Poularikas A. D. The Hilbert Transform The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas Boca Raton: CRC Press

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1 Poularikas A. D. The Hilber Trasform The Hadbook of Formulas ad Tables for Sigal Processig. Ed. Alexader D. Poularikas Boca Rao: CRC Press LLC,999

2 5 The Hilber Trasform 5. The Hilber Trasform 5.. Defiiio of Hilber Trasform where P sads for he Cauchy pricipal value of he iegral. Covoluio form represeaio 5. The Hilber Trasform 5. Specra of Hilber Trasformaio 5.3 Hilber Trasform ad Dela Fucio 5.4 Hilber Trasform of Periodic Sigals 5.5 Hilber Trasform Properies ad Pairs 5.6 Differeiaio of Hilber Pairs 5.7 Hilber Trasform of Hermie Polyomials 5.8 Hilber Trasform of Produc of Aalyic Sigals 5.9 Hilber Trasform of Bessel Fucios 5. Isaaeous Ampliude, Phase, ad Frequecy 5. Hilber Trasform ad Modulaio 5. Hilber Trasform ad Trasfer Fucios of Liear Sysems 5.3 The Discree Hilber Filer 5.4 Properies of Discree Hilber Trasform 5.5 Hilber Trasformers (coiuous) 5.6 Digial Hilber Trasformers 5.7 IIR Hilber Trasformers Refereces x( η) ( ) υ() {()} x P η η d P x = = H = d ηη η υη ( ) υη ( ) x () = { υ()} = P η η d = H P d η η v () = x () x () = υ() 999 by CRC Press LLC

3 Fourier rasform of ν() ad x() ad / (see Table 3..3) V( ω) = X( ω)[ jsg( ω)] F { jsg( ω) X( ω)} = υ( ) F jsg( ω) = Example If x ( ) = cos ω, he H{cos ω} = υ( ) The resul is due o he fac ha Example If x () = p() a he cosωη = P dη η cos[ ω( y )] = P + dy y cosωy = cosωp dy siω P y = si ω. cos ωy/ y is a odd fucio ad siωy dy y siωy P dy =. y a d v p P P d () = { a()} = ε η a η H η + ε η ε a + a = lim l( η ) l( η ) = υ( ) = l ε a + ε a Example If x ()= a he + a ah{ } = alim l a a =. Hece, if x o = cosa is he mea value of a fucio, he x () = xo + x (). Therefore H{ x + x ( o )} = H{ x( )}. This implies ha he Hilber rasform cacels he mea value or he DC erm i elecrical egieerig ermiology. 5.. Aalyic Sigal A complex sigal whose imagiary par is he Hilber rasform of is real par is called he aalyic sigal. ψ( z) = ψ(, τ) = x(, τ) + jυ(, τ), x ad υare real fucios 999 by CRC Press LLC

4 z = + jτ υ(, τ) = H{ x(, τ)} The fucio ψ() z = x(,) τ + jυ(,) τ is aalyic if he Cauchy-Riema codiios x υ x υ = ad = τ τ are saisfied. Example The real ad imagiary pars of he aalyic fucio α+ τ ψ( z) = /( α jz) = + j ( α+ τ) + ( α+ τ) + saisfy Cauchy-Riema codiios ad, hece, hey are Hilber rasform pairs. ψ() + ψ () ψ() ψ () x () = υ() = j ( τ = ) 5. Specra of Hilber Trasformaio 5.. Oe-Sided Specrum of he Aalyic Sigal x () + x( ) x () x( ) x () = xe() + xo() = + X( ) = Xr( ) + jxi( ) = xe( )cos d+ j ω ω ω ω xo( )siωd V( ω) = V ( ω) + jv( ω) = Specrum of he Hilber rasform r i V ( ω) = jsg( ω)[ jx ( ω)] = sg( ω) X ( ω) (see also 5..) r V( ω) = sg( ω) X ( ω) i r i i Example H{cos ω} = si ω, H{si ω} = cosω ad, herefore, j ω j ω j ( ω ) H{ e } = siω jcosω = jsg( ω) e = sg( ω) e oe: The operaor jsg( ω) egaive frequecies. provides a / phase lag for all posiive frequecies ad / lead for all 5.. Fourier Specrum of he Aalyic Sigal H{ x ( )} = υ( ); F{ x ( )} = X( ω); F{ υ( )} = jsg( ω) X( ω) 999 by CRC Press LLC

