Nikesh Bajaj. Fourier Analysis and Synthesis Tool. Guess.? Question??? History. Fourier Series. Fourier. Nikesh Bajaj
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1 Guss.? ourir Analysis an Synhsis Tool Qusion??? Digial Signal Procssing School of Elcronics an Communicaion Lovly Profssional Univrsiy Wha o you man by Transform? Wha is /Transform? Diffrn bwn an Diffrn bwn an Laplac Transform 3 By: ourir Jan Bapis Josph ourir - March May 83, Auxrr, ranc, Mahmaician, Egypologis, Rvoluionary Discovry (8). Scinific Avisor H claim ha any prioic signal can b rprsn by a sris of harmonically rla sinusoial. OURIER SERIES H also obain a rprsnaion of Aprioic signal, no as wigh sum of harmonically rla sinusoial, bu as wigh Ingral of sinusoial ha ar no a all harmonically rla. 5 -OURIER INTEGRAL or TRANSORM By: Sir Isaac Nwon (4 January March Pirr-Simon Laplac (749-87) Jan Bapis Josph ourir ( March May 83) 77 By: Hisory In 748 L. Eulr: Vibraing Sring Vrical flaion a any im is h linar combinaion of normal mos, h show i In 753 D. Brnouli argu sam on physical groun (No mah) bu was no accp. Eular iscar Trigonomric sris In 759 J. L. Langrang srongly criiciz Trig. Sris. 4 By: 6 is a Gra Mahmaical Pom Lor Klvin (William Thompson) Laplac Rjc ourir s Thsis Laplac Gnraliz ourir s on By:
2 x( prioic signal k x jk ( ) a k Whr cofficins ar a k T T x( jk prio, T 7 By: A Sum of Sin Wavs Asin(x 9 By: o Prioic signal x( jk ( ) k k -To/ -T T To/ T x a a ak k sin kt k sin k ot To ko 8 Lin spcra, iscr By: lins Wha is h Us of? Us in Analysis :LTI sysm HOW? Us in Synhsis w /T By: If T??? -To/ -T T To/ T will bcom Aprioic will bcom Coninus or Aprioic signal x(, T X ( j ) x( j Invrs x( X ( j) j coms in h picur By: By:
3 Dos S always xis for any prioic signal? Convrgnc Criria: Dirichl. Absoluly Ingrabl T x ( <. Thr shoul b fini numbr of Maxima an Minima in any prio. 3. Thr shoul b fini numbr of fini isconinuiis in any fini prio 3 By: Pair Invrs : j f ( j ) Synhsis : j ) f ( Exampl j Analysis Dos T always xis for any aprioic signal? Convrgnc Criria: Dirichl. Absoluly Ingrabl x ( < T. Thr shoul b fini numbr of Maxima an Minima in any fini inrval. 3. Thr shoul b fini numbr of fini isconinuiis in any fini prio. 4 By: Exampl - f( j j f j ( ) ( ) j j ( j j ) sin Exampl f( ) ) 3-3 j arg[)] f( j j j ) f ( ( j) j j f j j ( ) ( ) j ( ) j j) arg[j)].5 - =
4 Noaion Propris of ourir [ f ( ] j) Linariy Propris of a f( a f( a ( j) a ( j) Tim Rvrsal - [ j)] f ( f ( j) Tim Scaling f ( a j a a Tim Shifing f ( j [ f ( ] f ( j f ( j f ( ( j j f ( ( j j f ( f ( j f ( j) ( ) f j j [ f ( )] f ( ) j f ) j( ) f ( ( ) j f ( j f ( j ( j j j) j 4
5 rquncy Shifing (Moulaion) Symmry Propry j ( ) f j( ) [ f ( j ) j ] f ( f ( j j( ) j( ) Exampl: [ f ( ] j) [ f ( cos]? Sol) j j f ( cos f ( ( ) j j f ( cos] [ f ( ] [ f ( [ j( )] [ j( [ Exampl:.5 / w ( / j) / / ] )] f(=w (cos / j W ( j) [ w ( ] sin / sin ( ) sin ( ( j ) [ w ( )cos ] = =5 ) [ j] f ( ) f ( f ( j) j) f ( ) j j j j Inrchang symbols an Exampl: / w ( / [ j] / f(=w (cos / j W ( j) [ w ( ] sin / sin ( ) sin ( ( j ) [ w ( )cos ] Exampl: a f ( sin j)? Sol) W / ) / w ( / ( j) sin [ W ( j] sin ( ) w sin a a [ f ( ] w a( ) a 5
6 of f ( of Ingral 35 j an lim f ( f ( [ f '( ] f '( f ( j j f '( j j) j j) j f ( j Th Drivaiv of ourir Transform j [ jf ( ] j ) f ( j) j f ( j f ( j [ jf ( ] [ jf ( ] j By: 34 j an f ( f ( L f ( x) x j j ( f ( x) x lim ( [ '( ] [ f ( ] j) j( j) ( j) j) j Propris of Spaial Domain (x) rquncy Domain (u) Linariy c f x cgx c u cgu Scaling f ax u a a Shifing f x x iux u Symmry x f u Conjugaion f x u Convoluion f x gx u G u n f x Diffrniaion n i u u x n By: No ha hs ar riv using frquncy ( i ux ) 6
7 Pairs Proof of Convoluion Propry y ( x( ) h( ) 37 By: Taking ourir ransforms givs: Y( j ) x( ) h( ) Inrchanging h orr of ingraion, w hav j Y( j) x( ) h( ) By h im shif propry, h brack rm is -j j Y ( j) x( ) H ( j) H(j), so j H ( j) x( ) H ( j) X ( j) 39 By: 4 By: inal Pairs (I) j 4 By: No ha hs ar riv using angular frquncy ( iux ) 7
8 Problms Sysm Conncions T, S Pairs (I) No ha hs ar riv using 43 By: angular frquncy ( iux ) DTT DT 45 By: 46 By: Summry- 47 By: 48 By: 8
9 49 By: 5 By: 5 By: 5 By: ourir amily ourir amily Transform Tim rquncy ourir sris Discr ourir ransform Coninuous ourir ransform Discr-im ourir ransform Coninuous, Prioic Discr, Prioic Coninuous, Aprioic Discr, Aprioic Discr, Aprioic Discr, Prioic Coninuous, Aprioic Coninuous, Prioic 53 By: 54 By: 9
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