1/25/211 The Eige Fucio of Liear Sysems.doc 1/7 The Eige Fucio of Liear Sysems Recall ha ha we ca express (expad) a ime-limied sigal wih a weighed summaio of basis fucios: v ( ) a ψ ( ) = where v ( ) = for < ad > T. Say ow ha we covolve his sigal wih some sysem impulse fucio g ( ): ( ) ( ) ( ) L v = g v d ( ) ψ ( ) = g a d ( ) ψ ( ) = a g d Look wha happeed! Isead of covolvig he geeral fucio v ( ), we ow fid ha we mus simply covolve wih he se of basis fucios ψ. ( ) Jim Siles The Uiv. of Kasas Dep. of EECS
1/25/211 The Eige Fucio of Liear Sysems.doc 2/7 Q: Huh? You say we mus simply covolve he se of basis fucios ψ ( ). Why would his be ay simpler? A: Remember, you ge o choose he basis ψ ( ). If you re smar, you ll choose a se ha makes he covoluio iegral simple o perform! Q: Bu do I firs eed o kow he explici form of g ( ) before I ielligely choose ψ ( )?? A: No ecessarily! The key here is ha he covoluio iegral: ψ ( ) ( ) ψ ( ) L = g d is a liear, ime-ivaria operaor. Because of his, here exiss oe basis wih a asoishig propery! These special basis fucios are: jω e for T 2π ψ ( ) = where ω = T for <, > T Now, iserig his fucio (ge ready, here comes he asoishig par!) io he covoluio iegral: Jim Siles The Uiv. of Kasas Dep. of EECS
1/25/211 The Eige Fucio of Liear Sysems.doc 3/7 ω ω ( ) L = j j e g e d ad usig he subsiuio u =, we ge: jω jω( u) g ( ) e d = g ( u) e du ( ) jω j u ( ) ω ( ) ( ) = = e g u e du + jω jω u e g ( u) e du See! Does ha asoish! Q: I m asoished oly by how lame you are. How is his resul ay more asoishig ha ay of he oher supposedly useful higs you ve bee ellig us? A: Noe ha he iegraio i his resul is o a covoluio he iegral is simply a value ha depeds o (bu o ime ): ( ω ) ( ) jω G g e d As a resul, covoluio wih his special se of basis fucios ca always be expressed as: Jim Siles The Uiv. of Kasas Dep. of EECS
1/25/211 The Eige Fucio of Liear Sysems.doc 4/7 g e d L e G e ( ) = = ( ω ) jω jω jω The remarkable hig abou his resul is ha he liear ψ = exp jω resuls i precisely he operaio o fucio ( ) [ ] same fucio of ime (save he complex muliplier ( ) I.E.: ψ ( ) G ( ω ) ψ ( ) L = Covoluio wih ψ ( ) exp[ jω ] G ω )! = is accomplished by simply muliplyig he fucio by he complex G ω! umber ( ) Noe his is rue regardless of he impulse respose g ( ) (he fucio g ( ) affecs he value of G ( ω ) oly)! Q: Big deal! Are here los of oher fucios ha would saisfy he equaio above equaio? A: Nope. The oly fucio where his is rue is: j ( ) e ω ψ = This fucio is hus very special. We call his fucio he eige fucio of liear, ime-ivaria sysems. Q: Are you sure ha here are o oher eige fucios?? Jim Siles The Uiv. of Kasas Dep. of EECS
1/25/211 The Eige Fucio of Liear Sysems.doc 5/7 A: Well, sor of. Recall from Euler s equaio ha: j e ω = ω + ω cos j si I ca be show ha he siusoidal fucios cos ω ad si ω are likewise eige fucios of liear, ime-ivaria sysems. The real ad imagiary compoes of eige fucio exp jω are also eige fucios. [ ] Q: Wha abou he se of values G ( ω )?? Do hey have ay sigificace or imporace?? A: Absoluely! Recall he values G ( ω ) (oe for each ) deped o he impulse respose of he sysem (e.g., circui) oly: ( ω ) ( ) jω G g e d Thus, he se of values G ( ω ) compleely characerizes a liear ime-ivaria circui over ime T. We call he values ( ) G ω he eige values of he liear, ime-ivaria circui. Jim Siles The Uiv. of Kasas Dep. of EECS
1/25/211 The Eige Fucio of Liear Sysems.doc 6/7 Q: OK Poidexer, all eige suff his migh be ieresig if you re a mahemaicia, bu is i a all useful o us elecrical egieers? A: I is ufahomably useful o us elecrical egieers! Say a liear, ime-ivaria circui is excied (oly) by a siusoidal source (e.g., vs( ) = cosωo ). Sice he source fucio is he eige fucio of he circui, we will fid ha a every poi i he circui, boh he curre ad volage will have he same fucioal form. Tha is, every curre ad volage i he circui will likewise be a perfec siusoid wih frequecy ω o!! Of course, he magiude of he siusoidal oscillaio will be differe a differe pois wihi he circui, as will he relaive phase. Bu we kow ha every curre ad volage i he circui ca be precisely expressed as a fucio of his form: Acos ( ω + ϕ ) Q: Is his prey obvious? o Jim Siles The Uiv. of Kasas Dep. of EECS
1/25/211 The Eige Fucio of Liear Sysems.doc 7/7 A: Why should i be? Say our source fucio was isead a square wave, or riagle wave, or a sawooh wave. We would fid ha (geerally speakig) owhere i he circui would we fid aoher curre or volage ha was a perfec square wave (ec.)! I fac, we would fid ha o oly are he curre ad volage fucios wihi he circui differe ha he source fucio (e.g. a sawooh) hey are (geerally speakig) all differe from each oher. We fid he ha a liear circui will (geerally speakig) disor ay source fucio uless ha fucio is he eige fucio (i.e., a siusoidal fucio). Thus, usig a eige fucio as circui source grealy simplifies our liear circui aalysis problem. All we eed o accomplish his is o deermie he magiude A ad relaive phase ϕ of he resulig (ad oherwise ideical) siusoidal fucio! Jim Siles The Uiv. of Kasas Dep. of EECS