AN INTRODUCTION TO FOURIER ANALYSIS PROF. VEDAT TAVSANOĞLU

Transcription

1 A IRODUCIO O FOURIER AALYSIS PROF. VEDA AVSAOĞLU 994

2 A IRODUCIO O FOURIER AALYSIS ABLE OF COES. HE FOURIER SERIES Priodic Funcions h Fourir Sris Complx Form of Fourir Sris Dirichl Condiions h Effc of Symmry Propris of h Signal on Fourir Sris HE FOURIER RASFORM Inroducion h Rlaion Bwn h Fourir ransform and h Fourir Sris Cofficins Som Elmnary Funcions and hir Fourir ransforms Propris of h Fourir ransform Poisson Sum Formula h Fourir ransform of h Puls rain p( h Fourir ransform of f(p( Sampling and Inrpolaion An Exampl of h Applicaion of h Fourir ransform in Puls Ampliud Modulaion (PAM HE FOURIER RASFORM OF DISCREE-IME SIGALS umrical Evaluaion of h Fourir Ingral h Discr im Fourir ransform Som lmnary funcions and hir DF Propris of DF Fourir ransform of Priodic Discr im Signals: h Discr Fourir Sris (DFS h Discr Fourir ransform (DF Efficin Compuaion of DF: h Fas Fourir ransform (FF

3 A IRODUCIO O FOURIER AALYSIS.. Priodic Funcions A funcion f p ( is priodic wih priod if whr m is any ingr.. HE FOURIER SERIES f p ( + m f p ( I follows from his dfiniion ha if is h priod any ingr mulipls of his priod, i.. m, ar also priods of f p (. L wo priodic funcions f p ( and f p ( hav h priods and, rspcivly. hs wo priodic funcions hav a common priod, if and only if hr xis ingrs n and n such ha n n i follows ha: n n Considr h sgmn f( of h priodic funcion f p ( givn by fp( f ( whr is h priod of f p (. I follows ha f < > p ( f ( + n n.. h Fourir Sris Considr h linar combinaion of h sinusoidal funcions cos, sin 3

4 Sinc h common priod is funcions is priodic wih., any linar combinaion of such sinusoidal Considr now i.., fp( cos cos3 + cos5 3 5 mus b a raional numbr ha is a raio of wo ingrs. n n is a common priod of f p ( and f p (. h smalls common priod is h las common mulipl of and. Sinc h angular frquncis corrsponding o and can b givn as and w hav If, hrfor, Exampl: is a raional numbr hn fp ( and f p ( hav a common priod. h wavforms cos3 and cos3 hav a common priod as h raio 3 5, whr 3 6 and 3, 3 is a raional numbr. h common priod is givn by 6 5 h wavforms cos and cos do no hav a common priod as is no a raional numbr. I is vidn from h abov ha if h priods of f p ( and f p ( ar commnsura such ha 4

5 n n whr n and n ar ingrs, hn hir sum f p ( + f p ( is also priodic wih. h priods of cos3 and cos3 ar, rspcivly, priod of cos3 + cos3 is 5 6. and 5. hrfor h 6 On h ohr hand cos + cos is no a priodic funcion as ingr valus canno b found such ha n n whr. Exampl: Considr f( cos 3 which can b wrin as 3 fp( cos cos.cos cos + cos cos + cos.cos cos + cos3 + cos cos + cos3 4 4 cos is priodic wih ; cos3 is priodic wih, hrfor cos 3 is priodic 3 wih 3. h figur givn blow shows a plo of h priodic funcion givn by 3 3 f p ( cos cos + cos3. / 4 4 5

6 h mos gnral form of h linar combinaion of such sinusoidal funcions can b givn as follows: f( a + (a cos + bsin p Such a sum is calld Fourir sris. Each rm obaind for a paricular valu of is calld a harmonic. a cos + bsin is hrfor calld h h harmonic. For on obains h firs harmonic which is also calld h fundamnal. Is frquncy is calld h fundamnal frquncy and is priod is calld h fundamnal priod. Howvr, h fundamnal dos no ncssarily xis in h Fourir sris of vry signal as h following xampl dmonsras: Considr h signal f p ( cos3 + cos3 Sinc h common priod is 5 6 h fundamnal priod is and h fundamnal frquncy is /. As can b sn from fp(, howvr, ha h fundamnal dos no appar in fp( which is h Fourir sris and i conains h 5h and 6h harmonics only..3. Complx Form of Fourir Sris Using cos ( + sin (, 6

7 w obain a cos ow dfin (a + b sin b. c + + (a + b a,c a b,c a b h Fourir sris can now b wrin in h following form: f p ( c whr c ar h complx-valud Fourir cofficins and can b wrin in h form whr c c a + b c and ϕ ( b an ϕ. a ow considr h following qusion: Givn fp(, how can w find c? Firs considr h xponnial funcions and m. W can show ha for ingral valus of and m hs funcions consiu an orhogonal s ovr a full priod. ha is, + / / m d ( m+ / d d / ( m + for m ( m+ ( m+ [ ] for m c / / f p ( c 7

8 8 as ow muliplying boh sids of h Fourir sris by m, w obain m m p c f + ( ( ow ingraing boh sids and ling m - givs / / ( c d f p his xprssion, oghr wih h Fourir sris consius h following Fourir-sris pair: d f c p ( / / p c f ( f p ( can b rplacd by f(, as f( f p ( for < /. C can also b wrin in h form d ( sin f d ( cos f c / / / / p p ow also using c a b, w obain d (cos f a / / p d (sin f b / / p h Analysis Equaion (Fourir-Sris Cofficins h Synhsis Equaion (Fourir-Sris

9 ow considr fp( a + (a cos + b sin Dfining α (a b, w can wri + Also dfining lads o and a b fp( a + α( cos + sin α α a α f f p p cosϕ ( a ( a b, α + α + α sinϕ ( cos cos( cosϕ + ϕ sin sinϕ which is y anohr form of Fourir sris. o ha c a α ϕ for for.4. Dirichl Condiions o all priodic funcions hav a Fourir-Sris xpansion. For a funcion o b Fourir sris ransformabl, and for h sris o convrg o h funcion islf, h funcion mus saisfy som condiions. hs ar nown as h Dirichl Condiions. hs ar sufficin condiions bu no ncssary. hs condiions rquir ha wihin a priod:. Only a fini numbr of maximum and minimums can b prsn;. h numbr of disconinuiis mus b fini; h disconinuiis mus b boundd, i.. h funcion mus b absoluly ingrabl, which rquirs ha f ( d < 9

10 (a (b (c Signals ha viola Dirichl condiions: (a Signal wih infini disconinuiis. (b Boundd signal wih infini variaions. (c Signal wih infini numbr of coninuiis.5. h Effc of Symmry Propris of h Signal on Fourir Sris A signal f( can b wrin in rms of is vn par f ( and odd par f o ( as follows:

11 whr f( f ( + f o ( f (- f ( and f o (- f o ( ow considr f which rvals ha p ( ( a cos + b sin f f W can conclud: o ( ( a b cos sin If f p ( is vn hn b If f p ( is odd hn a If f ( f + hn f( posssss HALF-WAVE SYMMERY whr is h priod of f(. Considr c for a funcion wih half-wav symmry c f ( d f ( d f ( d + ow rplac by + in h firs ingral w obain c f( d f( d + ( f( d f ( d, odd, vn Conclusion: Wavforms having half-wav symmry conain odd harmonics only.

