Hilbert Transforms, Analytic Functions, and Analytic Signals

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Allan Hutchinson
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1 Hilbrt Transorms Analtic Fnctions and Analtic Signals Cla S. Trnr 3//5 V. Introdction: Hilbrt transorms ar ssntial in ndrstanding man modrn modlation mthods. Ths transorms ctil phas shit a nction b 9 dgrs indpndnt o rqnc. O cors practical implmntations ha limitations. For ampl th phas shiting o a low rqnc implis a long dla which in trn implis a comptational procss that maintains a long histor o th signal. Hilbrt transorms ar sl in crating signals with on sidd Forir transorms. Also th concpts o analtic nctions and analtic signals will b shown to b rlatd throgh Hilbrt transorms. 9 Dgr Phas Shitrs: W will tak a spctral approach and start with an idal 9-dgr phas shitr. To do this w wold lik a nction that will transorm a sinsoid to anothr sinsoid with th sam amplitd and rqnc bt simpl phas shitd b 9 dgrs. So rcalling th trigonomtric idntitis: Thn w dsir or transorm to do th ollowing: Cos t 9 Sin Sin t 9 Cos Cos Sin Sin Cos Ths two conditions will also cas a sinsoid with arbitrar phas to b shitd 9 dgrs. This ollows sinc it ma b dcomposd into a sin and a cosin and ach o ths componnts will b shitd b 9 dgrs. To show this start with a sinsoid o arbitrar phas θ and dcompos it into a sin and a cosin componnts. Cos t θ Cos Cos θ Sin Sin θ 3 Nt transorm phas shi th two componnts and rdc. Cos t 9 Cos θ Sin t 9 Sin θ Sin Cos θ Cos Sin θ Sin t θ Cos t θ 9 4

2 So w can s th ct o th transormation is a 9-dgr phas shit rgardlss o th original phas. W also dsir th transorm to b prssibl as a linar conoltion. Ths th transorm will also ob sprposition. Th linarit proprt was sd abo in th arbitrar phas cas. B modling or transorm this wa w can thn s a powrl thorm rom Forir analsis that qats conoltion in on domain to mltiplication in th othr. So lt s look at th transorm in th rqnc domain. Rcalling th Forir pairs or sinsoids: Cos ω Sin ω δ ω ω δ ω ω / 5 δ ω ω δ ω ω / Thn or rqnc domain phas shit rqirmnts bcom atr cancling ot th twos and moing or: [ δ ω ω δ ω ω ] ω [ δ ω ω δ ω ω ] [ δ ω ω δ ω ω ] H ω [ δ ω ω δ ω ω ] H 6 Rcalling th Dirac Dlta nction is a distribtion whos non-ro spport is or an ininitl narrow rgion w thn onl ha to compar 4 points to sol this st o qations. It is hlpl to look at positi rqncis sparatl rom th ngati ons. Doing this w onl compar two points at a tim. So or ω ω > w ind δ H ω δ or simpl H ω. Likwis or ω ω < w ind H ω. Finall or ω ω w ind δ H hnc H. Atr ptting ths cass togthr w ind th rqnc domain soltion or th 9- dgr phas shitr is: Th Signm nction is dind as: H ω Sgn ω 7 Sgn ω ω > ω ω < 8

3 Hilbrt Transorms Analtic Fnctions and Analtic Signals Now w nd th tim domain rsion o th phas shitr so w can prss th phas shitr as a conoltion. This is achid st b inding th inrs Forir transorm o H ω. Th rslt is: ω ωt t ωt h sgn ω dω dω π π π dω π t 9 So whn w wish to phas shit a nction or ampl b 9 dgrs w st conol it with / π. Rcalling th standard intgral orm or th conoltion o two nctions h. ξ h t ξ dξ Sinc conoltion is commtati w also ha: h ξ t ξ dξ So in trms o conoltion or phas shitr has th ollowing intgral orms ξ t ξ dξ dξ π t ξ π ξ Now that w ha a dinition or th orward Hilbrt transorm w ind th dinition or th inrs transorm. Th inrs Hilbrt transorm ma b thoght o as Hilbrt transorming a nction thr tims. I.. shiting 7 dgrs is th sam as shiting ngati 9 dgrs. Two shits rslt in a 8-dgr phas shit which is simpl ngation. So a ngati 9-dgr phas shit is simpl th ngation o a 9-dgr phas shit! So or th inrs Hilbrt transorm w st conol with - / π. Hilbrt Transorm Dinitions: Som paprs start b dining a Hilbrt transorm is as an Intgral transorm. Sinc w startd rom a phas shitr point o iw and thn w modld it as a conoltion w immdiatl ha two rprsntations or th dirct and inrs transorms. O cors an lmntar chang o ariabl conrts on rprsntation into th othr. 3//5 Pag 3 o 3