5 F{ ψ( )} = x( ) + jυ( ) = Ψ( ω) = X( ω) + jv( ω) = [ + sg( ω)] X( ω) ω > + sg( ω) = ω = ω < oe: The specrum of he aalyic sigal is wice ha of is Fourier rasform a he posiive frequecy rage < ω <. Example If ψ( ) = + j he F { ψ( )} = [ + sg( ω)] e ω + + where H{ /( )} /( ) F{ /( + = + ad + )} = e ω. 5.3 Hilber Trasform ad Dela Fucio 5.3. Complex Dela Fucio If we defie ( f) = ( f) + sg( f), he he fucio (see Fourier rasform properies [symmery] ad fucio, Chaper 3). 5.4 Hilber Trasform of Periodic Sigals 5.4. Hilber Trasform of Period Fucios A periodic fucio ca be wrie i rigoomeric form Therefore we obai ψ jω jω jω ( ) = ( f) e df = ( f) e df + sg( f) e df δ = δ() + j 5.3. Hilber Trasform of he Dela Fucio From (5.3.) implies H{ δ()} = x ( ) = C + C cos( ω + ϕ ), ω = / T, T = period p because he Hilber rasform of a cosa is zero (see 5..). o = o o v ( ) = H{ x ( )} = C si( ω + ϕ ) p p = o 999 by CRC Press LLC

6 A periodic fucio ca also be wrie i complex form xp()= α e = jωo Therefore, o v ( ) = H{ x ( )} = α H{ e } = jsg( ) e p p = jω = jωo 5.5 Hilber Trasform Properies ad Pairs 5.5. Hilber Trasform Properies TABLE 5. Properies of he Hilber rasformaio Origial or Iverse o. ame Hilber Trasform Hilber Trasform oaios υη ( ) x () = Time domai defiiios d or x () = η η υ() 3 Chage of symmery 4 Fourier specra F x ( ) = X( ω) = X( ω) + jx( ω); X( ω) = jsg( ω) V( ω); e x () or H [ υ ] υ() or xˆ( ) or H[ υ ] x () = x () + x (); o e o x( η) υ() = η dη υ() = x () υ() = υ () + υ () o e F υ( ) = V( ω) = V ( ω) + jv ( ω) V( ω) = jsg( ω) X( ω) e o For eve fucios he Hilber rasform is odd: Xe ( ω) = x e ( )cos( ω) d υo ( ) = Xe ( ω)si( ω) df o o For odd fucios he Hilber rasform is eve: Xo ( ω) = x o ( )si( ω) d υe ( ) = Xo ( ω)cos( ω) df o o 5 Lieariy 6 Scalig ad ime reversal 7 Time shif 8 Scalig ad ime shif 9 Ieraio e = eve; o = odd ax () + bx () aυ () + bυ () xa ( ); a> υ( a) x( a) υ( a) x ( a) υ( a) xb ( a) υ( b a) Fourier image H[ x ( )] =υ( ) jsg( ω) X( ω) HH [ [ x]] = x( ) [ jsg( ω)] X( ω) HHH [ [ [ x]]] = υ ( ) 3 [ jsg( ω)] X( ω) HHHH [ [ [ [ x]]]] = x( ) 4 [ jsg( ω)] X( ω) 999 by CRC Press LLC

7 TABLE 5. Properies of he Hilber rasformaio (coiued) Origial or Iverse o. ame Hilber Trasform Hilber Trasform Time derivaives Covoluio Auocovoluio equaliy 3 Muliplicaio by Firs opio x ( ) = υ ( ) υ ( ) ( ) = x Secod opio for τ = eergy equaliy 4 Muliplicaio of x (low-pass sigal) x (high-pass sigal) sigals wih ooverlappig specra x() x() x() υ() 5 Aalyic sigal ψ( ) = x() + jh [ x()] H[ ( )] ( ) 6 Produc of ψ() = ψ () ψ () H [ ( )] ( ) H [ ( )] aalyic sigals = H[ ψ( )] ψ( ) 7 oliear rasformaios xx ( ) υ( x) 7a 7b a c y = b + a S is suppor of x e () d x ( ) = () d d υ υ ( ) = x () d x() x() = υ() υ() x () υ () = x() τ x( τ) dτ = υ() τ υ( τ) dτ υ () x () x() υ() x( τ) dτ x ()= c x b + a b b y = a+ x x a ()= + c υ() = υ P b + a υ () b b = υ a + υ a ( a ) x () d oice ha he oliear rasformaio may chage he sigal x() of fiie eergy o a sigal x () of ifiie eergy. P is he Cauchy Pricipal Value. 8 Asympoic value as for eve fucios of fiie suppor: a xe() = xe( ) lim υo () = xe () d s 5.5. Ieraio Ieraio of he HT wo imes yields he origial sigal wih reverse sig. Ieraio of he HT four imes resores he origial sigal I Fourier frequecy domai, -ime ieraio raslaes he -ime muliplicaio by jsg(ω) Parseval s Theorem v () = H{()} x 999 by CRC Press LLC