12 Wavform Symmry a, b, ϕ c Harmonics on All Around -axis a c All Evn (around h ordina If 4a, f(- c has halfwav symmry b, ϕ cos rms only c -ral If 4a hn c xis only for odd All Odd only Odd (around h origin a, ϕ 9 Sin rms only c -imaginary All f + f ( Half-wav on Odd only Odd only and f + f( f ( f ( Odd and halfwav b, ϕ c -ral Odd only

13 h signal f( shown in h following has a hiddn half-wav symmry, in ha, whn h man valu a c is subracd from i h rs of h signal has a half wav symmry. EXAMPLES. c, a, b and f p (? Fundamnal priod : Fundamnal frquncy: c f ( d p a a f a d, a p a c f ( d d a a 3

14 a a a sin a sin a a sin a a a 4s, a s hn, c sin c sin sinc c c, c c, c3 c3, c4 c4, c5 c Sinc 4a in his xampl, f p ( c has half-wav symmry. hrfor only h odd harmonics xis. his is in addiion o h propry ha c s ar ral, which is a rsul of vn symmry. hrfor c c c3 c f(.5+ ( + ( a a 3 f p(.5 + cos cos

15 5. c, a, b, f p (? For ( ( p d A d d A d A A d A A d f d f c using vdu uv udv w can wri d d α α α α α α α α [ ] A A A c ( for A A for A A f

16 A A A ow subsiuing, w obain and considring ha, sin and + cos ( ( ( [ ] A A c cos cos c can b found from ( A d f c For, c for vn, c for odd. For odd valus of, w hav ( c. o a ino accoun ha c for vn valus of, w can rplac by +, hn w obain ( [ ] + + c for,,,...hrfor ( p c c c f ( which yilds ( ( cos 4 ( p A A f 3.

17 7 f p (: Fundamnal priod Fundamnal frquncy f p (: (Dashd lin Fundamnal priod Fundamnal frquncy hrfor for A A f cos cos ( in fac A f p cos ( whr ( rads: for all h qusion is: How can w xprss f p ( as a sum of sinusoidal wavforms? ( ( d A d A d A c cos ( ( ( ( A ( ( sin( sin( cos sin cos cos sin sin( + + cos sin cos cos sin sin( ( ( ( A A A c 4 cos 4 cos cos + + Finally, w obain ( A A A A f p cos 3 35 cos 5 cos 3 4 cos 4 4 cos (

18 . HE FOURIER RASFORM.. Inroducion Considr h Fourir sris cofficins c for h priodic squar wav: sin a c or sin a a which is indpndn of a c As incrass c rmains unchangd. As incrass h fundamnal frquncy dcrass and h discr frquncis bcom closr and closr and finally bcoms a coninuum. I will b shown in h following ha, by convring a priodic wavform o a nonpriodic on by incrasing h priod on obains h Fourir ransform pair from h Fourir-Sris pair. h Fourir sris pair: f ( f ( c c d 8

19 By muliplying and dividing h righ sid of h las quaion, w obain c f ( Dfining w hav c F( f ( F( ow if w a h priod o infiniy w can wri d F( F( hrfor HE FOURIER- RASFORM PAIR F( f ( f ( d F( dw FOURIER RASFORM IVERSE FOURIER RASFORM.. h Rlaion Bwn h Fourir ransform and h Fourir Sris Cofficins Considr h priodic signal f p ( givn by p ( f ( + n n f whr f( is a fini-duraion signal (FDS dfind by f p ( for < f ( ohrwis h Fourir cofficins for f p ( ar givn as 9

20 c f ( d Sinc h Fourir ransform of f( is givn by F ( i is vidn ha w hav f ( d c F( W can conclud ha h Fourir cofficins for h priodic signal f p (, which is mad up of FDS f(, is obaind by sampling F( in frquncy wih a sampling inrval of. I will b shown lar using Poisson Sum Formula ha his is also ru in h cas of an arbirary signal f(. Conclusion: As a rsul of his w can wri f p ( F(

21 Exampl: n p ( δ( + n p( is calld Puls rain and is a priodic wavform P ( δ( hrfor w obain P( + d p δ( + n ( n ( cos + cos + cos Som Elmnary Funcions and hir Fourir ransforms δ( funcion Dfin dg( f ( g( d f ( d dg( g( + Δ g( Lim d Δ Δ For, Δ w hav dg( g( g( Lim d f ( ow dfin h funcion Δ ( as

22 hn f ( p ( d f ( d hrfor Lim Lim g( g( dg( f ( d Lim d f ( Δ ( d f ( δ( d f ( f ( h dla funcion, δ(, is dfind as: δ( for f ( δ( d f ( whr f( is coninuous a origin h following propris ar obaind from his dfiniion. Propris of dla funcion:. δ( is a signal of uni ara vanishing vrywhr xcp origin: δ( d, δ( for. δ( can b wrin as a limi δ( Lim Δ (, whr Δ ( for < lswhr 3. δ( is h drivaion of h uni-sp funcion u( δ ( du( d h rcangular puls p (, h mor pracical puls z ( and δ( ar shown in h following figur along wih hir drivaivs. For sufficinly small, i is vidn ha p ( and z ( can b rplacd by δ(. his fac is also usifid by comparing h drivaivs.

23 Fourir ransform of δ( h Fourir ransform of δ( is givn by: Δ( δ( d Using h dfiniion of δ(: δ ( f ( d f ( and sing f( -, w hav f( hrfor w obain I { δ( } Δ( 3

24 Fourir ransform of L us firs valua h invrs ransform of δ( I { δ( } δ( d hrfor I {} δ( Physical inrpraion of I{δ(} and I{} δ( I{δ(} indicas ha, a suddn chang happning in h form of a spi in h imdomain signal conains all frquncy componns wih h sam wigh. As δ( is a horical signal of zro duraion, i conains vn infini-frquncy componns. In pracic w hav pulss of vry shor duraion. his mans ha h frquncy spcrum of such signals will includ xrmly high frquncis bu no infini frquncis. I{} δ( implis ha h frquncy spcrum of a consan signal, i.. a DC signal, conains zro-frquncy componn only. h Signum Funcion and is Fourir ransform h Signum funcion is dfind as sign ( - for > for < 4

25 his funcion is undfind for. w can approxima signum by a whr a is small. hn I { sign( } Lim a Lim a sign( Lim a a Lim a a a d + (a d + Lim + a a (a + sign( a sign( a a d d sign( + Lim a (a + d (a+ W obain: For I { sign (} For I{ sign (} h Uni-Sp Funcion and is Fourir ransform u ( for > for < u( is also undfind for. Using h sign(, w can xprss u( as u ( + sign( 5

26 hrfor I { u (} I + I sign( which yilds I { u (} δ( + o ha { u( } δ( I for. HE FOURIER RASFORM OF HE PULSE FUCIO h puls funcion is dfind as: ( p a I { p (} a - a a p ohrwis a ( d a a a [ ] [ ] a d sin a sin a a a sin ca a a a 6

27 HE FOURIER RASFORM OF HE CAUSAL EXPOEIAL FUCIO Considr h causal xponnial funcion givn as: α f( u( ; α> I { f (} α α u( d ( α + d ( α+ ( α+ d α ( α + α Sinc for, w obain F ( α + α + an α 7

28 .4. Propris of h Fourir ransform If I{f(} F( Linariy: I{ a.f ( + b.f ( } ai{f (} + bi{f (} a.f ( + b.f ( whr a and b ar consans. his propry is h dirc rsul of h linar opraion ingraion. im and Frquncy Scaling: I a a { f (a} F Proof: I { f (a} f (a. d h variabl chang a yilds a > : a < : I I a { f ( a } f ( a d a { f ( a } f ( a d f ( a d a Combining hs wo, w obain { f ( a } f ( a I d a Q.E.D. Spcial cas for a yilds 8

29 I { f ( } F( If f( is a ral funcion his mans I { f ( } F ( Dualiy (or symmry Propry: If I{f(} F( hn I{ F ( } f ( Proof: Considr h Invrs Fourir ransform formula f ( F( d Maing h doubl chang of variabls τ and υ f ντ ( τ F( ν dν and now h scond chang of variabls τ and υ f ( F( d Q.E.D. Exampl: Dualiy (or symmry propry 9

30 (a (b (a Illusraion of h symmry propry of h puls funcion (b Illusraion of h symmry propry of h dla funcion im Shifing: Proof: ± { f( } F( I ± { f ( ± } f ( d I ± h variabl chang ± yilds 3

