Notes on the Fourier Transform

Transcription

1 BE/EECS 56 4 FT Note Note on the Fourier Trnfor Definition. The Fourier Trnfor FT relte function to it frequenc oin equivlent. The FT of function i efine b the Fourier integrl: G F{ } e iπ for R. There re vriet of eitence criteri n the FT oen t eit for ll function. For eple the function co/ h n infinite nuber of ocilltion n the FT integrl cn t be evlute. If the FT oe eit then there i n invere FT reltionhip: iπ F { G } G e Uniquene: Given the eitence of the invere FT it follow tht if the FT eit it ut be unique. Tht i for function for unique pir with it FT: G Cvet. An eception to the uniquene propert i cl of function clle le or null function. An eple i the continuou function f {. Thi function n other like it hve the e Fourier trnfor f : F. Thu the uniquene eit onl for function plu or inu rbitrr null function. In prctice thee function re not relizble energle n thu for the purpoe of thi cl we will ue tht the FT i unique. Alternte FT Definition. In the bove erivtion the i the frequenc preter. There i nother coon FT efinition tht ue rin frequenc preter ω: iω G ω F{ } e G ω / π with n invere FT of: g F G ω ω G ω e i { } ω π Unit. If h unit of Q then will hve unit of ccle/q or Q -. Plee note tht uner our efinition of the FT thi i not n ngulr frequenc with unit of rin/q but jut plin Q -. Plee lo keep in in tht i the ine of vrition for eple we cn hve repreent velocit tht vrie function of ptil loction. The function h unit c/ but h unit c n G h unit of c/ but h unit of c -. Eple: Tie econ Ditnce c Teporl Frequenc - Hz ccle/ Sptil Frequenc c - ccle/c

2 BE/EECS 56 4 FT Note Setr Definition. We firt ecopoe oe function in to even n o coponent e n o repectivel follow: e o [ + ] [ ] thu e + o n e e n o o A function i Heritin Setric Conjugte Setric if: Re{ } e n I{ } o thu e + io g * Setr Propertie of the FT. There re everl relte propertie:. If i rel then G i Heritin etric e.g. G G*-.. If i rel n even G i rel n even. 3. If i rel n o G i iginr n o. 4. If i rel G cn be efine trictl b non-negtive frequencie. 5. If i iginr then G i Anti-Heritin etric e.g. G -G*-. Proof of. G e iπ [ e + o [coπ i in π] co i evenin i o e co π + e + o i o co π i o i e in π i e o in π E + i io co i even in in i o i - o E io E + io G * Q.E.D. Coent. One intereting conequence of the etr propertie i tht if i rel the onl one-hlf of the Fourier trnfor i necer to pecif the function thi follow fro propert. bove. ore pecificll i trictl eterine b G for ll non-negtive frequencie.

3 BE/EECS 56 4 FT Note 3 Coent on negtive frequencie. Conier rel-vlue ignl igine voltge on wire or the oun preure gint our erru the Fourier trnfor of thee i copletel pecifie b the non-negtive frequencie e.g. G- G*. We cn rgue tht we hve the concept of frequenc ocilltion/econ but it oen t rell ke phicl ene to tlk bout poitive or negtive frequencie. In thi ce we coul rgue tht the hving poitive n negtive frequencie i erel theticl convenience. Are there ce where negtive frequencie hve ening? Conier the bit in rill it cn turn clockwie or counter clockwie n ifferent rottionl rte. Here poitive n negtive frequencie hve phicl ening the irection of rottion. A we hll ee there re ce in eicl iging where thi itinction i iportnt for eple the gnetic oent in RI i ce where the ign inicte the irection of preceion. Convolution Definition. The convolution opertor i efine : The convolution opertor coute: g * h h - g * h h h h * - The elt function δ. The Dirc elt or iplue function i theticl contruct tht i infinitel high in plitue infinitel hort in urtion n h unit re: δ { n δ ot propertie of δ cn eit onl in liiting ce e.g. equence of function δ or uner n integrl. Soe iportnt propertie of δ : g n - δ with continuou t δ with continuou t δ with continuou t F{ δ } Delt function propertie. Firt two re technicll onl efine uner the integrl but we ll till tlk bout the. Siilrit tretching δ δ Prouct/Sifting δ δ Sifting δ 3