4 Hilbrt Transorms Analtic Fnctions and Analtic Signals Gin a nction thn its Hilbrt transorm is: ξ t ξ P dξ P dξ π t ξ π ξ 3 Th inrs Hilbrt transorm is likwis similarl dind. ξ t ξ P dξ P dξ π t ξ π ξ 4 Sinc th basic intgrals ar impropr ths ar to b alatd as Cach Principal Val CPV Intgrals. This implis a carl limiting procss takn smmtricall abot th singlarit which rslts in act cancllation. This basicall mans or th st ariant o th transorm: P ε R ξ ξ ξ 5 dξ Lim ξ t R R ε ξ t dξ ξ t ε dξ ξ t Hr psilon is an incrasingl small distanc rom th singlarit. A similar limiting procss is also sd with th nd orm. Sinc intgration is a linar opration w s that Hilbrt transormation is also a linar opration. This mans an nction pandd into a sm o sinsoids can b asil Hilbrt transormd b doing th appropriat 9-dgr phas shits on ach o th componnts. Unlik othr tps o transorms Hilbrt transorms la th nction in th sam domain as th original th Hilbrt transorm o a tmporal nction is itsl tmporal. An ampl o alating a CPV intgral Lt s alat: P d 6 So sing th limit approach w ind: P Lim Lim Lim Lim d ε ε d ε d ε ε d d d d ε ε ε ε ε 7 3//5 Pag 4 o 3

5 Hilbrt Transorms Analtic Fnctions and Analtic Signals A chang o ariabl was mad in th middl stp. A simpl ampl o inding a Hilbrt transorm ia conoltion: Lt s ind th Hilbrt transorm o Cos. So insrting Cos into th nd orm o th Hilbrt transorm intgral w obtain: Cos t ξ P dξ π ξ 8 Using th trigonomtric idntit or th cosin o a sm o angls w now ind: Cos t ξ Cos ξ Sin ξ P dξ Cos Sin dξ π ξ π ξ ξ 9 Now sing or aorit intgral tabl or an adroit intgration tchniq [s appndi A] obtain th ollowing idntitis. So now th soltion is as. Cos ξ dξ ξ Sin ξ dξ π ξ Cos ξ Sin ξ Cos dξ Sin dξ Sin π ξ π ξ Ths w ind Sin. Hnc th Hilbrt transorm o Cos is Sin. Likwis a similar procss ma b sd to ind th Hilbrt transorm o Sin is Cos. So this riis that conoltion with th krnl π dos indd prorm a 9-dgr phas shit opration. 3//5 Pag 5 o 3

6 Hilbrt Transorms Analtic Fnctions and Analtic Signals 3//5 Pag 6 o 3 Analtic Fnctions Th thor o Hilbrt transormation is intimatl connctd with compl analsis so w will now look at a rqirmnt or isotropic dirntiation o compl nctions. W will s th standard paradigm o sing a compl ariabl composd o two ral componnts combind in th ollowing wa: Hr and ar ral als and rprsnts. Similarl w can compos a compl ald nction b combining two ral ald nctions also standard: 3 Nt w will tili th standard dinition o a driati rom ral analsis and tnd it to th compl cas. This driati limit is: Lim ' 4 A diiclt with th dinition is th rslt ma dpnd on th dirction that takn b hnc w wold lik to know ndr what conditions th driati is path indpndnt. So w will tak two orthogonal paths and rqir th rslting limits to b th sam. This will ild path indpndnc and ha a driati dinition that gis consistnt rslts. Rcall. So w will irst lt and mo along th -ais. Lim ' 5 Nt will mo along th -ais and lt. Lim ' 6