31 ( ± ± ( d f ( d F( f o ha I ± { f ( ± } F( F( W conclud ha h im shifing has no ffc on h magniud of h Fourir ransform. I only changs h phas by ±. In fac ± { I{ f ( ± } arg F( { } ± + arg F( arg { } Frquncy Shifing: Proof: { (} ± ± ( I f f d ( ( f d F ( [ ] Exampl: h Fourir ransform of h Priodic Exponnial Funcion Rquird is { } ± I. According o frquncy shifing propry, w can wri I I { (} ± f F( [ ] If f( hn F ( δ( { ( } ± f F( [ ]. hrfor 3

32 I { } ±. δ( Exampl: h Fourir ransform of h Causal Priodic Exponnial Funcion L f( u( whr u( is h uni-sp funcion I { (} ±.u U( [ ] Sinc U I ( I{ u( } δ( + { (}.u δ( ± + ( Modulaion: I I F { f ( cos } [ F( [ + ] + F( [ ]] { f ( sin } [ ( [ + ] F( [ ]] Proof: I + { f ( cos } I f ( I f ( F + F ( [ ] ( [ + ] { } { ( } + I f Proof of h scond xprssion can b givn in a similar way. Exampl: h Fourir ransform of cos and sin L f( hn F ( δ( I I. Using h modulaion propry for f(, w obain { cos } δ ( + + δ ( { sin } δ( + δ( 3

33 Exampl: h Fourir ransform of h Causal Sinusoidal Funcions aing f( u( in Modulaion Propry w g I I U U +, w obain I cos.u( δ + Sinc ( δ( I U { cos.u( } U( [ + ] + U( [ ] { sin.u( } ( [ ] ( [ + ] { } ( + δ( + { sin.u( } δ( + δ( + 33

34 Exampl: α Evalua h Fourir ransform of ( cos cos.u( f α aing f ( u( I { α u( } F( α + Using h modulaion propry yilds α + I + α { cos.u( } ( ( ( α+ + α+ α + + h following circui is an applicaion for h abov 34

35 VC (s R + Ls E(s RLCs + Ls + R RC s + R s + L s + RC LC L α RC R L α R RC L VC (s H(s. E(s α s s + α + αs + α +. α s + α ( s + α + whr α + LC s L{ cos u( } s + s + α s + α + α L { cos u( } ( If ( δ(, E(s H(sV c (s. hrfor for ( δ( α h( vc( cos u( α α I{ h( } I{ cos u( } α α α + ( α + + α Exampl: Evalua h Fourir ransform of ( cos( + ϕ.u( g 35

36 α L f ( u(. ow wri g( in h following form: g( f (cos ( + ϕ f ( ϕ ϕ f ( + f ( Using h linariy propry w g I ( +ϕ ( +ϕ + ϕ { } { } { } ϕ g( I f ( + I f ( ow using h frquncy shifing propry w obain I ϕ ϕ { g( } F( [ ] + F( [ + ] α + ϕ + ( α + ( + ϕ I ( α + ϕ + ϕ + ( α + + { } ( α + cosϕ + g( ( α + + sin ϕ ϕ ϕ 36

37 Exampl: Find h Fourir ransform of h Radio Frquncy Puls ( cos givn as in h following: p a whr p a ( is I { (} p a sin a hrfor using h modulaion propry; w obain I { ( cos } p a sin a ( ( For a w obain h following: sin a + ( + ( + 37

38 38 Exampl: (a Evalua h Fourir ransform of f ( and f ( in Fig.. Fig.. (b Exprss f( in Fig. using f ( and f (. Using h rsul of par (a show ha h Fourir ransform of f( in Fig. is givn as ( a a sin a F Fig.. (c Using h rsul obaind in (b valua h fourir sris cofficins of h priodic funcion f p ( in Fig.3 and xprss f p ( as a sum of ral sinusoidal funcions. Fig.3. (a ( dv d u a d ( f F a + a d a a a

39 39 a a a + [ ] a a a + ( a a d a d a d ( f F [ ] a a a + (b f( f ( + a + f ( a { } ( ( ( + I F F F ( f a a [ ] [ ] a a a a [ ] a a 4sin a cos a a a sin a (c ( F c whr a. a a sin a a c 4 a sin c f ( f ( F ( F (

40 fp( sin ( fp( ( 4 + ( cos 4 + cos + cos3 + cos Dmodulaion: ( W now ha a carrir signal cos can b modulad by h modulaing signal f( by simply muliplying cos by f(, i.. h modulad signal is g( f( cos or G F ( ( [ ] + F( [ + ] ( F Sinc h carrir frquncy is no prsn in G(, his yp of modulaion is nown as doubl-sidband supprssd carrir (DSBSC. In ordr o rcovr h signal f( from h signal g(, g( is muliplid by cos, i.. cos is modulad by g(. W obain H F 4 + F 4 ( G( [ ] + G( [ + ] F ( [ ] ( + F( + F( [ + ] ( + F( [ ] + F( [ + ] if his signal is now passd hrough an idal low-pass filr, F( procss is calld dmodulaion. is rcovrd. his 4

41 A Muliplir Circui 4

42 (a (b Frquncy Muliplxing, Dmuliplxing and Dmodulaing Signals 4

43 im Diffrniaion: ( df I d n d f I d ( n F ( n ( F( Proof: Considr h invrs Fourir ransform xprssion: f ( ( F d ow diffrniaing boh sids of his wih rspc o, w obain ( df d [ ( ] F d and for h nh ordr diffrniaion w g n d f d ( n im Convoluion: n [( F( ] d In words: Convoluion in h im domain implis muliplicaion in h frquncy domain. Proof: f ( *h( f ( τ h( τ I dτ { f ( *h( } I f ( τ h( τ dτ f (( τ h τ dτ d Changing h ordr of ingraion, w can wri I I { f ( *h( } F(.H( { f ( *h( } f ( τ h( τ d dτ ( ( f τ h τ d d τ ow dfin τ u which yilds u + τ, and w obain 43

44 I τ { f ( *h( } f ( τ u dτ h( u du ( H ( Corrlaion: Proof: I { f ( h( } f ( τ h( τ dτ d f τ h τ d dτ f τ F Q E D ( ( ( h( τ d dτ ow dfin τ u which yilds τ u, and w obain u τ I { f ( h( } f ( τ h( u du dτ τ u ( τ dτ h( u du F( H( F( H ( f If f( h( hn w obain h quaion for auocorrlaion as in h following: Applicaion of I{ f ( *h( } F( H( im invarian sysm : Frquncy rspons of a linar For a linar im invarian (LI sysm h oupu, y(, is givn in rms of h inpu, f(, as follows: y { f ( h( } F(. H ( I { f ( f ( } F( F ( F( I ( f ( *h( f ( τ h( τ dτ whr h( is h impuls rspons of h LI sysm. Exampl: 44

45 L ( f. hn τ ( f ( *h( h( τ y dτ L τ u, u + τ, dτ - du y y u ( h( u du h( u u du ( H( For f (, clarly y ( H(. W will now obain h sam rsul using h convoluion propry. L ( f hn F( δ( Convoluion propry yilds y ( δ( H( Y ( Y( d δ( H( d ( H( y If ( cos hn y f F ( δ( + δ( + ( δ( H( + δ( + H( Y ( Y( d δ H ( H( d + δ( + H( d ( + H( 45

46 ϕ( Wri H( in h form ( H( ϕ( ( H( H as H ( H ( which yilds H hn H ( H(, ϕ ( ϕ( hrfor w obain y ( ( ( + ϕ ( + ϕ H + which yilds ( H( cos( + ϕ y Exampl: L h( u( and f ( cos and valua ( + + y. H ( ; H(, ϕ( an Sinc rad/s, w obain H ( ϕ an 45, ( ( cos( 45 y + Ingraion: g ( f ( τ dτ 46

47 g G ( f ( τ dτ f (( τ u τ dτ u( f ( ( U( F( ( + δ( U F U F ( ( ( + δ( F( G ( F( + δ( F( As F( is h DC componn in f(, in h cas of sinusoidal sady sa F( as hr is no DC componns availabl in h signals. On h ohr hand if hn δ. hrfor for w hav ( Exampl: G ( F(. h ingro-diffrnial quaion is givn as: R i ( ( di + L + i d C ( d v( aing h Fourir ransform of boh sids w obain 47

