# Hilbert Transform Relations

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2 ( () (vp ( ) vp d Here vp symbols deoe Cauchy pricipal value of he iegral ha follows his iegral defies liear coiuous form of, herefore vp / is а disribuio (or geeralized fucio) [4] hese disribuios repeaedly are applied i quaum mechaics: () vp, j v vp j v hey are he Fourier rasforms of he ui sep Y( [4] ad is mirror owards he ordiae axis Y( = Y( (he operaor reverses direcio of he ime, ad δ is Dirac dela fucio) If F is he Fourier operaor ( =, frequecy), i is well-ow ha F j sg( v), (3), sg( ),,, Y( Y( Y( ( sg( ) A direc prove of he firs equaio could be doe applyig Lobachevsi iegral [5], or periodizaig /: (4) 4 I ( a) F si( a d sg( a) j v e d j si( d sg( a), si(v ) d I (v) j sg( v) j Hilber rasform ca be defied as a covoluio of he sigal x( ad vp / ( ) (5) ( ( )) vp ( ) x x x d he Paley-Wieer codiio [6] is ecessary ad sufficie for exisig of Hilber rasform, ad he reverse rasform is give by III Coiuous case (sigals o = R/Z) Le x( is such a fucio, ha x( = x(+т), where is a real umber he so-called period he umber is always period; a umber opposed o he period is period oo

5 fucios For he coiuous case whe R (or i discree case, whe Z) hese are he auomorphisms: ρ: x( x( ), σ: x( x ( I () l, is he Kroecer s symbol Les he dimesio of he sigals ( vecors ) space is a eve umber, ad he discree dela (vecor) of Dirac has he form: δ [,,,, ] I ha case he sig vecor aalog of sg(v) from (4), has he form / () s ( ρ ) ρ δ For = 8 his vecor loos lie his: / Т s [,,,,,,, ] 8 If is odd he middle zero will disappears As i he coiuous case, whe σ sg( ν) sg( ν) sg( ν), his vecor is odd, ie σ s s he discree Fourier operaor has he form l j l (3) F e ρ δ δ ρ,l his operaor is uiary, ie Hermiia-cojugaed coicide wih is iverse oe [9]: F F * = herefore from () ad of hese depedeces (hey are demosraed i [8]; modulaio operaor? is defied afer (5), ad f is he -h colum of he discree Fourier operaor from (3)), F ρ? F; F δ ;? f, oe ca obai: * κ F ( j s ) cg ( ( )) ρ δ /, (4) κ ( ρ) cg ( ( ) ) ρ / he firs row of (4) is he impulse respose, ad he secod oe is he cyclic discree edomorphism (sysem fucio) of Hilber (a ideal cyclic Hilber rasformer or 9 degree phase shifer), ha is aisymmeric ad (ai-) commue wih σ, ie κ κ σ κ σ σ κ κ σ σ κ κ σ he magiude respose of his filer for = 6 is give o Figs ad he 8 verical lies of he grid are draw rough he pois wih abscises { /6}, for which he value is exacly db, ad he magiude respose is pure imagiary For he oher frequecies deviaios are big ad a real compoe appears he same behavior is, as i is show i [3], of he aalyzig filers of he Fas Fourier rasform (FF) Whe desigig of Hilber rasformers, he objecive is a equiripple 7

6 approximaio of he sig-fucio [, 3] Applyig of he (cyclic) FF wih such bad filers demosraes, ha his approach is o always obligaory I is of ieres he edomorphism (), ie double applyig of a Hilber filer Direc evaluaig from (4) seems isuperable he covoluio of he impulse resposes of wo serial filers ad he formula for he -h co-ordiae of a covoluio, derived i [8] gives us κ κ κ, κ ( κ ρ σ κ) ( F κ F ρ σ κ) (here ( а b ) is a ier produc of wo vecors [9, ]), / / (5) κ ( ρ) ρ?? / I previous equaio? diag(, w, w,,w ); w e, is he meioed before modulaio operaor [8] ad is he vecor of all s From i follows several impora coclusios: I) he operaor () is orhogoal projecor, as i is symmeric ad ( ())( ()) = () II) From (5) follows ha 3 () = (); he wo vecors, / are mapped from io he zero vecor (i frequecy area i s obvious) III) he geeralize iverse edomorphism of is ; he pseudoiverse of could be received if oe ca ae io cosideraio I и II ad ha he pseudoiverse of a orhogoal projecor P + co-iside wih he same projecor P []: 3 (6) κ ( ) ( ) ( ) ( )( ) IV) Le ( κ) { z:z κ( x)} is he rage of he edomorphism ha is a liear sub-space [9, ] I s well ow ha + is a orhogoal projecor io his subspace, herefore his projecor co-iside wih () Cosequely we have for he dimesios of his subspace: (7) dim ( ) dim ( ) r( ) From (5) follows, ha he race of he projecor is Тr( κ ) he erel [9] Ker() is o oly he zero vecor ad his gives a reaso o be see as a edomorphism bu o as a rasform, ha will require i o be a auomorphisms IV Eigevecors ad eigevalues of he Hilber edomorphism is preseed i he caoical bases { ρ δ } from he circula marix (4), ha s why is eigevecors co-iside wih he Fourier rasforms colums [8]: (8) κ F F Λ; F κ Λ F; κ κ; F F j 8

