The Signal, Variable System, and Transformation: A Personal Perspective

Transcription

1 The Sgal Varable Syem ad Traformao: A Peroal Perpecve Sherv Erfa 35 Eex Hall Faculy of Egeerg

2 Oule Of he Talk Iroduco Mahemacal Repreeao of yem Operaor Calculu Traformao Obervao O Laplace Traform SSB A A Example Of A Complex-Tme Syem Obervao O LTV Syem Tme Varable A New Perpecve /6/7

3 /6/7 3 Iroduco The clacal heory of varable yem baed o he oluo of lear ordary dffereal equao wh varyg coeffce. The varyg coeffce are uually fuco of a depede varable alo called he me varable. The me varable aumed o be real for phycal yem. m k k k k y b x a

4 Wha a Operaor Calculu? The fudameal dffereal equao of a LTV yem : Ue he operaor a d y dx D f x d dx The fudameal equao cover o: a D y f x a Ue he operaor D The fudameal equao for a yem a re cover o: a y x f x /6/7 4

5 Reul of Ug Operaor Calculu Obervao A a reul of ug he operaor calculu he homogeeou repoe ha a paer. The repoe of homogeeou equao: a y x a lear combao of expoeal: y x a e are roo of he operaor characerc equao: x a /6/7 5

6 Exedg he Operaor Calculu: Traformao Expad he fudameal equao: a d y dx f x f x x Aume a expoeal oluo by geeralzao of he homogeeou oluo: The fudameal equao yeld: y x x e x ae f x x where a roo of he operaor characerc equao: lm a lm f x e x x L { f } + x F f x e dx /6/7 6 x

7 Laplace Traform: Good or Bad? Iroduced by Laplace 77 ad appled moder ue by Olver Heavde. Obervao The Laplace raform obaed a a reul of exedg he cocep of he operaor calculu for olvg dffereal equao whch ca decrbe he fudameal equao of phycal dyamc yem. Obervao 3 The oluo ex f here are fe umber M ad σ uch ha : f σ x x < Me x.obervao 4 The depede varable x ca repree ay parameer of he yem; e.g. he me. Obervao 5 he roo of he characerc equao F Hece a complex umber or beer ad a complex varable geeral. Obervao 6 If x repree he depede me varable he by defo repree he complex frequecy he raform doma. Queo Ca fx be a complex fuco of x? Queo Ca x repree a depede complex varable? /6/7 7

8 Complex Tme Syem? The gle de bad SSB amplude modulao AM a example of complex-me yem. The SSB pecrum obaed by hfg he pecra by ad M ω M ω u ω M + ω ad c c ω ω M ω u ω repecvely a how. /6/7 8

9 : SSB Syem Co. + h m [ m jmh ] m h he Hlber raform of m I ca be how ha: m [ m + jm ] where m h π m τ dτ τ Fourer raform of m h : M h ω jm ωg ω M ω H ω where: j e H ω jg ω j j e Hω he rafer fuco of m + π j π a deal ω > ω < phae hfer ha produce he magary-me par of he real-me fuco m + π /6/7 9

10 SSB Syem co. H ω H ω Repreeao of he rafer fuco The USB gal : I me doma: O Φ ω H ω /6/7 π a deal π O π USB + c c ϕ USB ω M ω ω + M ω+ ω Subug for m + ad ϕ USB m m coω jω c + e + ω phae-hfer. m m e h equao reul : c m h ω c jω c

11 Example of a SSB Syem For a ω M ω πe fd m +. L - a + a { } { } - a ω M ω L πe The Fourer raform of he Hlber raform of M ω The Hlber raform : M h : [ ω ] aω aω e u e u ω jm ωg ω jπ ω - m F ω m h + a j a+ j + a { Mh } j a + j + a [ m + jmh ] /6/7

12 Characerzao of LTV yem Coder a gle-pu gle-oupu SISO lear dyamc yem characerzed by he fudameal dffereal equao of a LTV yem: a x m k b k y k k x LTV Syem y /6/7

13 Characerzao of LTV yem Co. Characerzao Operaor form: a D y b D x L D y K D x where: y he oupu repoe gal x he pu excao gal a yem varable parameer kow couou fuco of me b k yem me-varyg parameer kow couou fuco of me D he h dffereal operaor d /d L he yem oupu operaor kow bvarae polyomal of me ad dffereal operaor K he yem pu operaor kow bvarae polyomal of me ad dffereal operaor /6/7 3

14 Obervao o he LTV yem Obervao I geeral me clock of he gal ad yem are o ychrozed;.e. he me varable of he gal ad yem are depede of each oher. L D τ y K D τ x Obervao A ay a of here a repoe whch a pecfed fuco of τ. Obervao 3 A ay fxed τ here a repoe whch a pecfed fuco of. Obervao 4 The yem repoe a fuco of varao of obervao parameer ad applcao parameer τ. Obervao 5 A zero-pu SISO LTV yem decrbed by: L D τ y a lear yem ha aural frequece are varyg wh τ. I oher word he oluo of h equao are expoeal fuco of me wh varyg aural frequece a gve by: where α τ y c e α τ /6/7 4 a fuco of varable coeffce of he fudameal equao of he yem uder coderao.

15 /6/7 5 Exeo of he Operaor Calculu o Soluo of LTV Syem Obervao 6 Coderg he varace propery of x D K y D L τ τ wh repec o ad τ ad by aalogy wh he cae of LTI yem we erpre h equao a a wo-dmeoal yem model ad hall ue a wodmeoal operaor calculu.e. wo-dmeoal Laplace raform DLT o fd repoe. { } { } x D K y D L D D τ τ L L X K Y L X L K Y where: B K A L

16 /6/7 6 DLT Soluo of LTV Syem τ h x y If τ δ δ τ δ τ x we deoe he D rafer fuco H of a LTV yem a: A B L K H Where: { } d d e e H j H h j j j j + + σ σ σ σ τ π τ D - L H ad τ h are called he b-frequecy rafer fuco ad bvarae mpule repoe repecvely.

17 Specal Cae: LTI Syem δ h y I he cae of LTI yem coeffce reduce o he famlar rafer fuco Noe ha leg h τ H π jω + a H jω jω e K L + jω + τ dωdω /6/7 7 ad b are all coa; H b a reul he vere of he wo-dmeoal Fourer raform DFT of H jω jω a gve by:

18 A Word o Tme Varable The me varable aumed o be a real varable for phycal yem. Th aumpo faclae aaly ad yhe of fxed me-vara yem by allowg he Laplace raform echque o be ued. However he aumpo of real me how o be adequae for realzao of me-varyg yem he raform doma. /6/7 8

19 A New Perpecve The dcuo h preeao baed o a dffere po of vew. Pobly of yem realzao hrough a examao of he behavor of yem ha are fuco of a complex me-varable. Th approach allow effec a wo-dmeoal Laplace raform DLT echque o be ued for he me-varyg yem he ame maer ha he coveoal frequecy-doma echque are ued coeco wh fxed yem. /6/7 9

20 To Be Coued Or The Ed /6/7