48 R I C ( + L I( + I( + δ( I( V( R + L + C I ( V( which yilds I(. hrfor w obain I V ( C ( LC + RC Frquncy Convoluion: In words: Convoluion in h frquncy domain implis muliplicaion in h im domain. Proof: F I ( H( F( τ H( [ τ] dτ { F( H( } F( τ H( [ τ] dτ d L τ u, u + τ I τ u { F( H( } F( τ dτ H( u du [ f ( h ( ] Exampl: I I hrfor I I { f ( } F( On h ohr hand { f ( h( } F( H( { } δ( { f ( } F( * δ( Q E D 48

49 I i follows ha F For F { f ( } F( [ ] ( * δ( F( [ ] ( * δ( F( PARSEVAL S IDEIY AD HE EERGY HEOREM Considr h frquncy convoluion propry: I { f (h(} F( H( Which can xplicily b wrin as For, w obain f (.h(. d F( τ H( [ τ] dτ f (.h(.d F( τ H( τ dτ aing h conuga of boh sids of H H Rplacing by H ( ( h( h ( ( h d ( d w obain ow rplacing h( wih h * ( in h abov quaion and aing τ d f (.h (.d F d PARSEVAL S IDEIY ( H ( 49

50 ow aing h( f( and τ f ( d F( d EERGY HEOREM Fourir ransform Propris Opraion f( F( ransform f ( f ( Invrs ransform F( d F( Linariy af ( + bf ( af ( + bf ( Symmry F ( f ( ± im shifing f ( ± F( ± Frquncy shifing f ( F( [ ] Modulaion f ( cos F [ ] + F + f ( sin F [ ] F + Scaling f (a F a a im diffrniaion n d f ( n d d [ ( ( [ ]] [ ( ( [ ]] n ( F( Ingraion g ( f ( τ dτ G( F( + δ( F( im convoluion f ( *h( f ( τ h( τ dτ F ( H( Frquncy convoluion f ( h( F( H( F( τ H( [ τ] Parsval s Idniy f (.h (.d F( H ( d Enrgy horm E f ( d F( d dτ 5

51 .5. Poisson Sum Formula Considr an arbirary funcion f (. If I{ f ( } F( hn f n ( + n F( n n n whr Proof:. f for >, hn h lf sid of his formula is a priodic funcion If ( f p ( f ( + n n whr ( f ( fp for < < hrfor h Fourir cofficins of f p ( ar givn by F ( hnc h formula givn abov is us h Fourir xpansion of f p (. If ( f for > hn w can giv h following proof. Sinc δ( d n δ ( + n ow l us us his signal as an inpu o a LI sysm whos impuls rspons is f (. 5

52 3, w hav y3 ( y4( Sinc ( x ( x 4 which yilds h formula w wishd o prov..6. h Fourir ransform of h Puls rain p( h puls rain is dfind as p ( δ( n n which is a priodic funcion. hrfor w can a h Fourir sris which is givn as p ( S ; S W can now a h Fourir ransform of his sris wihich yilds I { p ( } P( δ( S On h ohr hand h Fourir ransform of ( δ( n S n p can b valuad as 5

53 P ( δ( n d n n his rsul can also b obaind by aing h frquancy domain Fourir sris of P p P ( δ( S ( δ( n n S. hrfor: n ( δ( n S Hnc h following abl: S S.7. h Fourir ransform of f(p( W now ha F ( I{ f (( p } F( P( Using P ( in his xprssion 53

54 F ( I{ f ( p( } F( δ( Illusraion of F ( I{ f ( p( } F( [ + n ] n ow using h formula givn abov, w can wri F Hnc w obain ( * δ( + F( [ + ] If h lf sid of his quaion is considrd o b sampld form of f ( hn h righ sid is h frquncy spcrum of h sampld vrsion of f (. ow l us valua h Fourir ransform of f ( f ( p( ingral: F ( I{ f (( p } F( [ + ] by h us of Fourir 54

55 I { f (( p } f ( δ( + n d f ( n n n n which, oghr wih h formula givn abov, yilds F n ( f ( n F( [ + ] n.8. Sampling and Inrpolaion In h numrical valuaion of h Fourir ingral w hav mniond h concp of slowly varying funcion. In h frquncy domain his concp corrsponds o bandlimidnss. Dfiniion: A funcion f ( is said o b band-limid if is Fourir ransform vanishs for largr han som consan: ( F for > C I will now b shown ha if a funcion is band limid, i is uniquly drmind in rms of is sampls. his mans ha alhough only h valus f ( n of h funcion f ( ar nown, h valus of f ( bwn n and (n+ for all n can b xrapolad from f(n n. I can b sn from h plo of h Fourir ransform of f ( p(, whr ( rain, ha a low pass filr can b usd o rcovr h original signal, f (, from (( p I { f (( p } F( [ + ] S p is h puls f. whr S is h sampling frquncy. I is vidn ha if S C if I f for < C { (( p } F( ( F for C. hrfor if F ( [ + ] is passd hrough an idal low-pass filr, ( rcovrd. S F can b 55

56 h Fourir ransform of h impuls rspons of an idal low-pass filr is dfind as Π ( for < ohrwis C h impuls rspons is hrfor givn by ( Π( Passing ( [ + ] h Fourir ransform F S d C C sin C d F hrough his low-pass filr yilds h oupu funcion f ( wih ( Π( F( [ + ] f ( can b obaind using h invrs Fourir ransform xprssion: f ( F( d S n Using F ( [ + S] f ( n and Π( U( U f S n n ( f ( n d S n n f n f S ( n ( n d ( n S f n ( n S S ( n S( n n sin S S ( ( n ( n S, w hav 56

57 Sinc S, w obain f ( f ( n n sin ( n ( n his is an inrpolaion formula for f ( using h discr valus ( n Sampling and inrpolaion is summarizd in h following: f. 57

58 his rsul can b givn in h form of h following horm: Sampling horm If h highs frquncy conaind in a signal f ( is C and h signal is sampld a a ra S C 58

59 hn f ( can b rcovrd from is sampl valus, f ( n whr f ( f ( n n sin ( n ( n S is h sampling frquncy. If S C, i.. F S FC, F S is calld h yquis ra and F S, as is calld h yquis frquncy. o ha F S is h maximum frquncy conaind in h signal, in ohr words F S is h signal band-widh..9. An Exampl of h Applicaion of h Fourir ransform in Puls Ampliud Modulaion (PAM In PAM a priodic squar wav is usd as h carrir signal. h ampliud of rcangular pulss in his wavform ar modulad by h sampl valus of an analogu mssag signal. his is shown in h following illusraion. o ha in PAM h puls rpiion frquncy in h carrir signal is h sam as h s is xprssd as sampling ra. h PAM wav ( s whr ( givn by ( f ( n( h n n f is h analogu mssag signal, ( h (, < < p, ohrwis h rprsns h rcangular puls and 59

60 Hr is h sampling ra. ow l f ( f ( n( δ n. n W will firs show ha ( F ( H( S *. o prov his i suffics o show ha s ( f ( *h( Convolving ( f (h im convoluion propry. f wih h (, w g ( *h( f ( τ h( τ dτ f ( n( δ τ n h( ττ d n ( n ( τ n h( τ dτ f δ n δ Sinc ( τ n h( τ dτ h( n, w obain ( *h( f ( n h( n s( f Q E D n F S w obain ow using ( F( [ + n ] S n ( F( [ + n ] H( n S A his sag l us now loo a h plos of H ( and { H( } arg. 6

61 ow considr passing s ( hrough an idal low-pass filr for rconsrucion. h following illusras his: 6