4 BE/EECS 56 4 FT Note 4 Convolution * δ δ * * δ δ * Fourier Trnfor Theore. There re n Fourier trnfor propertie n theore. Thi i prtil lit. Aue tht F { } G F { h } H n tht n b re contnt: Linerit F { + bh } G + bh Siilrit tretching Shift F { } G F{ } G e iπ Convolution F { * h } G H Prouct F { h } G * H Cople oultion F i π g e } G { oultion F{ coπ } [ G + G + ] Rleigh Power F{ inπ } G h * G [ G G ] i + Cro Power g H * Ai Reverl F { } G * Cople Conjugtion F{ g * } G * Autocorreltion F { * } G G * G Revere Reltionhip F{ G } Differentition { } iπg F oent i F{ } G π DC Vlue G Soe coon FT pir: g G δ δ coπ [ δ + δ + ] inπ rect { inc tringle { < < [ δ δ ] i + inπ inc π rect inc 4

5 BE/EECS 56 4 FT Note 5 gn { e π e π < iπ e + π e for > ; otherwie iπ + π J π cob rect / π cob The cob function cob. The pling or cob function i trin of elt function: n cob δ n The Fourier trnfor of cob i: F { cob } cob Proof. iπn F{ cob } F δ n e F n n The RHS of the bove epreion cn be viewe the eponentil Fourier erie repreenttion of perioic function F with perio n α n for ll n. Recll the Fourier erie epreion re: F iπn α e whereα F e. n n iπn Now let G rectf one perio of F n thu α n F e iπn G e n iπn F G. Now oberve tht F { G } One function tht tifie thi reltionhip i G δ. Thu one poible Fourier trnfor of cob i: F δ cob B uniquene of the Fourier trnfor thi i the unique Fourier trnfor of cob. n 5

6 BE/EECS 56 4 FT Note 6 Spling n repliction b cob. The cob function cn be ue to ple or etrct vlue of continuou function. Spling with perio cn be one : n δ n δ n n n. cob δ n n B the tretching n ifting propertie of the elt function. A function cn be replicte with perio b convolving with cob function: n [ * δ n ] [ * n ] *cob δ n B the tretching n convolution propertie of the elt function. n n Spling Theor. When nipulting rel object in coputer we ut firt ple the continuou oin object into icretize verion tht the coputer cn hnle. A ecribe bove we cn ple function t frequenc f / uing the cob function: n g cob n δ n. The Fourier trnfor i: G G * cob G * G * δ G f δ f Thu pling in one oin le to repliction of the pectru in the other oin. The pectru i perioic with perio f. Tpicll onl frequencie le thn f / cn be repreente in the icrete oin ignl. An coponent tht lie outie of thi pectrl region f f / reult in liing the i-ignent of pectrl infortion. / G Originl Spectru G Replicte Spectr -f/ Aliing f/ 6

7 BE/EECS 56 4 FT Note 7 The Whittker-Shnnon pling theore tte tht bn liite function with iu frequenc cn be full repreente b icrete tie equivlent provie the pling frequenc tifie the Nquit pling criterion: f If thi i the ce then the originl pectru cn be etrcte b filtering n b uniquene of the FT the originl ignl cn be recontructe. To recontruct the originl ignl we ppl recontruction filter H rect / f rect : Gˆ G H G rect G if there i no liing In the oin thi reult in inc interpoltion: ˆ g * inc n n δ n *inc n inc If the Nquit criterion i et then g ˆ. n n gn Info fro ll ple contribute to thi point D Liner Ste Conier te S[. ] with n input function f n n output or repone function S[f]. Thi te i liner if n onl if: S [ α f + βf ] αs[ f ] + βs[ f ] αg + βg for ll α β f n f. ore generll the uperpoition of n rbitrr et of input function will iel net repone tht i the uperpoition of repone to ech input function lone. 7