7 Hilbrt Transorms Analtic Fnctions and Analtic Signals 3//5 Pag 7 o 3 Th rqirmnt that th two limits ar to b th sam rslts in th Cach-Rimann rlations. Ths ar ond b stting th two limits qal to ach othr and qating th ral and imaginar parts. Th C-R rlations ar: 7 I a compl nction obs th C-R rlations and has dirntiabl componnts it is said to b analtic. A nction ma b analtic or st part o its domain. Howr a nction can t b analtic at st a singl point i it is analtic at a point thn it also mst b analtic in an opn nighborhood arond that point. Sinc w composd or compl nction b smming togthr two nctions on prl ral and th othr prl imaginar it wold sm that or compl nction has two dgrs o rdom. Howr th nction s bing analtic crats an intrsting rstriction. I w rcall that and and thn lt s look at th driati with rspct to. So sing th chain rl w ind th ollowing pansion: 8 Nt s 9 So or driati now has th orm: 3 I th nction obs th C-R rlations thn th partial driati is simpl ro! So in a sns analtic nctions ar indpndnt o. Anothr nat qalit o analtic nctions is th componnts and ach sol Laplac s qation. Spciicall and. Ths ollow dirctl rom dirntiating th C-R rlations. Starting with th C-R rlations:

8 Hilbrt Transorms Analtic Fnctions and Analtic Signals and 3 Atr dirntiating th irst rlation with rspct to and th scond rlation with rspct to w ind: 3 and Now qating th mid partials w ind so w s that Laplac s qation is satisid. A similar procss can b applid to mak th cas or. Sinc both componnts sol Laplac s qation thir sm will also so w ha th ollowing nat thorm or analtic nctions: 33 In or proo o this w assmd th qialnc o mid partials and th istnc o highr ordr driatis. In adancd tts on compl analsis ths will b shown to b alwas tr or analtic nctions. An Analtic Fnction Eampl Eampl: Show ind: is analtic. Using th ponnt addition rl and Elr s idntit w Cos Sin Cos Sin 34 So w ha: Cos Sin 35 Now ind th or partial driatis: Sin Sin Cos Cos 36 Plgging ths into th C-R rlations on inds th ar obd or all and. Sch a nction is analtic rwhr and is somtims calld an ntir nction. 3//5 Pag 8 o 3

9 Hilbrt Transorms Analtic Fnctions and Analtic Signals Analtic Signals Rlatd to th concpt o analtic nctions is th ida o an analtic signal. On wa to mak on is b alating an analtic nction along on o th as. For or ampl nction I will s th ral ais. Or abo analtic nction has th corrsponding analtic signal: Cos Sin 37 Th choic o ais dpnds on th analtic nction. I th wrong choic is mad th rslt will not b an analtic signal. How to chos which ais will b mad clar shortl. Analtic signals ha sral proprtis that pro important in signal procssing. Th irst proprt is analtic signals ha on-sidd Forir transorms. A scond proprt is analtic signals ob a gnraliation o Elr s idntit. A third proprt is th analtic signal s imaginar portion is th Hilbrt transorm o its ral portion. Assming or third proprt is tr thn gin an analtic signal ψ t its Forir transorm is Ψ ω U ω V ω. Nt s th spctral orm o th Hilbrt transormation to ind: Ψ ω U ω Sgn ω U ω Sgn ω U ω 38 And w can asil s that Ψ ω whn ω <. So this signal has no ngati rqnc componnts. Likwis starting with th congat ψ w ind its spctral rprsntation to b Ψ ω Sgn ω U ω which has no positi rqnc componnts. Th componnts o analtic signals also ob a gnraliation o Elr s idntit. For ampl: ψ ψ ψ ψ 39 And o cors: ψ t ψ 4 wt For Elr s idntit st lt ψ Cos ω Sin ω. 3//5 Pag 9 o 3

