77 Chap 5 Coninuous-im ourir ransform and Applicaions 5 Illusraiv Dfiniion of ourir ransform In his chapr, ill dvlop h basis for ourir analysis of non-priodic signals, hich is h only group of signals maningful in nginring and ral-lif applicaions radiionally, ourir analysis is prsnd by giving h dfiniions as did for Laplac ransforms in Chapr 3 Hr, ill approach i as a limiing bhavior of ourir sris analysis for priodic signals o do ha, l us considr an apriodic non-priodic signal x and form is priodic xnsion x by rpaing i vry sconds, hr is h priod x x Guard Inrval - I is clar from abov ha h xpansion signal can b rin as: x x x + n if < x 5 for vry in gr n h ourir sris rprsnaion for his n priodic signal is simply: j : x 5a Synhsis Equaion j Analysis Equaion: x d 5b < > In h limi as, obsrv ha d, hich is an infinisimally small quaniy in frquncy-domain and i implis ha, a coninuous variabl and d df inally, ih his limiing bhavior, h summaion in 5a bcoms an ingral as follos: d j j j x d x d x d 53 d and j j d j x x d d hs nos ar Husyin Abu, Augus 998 54
φ 78 Wih his rsul, hav implicily dfind h ourir ransform rlaionships for x j Analysis Equaion: x d { x } 55a j Synhsis Equaion: x d { } 55b hs o quaions ar calld ourir ransform pair and normally shon by: x In gnral, is a complx funcion of h ral-valud frquncy variabl and i is rin in rms of magniud and phas rms: jφ 56 hrfor, plo on curv for h magniud and anohr on for h phas of a givn xprssion φ Dirichl Condiions: ourir ransform xiss if: x is absoluly ingrabl: x d < x is a ll-bhaving funcion ha is, only a fini numbr of jumps of fini siz, minima and maxima occur ihin any fini inrval: < < Exampl 5 ind h ourir ransform of a rcangular puls ga funcion, rcangular imindo x τ -τ/ τ/ 4/τ /τ /τ 4/τ 6/τ 8/τ hs nos ar Husyin Abu, Augus 998
79 τ / τ / j j j j / + j / x d d [ ] j j τ / τ / 57 τ τ τ τ [ j Sin ] Sin τ Sinc τ Sa j I is asy o obsrv ha h rsuling spcrum is ral and symmrical for all valus of frquncy his implis ha h phas rspons is zro: φ for all h magniud is dcrasing as a funcion of / Approximaly, 9% of nrgy conn of his spcrum is undr h main lob of h plo: / τ < < / τ In ohr ords, h nrgy undr all h ail lobs is abou % Exampl 5 ind h ourir ransform of a dla funcion W ill solv his problm using h Sifing horm dfiniion of a dla funcion: { V j } V d V { } 59 Similarly, can us h invrsion formula o hav: j d 5 rom his final rsul can dduc ha h ourir ransform of a consan is a dla funcion in frquncy-domain: V } V 5 { Exampl 53 ind h ourir ransform of a complx xponnial harmonic funcion: j { j } j j j d d 5 On h ohr hand, h ourir ransform of a on-sidd dcaying xponnial funcion is: a x u and a > a j a+ j u d d a + j hs nos ar Husyin Abu, Augus 998 53 Exampl 54 ind h ourir ransform of a priodic signal x ih a priod As no from h prvious chapr ha h priodic funcions hav ourir sris rprsnaion: x j L us a ourir ransform of his quaion rm-by-rm:
hs nos ar Husyin Abu, Augus 998 8 } { } { x j 54 hrfor, h ourir ransform is a squnc of impuls funcions rgardlss of h acual shap of h signal x In ohr ords, h ourir ransform of all priodic funcions is a family of impulss Wha ma hm diffrn for various x shaps ar h valus of h cofficins { } Exampl 55 Using h rsuls from h prvious xampl, ar asd o find h ourir ransform of an impuls rain x x j j > < > < 55 L us subsiu hs cofficin valus in o h rsul abov o obain: 56 hus, conclud ha h ourir ransform of an impuls rain is anohr impuls rain in h frquncy-domain ih diffrn srnghs in h cofficin s 5 Propris of ourir ransforms Linariy: h ourir ransform is a linar ransform bg a bg f a + + 57 Symmry: If x is a ral signal hn or in polar form: j j and φ φ 58 hrfor, hav an vn-symmry of h ampliud spcrum and an odd-symmry for h phas spcrum x: : / / / / - - / / 4 / /