8 F{ υ( )} = V( ω) = jsg( ω) X( ω) V( ω) = jsg( ω) X( ω) = X( ω) sice Orhogoaliy E = x () d = X( ω) df = eergy of x() x E = V( ω) df = X( ω) df = E υ x υ( ) () = x d Fourier Trasform of he Auocovoluio of he Hilber Pairs F{() x x ()} = X ( ω) F{ υ( ) υ( )} = [ jsg( ω) X( ω)] = X ( ω) x () x () = x( τ) x ( τ) dτ = υ( τ) υ( τ) dτ = υ() υ() x () x () = υ ( ) υ ( ) Hilber Trasform Pairs TABLE 5. Seleced Useful Hilber Pairs o. ame Fucio Hilber Trasform sie cosie 3 Expoeial 4 Square pulse 5 Bipolar pulse 6 Double riagle 7 Triagle, ri() 8 Oe-sided riagle si( ω) cos( ω) cos( ω) si( ω) ω jsg( ω) e jω e j () a ( )sg( ) a l + a a l ( a/ ) ( )sg( ) a l ( a/ ) / a, a, > a a l + l + a a a ( / )l a + a 999 by CRC Press LLC

9 TABLE 5. Seleced Useful Hilber Pairs (coiued) o. ame Fucio Hilber Trasform 9 Trapezoid Cauchy pulse a a + b a+ b a a l ( )( ) + l + l ( ) + b a ( a )( b ) b a b ( a+ ) a + Gaussia pulse Parabolic pulse 3 Symmeric expoeial e f e si( ω) df ; ω = f (/ a), a e a [ ] a (/ a) l + a a a si( ω) df a ω 4 Aisymmeric expoeial sg( e ) a a a ω cos( ω) df 5 Oe-sided expoeial ( e ) a asi( ω) ωcos( ω) df a ω 6 Sic pulse si( a) si ( a / ) cos( a) = a ( a / ) a 7 Video es pulse cos ( / a); a a si[ ω /( a)], > a si( ω ) df 4a ω ω 8 Specra of a ( ) ad cos( ω ) si( ω ) cos( ) a ( )cos( ω ω ) a ( ) si( ) a ( ) overlappig ω cos( ω ) 9 Bedrosia s heorem a ( )cos( ω ) a ( )si( ω ) A cosa a zero Hyperbolic Fucios: Approximaio by Summaio of Cauchy Fucios (see Hilber Pairs o. ad 45) o. ame Fucio Hilber Trasform Tage hyp. Par of fiie eergy of ah ah( ) = ( η+ 5. ) + η= η= sg( ) ah( ) δ() + ( η + 5. ) ( η+ 5. ) + η= ( η + 5. ) ( η+ 5. ) + 3 Coage hyp. 4 Secas hyp. 5 Cosecas hyp. coh( ) = + η= sech( ) = ( ) ( η ) η= cosech( ) = ( ) η= ( η) + ( η + 5. ) ( η+ 5. ) + ( η ) ( η) + δ() + η= ( ) η= ( η ) δ() + ( ) η= η ( η) + ( η+ 5. ) + ( η ) η ( η) by CRC Press LLC