10 Hilbrt Transorms Analtic Fnctions and Analtic Signals To show th Hilbrt transorm proprt o analtic signals w will ha to alat a not so obios intgral. W want to alat th ollowing intgral or th 4-pic contor shown in igr : ψ 4 d τ C Figr Th or pics ar: th otr smicircl with radis R th lt horiontal lin sgmnt th right horiontal lin sgmnt and th innr smicircl with radis psilon. Sinc w ar assming ψ is analtic on and insid o c and th path taks a dtor arond th singlarit th Cach-Gorsat thorm sas this qals ro. So nt w pand this intgral into 4 path intgrals whos sm is ro. Ths w ha: ψ ψ ψ ψ d τ τ τ τ d d d A ε R R ε B 4 Sinc w will look at th limiting cas whr th limits ar sch that th otr smicircl bcoms ininitl larg and th innr smicircl likwis bcoms ininitl small w ar abl to combin th middl two intgrals into a CPV intgral. Hnc d P d A ψ ψ ψ d τ τ τ B 43 Earlir I mntiond how th choic o ais mattrs. This is whr w choos th ais so that Jordan s Lmma ma b applid to th st intgral to mak it ro. For som analtic nctions w will s th ais instad o th ais. Th last intgral somtims calld 3//5 Pag o 3

11 Hilbrt Transorms Analtic Fnctions and Analtic Signals a dtor intgral sinc it is sd to hop arond th singlarit alats to πψ τ. This intgral is ond b sing a ariation o Cach s intgral ormla whr in th limit o th path s radis going to ro th intgral s al is θψ τ whr θ is th angl masrd ccw sbtndd b th path rlati to th singlarit. So plgging in ths two rslts w ind: P ψ d τ πψ τ 44 Now rcalling ψ w ind: P d τ P d π τ τ τ 45 So b splitting this ot into two ral ald qations w ind: τ P π τ P π d τ d τ 46 Ths ar simpl th Hilbrt transorm rlations btwn τ and τ. Appndi A: Intgration tricks Itratd Intgration b Parts Lt s sa w wish to intgrat d. Sinc this is an intgral o a prodct o nctions w know that intgration b parts wold b th standard approach. Howr with this ampl w will ha to do this mltipl tims. Thr is an approach calld itratd intgration b parts that is asil applid. And th mthod is actall asir to rmmbr. Basicall w sparat th intgrand into two parts that bcom th hadings o two colmns. Each ntr in th lt colmn is ormd b taking th driati o th ntr right abo it. Similarl ach ntr in th right colmn is mad b taking th intgral o th ntr right abo it. Usall th colmns ar illd ntil th bottom lt hand ntr is 3//5 Pag o 3

12 Hilbrt Transorms Analtic Fnctions and Analtic Signals ro. Th intgral is mad p o th sm o diagonal ppr lt to lowr righ prodcts with altrnating sign pls th intgral o th prodct o th ntris on th bottom row takn with a contind altrnating sign. I th rows ar illd in ntil th bottom row is ro thn this last intgral is not ndd. An ampl will mak this itratd procss clar. For d w bild th ollowing tabl: So now combining trms with altrnating sign w ind: d d 4 C 48 So on can crtainl s this mthod s icac. Sinc this ampl is on whr th lt hand trm is a polnomial th rows ar illd in ntil th bottom row s prodct is ro. Howr som intgrals ha trms that will not go awa rgardlss o th nmbr o s tims a nction is dirntiatd. For ampl lt s ind: Sin d so w bild th tabl lik this: s s Sin 49 s s s Cos Sin So now w ind: s s s Cos s Sin s Sin s Sin d d C 5 This is on o th cass whr th intgration procss taks o arond a loop and nds p almost whr o bgan. Ecpt th bottom intgral works ot to b an indpndnt nction tims th original intgral. So simpliing th algbra w ind: 3//5 Pag o 3

13 Hilbrt Transorms Analtic Fnctions and Analtic Signals s s s Sin d Cos s Sin C 5 Which whn th intgral is ll rsold ilds: Cos s Sin C s 5 s Sin d s W will soon ha nd or th rlatd smi-ininit dinit intgral which w can asil alat sing th abo rslt. Sin s d s 53 / Sbstittion Now lt s look at a mthod or alating th intgral o a sinc nction. Spciicall Sin d. First raliing th intgrand is n w can rdc it to a smi-ininit intral and thn prorm th ollowing trick sbstittion. s how to do th intgral. st ds. So w can now s d d Sin dds Sin Sin s ds Tan s ] π 54 3//5 Pag 3 o 3

Computer Vision. Fourier Analysis. Computer Science Tripos Part II. Dr Christopher Town. Fourier Analysis. Fourier Analysis

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