62 I is vidn ha h oupu of h idal low-pass filr is qual o F ( H(. his is quivaln o passing h original analogu signal f ( hrough a low-pass filr of ransfr funcion H (. Sinc H P P ( sinc. P I mans ha, by using puls-ampliud modulaion w inroduc ampliud disorion and a dlay of P. I is vidn ha, in ordr o rduc boh of hs ffcs w hav o us as narrow a puls as possibl in h carrir signal. 6

63 3. HE FOURIER RASFORM OF DISCREE-IME SIGALS In scion.7 h Fourir ransform of h δ - sampld form of a coninuous-im signal f ( was givn. f ( is obaind by using h puls rain p ( as follows: ( f (( p f I was shown ha h Fourir ransform is of h form I { f ( } F ( f ( n n n W will now show ha a similar rsul is obaind in h numrical valuaion of h Fourir ingral using rcangular approximaion. 3.. umrical Evaluaion of h Fourir Ingral Considr F ( f ( d Using h rcangular approximaion, w can wri F ( f ( n n n his xprssion givs a vry good aproximaion o h Fourir ingral, i.. o F ( whn f ( is a slowly varying funcion of im. In ohr words, is sufficinly small for f n. f ( o b spcifid by is sampl valus ( 3.. h Discr im Fourir ransform ow w will dmonsra ha h summaion givn abov can b obaind in wo mor ways. h following has alrady bn shown: F ( I{ f ( p( } f ( δ( + n d f ( n n n n f ( h ohr on uss h sampls f ( n o dfin a nw Fourir ransform 63

64 whr ω ( d( Fd f n d n ( ( nω f n f n and ω I is vidn ha d ω ( ( F F 64

65 ow considr h invrs ransform problm, i.. givn d ( f d ( n? Considr ω ( d( Fd f n n nω and muliply boh sids of his quaion by ω m ( d( Fd ω f n n ω( mn m ω. W obain F ω, how can w valua ow ingra boh sids of his quaion wih rspc o h frquncy ω bwn - and. W obain ω m d( ω ω d( n ω( mn F d f n dω Sinc ω( mn m n dω m n w obain ω nω fd( n Fd( dω ω ( d( Fd f n n nω ω nω fd( n Fd( dω DISCREE-IME FOURIER RASFORM PAIR hs formula rprsn h Fourir ransform of discr-im signals. On h ohr hand rsul of h approximaion of h Fourir ingral for coninuous-im signals hy in h sam im rprsn h numrical valuaion of Fourir ransform for coninuous im signals. ow l us compar h coninuous- and discr-im Fourir ransform formula. 65

66 F As ( n ( f ( d ω ( d( Fd f n n nω f d rprsn numbrs, d ( n ω. hrfor d ( funcions ( F ω is a linar combinaion of priodic xponnial F ω is priodic wih ω. his is, howvr, no h cas for F, alhough f ( is also a priodic xponnial funcion of. For vry h ingraion dos no yild a priodic funcion of. W can now xnd h char givn arlir in h following form: 66

67 Dnoing priodic, apriodic, coninuous and discr by P, AP, C and D, rspcivly, on obains h following corrspondnc: im- domain AP + D AP + C P + C Frquncy-domain C + P C + AP D +AP As ω is a coninuous variabl F ( ω sinc ω is a priodic funcion of ω and sinc a priodic funcion of ω wih priod. F ( ω is a coninuous funcion of ω. On h ohr hand ω is h argumn of ( F ω, F ( ω is normally invsigad in h rang from ω o ω. In fac sinc ω is priodic wih, all valus of ω for valus of ω byond - and ar h sam for hos wihin h rang ω. hrfor, for a discr-im signal f ( n h frquncy rang is wid and is normally an as ω. On h ohr hand sinc w hav ω * ω ( ( F F ω ω ( F ( F ω ω { ( } arg F( arg F { } hrfor w only nd o valua F ( ω for ω. ( obaind using ( +α * ( α ( F F ( 3.3. Som lmnary funcions and hir DF h Uni Sampl: is F ω for ω can b h uni sampl is dfind as δ( n for for n n 67

68 h DF of δ ( n is givn by hrfor ω ( ( nω ( ( ( Δ δ n n ω { ( } ( Iδ n Δ x n δ n X w h Uniy Squnc: h uniy squnc is givn as f ( n δ( n h DF of f ( n is givn as ω ( F n ow a h IDF of nω ω ( δ( ω H whr ω is a coninuous variabl and δ( ω is h Dirac s impuls funcion no h uni sampl. IDF yilds nω h( n δ( ω dω foralln hrfor ( n f ( n h 68

69 ω nω ( δ( ω F n h Uni-Sp Squnc: h uni-sp squnc is dfind as u ( n for for n n < Hnc u ( n u( n δ( n I hrfor { u ( n u( n } I{ u( n } I{ u( n } { ( } ( I u n U ω ω ω 3ω ω ω 3ω ω { u( n } U( ω I { ( } ( ω ω { } ( ( I u n I u n U ow l U( ω b dcomposd as whr ω ω ω ( ( + ( U U U ( ( U ω for ω U ω for ω W hav 69

70 ω ω ( ( ( ω U U + A soluion can b givn as U U C ω ω ω ( ( δ( ω ω ( + Cδ( ω U On h ohr hand W can wri ω nω { ( } ( n( ω ω u n u n u( n U( I n n ωn ωn u( n dω+ Cδ ( ω dω ω ω ( n ωn u( n dω+ Cδ ( ω dω ω C u( dω+ ω C u( dω+ ω Adding boh sids ω ω + C dω+ ω ω + C + C I +δ ω ω ω { u( n } U( ( 7

71 h Exponnial Squnc: n L f ( a u( n n nω ω { f( } a u( n ( a I n n n L { } n a ω α hn I f ( n α S n n α + α + α + + α n αs α α α + α + n + α + α + + ( α S α α S α + If α < hn ω α a < and for α + hrfor, w obain and S α I ω a n ω { a u( n } F( ω ( F acosω + a sin ω + a acosω ( { ( } ( arg F asin Φ ω an acos ω ω ω 7

72 h maximum and minimum valus for F ( ω ar obaind for ω and ω. For ω F( For ω ( ; a F + a Discr im Rcangular Puls: p ( n for n L ohrwis 7

73 p ( n can b xprssd in rms of h uni-sp squnc u ( n, whr u W hav ( n for for n n < p ( n u( n u( n L ω { ( } ( ( ( I p n P u n u nl n nω ω ( P Lω Lω Lω L Lω nω ω ω ω ω n ( L ω Lω sin ω sin ω P( For Applying L Hopial s rul, w obain ( P L Lω cos L ω cos ω ( P Lω sin ω sin Lω sin ω ( L ω arg{ P( } ( arg ϕ ω + ω sin Lω sin if ( ω > arg ω ω sin if ω< ( whr ( 73

74 Magniud and phas of D.. Fourir ransform of h discr-im rcangular puls In gnral, for a rcangular window of siz L, i.., a sring of L impulss bginning a zro and ndin a L-, w s h following: a. h magniud plo has zro spacd /L apar wih h main lob around and sid lobs rducing in ampliud as ω movs oward. b. h slops of all lins in h angl plo ar (L / wih nd of firs lin a (L /L and nd of h h lin a (L /L. h angl plo is anisymmric abou and picwis linar. 74

75 A discr-im shif invarian sysm wih impuls rspons qualing h rcangular window acs li a low-pass filr passing signals wih digial frquncis roughly bwn and /L and anuaing signals wih frquncis bwn /L and. h imporan propry ha h phas is linar hroughou h passband is usful in FIR digial filr dsign. h Discr im Sinc W will a his problm by showing ha h IDF of h rcangular puls in h coninuous-frquncy domain is h discr-im sinc. Considr for ω ω ω C < F ( ohrwis h IDF of F ( ω is obaind as follows: a. n b. n ω ωn f ( n F( dω ( f n ( ωc ωn dω ωc ωc ω dω ωc ωc C ωc ω ωcn ωcn f n dω n ωc sin ωcn n ω n C 75

