue: x, hen y y h x d h x d x max max OR However, his is no always he case. Posiive Feedbac causes insabiliy h d h d 37
Wha is needed or BIBO For a coninuous ime sysem, he poles o Hp mus lie wihin he le hand complex plane such ha Re s i < where s i are he poles o Hp. his will assure ha he ree response will be damped and no grow exponenially. HIS IS WHY WE SUDIED SOLUIONS OF LINEAR ODE IN ERMS OF SOURCE-FREE AND SOURCE COMPONENS. HIS IMPLIES HA Hp AND HE IMPULSE RESPONSE, h, MAY BE RELAED. 38
Sabiliy o Sysems Discree Sysems I a sysem is sable, i.e., Bounded Inpu, Bounded Oupu BIBO, he ollowing mus be rue: x[n], hen y[n] y[n] h[m]x[n m] x max h[m] OR h[m] However, his is no always he case. Posiive Feedbac causes insabiliy 39
Wha is needed or BIBO For a discree ime sysem, he eigenvalues o hn [ ] mus lie wihin he uni circle n such ha z where z are he eigenvalues o h[ n] Az. i i i i i his will assure ha ree response will no diverge and hn [ ] Using he ormula or he parial sums o a geomeric series, where N is he number o roos o he Characerisic Equaion L L N N L N L N n n i lim hn [ ] lim zi lim zi L L L L n n i i n i zi i z i z, lim lim i L N N L n i zi L L n i i zi z z angle z i i i L L z z Langle z i i i L n n i A z i n i <. z lim ; provided zi z L lim z lim{ z Langle z } lim{ z } { Langle z } i i i i L L L L lim{ z } ; z andlim{ z } ; z L i i i i L L L i i 4
Homewor Convoluion Veriy your all your resuls o hese convoluion problems using Malab and is conv uncion. Problem Assume ha a sysem response is given by he ollowing: h Sech he response o a u, b u-u-a or a=.5, a=, and a=5, and c evaluae e - u a = and = Problem Assume ha a sysem response is given by he ollowing: h /3 3 Evaluae he response o e - u a = and =3
Homewor Sabiliy Deermine he sabiliy o he ollowing sysems wih poles in he complex plane, describe he orm o he ransien response: Imaginary axis Imaginary axis Imaginary axis Real axis Real axis Real axis C.3., C.3. 4