8 BE/EECS 56 4 FT Note 8 Aitionll if n input i cle e.g. b α then the output will lo be cle b the e ount. Be on the ifting propertie of the elt function we know tht f f δ f * δ which i the uperpoition of n infinite nuber of weighte n hifte elt function. Be on linerit the output of thi te S[f] i: g S f δ f S[ δ ] The te operte on function of n f i contnt cling fctor. S[δ- ] i pecil function know the ipule repone n cn i efine : [ δ ] h ; S i the repone to n ipule locte t n g f h ; i known the uperpoition integrl. Thi repreenttion for the output i vli for n liner te. Now conier te tht i hift invrint or tie invrint for function of tie. We efine te being hift invrint if n onl if: - S[f-] for ll g n. For liner hift invrint te the ipule repone cn be written : The uperpoition integrl then becoe: [ δ ] h ; h h ; S f h f * h the convolution of the input function with the ipule repone h. For liner hift invrint te or liner tie-invrint te onl we cn then conier the Fourier oin equivlent: G FH Where H i known the trnfer function or te pectrl repone. 8

9 BE/EECS 56 4 FT Note 9 Note on the D Fourier Trnfor Definition. The D Fourier Trnfor FT relte function to it frequenc oin equivlent. The FT of function i efine b the D Fourier integrl: G u v There i lo n invere FT reltionhip: i π u+ v F{ } e i π u+ v F { G u v} G u. v e uv Uniquene: Given the eitence of the invere FT it follow tht if the FT eit it ut be unique. Tht i for function for unique pir with it FT: G u v D FT in Polr Coorinte. We conier pecil ce where the functionl for of i eprble in polr coorinte tht i rθ g R rg Θ θ. Since g Θ θ i perioic in θ it h Fourier erie repreenttion: It cn be hown tht F inθ Θ θ n e. n g inθ n inφ { g r e } i e D R πg R r J n πrρ rr where the prt uner the integrl in known the Hnkel trnfor of orer n n J n i the n th orer Beel function of the firt kin: π i inϕ nϕ J n e ϕ. π π iπ u+ v iπrρ co θ φ Derivtion of the Hnkel trnfor reltionhip relie on e e. Thu the D FT in polr for i: n inφ { g R r gθ θ } n i e G ρ φ F πg r J πrρ rr n For the pecil ce of circulr etr of g tht i rθ g R r then: G ρ φ G ρ π g R r J πrρ rr which i lo circulrl etric function. The invere reltionhip i the e: g R r π G ρ J πrρ ρρ R n Setr Propertie of the FT. If i rel then Guv i Heritin Setric tht i Guv G*-u-v. If i rel n even tht i -- then Guv i lo rel n even. Finll ecribe bove if we hve rel n circulrl etric function rθ g R r the Gρφ Gρ rel n circulrl etric function. 9

10 BE/EECS 56 4 FT Note The elt function δ. The elt function in two i equl the to prouct of two D elt function δ δ δ. In nner iilr to the D elt function the D elt function h the following efinition: n δ { n δ otherwie ot propertie of δ cn be erive fro the D elt function. There i lo polr coorinte verion of the D elt function: δ δ r / πr. Fourier Trnfor Theore. Let n b re non-zero contnt n F{} Guv n F{h} Huv. Linerit F { + bh } G u v + bh u v gnifiction F { g b } u v iπ u+ vb Shift { } b G F b G u v e iπ + b Cople oultion F{ e } G u v b Convolution/ultipliction { ** h } b ** h h F G u v H u v Seprbilit F { h } G u v ** H u v g g F{ } F D { g } F D { g } Power G u G v G u v uv h * Ai Reverl F { } G * u v Soe coon D FT pir: g G u v δ u v δ b i π u+ e δ vb πr e co π π e e coπ e πρ e G u v H * u v uv π u πv e π [ δ u + δ u + ] δ v rect rect δ uinc v rect rect b u v inc inc r Jπρ circ r { jinc ρ r > ρ b b

11 BE/EECS 56 4 FT Note r cob cobcob ρ cobuv cobucobv The cob function in D cob. The D pling or cob function i efine cobcobcob n h the D FT F{cob} cobuv. Forll the D cob function i efine : n cob δ n In nner iilr to the D ce we cn prove tht Fourier trnfor of the D cob function i lo D cob function given in the bove tble. Spling Theor in D. In nner iilr to pling in D pling in D cn be oele ultipling function tie the D cob function. With ple pcing of n in the n irection the ple function i: g cob δ n n n δ n n δ n n The icrete oin equivlent i g n n g n. In the Fourier oin the reult i: G u v G u v ** cob u v G u v ** n n G u δ u Thu pling in one oin le to repliction of the pectru in the other oin. Spcing of the replicte pectr i //. The Whittker-Shnnon pling theore in D tte tht bn liite function with iu frequencie n cn be full repreente b icrete tie equivlent provie the pling frequenc tifie the Nquit pling criterion: n n n v Uner thee circutnce there i no pectrl overlp or liing the originl pectru n b uniquene of the FT the originl ignl cn be recontructe. v