76 3.4. Propris of DF Linariy { } ( If I f ( n F ω and ( { f } ( n F ω I hn ω ω { af( n bf( n } af( bf( I + + im and Frquncy Scaling { f( an } F ω a I Proof: ω n { ( } ( ( I f an f an f m n m ω m a Spcial cas for a - yilds { f( n } F( ω I 76

77 im Shifing ω n { f( n n } F( ω I Proof: { ( } ( I f n n f nn n ωn n n m ω m ω n { f ( n n } f( m I m Exampl: Evalua h DF of L L+ ps ( n u n+ u n p S (n is h shifd vrsion of p( n u( n u( n L In fac L ps ( n p n+. According o h im-shifing propry w can wri S L ω ω ω ( ( P P Sinc L ω ( P ω Lω sin ω sin w obain 77

78 P S ω ( Lω sin ω sin Spcral characrisics of symmrical rcangular puls 78

79 im Convoluion horm { } ( L I f ( n F ω and h( n { } H( ω I. Dfin hn Proof: y n f n *h n f h n ( ( ( ( ( ω ω ω { y( n } Y( F( H( I ω ( ω n ( Y y n f ( h( n n n ω n Inrchanging h ordr of summaion yilds ω ( ( ( Y f h n n ωn hrough h chang of variabl n m, w obain ω ( ( ( m ω Y f h m ω f ( h( m ω F ω ( H ( m ωm ω m An Applicaion of h Convoluion horm: h Rspons of an LI Sysm o a Sinusoidal Inpu L h inpu b of h form ( f n ±ω n whr ω. W wish o valua h oupu y(n of an LI sysm wih h unisampl rspons h(n o f(n. 79

80 h convoluion horm yilds ω ω ω ω ( ( ( ( ( Y F H δ ωω H ω aing h invrs ransform If f ( n ω ωn y( n δ( ωω H( dω yilds rspons ω ( ( y n H n ω ω n yilds h obvious rsul ω ( ( ω n. y n H ( n f n ω +θ θ On h ohr hand for (, w hav f ( n ω and for ( ow l θ ω ( ( y n H ω n ( n f n ω +θ, w hav θ ω ( ( ω n. y n H ( ( ω +θ f n cos n n hnc Using ± ( ω n+θ ±θ I { } δ( ω ω w g 8

81 { ( } θ θ cos n ( ( I ω +θ δ ωω + δ ω+ω θ θ ω ωn y( n δ( ωω + δ( ω+ω H( dω θ ω ω θ ω H( + H( n ω n Sinc and ( ( ( ( H H ϕ ( ω ω ω H H ϕ ( ω ω ω ω ( ( H h n n ω n w hav ω ω ( ( H H hnc ω ω ( H( H and ϕ( ω ϕ( ω Using hs in y(n yilds ω ( ω ( n+θ+ϕ ω n+θ+ϕ y( n H( + ω ( ( ( ω +θ+ϕ( ω y n H cos n Exampl: f n 3cos n L ( Evalua y(n for h following LI sysm if ( h n,,. 8

82 ω, θ 3 6 H cos ω ω ω ω ω ω ω ( ( + ω ω ( + ω ( H cos ϕ ω ω For ω H +, ϕ y( n cos n 3 6 h Corrlaion horm { } ( If I f ( n F ω and h( n whr r ( { } H( ω I hn I rfh n f h n F H fh ω ω ( ( ( ( ( n is calld h corrlaion bwn f(n and h(n Proof: Comparing r ( fh hrfor fh ( ( ( r f *h ω ω { rfh ( n } F( H( I n wih h convoluion givn abov w conclud ha 8

83 Parsval s horm { } ( If I f ( n F ω and h( n Proof: { } H( ω I, h righ hand sid of his quaion can b wrin in h form: ω n * ω * ω ω n * f ( n H ( dω f ( n H ( dω f ( n h ( n n n n In h spcial cas whr f(n h(n Parsval s rlaion rducs o * ω * ω ( ( ( ( n f n F dω ω ( ( n f n h n F H dω h lf hand sid of his xprssion is simply h nrgy E f of h signal f(n. his is also qual o h auocorrlaion of f(n valuad a. h ingrand on h righ hand sid is h nrgy dnsiy spcrum and h ingral ovr h inrval ω yilds h oal nrgy of h signal. hrfor w can wri: E r f n F dω R ω dω ( ( ω ( ( f ff ff n Frquncy Convoluion horm { } ( L I f ( n F ω and h( n { } H( ω I. hn whr I ω ω { f( n h( n } F( *H( ( ω ω θ ( ωθ ( ( ( F *H F H dθ 83

84 Proof: θ n ( ( ( θ ( ωθ ( F H f n hm n m ( ωθ m n m ( ( f n h m ( ω n+ mn θ ω n ( f ( n h( n + f ( n h( m n n m n I{ f ( n h( n } G, I { f( n h( n } dθi f( n h( n { } ( ωθ ω n mnθ G ( ωθ, dθ Q.E.D. h Frquncy Shifing Propry n ( { ±ω f( n } F ω ω I ( Proof: n { ( } n n ( ( ±ω ±ω ω ω ω n I f n f n f n F n n ( ω ω ( n ± ω Exampl: h DF of h Priodic Exponnial funcion aing f ( n n (h uniy squnc w hav hrfor ω ( δ( ω F ±ω n { } ( I δ ω ω 84

85 or Sinc ±ω n { } ( I δ ω ω for ω and priodic wih ( ω n+θ θ ωn w g ± ( ω n+θ ±θ I { } δ( ω ω for ω and priodic wih Modulaion Propry ( ( [ ω+ω] ( ωω ( [ ω+ω] [ ωω ] I{ f( n cosω n} F + F [ ] I{ f( nsin ω n} F F hs can b obaind using h frquncy shifing propry and h xponnial xprssions for cos ω n and sin ω n. Exampl: h DF of cos ω n and sin ω n. L f ( n n propry:. Sinc ( ( F ω δ ω ; ω w obain from h modulaion { cos n} ( ( { sin n} ( ( I ω δω+ω +δωω I ω δω+ω δωω Exampl: h DF of h Causal Sinusoidal Squncs aing f(n u(n in modulaion propry w g W hav [ ωω] ( ( ω+ω [ ωω] ( ( [ ω+ω ] I{ cosω nu( n } U + U [ ] I{ sin ω n u ( n } U U ω ( +δ( ω U + ω 85

86 Using his abov yilds ( ( ( ω cosω I{ cos ω n u ( n } + δ ωω + δ ω+ω ω ω cosω + ( ( ( ω cosω I{ sin ω n u ( n } + δ ω+ω δ ωω ω ω cosω + n Exampl: Evalua h DF of g( n a cos( ω n+ϕ u( n L f ( n a n u( n Wri ( ω n+ϕ ( ω n+ϕ + g( n f( n cos( ω n+ϕ f( n ϕ ωn ϕ ωn f( n + f ( n Using h linariy propry ϕ I { g( n } G( I f( n + I f n n ϕ n { } { ( } ω ω ω [ ω+ω] ( ( ω ϕ [ ωω ] ϕ G( F + F ω ( F a ω 86

87 ϕ ϕ ω ( + ( ωω ( ω+ω G a a Diffrniaion in Frquncy { } F( ω If f ( n I hn ω ( df I { nf( n } dω Proof: ω ( df ωn n f ( n Q.E.D. dω n 3.5. Fourir ransform of Priodic Discr im Signals: h Discr Fourir Sris (DFS h DF of a fini-duraion squnc (FDS f(n is givn by ω ( ( F f n n ω n which is a coninuous funcion of ω. ow l us a discr valus of ω in a priod F ω a h frquncis givn by F ω. In ohr words sampl ( of ( 87

88 ω for,,,.., W obain: F f ( n,,,.., n n ow considr h invrs problm; i.. givn F find f(n. Firs muliply boh h lf and righ sids of h quaion abov by. m and sum ovr m n m ( F f( n f ( n n n m n ( for n m ( ingr ± ohrwis h firs cas is obvious. h scond can b xplaind as follows: ( nm ( nm ( nm ( nm ( nm for nm ± ( nm hrfor for hrfor ( ( nm f n f n δ n m f m n n m F f ( m ( ( ( n m ( 88

