1Alexander-Sadiku Alexander-Sadiku Fundamentals of Fundamentals of Electric CircuitsElectric CircuitsChapter 18Chapter 18Fourier TransformFourier TransformCopyright The McGraw-Hill Companies, Inc. Permission require !or reprouction or isplay."#ourier Trans!orm#ourier Trans!ormChapter 1$Chapter 1$ 1$.1 %e!inition o! the #ourier Trans!orm 1$." Properties o! the #ourier Trans!orm 1$.& Circuit 'pplications& 1$.1 %e!inition o! #ourier Trans!orm (1)1$.1 %e!inition o! #ourier Trans!orm (1)* It is an integral trans!ormation o! f(t) !rom the time omain to the !requency omain F()* F() is a comple+ !unction, its magnitue is calle the amplitude spectrum, while its phase is calle the phase spectrum.Gi-en a !unction f(t), its !ourier trans!orm enote .y F(), is e!ine .y ) ( ) ( dt e t f Ft j/ 1$.1 %e!inition o! #ourier Trans!orm (")1$.1 %e!inition o! #ourier Trans!orm (")0+ample 1%etermine the #ourier trans!orm o! a single rectangular pulse o! wie an height ', as shown .elow.A rectangular pulse1 1$.1 %e!inition o! #ourier Trans!orm (&)1$.1 %e!inition o! #ourier Trans!orm (&)2sin222 /2 /) (2 / 2 /2 /2 / c Aje e AejAdt Ae Fj jt jt j

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2olution3Amplitude spectrum of the rectangular pulse4 1$.1 %e!inition o! #ourier Trans!orm (/)1$.1 %e!inition o! #ourier Trans!orm (/)0+ample "35.tain the #ourier trans!orm o! the 6switche-on7 e+ponential !unction as shown .elow.8 1$.1 %e!inition o! #ourier Trans!orm (1)1$.1 %e!inition o! #ourier Trans!orm (1)2olution3 j adt edt e e dt e t f Fet u e t ft j at j jat t jatat+ ' + 1 ) ( ) (Hence,0t , 00 t ,) ( ) () ($ 1$." Properties o! #ourier Trans!orm (1)1$." Properties o! #ourier Trans!orm (1)[ ] ) ( ) ( ) ( ) (2 2 1 1 2 2 1 1 F a F a t f a t f a F + +9inearity3 I! F1() an F2() are, respecti-ely, the #ourier Trans!orms o! f1(t) an f2(t)0+ample &3 [ ] ( ) ( ) [ ] [ ] ) ( ) (21) sin(0 0 00 0 + j e F e Fjt Ft j t j: 1$." Properties o! #ourier Trans!orm (")1$." Properties o! #ourier Trans!orm (")[ ] constant a is, ) (1) ( aaFaat f FTime 2caling3 I! F () is the #ourier Trans!orms o! f (t), thenI! ;a; 1$." Properties o! #ourier Trans!orm (&)1$." Properties o! #ourier Trans!orm (&)[ ] ) ( ) (00F e t t f Ft j Time 2hi!ting3 I! F () is the #ourier Trans!orms o! f (t), then0+ample /3 [ ]jet u e Fjt+ 1) 2 (2) 2 (11 1$." Properties o! #ourier Trans!orm (/)1$." Properties o! #ourier Trans!orm (/)[ ] ) ( ) (00 F e t f Ft j#requency 2hi!ting ('mplitue Moulation)3 I! F () is the #ourier Trans!orms o! f (t), then0+ample 13 [ ] ) (21) (21) cos( ) (0 0 0 + + F F t t f F1" 1$." Properties o! #ourier Trans!orm (1)1$." Properties o! #ourier Trans!orm (1)) ( ) ( s F j t udtdfF 1]1

Time %i!!erentiation3 I! F () is the #ourier Trans!orms o! f (t), then the #ourier Trans!orm o! its eri-ati-e is0+ample 43 ( ) j at u edtdFat+1]1

1) (1& 1$." Properties o! #ourier Trans!orm (4)1$." Properties o! #ourier Trans!orm (4)) ( ) 0 () () ( FjFdt t f Ft1]1

Time Integration3 I! F () is the #ourier Trans!orms o! f (t), then the #ourier Trans!orm o! its integral is0+ample 83 [ ] ) (1) ( + jt u F1/ 1$." Properties o! #ourier Trans!orm (8)1$." Properties o! #ourier Trans!orm (8)[ ] ) ( * ) ( ) ( F F t f L ?e-ersal3 I! F() is the #ourier Trans!orms o! f (t), then re-ersing f(t) a.out the time a+is re-erses F() a.out !requency. 0+ample $3 [ ] [ ] ) ( 2 ) ( ) ( 1 + t u t u F F11 1$." Properties o! #ourier Trans!orm ($)1$." Properties o! #ourier Trans!orm ($)[ ] [ ] ) ( 2 ) ( ) ( ) ( f t F F F t f F%uality3 I! F() is the #ourier Trans!orms o! f (t), then the #ourier trans!orm o! F(t) is 2f(-).12) (then, ) ( If2+ Fe t ft0+ample :3 [ ] +e f F tF(t)2 ) ( 2 then12If2Duality property14* I! X(), H() an Y() are the #ourier trans!orms o! x(t), h(t), an y(t), respecti-ely, then* It is e!ine as 1$." Properties o! #ourier Trans!orm (:)1$." Properties o! #ourier Trans!orm (:)[ ] ) ( * ) (21) ( ) ( ) ( X H t x t h F Y * In the -iew o! uality property o! #ourier trans!orms, we e+pect[ ] ) ( ) ( ) ( * ) ( ) ( X H t x t h F Y ) ( * ) ( ) ( or ) ( ) ( ) ( t h t x t y d t h x t y 18* #ourier trans!orms can .e applie to circuits with non-sinusoial e+citation in e+actly the same way as phasor techniques .eing applie to circuits with sinusoial e+citations.* @y trans!orming the !unctions !or the circuit elements into the !requency omain an taAe the #ourier trans!orms o! the e+citations, con-entional circuit analysis techniques coul .e applie to etermine unAnown response in !requency omain.* #inally, apply the in-erse #ourier trans!orm to o.tain the response in the time omain. 1$.& Circuit 'pplications (1)1$.& Circuit 'pplications (1)Y() = H()X()1$0+ample 1>3 #in v0(t) in the circuit shown .elow !or vi(t)=2e-tu(t) 1$.& Circuit 'pplications (")1$.& Circuit 'pplications (")1:2olution3 1$.& Circuit 'pplications (&)1$.& Circuit 'pplications (&)) ( ) ( 4 . 0 ) ( gives ansform Fourier tr inverse the Taking). 0 )( ! (1) ( "Hence,2 11) ( ") ( ") ( is circuitthe of functiontransferThe

!2) ( " is signa# in$utthe of ansform Fourier tr The!. 000i0it u e e t vj jjHjt t + ++ +