Hilbert Transform Relations


 Austin Malone
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1 BULGARIAN ACADEMY OF SCIENCES CYBERNEICS AND INFORMAION ECHNOLOGIES Volume 5, No Sofia 5 Hilber rasform Relaios Each coiuous problem (differeial equaio) has may discree approximaios (differece equaios) Gilber Srag (SIAM Review, Vol 4, 999, No, 3547) Bozha Zhechev Isiue of Compuer ad Commuicaio Sysems, 3 Sofia Absrac : I his paper he Hilber rasform differe cases for coiuous, periodic ad discree sigals are aalyzed he mai aeio is paid o he properies of he discree cyclic rasform he eigevecors ad eigevalues of his rasform, projecors oo he regio of he values, pseudoiverse edomorphism, ad coecios wih aoher varias are foud he properies of he magiude respose of he differe Hilber s filers are demosraed Keywords: digial sigal processig, iverse filerig, badpass sigals, siglesidebad modulaio, image processig, discree Fourier rasform (DF), fas rasforms, ivaria spaces, pseudoiverse I Iroducio he Hilber rasform (or more correcly edomorphism) is applied i may areas: geeraig of siglesidebad sigals, iverse filerig, image processig, speech processig, radiolocaio, compressig ad ec [,, 3] A purpose of his paper is o represe compleely whe sigals are defied o he se of he real umbers R, ieger umbers Z, oedimesioal orus R Z ad complee residue sysem modulo, Z Z [8] his approach gives possibiliies o obai he basic properies ha are difficul o be aalyzed separaely II Coiuous case (sigals o R) Le sigals domai be he real lie R he fucio / is o summable i he viciiy of he poi =, bu i s well ow [4] ha if has limied regio of suppor ad is a leas oe ime differeiable a he begiig of he coordiaes, his iegral exiss: 3
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 wellow 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 PaleyWieer 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 socalled period he umber is always period; a umber opposed o he period is period oo
3 ad he sum of wo periods is agai a period hus he periods are some subgroup of he addiive group of he real umbers R (see he defiiios a he ed of he paper) I is he socalled group of he periods If x( is coiuous fucio is group of periods is closed subgroup of R Bu here exis oly hree closed subgroups of R: he subgroup, reduced o A fucio wihou periods differe of is aperiodic A whole group R; a fucio ha has as a period every real umber is a cosa 3 he se of muliples, >, ieger from Z If he group of periods belogs o oe of he laes wo cases, x( is periodic; he umber of he hird oe is so called mai period of x( Le is a circle wih a ceer O ad legh i he plae Ox Every fucio x( o ca be coeced wih a fucio x ~ o R, if x ~ ( = x(m), where M is a poi of wih a curviliear abscissa s = he begiig of he referece is he poi A of o he axis O ad direcio is couerclocwise he fucio x ~ is periodic wih period Ad vice versa, if x ~ is periodic fucio o R wih period, i ca be received wih previous procedure from oe ad oly fucio x he mappig x x ~ is a isomorphism bewee he fucios o ad R Oe of he reasos for iroducig of he periodic fucios is ha he fucios o he rigoomeric circle ca be cosidered as fucios of he agle θ wih a period of [4] Le φ is a fucio o R, ha could be made periodical i his way: ~ (6) ( ) ( l) l If his fucioal series coverges (for isace if j is wih limied suppor, ~ will be periodic fucio wih period Whe j ( = / ad his equilaeral hyperbola i (6) is coiled o he ui circle, i will be uiformly coverge series: (7) ~ ( l l 4 l l l he ex ideiy ca be proved by iducio [7, p 37]: (8) cg( l cg l cg g l l l l cg cg For he firs erm of he secod row of (8) coverges o /, ad he lh erm o l Hece he fucio series i (7) is expasio of (cg(/))/ Aoher proof of his expasio (Euler s expasio) ca be foud i [5] We have from here for he Hilber rasform of a periodical fucio x( wih period p: 5
4 (9) ( x( ) x( d x( ) d (( ) ) ( ) x( d ( ) x( )cg d his resul ca be foud i [, p78; 3, p 68] From (4) ad (5) ad he basic propery of he Fourier operaor o rasform covoluio io algebraic muliplicaio [4], ad from F(cos( ) ( ( v ) ( v )), F(si( ) ( ( v ) ( v )) j (here d() is a Dirac dela fucio [4]), follows: (a) (cos() = si( his resul ca be obaied from (6) oo, because cos( is а periodic fucio wih mai period p I ca be foud oo, ha (b) (si() = cos( hese wellow ad ofeapplied formulae (a) ad (b), are he mos impora relaios of he Hilber edomorphism (i his case acs as a iegral operaor) hey refer o every pair {cos(p, si(p}, From hem follow may ieresig resuls Hilber rasform coecs real ad imagiary par of he frequecy respose of a causal sysem, gai ad phase of such a sysem, he evelope ad phase of badpass sigals ad ec [, 6] IV Discree sigals (sigals o Z Z) IV Geeral properies of he discree (cyclic) edomorphism of Hilber Le s iroduce hese wo operaors (liear represeaios of he geeraors of a dihedral group D ) [8]: = [,l ], = [,l ],, l =,,, (mod ) (), 4 4 hese are respecively he righshif operaor ad sig operaor σ for he sigals o Z Z he las se ca be preseed as verexes of he iscribed i circle (wih legh = ) regular polygo, received afer coilig o i of R, ad herefore of Z oo I ha way oe ca cosruc he class of he discree periodic 6