89 f ( n F n I is vidn from his xprssion ha f(n is priodic wih alhough w hav sard off wih a fini-duraion squnc f(n. h diffrnc bwn h IDF xprssion ω ωn f ( n F( dω and f ( n F n lis in h fac ha, alhough h ingrand in h firs xprssion is a priodic funcion of n for vry ω h rsul of h ingral is no. Bu i is vidn ha scond xprssion is a summaion of priodic xponnials wih ingr mulipl frquncis, hrfor yilds f(n priodic wih n. ow considr h priodic squnc f p (n ha is mad up of h fini-duraion squnc f(n as follows: P ( ( + f n f n m m whr is h duraion of f(n and h priod of f p (n. (All f(n having a duraion shorr han can b mad o hav a duraion by zro padding. Apply h DF xprssion o f(n and sampl in frquncy a ω F n ( f n n h invrs yilds h fini duraion squnc f(n which is only valid for n n f ( n F n or if h im-duraion rsricion is lifd w obain 89

90 n fp ( n F n h following is calld h Discr Fourir Sris Pair for Priodic Squncs. Discr Fourir Sris Cofficins Discr Fourir Sris Exprssion n P P( n F f n n, n fp( n FP n, Analysis Equaion Synhsis Equaion Exampl: Find h DFS cofficins of ( ( ( fp n cos n n, n n FP fp n cos n n n cos n + n n n n n ( F P ( + n n( + + n n n for ± m+ n n( ( n ohrwis ( Similarly n n + ( for ± m n ( + ohrwis ( + 9

91 hrfor n( δ ± m+ n ( [ ] n( + δ ± m n ( [ ] P + δ δ + + δ δ + ( ( ( ( F + δ ( + + δ( + δ( + + δ( + Expansion ino Fourir sris using on priod of FP fp ( n ( ( δ + δ + n ( n + n n n n ( n n P ( f n cos + n n 9

92 W can also a h summaion in h rang. In his cas w hav n n fp( n FP δ ( + +δ( n n cos + n Exampl: Find h DFS cofficins of h priodic squnc f p (n dfind by f P ( n n for n 3 wih priod 4, i.. f ( n 4m f ( n Soluion: P + for all ingr m. P Using FP( fp( n n n w can wri 3 n 3 n F P ( n n F P ( FP ( FP ( ( n

93 5 5 5 FP ( F 6 6 P F 4 F P ( ( ( ( P If h duraion L of f(n is shorr han h priod of f p (n hn w can incras h duraion of f(n o by zro padding. In pracic f(n is us a window from h signal undr considraion and F( is also a window from h sampld vrsion of h DF of f(n. 93

94 3.6. h Discr Fourir ransform (DF From compuaional poin of viw; w can only dal wih fini-duraion squncs boh in h im- and frquncy-domains. hrfor o carry ou a frquncy analysis on a im-domain signal only a window from h original signal can b considrd. On h ohr hand, in h frquncy domain again only a limid par of h spcrum can b considrd. L us now considr onc again h DFS pair. Considr also on priod of f p (n, i.. f(n. As w hav alrady poind ou FP F W can also wri n f ( n F for n hrfor w dfin h following: DF IDF ( ( n n F f n n f ( n F( n his is calld h DF pair. If a priodic squnc is mad up of f(n such ha P ( ( + f n f n m m hn h DF is on priod of FP and f(n obaind by IDF is on priod of f p (n. 94

95 Rlaionship Among Rcord Lngh, Frquncy Rsoluion and Sampling Frquncy Sampling implis ω S, whr S is h sampling inrval. For an -poin DF, h DF cofficins F( ar valuad a h discr frquncy poins ω. hrfor h frquncy spacing bwn wo conscuiv DF cofficins is Δω. h corrsponding spacing in h analogu frquncy can b found from Δω Δ S Δω Δ or S FS Δ F S R Δ F S whr. F S is h sampling frquncy.. R S is h rcord lngh. 3. Δ F is h frquncy rsoluion. If x(n conains frquncy componns closr han Δ F hrz, hn h DF dos no rprsn hs as spra and disinc frquncis. In ordr o rsolv clos- oghr frquncis w nd o ma Δ F sufficinly small. his is rfrrd o as incrasing h rsoluion. h rlaionship givn by Δ F S R clarly shows ha, in ordr o incras h rsoluion h rcord lngh R should b incrasd. If h sampling frquncy is hld consan howvr, o incras rsoluion on has o incras. 95

96 3.7. Efficin Compuaion of DF: h Fas Fourir ransform (FF Considr ( ( F f n n n Dfin W and h vcors and h marix ( ( (, F F(,F(,,F( f f,f,,f [ ] W n ow w can wri F f n is an x complx valud marix. hrfor using sampls from h signal o compu frquncy componns rquirs complx muliplicaions and complx addiions. h DF hrfor bcoms unpracical for larg valus of. For xampl, for 4, w would rquir 6 complx muiplicaions and addiions. Som fas Fourir ransform algorihms hav bn invnd. Hr w will prsn h dcimaion in-im (DI FF. h DI-FF algorihm is basd o nhaving a numbr of poins r whr r is an ingr. h undrlying rason for his is ha his algorihm rlis on spraing h -poin squnc f(n ino wo squncs of lngh h vn- and h odd-indxd sampls, rspcivly. Considr ( ( F f nw n n corrsponding o 96

97 his can b wrin as n ( ( + ( F f nw f nw n vn n odd ( ( ( n ( ( + + F f n W W f n W n n n n Sinc W w hav W W ow dfin ( f( n, h( n f ( n g n + ; n,,,, 97

98 w g n ( ( + ( F gnw W hnw n n n h summaion in his xprssion rprsn wo -poin DF s which ar hrfor ( ( G g n W n ( ( H h n W n n n ( ( + ( F G WH Bing -poin DF s, G( and H( ar priodic wih. On h ohr hand + W W W W as W hrfor F + G( WH( for his xprssion and h on givn for F( consiu h rcomposiion of -poin DF s ino -poin DF. 98

99 Considr now h wo quaions: F ( G ( + WH ( for F + G( WH( for h compuaion of G( rquirs complx muliplicaions. h sam applis o h compuaion of H(. Anohr complx muliplicaions ar rquird o compu WH. h oal complx muliplicaions o compu h -poin DF is hrfor ( + +. For larg valus of his is qual o which mans ha h numbr of complx muliplicaions has bn rducd by a facor of. 99

100 W can now furhr dcompos h -poin DF s G( and H( ino wo 4 -poin DF s by spraing g(n and h(n ino wo squncs of lngh 4 corrsponding o vn- and odd-indxd sampls rspcivly, as follows: ( ( G g n W n n 4 4 n g( n( W W g( n ( W + + n n n whr g( n f( 4n g( n+ f ( 4n+ and W W 4 ow dfin w hav g( n p( n g( n+ q( n 4 ( ( P p n W n 4 ( ( Q q n W n n 4 4 n and G( P( + WQ( P( + W Q( 4 Whr P( and Q( ar wo 4 -poin DF s which ar priodic wih a priod of 4. In xacly h sam way w can dcompos H( as

101 H( U( + W V( 4 whr U( and V( ar givn by 4 -poin DF s which ar priodic wih a priod of 4 and ar 4 ( ( U u n W n 4 ( ( V v n W n n 4 n 4 whr h( n u( n h( n+ v( n hrfor h idnificaions ar mad as follows: f ( g( p( f ( h( u( f ( g( q( f ( 3 h( v( f ( 4 g( p( f ( 5 h( u( f ( 6 g( 3 q( f ( 7 h( 3 v( f ( 8 g( 4 p( f ( 9 h( 4 u( f ( g( 5 q( f ( h( 5 v( P(, Q(, U( and V( ar hrfor priodic wih priod. On h ohr hand 4

