15/03/1439. Lectures on Signals & systems Engineering

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1 Lcturs o Sigals & syms Egirig Dsigd ad Prd by Dr. Ayma Elshawy Elsfy Dpt. of Syms & Computr Eg. Al-Azhar Uivrsity aymalshawy@yahoo.com A sigal ca b rprd as a liar combiatio of basic sigals. Th rspos of LTI to ay iput cosiig of liar combiatio of basic sigals is th liar combiatio of th idividual rsposs to ach of th basic sigals. Covolutio Sum & Covolutio Itgral rpr a sigal as liar combiatio of shiftd impulss. Fourir sris ad trasform uss a complx xpotial sigals with diffrt frqucis will b usd iad of shiftd impulss ( Dlayd or advacd). Rpratios of Priodic Sigals Ja Bapti Josph Fourir, bor i 768, i Frac. 807,priodic sigal could b rprd by siusoidal sris. 89,Dirichlt providd prcis coditios. 960s,Cooly ad Tuy discovrd fa Fourir trasform. 3 4

2 Importac of complx xpotials i LTI sym: Th rspos of a LTI sym to a complx xpotial iput is th sam complx xpotial with oly a chag i amplitud. Th Rspos of LTI Syms to Complx Expotials LTI. Cotiuous-tim sym s ( t ) h( ) y( x( * h( x( t ) h( ) d s d h( ) d s) x( s) ht yt Eigvalu Th complx amplitud factor s) or z) is a fuctio of th complx variabl s or z. Eig fuctio H s H s h t dt 5 6 Th Rspos of LTI Syms to Complx Expotials LTI. Discrt-tim sym y [ ] x[ ]* h[ ] x[ ] h[ ] Eig fuctio h[ z z ] z h[ ] z z) x[ ] z) z H z z z h y Eigvalu H z hz 7 For th cotiuous LTI syms cosidr th iput : From th igfuctio proprty, th rspos to ach part is: Ad from th suprpositio proprty: 8

3 (3) Iput as a combiatio of Complx Expotials Cotiuous tim LTI sym: N x( a N y( ah ( s ) Discrt tim LTI sym: N x[ ] a z N y[ ] a H ( z s t ) z 9 Exampl 3. Cosidr a LTI sym : X( yt xt 3 Impuls rspos j xt t H s = 3s 0 Exampl 3. Cosidr a LTI sym : X( yt xt 3 Impuls rspos: X( xt cos4t cos7t H s = 3s Exampl : Cosidr a LTI sym for which th iput xt cos t ad th impuls rspos t h t u dtrmi th output y t t H s j4t j j4t j j4t H s j7t yt cos4 t 3 cos7 t 3 xt 3 j j7t j j7t y t = j j4t + j j4t + j j7t + j j7t Try to solv it 4 4 y t j j j t j t 3

4 Rpratio of CT Priodic Sigals If th iput to a LTI sym is rprd as a liar combiatio of complx xpotial sigal Th Th output also ca b rprd as liar combiatio of th sam complx xpotial sigal Output compot = iput compot X igvalu 4 Rpratio of CT Priodic Sigals 3.3. Liar Combiatios of Harmoically Rlatd Complx Expotials () Gral Form Th of harmoically rlatd complx xpotials: x t = + = a jw 0t = + = a j( π T )t (, 0,, j 0t j( / T ) t Fudamtal priod: T ( commo priod ) 5 6 4

5 So, arbitrary priodic sigal ca b rprd as x t j t a 0 ( ) ( Fourir sris ) 3 Exampl 3. j t x t 3 a a0 a /,, a a 3 / 4 / 3 a Cofficits Spctral Cofficits K= K= K=N j0 t j0t : Fudamtal compots j0t j0t : Scod harmoic compots, jn0 t jn 0t, : Nth harmoic compots 7 8 ad Fourir Basis Fuctios Ay iput sigal ca rprd as a liar combiatio of shiftd impulss (for ithr DT or CT sigals) How (ipu sigals ca b rprd as a liar combiatio of Fourir basis fuctios (LTI igfuctios) which ar purly imagiary xpotials Ths ar ow as cotiuous-tim Fourir sris Th bass ar scald ad shiftd siusoidal sigals, which ca b rprd as complx xpotials x( = si( + 0.cos( + 0.si(5 9 EE-07 SaS, L7 jt 0/6 x( 5

6 Dtrmiatio of Rpratio j0t xt a j t 0 0 a x t dt T T0 Sythsis quatio Aalysis quatio a Cofficits Spctral Cofficits 6

7 Figur 4. a T 4T b T 8T c T 6T T 0 / T 6 Covrgc of Fourir sris 7 7

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10 3 Why is Fourir Thory Importat (ii)? Fourir trasforms map a tim-domai sigal ito a frqucy domai sigal Simpl itrprtatio of th frqucy cott of sigals i th frqucy domai (as opposd to tim). Dsig syms to filtr out high or low frqucy compots. Aalys syms i frqucy domai. Ivariat to high frqucy sigals EE-07 SaS, L7 40/6 0

11 Why is Fourir Thory Importat (iii)? If F{x(} = X(j) is th frqucy Th F{x (} = jx(j) So solvig a diffrtial quatio is trasformd from a calculus opratio i th tim domai ito a algbraic opratio i th frqucy domai (s Laplac trasform) Exampl d y dy 3y 0 dt dt bcoms Y ( j) jy ( j) 3Y ( j) 0 ad is solvd for th roots (N.B. complmtary quatios): j 3 0 ad w ta th ivrs Fourir trasform for thos. j t Cotiuous LTI sym ht s) h( ( s j) ht j ) h( j dt t dt Hs ad LTI sym Eigfuctios of LTI Sym Sym Fuctio H j jt z j Frqucy Rspos Discrt LTI sym h[] H ( z) h[ ] z ( z h[] j ) j ) h[ ] H j Hzz j j EE-07 SaS, L7 4/6 j0t xt a a Liar Combiatios of Eigfuctios H ad LTI Syms j t j ht dt h t j t t a H j 0 y 0 Priodic Sigal Frqucy Rspos of LTI Sym H j a ( ) jh ( j0 ) 0 j0 ) j0 ) " gai" icludig both amplitud & phas H mu b wll dfid ad fiit. ( j) Exampl Frqucy rspos: Iput sigal: Fid out b of output sigal x( 3 3 a b ) jt y( j) j0 a 0 Impuls rspos: 3 3 j b j 0 t t h( u( d j

12 Exampl Cosidr a LTI sym with iput th uit impuls rspos Rpratio of output xt a j t xt t 4t ht ut yt, dtrmi th jt ( ) a x( dt T T j t j t yt b ah j j) x( j t dt j 4 b 0 is v j 4 is odd 45