10 TABLE 5. Seleced Useful Hilber Pairs (coiued) o. ame Fucio Hilber Trasform Hyperbolic Fucios by Iverse Fourier Trasformaio; ω = f o. sg( ) ah( a/ ) Re a > coh( ) sg( ) sec h( a / ) Dela Disribuio, /() Disribuio ad is Derivaives: Derivaio Usig Successive Ieraio ad Η Differeiaio Ieraio cos( ω) df asih( ω / a) ω coh( ω / a) ω cos( ω) df a ω df acosh( ω /( a) si( ) cos ech( a / ) ah( ω /( a))cos( ω) df a ω sec h ( a / ) ω df asih( ω /( a)) si( ) Η If x () v () he x ( ) v ( ) H [ v ( )] HH [ u ( )] x ( ) Operaio x () v () 3 3 Ieraio 33 Differeiaio 34 Ieraio 35 Differeiaio 36 Ieraio 37 Differeiaio 38 Ieraio 39 δ( ) /( ) /( ) δ( ) δ ( ) /( ) /( ) δ ( ) () δ 3 /( ) 3 /( ). 5 δ () δ 4 6 /( ) 4 /( ) ( / 6) () δ x () δ() x( ) /( ) The procedure could be coiued. Equaliy of Covoluio δ() δ() δ() = δ() δ ( ) δ() = δ ( ) = δ ( ) δ ( ) δ ( ) = () δ = () δ 6 () δ δ() = () δ = () δ δ ( ) = () δ = 4 3 Approximaig Fucios of Disribuios (see o. 3 o 37 of his able) δ( a, ) d= a ( / a) x () v () l( a + ) θ( a, ) d= 999 by CRC Press LLC

11 TABLE 5. Seleced Useful Hilber Pairs (coiued) o. ame Fucio Hilber Trasform a δ( a, ) = a + a δ ( a, ) = ( a + ) θ 6a ( a δ a, ) = 3 ( a + ) 4a 4a ( δ a, ) = 4 ( a + ) 3 ( a, ) = a + Derivaio Usig Successive Ieraio ad Differeiaio (see he iformaio above o. 3) a θ ( a, ) = ( a + ) 6a ( θ a, ) = ( a + ) a 6a ( θ a, ) = 4 ( a + ) 4 Operaio 49 5 Ieraio 5 Differeiaio 5 Ieraio 53 Differeiaio 54 Ieraio ame 55 Samplig sequece 56 Eve square wave 57 Odd square wave 58 Squared cosie 59 Squared sie 6 Cube cosie 6 Cube sie Trigoomeric Expressios x () v () si( a) cos( a) si ( a / ) = cos( a) si( a) δ( ) + si( a) + cos( a) aδ( ) cos( a) a si( a) δ ( ) + si( a) a cos( a) δ a ( ) cos( a) a a si( a) 3 () δ + δ () + 3 Seleced Useful Hilber Pairs of Periodic Sigals = δ( T) x () p ν () p T = cos[( / T)( T)] sg[cos( ω)], ω = / T ( / )l a( ω / + / 4) sg[si( ω)], ω = / T ( / )l a( ω / ) cos ( ω) 5. si( ω) si ( ω). 5si( ω) cos 3 ( ω) 3 4 si( ω) + 4 si( 3ω) si 3 3 ( ω) cos( ω) + 4 cos( 3ω) cos 4 ( ω) 4 si( ω) + si( 4ω) si 4 ( ω) si( ω) + si( 4ω) cos ( ) ω si( ω) + si( 3ω) + si( 5ω) cos ( ω) 3 si( ω) + 3 si( 4ω) + 3 si( 6ω) cos( a + ϕ)cos( b + Ψ) cos( a + ϕ)si( b + Ψ) < a < b ϕ,ψ =cosas 999 by CRC Press LLC

12 TABLE 5. Seleced Useful Hilber Pairs (coiued) o. ame Fucio Hilber Trasform 67 Fourier Series Xo + X cos( ω + ϕ ) = 68 Ay periodic fucio x T = geeraig fucio = x T X si( ω+ ϕ ) k= ( ) co[( / T)( kt)] T xt () δ ( kt) k= 5.6 Differeiaio of Hilber Pairs 5.6. Differeiaio Pairs Example H{ ( x)} = ν ( ) d x() d ν() H = d d 5.6. Derivaive of Covoluio d H{()} δ = ; H{ ( δ )} = d = H{()} x = H ν() () x () ν = d d H{ ( x)} = H () ( ) x () d = ν ν d (see 5.6. ad 5.5.5) = = x () ν() H{ ( x)} = H ν ( ) ( ) x ( ) ν = Fourier Trasform of Hilber Trasform ν( ) = x ( ), F{ ν( )} = jsg( ω) X( ω) F{ ( ν )} = jω[ jsg( ω) X( ω)] = ωsg( ω) X( ω) 999 by CRC Press LLC