12 BE/EECS 56 4 FT Note To recontruct the originl ignl we ppl recontruction filter H u v rect urect v. Gˆ u v G u v H u v G u vrect urect v G u v if there i no liing In the oin thi correpon to inc interpoltion in D inc gˆ g ** inc inc inπ : π n δ n n **inc n inc n inc n n inc inc n g n inc The lt line eontrte how the originl continuou ignl cn be retrieve fro the icrete ple verion of.

13 BE/EECS 56 4 FT Note 3 Eple of Fourier Trnfor: D t in Low ptil freq High ptil freq ige oin t ige oin t ige oin D t in Low ptil freq High ptil freq Fourier oin t Fourier oin t Fourier oin 3

14 BE/EECS 56 4 FT Note 4 4

15 BE/EECS 56 4 FT Note 5 5

16 BE/EECS 56 4 FT Note 6 rectrec Guv incuincv cling gnifiction propert cling gnifiction propert hifting propert oultion Ige AbFourier RelFourier IFourier 6

17 BE/EECS 56 4 FT Note 7 incinc Guv recturectv pling pttern with In the Fourier trnfor we hve the repliction pttern with pcing / / pling pttern with < In the Fourier trnfor we hve the repliction pttern with pcing / > / pling pttern with << Thi h liing in the v irection Ige Dt Fourier Dt 7

18 BE/EECS 56 4 FT Note 8 D Liner Ste Conier n iging te with n input ige I tht goe through oe te S[. ] n prouce n output ige I e.g. I S[I ]. Severl propertie tht we re interete in re: Linerit which two prt uperpoition n cling. Thu te i liner if n onl if: S[ α + βh ] αs[ ] + βs[ h ] For liner te we cn efine n ipule repone the output of te for n ipule locte t poition : h ; S[ δ ] n in generl we cn efine the output given oe input ige I uing the uperpoition integrl: I I h ; Spce Invrince. A te i pce invrint if n onl if: I --b S[I --b] for ll b n I. Alterntel te i pce invrint if n onl if the ipule repone cn be epree in ter of the hift of the elt function: h S[ δ ]. Thu h S[ δ ] i ll tht i neee to pecif the te. The uperpoition integrl becoe: I I h I ** h where ** inicte D convolution. 8

19 BE/EECS 56 4 FT Note 9 Eple of D Iging Ste. Here we conier pinhole iging te: I thi te liner? Auing tht the perture i open or cloe n tht light lw trvel in tright line through the hole then e the te i liner. We houl then be ble to eterine the ipule repone. Let firt conier two ifferent gnifiction fctor one for the object n one for the pinhole perture. For object gnifiction we igine the pinhole in infinitel ll: For hift of in the input we will get hift of input ource gnifiction ter b. b in the output plne. Thu we efine n Conier then te for elt function t poition : h ; S[ δ ] Cδ where the elt function pper cle b C n t loction. I thi te pce invrint? No the bove epreion cnnot be written function of n. Now given n input ige I we cn till eterine the output ige uing the uperpoition integrl reeber the te i till liner: 9

20 BE/EECS 56 4 FT Note ; I C I C I C h I I δ δ The output i cle n gnifie verion of the input ige. For the pinhole gnifiction igine tht the pinhole h riu of R then the riu in output plne woul be R + b ieling n perture gnifiction function of b +. Now uppoe we h oe perture function thi will now be gnifie in the output plne. Thu the ipule repone will tke on for iilr to: ; C h n the ouput ige will tke on the for: ** I C I C I C I Here the output ige i the convolution of the cle n gnifie verion of the input ige n the pinhole function. Though thi te i not pce invrint we were cn till ble to write the output in the for of convolution though not convolution of the input ige but with gnifie verion of the input ige.