Discrete Hilbert Transform. Numeric Algorithms

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1 Volum 49, umbr 4, Discrt Hilbrt Transform. umric Algorithms Ghorgh TODORA, Rodica HOLOEC and Ciprian IAKAB Abstract - Th Hilbrt and Fourir transforms ar tools usd for signal analysis in th tim/frquncy domains. Th Hilbrt transform is applid to casual continuous signals. Th majority of th practical signals ar discrt signals and thy ar limitd in tim. It appard thrfor th nd to crat numric algorithms for th Hilbrt transform. Such an algorithm is a numric oprator, namd th Discrt Hilbrt Transform. This papr maks a brif prsntation of known algorithms and proposs an algorithm drivd from th proprtis of th analytic complx signal. Th mthods for tim and frquncy calculus ar also prsntd.. ITRODUCTIO Signals can b classifid into two classs: analytic signals (for instanc x( t) = Asinωt ), and xprimntal signals (masurd signals). Th last catgory rprsnts ral signals and is of grat importanc in applications. An xprimntal signal rprsnts a signal obsrvd during a limitd intrval of tim. It is a sampl of th original signal, which charactrizs a physical procss of intrst. Th xprimntal signal can b a continuous tim signal (analogical), or a digital signal (discrt). Th practical limitations of th systms usd to analyz analogical signals impos that th xprimntal analogical signals had a limitd frquncy band [],[]. If th original signal dosn t hav a limitd band, a low-pass filtration nds to b applid in ordr to obtain th xprimntal signal which will b analyzd. Th rul also applis to sampld signals, which nd to hav a limitd frquncy band too. As a rsult, bfor acquisition, th xprimntal analogical signal will b lowpass filtrd. Th acquisition frquncy nds to b two tims th biggst frquncy of th signal s spctrum in ordr to avoid th aliasing procss th yquist condition. Th discrt signal will b analyzd on a computr systm, which implis its digitization (th digital signal is th discrt signal convrtd in binary format, accordingly to th adoptd analog/numric convrsion; in most of th cass, th signal acquisition hardwar also dos th digitization of th signal sampls). Th rsultd digital signal has th gratst importanc in numric analysis oprations. Som othr rmarks nd to b mad. Sinc th sampld signal has a limitd lngth, it nds to ithr () hav a infinit frquncy spctrum, or () b a priodic signal. In cas (), th sampling dosn t rspct th yquist condition. Or, in cas () w choos to rprsnt th signal as a priodic on, with an xtndd priod. In both cass, th digital signal cannot xactly rprsnt th original physical procss. In th cas of th Hilbrt transform, it s a known fact that th signal x(t) nds to b causal (that is x(t)=, for t < ). Th sampld signal x[ is in this cas a non-priodic squnc, ral and causal. In such a cas, w can talk of a discrt Hilbrt transform applid to th squnc x[. Th complx analytic signal associatd to th x[ squnc has th spctrum diffrnt 8 Mdiamira Scinc Publishr. All rights rsrvd.

2 486 ACTA ELECTROTEHICA from zro only for th intrval of positiv frquncis. Whn x(t) is a priodic signal, x[ is a priodic squnc and w cannot talk of causality (th priodic trm implis th squnc xtnsion from to + ). A calculus algorithm for th Discrt Hilbrt Transform in this cas imposs th condition that th Discrt Fourir Transform of th complx analytic squnc to b qual to zro in th intrval of ngativ frquncis. And of cours, for positiv frquncis, th spctrum of th analytic squnc to b two tims th spctrum of th signal x[. In this cas, th Hilbrt transform can b usd with all its known advantags rgarding th causal signals. Th nxt paragraphs prsnt th mthods for calculating th Discrt Hilbrt Transform.. HILBERT TRASFORM I COTIUOUS TIME To start, w prsnt first th thory of th Hilbrt transform, dfinitions, proprtis [], []. Lt s considr a ral masurmnt signal: () x( t) L () () Whr L is th signal class with intgral squar. Th Hilbrt transform of th signal x(t) is: x τ x t { x t v p dτ π ( ) ˆ( ) = H ( ) = () t τ v.p. x ( τ ) t ε x( τ ) x( τ dt = lim + ε dτ ) dτ, t τ t τ t+ ε t τ whr v.p. rprsnts th functional namd principal valu. x (t) is impropr namd th conjugat of x(t). () W also hav x (t) L. x(t) is th invrs Hilbrt transform of x (t) : - x t x t { x t v p dτ π ( ) ( ) = H ( ) =.. (3) t τ Lt s obsrv that x (t) is dtrmind by th convolution of x(t) with th signal : π t x( t) = x( t). (4) π t Th abov rlation allows th calculus of th spctral dnsity of x (t) : X ˆ ( jω) = F{ x(t) = F{ x(t) F π t (5) Or X ˆ ( jω) = X ( jω) F (6) π t Sinc: F = j sgn ( ω) π t It rsults: X ( jω) = X ( jω)[ j sgn ω] (7) Or: j X ( jω), ω > X ( jω) = (8) j X ( jω), ω < As a rsult, th spctral dnsity function of th x(t) signal s conjugat is obtaind by changing th phas of th spctral dnsity for X ( jω ) by ± π/. It rsults: H x(t) = xt ( ) = F - Xˆ ( jω) (9) Th invrs Hilbrt transform is dfind in rlation (3). W can writ: - x ( t) = H { x ( t) = H{ xˆ (t) () Taking into account rlation (8) it rsults: x(t) = H - F { jx(jωj ˆ { x(t) ˆ = - F { jx(jωj,ω > (),ω < Th analytic signal x w build th analytic signal z (t) : z ( t) = x ( t) + j x ( t) () W obsrv that: Z ( jω ) = F { z(t) = X (jω) + j X (jω) (3) Rfrring to rlation (7) w obtain: Having th pairs x (t) and ( t) = H{ x(t)