13 5.7 Hilber Trasform of Hermie Polyomials 5.7. Hermie Polyomials ad heir Hilber Trasform d H() = ( ) e e =,,, L, < < d H () = H () ( ) H () =,, L ω / 4 f F{ e } = e = e f f jω v () = H{()} x = H{ e } = F { V( ω)} = jsg( ω) e e df = e siωdf H{ e } e f = ω cosω df 5.7. Table of Hilber Trasform of Hermie Polyomials TABLE 5.3 Hilber Trasform of Weighed Hermie Polyomials [oaio: x = exp( )] Hermie Polyomial Hilber Trasform Eergy Hx H( Hx) E Eergy = () x ω exp( f )si( ) df / ( ) x ωexp( f )cos( ω) df / ( 4 ) x ω exp( f )si( ω) df 3 / ( 8 3 ) x ( ) x 3 ω ω exp( f )cos( ) df 5 / 4 ω ω exp( f )si( ) df 5 / ( ) x ω exp( f )cos( ω) df 945 / H x = ( ) [ H ( ) ( ) ω exp( f )si ω+ df ( ) H ( )] x H d = [ H( xh )] d = 3 5 L ( ) /, 999 by CRC Press LLC

14 5.7.3 Hilber Trasform of Orhoormal Hermie Fucios (see Chaper ) Hilber Trasform of Orhoormal Hermie Fucios TABLE 5.4 Hilber Trasforms of Orhoormal Hermie Fucios (Eergy = ). oaios: H{ h ( )} = ν ( ) / / h () = (!) e H () =,,, L Hermie Fucios Hilber Trasforms h υ Recurre oaio 3 h5 = / 5 h4 4/ 5 h3 υ5 = / 5 υ4 4 5 υ3 b /... ( )! () ( ) ( ) ( = h d )!! ν τ τ ν! / / h ( ), h ( ), L h, h, L; υ ( ), L υ, υ, L o o o o 5. / 5. g e f ( ) = si( f) df; a= e ; b= h = a υ = bg() h = h υ = υ b h = h / h υ = υ / υ b h3 = / 3 [ h h] υ3 = / 3 υ υ h = / h 3/ 4 h υ = / υ 3/ 4 υ h 4 3 = ( )! h! ( )! ( )! h + υ 4 3 = ( )!! [ υ ( )! h () τ dτ] ( ) υ! () orecurre oaio h = a bg() h h h h = a b[ g( ) ] 3 4 = a 8 ( 4 ) b[( ) g( ) ] = a 3 48 ( 8 ) / b ( ) g( ) + = a ( + ) 3 4 4/ 3b ( ) g( ) by CRC Press LLC

15 TABLE 5.4 Hilber Trasforms of Orhoormal Hermie Fucios (Eergy = ). (coiued) oaios: h ( ), h ( ), L h, h, L; υ ( ), L υ, υ, L Hermie Fucios Hilber Trasforms h () υ () 4 a h = ( + ) ( 4 ) / b ( +. ) g( )... h () = a H, ()! o o o o 5. / 5. g e f ( ) = si( f) df; a= e ; b= h dτ b b 3/ 4 b H () = H () ( ) H () 5.8 Hilber Trasform of Produc of Aalyic Sigals 5.8. Hilber Trasform of Produc of Aalyic Sigals: From H{ ψ( )} = H{ x( ) + jυ( )} = H{ x( ) + jh{ x( )}} = H{ x( )} jx( ) = υ() jx() = j( x() + jυ()) = jψ() we obai H{ ψ() ψ()} = jψ() ψ() = ψ() H{ ψ()} = ψ() H{ ψ()} sice he produc ca be cosidered as a aalyic fucio ψ() The h Produc of a Aalyic Sigal H{ ψ ( )} = ψ( ) H{ ψ( )} = jψ ( ) H{ ψ ( )} ψ = ( ) H{ ψ( )} = jψ ( ) Example Because H{( ) j } = j( j), we obai H{( j) } = ( j) ( j( j) ) = j( j) 5.9 Hilber Trasform of Bessel Fucios 5.9. Hilber Trasform of Bessel Fucio: ˆ ( ) H{ ( )} ˆ J ( ) J J ( ) si( si ) = = = ϕ ϕ dϕ =! = 999 by CRC Press LLC