8 Here Y is he discree ui sep; is zero ad / compoes are? (he symbol deoes compoewise muliplicaio of wo vecors or Schur s muliplicaio) Тhe filer + j exracs, i coiuous case, he upper sidebad, bu i he discree oe his is o eirely he same, maily because of he form of Y I ca be desiged a his oe filer: β ρ ( κ ) j κ; β ; β * β he auomorphisms β is a ivoluio (is square is ideiy) ad i is a Hermiia morphism ie coicide wih is Hermiia-cojugaed I ha case for he wo orhogoal projecors (+β)/ ad ( β)/ i will be rue ha: β F diag (, ) F, β F diag (, ) F he firs projecor cus off he upper / co-ordiaes of he specrum of a sigal, ad he secod oe he lower / co-ordiaes I ca be show ha hese wo filers paricipae i cosrucig of he full recursive form of FF [8, 3] he real filer +, i coras o, is a auomorphisms, ie here exiss iverse oe, which permis recosrucio of he ipu sigal he iverse filer is ( ) ( ) Every oe of he orhogoal sigals x ad x ca be exraced wih his filer from he mixure of hem he mos impora propery of he Fourier rasform is, as i is well ow, ha i rasforms covoluio io muliplicaio ad vice-versa muliplicaio io covoluio [4] For he sigals o Z Z his propery loos lie: F ( x y) F x F y, F ( x y) F x F y Le c F c, s are he -h colums of C ad S from () I s easy o be show ha ( ρ ρ ) δ; F s ( ρ j ρ ) δ; ( ρ Applyig of hese relaios ad formig of he real sigal z c x κ ( c ) κ ( x) c x s κ ( x), gives (Y is he discree ui sep): F z ( ρ (( σ Y ) X ) ρ ( Y X )) δ) x ρ hese formulae represe discree varia of he well-ow scheme of Harley for modulaio wih a sigle sidebad x

9 V Discree sigals (sigals o Z) he Hilber edomorphism could be obaied for sigals o Z from he former case he las row of (4) ca be represe i he form (we assume ha is divisible of 4; his does decrease geeraliy): κ( ρ) / 4 cg ( ( )) ρ / 4 / cg ( ( ) ) ρ If we chage i he secod sum he variable ( ), i ca be obaied he form (oe ca see he ocausaliy ad ai-symmery of he Hilber s filer): ( ) () κ ( ρ) cg ( ( ))( ρ ρ ) Whe goes o ifiiy / 4 (3) lim ( cg ( ( ))) ( ) We obai he edomorphism of Hilber for sigals o Z: ( ) (4) κ ( ρ) ( ρ ρ ) ( ) Here ρ ges he meaig of a righ shif operaor (delayig) defied afer (5), where is a umber from Z his resul could be obaied direcly if we cosider he expasio of he impulse respose of he Hilber filer (defied as a fucio o he group of he ieger umbers Z) by he group of characers of Z, isomorphic of he group of he oe-dimesioal orus a corollary of he so-called Poryagi dualiy) [, ] If he frequecy respose has by defiiio o he ierval [?,? ] (icludig fully he ui circle, w = p) he form (5) j sg( ν) j, ν /,, ν,, / ν, he for he -h coefficie of he impulse respose will be obaied / si ( ) j ν j ν (6) κ j e dν j e dν,, /, his is he -h coefficie of he expasio i (4) his very resul is give i [, p79] he problems begi from here for desigig of appropriae filer wih fiie impulse respose O he Fig 3 is give magiude respose of he filer from (4) wih he firs four erms of he series he covergece of his series o he sgfucio is o uiform bu mea square Comparig of Figs ad 3 shows ha by

10 equal umber of erms he cyclic filer has smaller ripples More over, i goes very / accuraely rough he ui (or db) io he pois of samplig { / } VI Coclusio his paper deals wih a ew approach o he differe relaios of he Hilber rasform rough iroducig of he differe regios of defiiio were revealed some sides of he relaio coiuous-discree sigals Discree cyclic Hilber rasform was aalyzed he eigevecors ad eigevalues of his rasform, pseudoiverse rasform, projecors io he regio of he values, was foud I s show ha roaes o 9 he ivaria spaces of he dihedral group he magiude respose of he cyclic Hilber rasformer, show o he Fig ad Fig possess ieresig properies I is wih big ripples bu i he samplig pois i s very accurae he obaied properies of his rasform permis o be aalyzed coecios of he (cyclic!) Fas Fourier rasform wih he Hilber rasform, ha will be a objec of aoher wor Ampl db Discree Hilber rasform Rad Freq w Fig Magiude respose; Eq () has 4 erms ( = 6) Ampl db Rad Freq w Fig Magiude respose ear ; Eq () ( = 6)

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