Continuous-Time Fourier Transform. Transform. Transform. Transform. Transform. Transform. Definition The CTFT of a continuoustime
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1 Ctiuus-Tim Furir Dfiiti Th CTFT f a ctiuustim sigal x a (t is giv by Xa ( jω xa( t jωt Oft rfrrd t as th Furir spctrum r simply th spctrum f th ctiuus-tim sigal dt Ctiuus-Tim Furir Dfiiti Th ivrs CTFT f a Furir trasfrm X a ( jω is giv by j x t X jω Ω t a ( a( dω Oft rfrrd t as th Furir itgral A CTFT pair will b dtd as CTFT x ( t X ( jω a a 3 Ctiuus-Tim Furir Ω is ral ad dts th ctiuus-tim agular frqucy variabl i radias I gral, th CTFT is a cmplx fucti f Ω i th rag < Ω < It ca b xprssd i th plar frm as jθ ( Ω X ( Ω ( Ω a a j Xa j whr θ ( Ω arg{ ( jω} a X a 4 Ctiuus-Tim Furir Th quatity X a ( jω is calld th magitud spctrum ad th quatity θ a (Ω is calld th phas spctrum Bth spctrums ar ral fuctis f Ω I gral, th CTFT X a ( jω xists if x a (t satisfis th Dirichlt cditis giv th xt slid Ctiuus-Tim Furir Dirichlt Cditis (a Th sigal x a (t has a fiit umbr f disctiuitis ad a fiit umbr f maxima ad miima i ay fiit itrval (b Th sigal is abslutly itgrabl, i.., x a ( t dt < Ctiuus-Tim Furir If th Dirichlt cditis ar satisfid, th Xa ( jω jωt dω cvrgs t x a (t at valus f t xcpt at valus f t whr x a (t has disctiuitis It ca b shw that if x a (t is abslutly itgrabl, th Xa( jω < prvig th xistc f th CTFT 5 6
2 7 Ergy Dsity Spctrum E x Th ttal rgy f a fiit rgy ctiuus-tim cmplx sigal x a (t is giv by Ex xa( t dt xa( t x* a( t dt Th abv xprssi ca b rwritt as jωt Ex xa( t X* jω dω a( dt 8 Ergy Dsity Spctrum Itrchagig th rdr f th itgrati w gt E x X Ω a * ( j xa( t X jω X a * ( a X jω dω a ( ( jω dω jωt dt dω 9 Ergy Dsity Spctrum Hc x( t dt X a ( jω dω Th abv rlati is mr cmmly kw as th Parsval s rlati fr fiitrgy ctiuus-tim sigals Ergy Dsity Spctrum Th quatity X a ( jω is calld th rgy dsity spctrum f x a (t ad usually dtd as Sxx( Ω Xa( jω Th rgy vr a spcifid rag f frqucis Ωa Ω Ω b ca b cmputd usig Ωb ( Ω dω x, r S xx Ωa Bad-limitd Ctiuus-Tim Sigals A full-bad, fiit-rgy, ctiuus-tim sigal has a spctrum ccupyig th whl frqucy rag < Ω < A bad-limitd ctiuus-tim sigal has a spctrum that is limitd t a prti f th frqucy rag < Ω < Bad-limitd Ctiuus-Tim Sigals A idal bad-limitd sigal has a spctrum that is zr utsid a fiit frqucy rag Ωa Ω Ω b, that is, Ω < Ω X Ω a a ( j, Ωa < Ω < Hwvr, a idal bad-limitd sigal cat b gratd i practic