16 ˆ J () = siωdω / ( ω ) ψ = + () J () jj ˆ () ˆ dω J ( ) = ( ω ) si( ), ˆ () ωdω = J () = ( ω ) / / cos( ) = The parehesis i he expoe idicaes umber of differeiaios wih respec o ime Hilber Trasform Pairs of Bessel Fucios: TABLE 5.5 Hilber Trasform of Bessel Fucios of he Firs Kid Bessel Fucio Fourier Trasform Hilber Trasform J () C ( f) ˆ J () = H [ J ()] J () C = ; ω < 5. C ( f)si( ω) dω ( ω ) = ; ω > J () C j C = ω C ( f) cos( ω) dω J () C C = ( ω ) C ( f) si( ω) dω J () 3 3 C = j( 4ω 3ω) C J () 4 C 4 C 4 = ( 8ω 8ω + ) J () 6 C C = ( 3ω 48ω + 8ω ) C 6 ( f) si( ω) dω... T ( ω ) = cos[ cos ( ω )] is he Chebyshev polyomial 3 C ( f) cos( ω) dω for =,,4, for =,3,5, 3 C ( f) si( ω) dω J () C5 = j( 6ω ω + 5ω) C C 5( f) cos( ω) dω J () C j T C = ( ) ( ω) ( ) / ( ) 4 ( + )/ C ( f) si( ω) dω C ( f) cos( ω) dω 5. Isaaeous Ampliude, Phase, ad Frequecy 5.. Isaaeous Agular Frequecy ϕ() ψ() x() jυ() A() e j = + = = A()cos ϕ() + A()si ϕ() 999 by CRC Press LLC

17 A () = x () + υ (), ϕ() = a 5. Hilber Trasform ad Modulaio 5.. Modulaed Sigal (see 5..) 5.. Isaaeous Ampliude ad Agular Frequecy (see 5..) 5..3 High-Frequecy Aalyic Sigals (Φ = ) υ() x () ϕ () = Ω() = F() isaaeous agular frequecy Ω() ϕ () F () = isaaeous frequecy = = d υ() x () ( υ ) υ() ( x) Ω( ) = a = d x () x () + υ () ψ() = A γ() e e ψ x o () = x() + jxˆ( ) ψ x () = x jφ jω () + ψ () m m A () = ψ x () = [ x () + x ˆ ()] d ω x ( ) =± a d ψ ψ upper lower SSB x ˆ( ) x () x / () = upper sidebad = ψ () e () = lower sidebad = ψ () e x () = x()cos Ω m xˆ( )siω x x jω jω where x ( )cos( Ω ) ad x ˆ( )siω represe double sidebad (DSB) compressed carrier AM sigals. 5. Hilber Trasform ad Trasfer Fucios of Liear Sysems 5.. Causal Sysems Hs () = A( αω, ) + jb( αω, ), σ= α+ jω B( ) A( ω) = λ P d λ ω λ 999 by CRC Press LLC

18 5.. Miimum Phase Trasfer Fucio Hϕ( jω) has all he zeros lyig i he lef half-plae of he s-plae. The miimum phase rasfer fucio is aalyic ad is real ad imagiary pars for a Hilber pair 5.3 The Discree Hilber Filer 5.3. Discree Hilber Filer A( ) B( ω) = λ P d λ ω λ H( jω) = H ( jω) H ( jω) H H ϕ ap ϕ ap ( jω) = miimum phase rasfer fucio ( jω) = all-pass rasfer fucio jϕω ( ) H ( jω) = H( jω) e = A ( ω) + jb ( ω) ϕ H{ A( ω)} = B ( ω) ϕ ϕ ϕ j k =,, L, Hk ( ) = k = ad k = j k = +, +, L, ( = eve) Hk ( ) = jsg k sg( k ), k =,, L, ( = eve) 5.3. Impulse Respose of he Hilber Filer jw hi ( ) = Hke ( ) = jsg k sg( k ) e k= k= i i ki = si( w) = si co, i =,, L,, w = ( eve) k= DHT of a Sequece x(i) i he Form of Covoluio i i υ() i = x() i h() i = x() i si co, i =,, L, = circular covoluio υ() i = h( i r) x( r), i =,, L, ( eve) r= jw 999 by CRC Press LLC