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 Hermiiacojugaed 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 sigfucio [, 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 coordiae 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 + coiside 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 subspace [9, ] I s well ow ha + is a orhogoal projecor io his subspace, herefore his projecor coiside 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 coiside wih he Fourier rasforms colums [8]: (8) κ F F Λ; F κ Λ F; κ κ; F F j 8
7 Here is a diagoal marix of he eigevalues Bu he Fourier rasform of he impulse respose of (from (4)) has by defiiio he form: (9) F κ F κ( ρ) δ Λ F δ Λ j s herefore he diagoal marix of he eigevalues has he form () Λ diag ( j s ) diag(, j, j,, j,, j, j,, j) he edomorphism is a aisymmeric edomorphism, ad hece i has pure imagiary eigevalues ad if l is a eigevalue, ha eigevalue is l Whe is eve, he zero ad he / rows of have ad whe is odd he middle drops ou Le F = C js, ad hese vecors are colums of C, S(C = C; S = S) respecively: { c,s;,,, }; { c c; s s,,, / } he from (8) ad () i follows: κ c s,, () κ s c, I ca be see here, ha roaes o 9 he pairs of orhogoal bases vecors { c, s } of he wodimesioal subspaces of dihedral group D [8] (9degree phase shifer) excep he zero ad (/)h oedimesioal subspaces, ha have basis respecively { c,c / } ad are mapped io zero he edomorphism roaes o 9 every vecor wih real coordiaes x : ( x κ x) ( κ x x) ( κ x x) ( x κ x) From here x κ x ad if x do have compoes from he erel Ker(), his will be pure roaio Oherwise besides roaio of 9, he vecor will be shored because of is erel compoes If i is from he erel, i will be mapped io he zero Equaios () specify how does his roaio ad shoreig become I s received from (9) ad () he orm [9, p33] of as he maximal (real) eigevalue of he symmeric marix : κ κ x λ κ κ F ( κ ) F F ( Λ max κ (κ κ), herefore shore he sigals, ad i he bes case i saves heir eergy (orm) he resuls of his par have may ieresig applicaios he complex filer (+ j) forms he socalled aalyic sigal [,, 3, 6] I has, as follows from (9) ad (), oly oesided Fourier rasform: F( j κ) x ( j Λ) F x Y X; F x X ; Y /,,,,, /,,,, 9 x ),
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 Hermiiacojugaed 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 / coordiaes of he specrum of a sigal, ad he secod oe he lower / coordiaes 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 viceversa 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 wellow 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 aisymmery 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 oedimesioal orus a corollary of he socalled 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 coiuousdiscree 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)
11 Ampl db Discree Hilber rasform  dom Z Rad Freq w Fig 3 Magiude respose; firs 4 erms of Eq (4) Some defiiios he group G is a se G wih biary operaio G G G, oed as ( a, b) ab ad such, ha: I is associaive Ideiy eleme ug exiss, ie ua = a = au for every a G 3 For every eleme a G a iverse eleme a' G exiss, ad aa' = u = a'a If G ad H are groups, he morphism :G H of hese groups is a fucio from G o H, which is morphism of heir biary operaios, ie (ab)=(a)(b) for all a,bg R e f e r e c e s O p p e h e i m, A, R S c h a f e r, J B u c Discreeime Sigal Processig Preice Hall, d ed, 999 P a p o u l i s, A Sysems ad rasforms wih Applicaios i Opic McGrawHill, P r o a i s, J, D M a o l a i s Digial Sigal Processig PreiceHall, I, S c h w a r z, L Meodes Mahemaiques Pour les Scieces Physiques Paris, Herma, 96 5 Ф и х т е н г о л ь ц, Г М Курс дифферециального и интегрального вычисления Москва, Наука,968 6 M a s o, S J, H J Z i m m e r m a Elecroic Circuis, Sigals ad Sysems Joh Wiley & Sos Ic, 96 7 G r a h a m, R, D K u h, O P a a s h i Cocree Mahemaics a Foudaio for Compuer Sciece AddisoWesley, Z h e c h e v, B Ivaria spaces ad fas rasforms I: IEEE ras o Circuis ad Sys II: Aalog ad Digial Sigal Proc, February 999, p 6 9 S r a g, G Liear Algebra ad is Applicaios Academic Press, 976 A l b e r, A Regressio ad he MoorPerose Pseudoiverse Academic Press, 97 S e r r e, JP Represeaios Lieaires des Groupes Fiis Herma, Paris, 967 M o r r i s, SA Poryagi Dualiy ad he Srucure of Locally Compac Abelia Groups Lodo, New Yor, Cambridge Uiversiy Press, Z h e c h e v, B Fas rasforms aalysis I: I Coferece Auomaics ad Iformaics, Sofia, Bulgaria, 3 May, I69, I7 3
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