3 Bad-limitd Ctiuus-Tim Sigals Bad-limitd sigals ar classifid accrdig t th frqucy rag whr mst f th sigal s is cctratd A lwpass, ctiuus-tim sigal has a spctrum ccupyig th frqucy rag Ω Ω p < whr Ω p is calld th badwidth f th sigal Bad-limitd Ctiuus-Tim Sigals A highpass, ctiuus-tim sigal has a spctrum ccupyig th frqucy rag < Ω p Ω < whr th badwidth f th sigal is frm Ω p t A badpass, ctiuus-tim sigal has a spctrum ccupyig th frqucy rag < ΩL Ω ΩH < whr ΩH ΩL is th badwidth Dfiiti - Th discrt-tim Furir trasfrm (DTFT X ( f a squc is giv by X ( x j [ I gral, X ( ω is a cmplx fucti f th ral variabl ω ad ca b writt as X ( X ( j ω + j X ( r im 6 Dfiiti - Th discrt-tim Furir trasfrm (DTFT X ( f a squc is giv by X ( x j [ I gral, X ( ω is a cmplx fucti f th ral variabl ω ad ca b writt as X ( X ( j ω + j X ( r im j ( X r( ω ad X im ar, rspctivly, th ral ad imagiary parts f X (, ad ar ral fuctis f ω X ( ca altratly b xprssd as X ( j ω X ( jθ( ω whr θ( ω arg{ X ( } X ( ω is calld th magitud fucti θ(ω is calld th phas fucti j Bth quatitis ar agai ral fuctis f ω I may applicatis, th DTFT is calld th Furir spctrum Likwis, X ( ad θ(ω ar calld th magitud ad phas spctra 7 8 3
4 Fr a ral squc, X ( ω ad X r ar v fuctis f ω, whras, θ(ω ad X ( j ω im ar dd fuctis f ω Nt: X ( j ω X ( jθ( ω+ k X ( j ω jθ( ω fr ay itgr k Th phas fucti θ(ω cat b uiquly spcifid fr ay DTFT j ( j ω Ulss thrwis statd, w shall assum that th phas fucti θ(ω is rstrictd t th fllwig rag f valus: θ( ω < calld th pricipal valu 9 Th DTFTs f sm squcs xhibit disctiuitis f i thir phas rspss A altrat typ f phas fucti that is a ctiuus fucti f ω is ft usd It is drivd frm th rigial phas fucti by rmvig th disctiuitis f Th prcss f rmvig th disctiuitis is calld uwrappig Th ctiuus phas fucti gratd by uwrappig is dtd as θ c (ω I sm cass, disctiuitis f may b prst aftr uwrappig Exampl-Th DTFT f th uit sampl squc δ[ is giv by ( δ[ δ[] Exampl- Csidr th causal squc α µ [, α < Its DTFT is giv by as X ( α α µ [ α ( α α α < 3 4 4
5 5 Th magitud ad phas f th DTFT X ( /(.5 ar shw blw Magitud ω/ Phas i radias ω/ 6 Th DTFT X ( f a squc is a ctiuus fucti f ω It is als a pridic fucti f ω with a prid : X ( j( ω + k j k x ] j( ω + k [ X ( 7 Thrfr X ( rprsts th Furir sris rprstati f th pridic fucti As a rsult, th Furir cfficits ca b cmputd frm X ( usig th Furir itgral X ( dω Ivrs discrt-tim Furir trasfrm: X ( dω Prf: l] l l dω 8 Th rdr f itgrati ad summati ca b itrchagd if th summati isid th brackts cvrgs uifrmly, i.. X ( xists Th l l] dω l l] si ( l l] ( l ( l ω d l l 9 Nw si ( l, l ( l, l δ[ l] Hc si ( l x [ l] l] δ[ l] ( l l l 3 5
6 3 Cvrgc Cditi - A ifiit sris f th frm X ( may r may t cvrg Lt X ( 3 Th fr uifrm cvrgc f X (, lim X ( X ( Nw, if is a abslutly summabl squc, i.., if x [ < 33 Th X ( < fr all valus f ω Thus, th abslut summability f is a sufficit cditi fr th xistc f th DTFT X ( Exampl-Th squc α µ [ fr α < is abslutly summabl as α µ [ α < α ad its DTFT X ( thrfr cvrgs t /( α uifrmly 34 Sic a abslutly summabl squc has always a fiit rgy Hwvr, a fiit-rgy squc is t cssarily abslutly summabl, Exampl- Th squc /,, has a fiit rgy qual t E x 6 But, is t abslutly summabl