19 5.3.4 DHT of a Sequece x(i) via DFT F {()} xi = Xk () D Vk ( ) = jsg k sg( k ) X ( k ) υ( i) = F { V( k)}, i, k =,,, L, ( eve) D F D discree Fourier rasform, F D iverse discree Fourier rasform Discree Hilber Filer whe is odd j k =,, L, Hk ( ) = k = j k = +, +, L, Also ( )/ hi ( ) = si( ki / ), i =,, L, hi () k= cos( i) cos( i/ ) co i = ( ) 5.4 Properies of Discree Hilber Trasform 5.4. Parseval s Theorem Exi {()} = xi () = Xk () i= Exi { ( )} E{ υ( i)} The reaso is ha he DC erm (average value of x(i)) is elimiaed i he DHT Discree Hilber Trasform x DC = xi () = X( ) i= k= H D { x ( i)} = υ( i) where x AC (i) is he aleraig par of x(i). AC x () i = x() i x AC DC 999 by CRC Press LLC

20 5.4.3 Eergies (powers) of x AC ad υ(i) X ( / ) xac() i = υ () i + ( eve) i= i= where he special erm X is zero, he wo eergies are equal. Example If x(i) = δ(i) ad = 8 we obai (see 5.3.3) υ( i) = δ( i) 4 si ( i/ )co( i/ ) Figure 5. shows he desired compoes ad rasforms. The x DC = /8 =.5 ad he eergies are: E { xi ( )} =, Ex { AC( i)} = =. 875, ad E i {()} υ = = = FIGURE 5. (a) The sequece x(i) cosisig of a sigle sample δ(i), (b) is specrum X(k) give by he DFT, (c) he samples of he discree Hilber rasform, (d) he correspodig specrum V(k), (e) he samples of he AC compoe of x(i), ad (f) he correspodig specrum X AC (k) Shifig Propery: See j mk/ F D ± = ± {( xi m)} e Xk () / υ( i) { jsg k sg( k ) e j mk = D ± F X ( k )} 999 by CRC Press LLC

21 5.4.5 Lieariy: H D { ax () i + bx ()} i = aυ () i + bυ () i Complex Aalyic Discree Sequece: ψ() i = x() i + jυ(), i υ() i =H D { x()} i H D{ ψ ( i )} = X ( k ) + j [ j sg k sg( k)] X( k), k =,, L, ( eve ) 5.5 Hilber Trasformers (coiuous) 5.5. Hilber Trasformer (quadraic filer) j ( f) H( jf) = F ϕ = H( f) e = jsg f j f > H( jf) = f = j f < ϕ( f ) = arg H( jf ) = sg f 5.5. Phase-Splier Hilber Trasformers Aalog Hilber rasformers are mosly implemeed i he form of a phase splier cosisig of wo parallel all-pass filers wih a commo ipu po ad separaed oupu pors, each havig he followig rasfer fucio respecively. j ϕ ( f) j ϕ ( f) Y( jf) = e, Y ( jf) = e wih δ( f) = ϕ( f) ϕ( f) = / for all f > All-Pass Filers H( jω) = R jx( ω) R+ jx( ω) ω = f ϕω ( ) = arg{( R jx( ω)) } = a RX( ω) R X ( ω) See Figure 5.a. 999 by CRC Press LLC

22 FIGURE 5. A all-pass cosisig of (a) a low-pass ad a complemeary high-pass, (b) a firs-order RC lowpass ad complemeary CR high-pass, ad (c) a secod-order RLC low-pass ad complemeary RLC high-pass. If X( ω) =, he (see Figure 5.b) ωc If X( ω) = ωl / ωc, y ϕ( y) = a, y ωrc ωτ y = = he (see Figure 5.c) ( y ) qy ϕ( y) = a, y ω/ ωr, ωr / ( y ) q y = = LC q = ω RC = R C/ L r 999 by CRC Press LLC

23 5.5.4 Desig Hilber Phase Spliers Example Filer wih wo firs-order all-pass filers i each brach. The phase fucio for he firs brach is (see Figure 5.3) y ay ϕ( f ) = a a, y frc y + ay = FIGURE 5.3 Phase Hilber splier wih wo all-pass filers. Fid a o ge he bes lieariy of ϕ ( f ) i he logarihmic scale. Small chages of a iroduce a rade-off bewee he RMS phase error ad he pass-bad of he Hilber rasformer. Fid shif parameer b o yield he miimum RMS phase error by ( f ) = a a by + ϕ Figure 5.4 shows a example wih a =.8 ad b =.4 givig he ormalized edge frequecies y =. 6 ad y = 3 ( f / f = 8. 75, or more ha 4 ocaves) wih ε RMS = Digial Hilber Trasformers 5.6. Digial Hilber Trasformers Ideal discree-ime Hilber rasformer is defied as a all-pass wih a pure imagiary rasfer fucio. aby aby jψ He ( ) = H( ψ) + jh( ψ) H ( ψ) = r for all f j < ψ < jψ He ( ) = jhi ( ψ) = ψ =, ψ = j < ψ < Equivale oaio r i He ( ) = jsg(si ψ) = sg(si ψ) e = H( ψ) e jψ j / j arg H ( ψ ) < ψ < H( ψ) = sg(si ψ) = ψ =, ψ = 999 by CRC Press LLC