102 W W W W as 4 W. hrfor w can wri G( P( + W Q( for 4 G + P( W Q( for 4 4 H( U( + W V( for 4 H + U( W V( for 4 4 h compuaion of G( rquirs + complx muliplicaion which als 4 4 applis o h compuaion of H(. hrfor for h compuaion of F( w now nd complx muliplicaions. Anohr complx muliplicaions ar rquird o compu F( from G( and H( bringing h oal numbr o 4 + Comparing his wih +, h oal numbr of complx muliplicaions ha wr rquird a h prvious sag his amouns o a rducion by a facor of and, a facor of 4 whn compard wih h dirc compuaion of DF. his procdur of spraing h squncs can b rpad unil h rsuling squncs ar rducd o on-poin squncs whr h final sag consiss of -poin DFs of hs on-poin squncs. So far w hav sn ha h firs dcomposiion of daa rsuls in -poin DFs and on sag for h rcomposiion of hs DFs ino on -poin DF. hn h scond

103 dcomposiion of daa yilds 4 4 -poin DFs and on sag consising of rcomposiions of 4 -poin DFs ino -poin DFs. his dcimaion of daa can b rpad again and again unil on obains -poin DFs and on sag having 4 rcomposiions of -poin DFs ino 4 4-poin DFs. I is vidn ha vry sag corrsponds o dividing by and his can b rpad unil on obains which is h r numbr of poins in h DFs a h final sag. Sinc, and hrfor h r numbr of rcomposiion sags is r. W can conclud h FF of an -poin squnc consiss of r rcomposiion sags and on sag of oal numbr of sags is r log. -poin DFs. hrfor h In gnral ach rcomposiion sag rprsns h rcomposiion of i i -poin DFs ino i i -poin DFs whr i is divisibl by wih i<. W will show lar ha ach sag consiss of complx muliplicaions and complx addiions. hrfor h oal numbr of complx muliplicaions in an FF will b log and ha of complx addiions will b log. h improvmn on h dirc compuaion of DF is: for muliplicaions for addiions υ mul log log υ add log log mul υ and add υ ar spd improvmn facors for complx muliplicaions and addiions, rspcivly. h abl givn blow givs a comparision of numbr of complx muliplicaions in h radix dcimaion-in-im FF and in h dirc copuaion of DF. 3

104 umbr of Complx Muliplis for DF and radix- FF Sandar DF FF υ mul , , , ,48, : : : : ow h FFs for, 4 and 8 will b prsnd:. ( ( ( ( n n F f nw f W + fw, whr G( f ( W f( H( f( ( W H( G + hrfor G( and H( ar h -poin squncs f( and f(, rspcivly. W hav: ( ( x ( ( ( x ( F f + W f F f + W f Sinc W w obain F ( f( + f ( F ( f( f( 4

105 his signal-flow graph srucur is nown as h burfly ( ( F whr f nw n 4 n ( ( ( ( 3 f W + f W + f W + f 3 W Dirc compuaion of DF f W + f W + f W + f 3 W 3 ( 4 ( 4 ( 4 ( 4 ( ( ( ( f W4 + f W4 + W4 f W4 + f 3 W 4 G H ( ( g( n f( n h( n f ( n+ n, hnc g( f ( h( f ( g ( f( h ( f( 3 ( ( + ( ( ( + ( G g W g W H h W h W G( and H( ar -poin DFs. hrfor w can wri: ( ( + ( 4-poin DF of f n -poin DF of g n W -poin DF of h n 4 ( ( + ( F G WH 4 ( ( + 4 ( ( + ( ( ( + ( ( + ( F G WH G H F G WH G WH 4 4 5

106 ( ( + 4 ( ( ( ( ( + 3 ( ( ( F G WH G H F3 G3 WH3 G WH 4 4 as G( + G(, H( + H( and W W W W Exampl: Us h abov graph o compu h FF of n for n 3 f ( n ohrwis f n,,, 4 8 i.. ( Soluion: 5 G( f( + f( G ( f( f( H( f( + f ( H ( f ( f( F ( G ( + H (

107 3 ( ( ( F G+ WH F ( G ( H ( ( ( ( F3 G WH F f W + f W + f 4 W + f 6 W vn n ( ( ( ( ( + ( + ( + ( + ( ow dfining f W f 3 W f 5 W f 7 W odd n ( f( n, h( n f( n g n and considring ha + ; n,,, 3 W W 8 4 w can wri 7

108 ( 4-poin DF of g n 3 F g W g W g W g 3 W ( ( + ( + ( + ( ( ( ( ( 3 + W 8 h W4 + h W4 + h W4 + h 3 W 4 4-poin DF of h n ( ( + ( F G WH 8 Rcomposiion of wo 4-poin DFs ino on 8-poin DF For,,.., 7 w obain ( ( ( + 8 ( ( + ( ( ( + 8 ( ( ( + 8 ( 3 ( ( + 8 ( ( ( ( ( + 8 ( ( ( ( ( ( ( + 8 ( ( 8 ( 6 6 ( ( + 8 ( ( + 8 ( ( 8 ( ( ( + 7 ( ( + 7 ( ( 3 ( F G WH G H F G WH F G WH F3 G3 WH3 F4 G4 WH4 G WH G H F5 G5 WH5 G WH G WH F6 G6 WH6 G WH G WH F7 G7 WH7 G3 WH3 G3 WH W can now dcompos ach of h 4-poin DFs o wo -poin DFs: 8

109 3 ( ( 4 + ( 4 + ( 4 + ( 4 ( + ( + ( + ( G g W g W g W g 3 W Dfining g W g W W g W g 3 W ( p( n, g( n q( n g n and considring + ; n, W 4 W w g ( ( + ( + ( + ( G p W p W W4 q W q W P( Q( poin DFof p(n poin DFof q(n o ha p( f( q( f( p ( f( 4 q ( f( 6 ow considr H( 3 ( ( 4 + ( 4 + ( 4 + ( 4 ( + ( + ( + ( H h W h W h W h 3 W Dfining hw hw W hw h3w ( u( n, h( n v( n h n and considring + ; n, W 4 W w g ( ( + ( + ( + ( H u W u W W4 v W v W U( V( poin DFof u(n poin DFof v(n 9

110 o ha u( f ( v( f ( 3 u ( f( 5 v ( f( 7 ow using W 4 W8, W + 4 W4 and P(+ P(, Q(+ Q(, U(+ U(, V(+ V(, ( ( + ( ( + ( G P W Q P W Q 4 8 ( ( + 4 ( ( + ( ( ( 4 ( ( 8 ( ( ( 4 ( ( ( 3 ( ( ( ( ( G P W Q P Q G P+ WQ P+ WQ G P + W Q P Q G 3 P 3 W Q 3 P W Q Rcomposiion of wo -poin DFs ino on 4-poin DF ( ( + ( ( + ( H U W V U W V 4 8 ( ( + 4 ( ( + ( ( ( 4 ( ( 8 ( ( ( 4 ( ( ( 3 ( ( ( ( ( H U W V U V H U+ WV U+ WV H U + W V U V H 3 U 3 W V 3 U W V Rcomposiion of wo -poin DFs ino on 4-poin DF

111 Sinc W8 h muliplicaions wih W8 a h firs sag has no ffc. In fac in h drivaion of FF hs muliplicaions hav no bn ncounrd. hy hav only bn inroducd for h sa of gnraliy and from programming poin of viw. As o h ordr of squnc afr i is dcimad r ims w ma h following obsrvaion: Considr h cas whr 8. Afr h firs dcimaion w obain h daa squnc

112 f(,f(,f( 4,f( 6,f(,f( 3,f( 5,f( 7 and h scond dcimaion yilds f(,f( 4,f(,f( 6,f(,f( 5,f( 3,f( 7 his shuffling of h inpu daa squnc has a wll dfind ordr. As shown in h following abl h inpu daa squnc obaind afr r dcimaions can b obaind by bi rvrsing h binary quivalns of h indics of h original inpu daa. Indics Dcimal Binary Birvrsd Dcimal h following abl shows how h inpu daa is shuffld y ach conscuiv dcimaion. n n n n n n n n n