7 37 T rprst a fiit rgy squc that is t abslutly summabl by a DTFT X (, it is cssary t csidr a masquar cvrgc f X ( : whr lim X ( X ( X ( dω Hr, th ttal rgy f th rrr j ω X ( X ( must apprach zr at ach valu f ω as gs t I such a cas, th abslut valu f th j rrr ( ω X X ( may t g t zr as gs t ad th DTFT is lgr budd Exampl-Csidr th DTFT, ω ωc H LP (, ωc < ω shw blw ωc H LP( ωc ω 4 Th ivrs DTFT f H LP ( h LP [ ω c j ω d ωc ω is giv by c c si ω c, j j < < Th rgy f h LP [ is giv by ω c / h LP [ is a fiit-rgy squc, but it is t abslutly summabl As a rsult h LP [ ds t uifrmly cvrg t H LP ( fr all valus f ω, but cvrgs t H LP ( i th ma-squar ss si ωc Th ma-squar cvrgc prprty f th squc h LP [ ca b furthr illustratd by xamiig th plt f th fucti si ω c H LP, ( fr varius valus f as shw xt 4 4 7
8 Amplitud Amplitud N ω/ N Amplitud Amplitud N ω/ N As ca b s frm ths plts, idpdt f th valu f thr ar rippls i th plt f H LP, ( arud bth sids f th pit ω ω c Th umbr f rippls icrass as icrass with th hight f th largst rippl rmaiig th sam fr all valus f ω/ ω/ As gs t ifiity, th cditi lim H LP ( H, ( dω hlds idicatig th cvrgc f H LP, ( t H LP ( Th scillatry bhavir f H LP, ( apprximatig H LP ( i th masquar ss at a pit f disctiuity is kw as th Gibbs phm LP 46 Th DTFT ca als b dfid fr a crtai class f squcs which ar ithr abslutly summabl r squar summabl Exampls f such squcs ar th uit stp squc µ[, th siusidal squc cs( ω + φ ad th xptial squc Aα Fr this typ f squcs, a DTFT rprstati is pssibl usig th Dirac dlta fucti δ(ω 47 A Dirac dlta fucti δ(ω is a fucti f ω with ifiit hight, zr width, ad uit ara It is th limitig frm f a uit ara puls fucti p (ω as gs t zr satisfyig lim p ( ω dω δ( ω dω p (ω ω 48 Exampl-Csidr th cmplx xptial squc Its DTFT is giv by X ( ( ω ω + k δ k whr δ(ω is a impuls fucti f ω ad ω 8
9 49 Th fucti X ( ( ω ω + k δ k is a pridic fucti f ω with a prid ad is calld a pridic impuls trai T vrify that X ( giv abv is idd th DTFT f w cmput th ivrs DTFT f X ( 5 Thus x [ δ( ω ω + k dω k δ( ω ω dω whr w hav usd th samplig prprty f th impuls fucti δ(ω 5 Cmmly Usd DTFT Pairs Squc δ[ DTFT ( ω + k δ k ( ω ω + k δ k µ [ ] + δ( ω + k k µ [, ( α < α 5 DTFT Prprtis Thr ar a umbr f imprtat prprtis f th DTFT that ar usful i sigal prcssig applicatis Ths ar listd hr withut prf Thir prfs ar quit straightfrward W illustrat th applicatis f sm f th DTFT prprtis Tabl 3.: DTFT Prprtis: Symmtry Rlatis Tabl 3.: DTFT Prprtis: Symmtry Rlatis 53 : A cmplx squc 54 : A ral squc 9
10 Tabl 3.4:Gral Prprtis f DTFT 55
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DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1
![DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1](/thumbs/79/79094742.jpg "DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1")
DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT
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