8 abl 5 ourir ransform Pairs for Slcd Signals Exampl 56 Using h propris of ral signals can urn ourir ransform ino a sin or a cosin ransform In paricular, discr vrsion of h cosin ransform has found a na applicaion ara in imag comprssion basd on JPEG and MPEG chniqus hs comprssion chniqus and hir drivaivs amp o xploi h symmry propris of imagry in h frquncy-domain hs nos ar Husyin Abu, Augus 998
8 j d x Cos d j x Cos d + x x Cos d x [ Cos x Sin d Sin is odd fn jsin ] d 59 As s from his las rsul ha h ourir ransform urns ino a ourir Cosin ransform for ral signals 3 im-shifing Dlays: Givn: x hn j j φ x 5 Nos: Ampliud spcrum: is unaffcd by a dlay im-shif of unis Hovr, h phas spcrum is linarly shifd by radians from h phas valu φ of h original spcrum his linar phas-shif propry mas h im-dlay issu vry usful in many applicaions 4 im-scaling Rsoluion Chang: Givn h ourir pair: x hn a imscal chang ill corrspond o: x a 5 a a Nos: or valus of a largr han "" his ors as a comprssor ha is, i couns slor han on im-uni a a im On h ohr hand, for a smallr han "" i ors as an xpandr o incras h rsoluion in im-domain h rsuling bhavior in frquncy-domain is jus h opposi Exampl 57 Using h abov propry find h ourir ransform of x a rc Rcalling { rc } Sinc and h rsoluion chang g: { a rc sin c 5 a a a hs nos ar Husyin Abu, Augus 998
83 5 Drivaivs and Ingrals:-Scaling: Givn h ourir pair: x hn hav: d a x j 53a d n d n b x j 53b n d c x τ dτ + 53c j d If hr is no DC rm, i, hn x τ dτ 53d j Exampl 58 Using h abov propry find h ourir ransform of a uni-sp funcion: u u Consan sgn 5 5-5 + As s from h abov figur, h uni-sp funcion can b rin as: u + Sgn 54 Bu also no ha: d { Sgn } quivalnly : j { Sgn } 55 d hich rsuls in: { Sgn } and { } 56 j { u } + 57 j 6 Enrgy in Apriodic Signals Parsval's horm: Rcalling nrgy in im-domain: j E x d [ x x x d j x [ d] d hich implis ha h nrgy is consrvd during ourir ransform and i can b summarizd by: d] d d d hs nos ar Husyin Abu, Augus 998
84 E x d x x d d d 58 Similarly, h nrgy conn of a signal ihin a fini band of frquncy: ill b: E x d 59 Exampl 59 Compu h porion of h nrgy for an xponnially dcaying signal in h frquncy inrval: 4 4 W no h ourir ransform for his signal boh from a prvious xampl and h ourir abls 5 ha: x u + j + Whn subsiu his ino 58 and 59 g: E d rom in graion abls + 4 4 4 E x d d arcan 4 + hrfor, h porion of h nrgy in his fini band is 844% 4 7 Convoluion Opraion: x Inpu signal Sysm y orcing funcion h Rspons H Y Oupu Signal y x h h * x x τ h τ dτ 53a jφ Y H Y 53b Y Y and Y + H 53c Nos: Oupu magniud spcrum is h produc of magniud spcrum of h inpu and h frquncy rspons of h sysm h phas spcrum is h sum of h phas spcra of h inpu and h sysm Exampl 5 Compu h uni-sp rspons o a sysm ih: h u Y H { x } { h } a hs nos ar Husyin Abu, Augus 998
85 Y [ + ] j a + j a + j + j a + j dircly from ourir abls L us no ha only hn hrfor, h firs rm in h las xprssion is s simply: rm Using his and parial fracion xpansion rsuls in: a A B Y + + a j a + j Bj + A a + j A and B a a inally, h oupu in h frquncy-domain ill b: / a / a Y + a j a + j and h invrs ourir ransform yilds h ansr: a y { } u a 8 Modulaion horm: x m M y Y y x m Y [ * M ] 53 Y ψ H ψ d 53 In ohr ords, o rplac a im-domain muliplicaion, mus prform a frquncy-domain convoluion As xpcd, ill amp o prform im-domain muliplicaion as much as can o acl modulaion ass Exampl 5 Givn a signal ih a bas-band spcrum frquncy conn of h signal in is naural habia, modula i ih an impuls rain of priod: / hs nos ar Husyin Abu, Augus 998
86 Basband Signal p impuls rain - -3 - - 3 4 h oupu of h modulaor mixr is simply: xs x p x n x n n 533 n n Hr, h impuls rain acs li a sampling funcion, i, isolaing h signal a a givn poin Equivalnly, in h frquncy-domain: P { p } { n } 534 n * P [ * ] S 535 I is clar from his rsul ha h oupu spcrum is a priodic family of h inpu spcra shifd o h nighborhood of ± ± s Bandpass modulad signal - - / - / hs nos ar Husyin Abu, Augus 998
87 abl 5 Propris of Coninuous-im ourir Pairs hs nos ar Husyin Abu, Augus 998