24 FIGURE 5.4 The phase fucios ad he phase error of he Hilber rasformer of Figure 5.3. arg[ H( ψ)] = sg(si ψ) ψ = f, f = f / f, f = samplig frequecy s s ocausal impulse respose of he ideal Hilber rasformer is i hi ( ) = si i =, ±, ±, L i 5.6. Ideal Hilber Trasformer Wih Liear Phase Term jψτ je < ψ < jψ He ( ) = ψ =, ψ = j( ψ ) τ je < ψ < si ( i τ) hi () = i =, ±, ±, L i τ hi () = h( i) i=,,, L 999 by CRC Press LLC

25 5.6.3 FIR Hilber Trasformers: Figure 5.5 shows a ocausal impulse respose Hilber rasformer ad is rucaed ad shifed versio so ha a causal oe is geeraed. FIGURE 5.5 Impulse resposes of (a) he ideal discree ime Hilber rasformer (see 5.6.) ad (b) a FIR Hilber rasformer give by he rucaio ad shifig of he impulse respose show i (a). Causal Filer Impulse Respose Trasfer fucio = Hi ( ) = h( i) z i = i h i hi () i i, i,,, + = = + = L L j j j i j ψ ψ ψ ψ f He ( ) = e hie ( ) = e jhi ( )si( ψi), ψ = f i= Ampliude of Hilber Trasformer (see Figure 5.6) i= s jψ Ge ( ) = hi ( )si( ψi) i= 999 by CRC Press LLC

26 FIGURE 5.6 The G(e jψ ) fucio of a FIR Hilber rasformer (ampliude). ormalized Dimesioless Pass-bad Hilber Trasformer Wψ = ψ ψ =, ψ, ψ = edge frequecies W [ H ] = f f z s 5.7 IIR Hilber Trasformers 5.7. IIR Ideal Hilber Trasformer (see Figure 5.7) H () z = + z G( z ) ideal half bad filer (see Figure 5.7a) HB Gz ( ) = all pass filer wih ui magiude H () z = z G( z ) ideal IIR Hilber rasformer H Fz () = z Gz ( ), z= e jψ (see Figure 5.7b) jψ jψ jφ ( ψ) jφ( ψ) G Fe ( ) = e e = e Φ( ψ) = 5. [sg(si( ψ)) sg ψ] Φ ( ψ) = Φ( ψ) + ψ G (see Figure 5.7e) jψ jψ jφg ( 5. + ψ) jψ j5 (. + ψ) H ( e ) = e e, z = e, z = e H arg{ z G( z )} = ψ + Φ ( 5. + ψ) G (see Figure 5.7g) IIR Hilber rasformer has a equi-ripple phase fucio ad exac ampliude. A ocausal rasfer fucio may have he form Hz ()= z i= az z i a i 999 by CRC Press LLC

27 FIGURE 5.7 Sep-by-sep derivaio of he IIR rasfer fucio of a Hilber rasformer Z G( z ), sarig from he rasfer fucio of he ideal half-bad filer give by + Z G(z ) Example Le ψ =. low-frequecy edge, ψ = 98. = high-frequecy edge ( =. ), phase equiripple ampliude Φ.. Because δ = si(. 5 Φ), δ =. 57. Usig he procedure from Asari (985), we fid a() = , a( ) =. 655, a() 3 =. 9467, ad a( 4) = Iserig a i s, i H(z) above, we fid he phase fucio. 999 by CRC Press LLC

28 Refereces Asari, R., IIR discree-ime Hilber rasformers, IEEE Tras., ASSP-33, 46-5, 985. Erdelyi, A., Tables of Iegral Trasform, McGraw-Hill Book Co. Ic., ew York, Y, 954. Hah, Sefa L., Hilber Trasforms, i Trasforms ad Applicaios Hadbook, Ed. Alexader D. Poularikas, CRC Press Ic., Boca Rao, FL, by CRC Press LLC