3 Volum 49, umbr 4, Z( jω) = X( jω) + j[ j sgn ω] X ( jω) = = X( jω)[+ sgn ω] = X( jω) u( ω ) (3.a) whr u (ω) is th unit stp function. It s usful to obsrv that: X ( jω) = [ Z ( jω ) + Z ( jω )] (4) X ( jω) = [ Z ( jω) Z ( jω) ] (5) j 3. DISCRETE HILBERT TRASFORM. CALCULUS ALGORITHMS. Dfinitions Having th signal x(t) dfind on th tim intrval [, t ], using a sampling priod T, w obtain th discrt signal x[ : xn [ ] = xnt ( ), n, (6) t Whr: T =. Th sampling frquncy f is chosn so f that th frquncy is gratr or qual to th last significant frquncy from th spctrum of x(t). W considr th discrt frquncy stp f π f =, ω f rspctivly. Th discrt Fourir transform (DFT) is: TFD - { x([ X[k] = x[, k, (7) = n= And th invrs discrt Fourir transform DFT - is: TFD - = -jnk π { X[ [k] x[ = X[ k], k, k = (8) Th sampl of th spctral dnsity corrsponding to frquncy k ω is dtrmind with th rlation: X ( j k ω ) = T X [ k] whr X(jω) is th Fourir transform in continuous tim. On th othr hand: Xk [ ]* = X [ k] = X[ k]. (9) jnk π Which shows that th sampl X [ k ] = X [ k ] has a corrspondnt sampl of th spctral dnsity, with th ngativ frquncy X ( kω ). For vn, th sampls X [], X[],..., x ar namd positiv harmonics, whil th sampls: X[ + ], X[ + ],..., X[ ], X[ ] X, X,..., X[ ], X[ ] ar namd ngativ harmonics. For odd, th sampls X[], X[],..., x ar calld positiv harmonics, whil th sampls: + + X[ ], X[ + ],..., X[ ], X[ ] X, X,..., X[ ], X[ ] ar calld ngativ harmonics. Th X[] componnt is th continuous componnt, whil th X ( - vn) is th yquist componnt. It is found whn th numbr of sampls,, is vn a situation frquntly found bcaus DFT is implmntd using an algorithm for which is vn. Also, Th X componnt is th continuous on. Similarly to rlation (), th discrt Hilbrt transform is dfind: H x[ = x[ = TFD X[ k] () Whr for vn: jx [ k], k =,, vn ˆ Xk [ ] = (a) jx [ k], k = +,, vn

4 488 ACTA ELECTROTEHICA W obsrv that th continuous and yquist componnts ar xcludd (for k = and k = ). Whil for - odd: jx[ k], k,, odd ˆ = Xk [ ] = (b) + jx [ k], k =,, odd Whr th continuous componnt is xcludd. Calculus Algorithms Rlation () can b writtn: x[ = H{ x[ = TFD { js[ k] X[ k] () Whr:, k =,, vn S[ k] =, k =, k = vn (3a) k = +,, vn And for - vn, k =,, odd Sk [ ] =, k=, odd (3b) + k =,, odd S[k] is a window which filtrs th intrst componnts of th Hilbrt transform. Obsrvations: Th squnc Sk [ ] can b obtaind: ) Using th function: π S[ k] = sgn sin k ; k =, (4a) ) Or th function []: S [k] = sgn[k]sgn[ ] (4b) Th following mthods of calculating th discrt Hilbrt transform rsult: 3.. Th invrs discrt Fourir transform algorithm Is basd on rlations () (.a) : ) W dtrmin th discrt Fourir transform of th numric squnc x [: X [ k] = TFD x[ ) W st th continuous componnt to zro: X[]= 3) If th lngth of squnc X[k] is vn, w st th yquist componnt to zro: X = 4) Th squncs X[k], k,, vn; or k =,, odd. (positiv harmonics) ar multiplid by j. 5) Th squncs + X [ k], k +,, vn; or k =,,, odd (ngativ harmonics) ar multiplid by +j. 6) W calculat th discrt Hilbrt transform using rlation (). 3.. Windowing th positiv and ngativ frquncis algorithm Is obtaind by applying a window to th positiv and ngativ spctral componnts, xcpt for th continuous and th yquist componnts. This algorithm is basd on rlations () (4): ) is chosn, th numbr of sampling points. ) S[k] is dtrmind, using rlation (4.b). 3) Th discrt Fourir transform of th numric squnc x[ is calculatd: X [ k] = TFD x[ 4) W dtrmin Xk ˆ [ ] = jskxk [ ] [ ] 5) And thn w dtrmin x[ = H x[ = TFD js[ k] X[ k] This algorithm is, at first viw, similar with th prvious on, xcpt for th last two stps.

5 Volum 49, umbr 4, Th convolution algorithm ) W dtrmin sn [ ] = TFD jsk [ ] (5) ) It rsults: x[ = x[ s[ That is: x[ x[ m] s[ n ] = m m= (6) (7) In [], s[ has this xprssion: π n π n s[ = sin cot, for vn (8) And s[ qual to zro for n=,,4,.... Whil π n cos( π n) sn [ ] = cot( ) for odd π n sin( ) (9) And s[ dosn t bcom qual to zro for n vn or odd. Lt s not that s [ k] = s[, n =,. This algorithm sms to b computd in a shortr tim. In fact, it rquirs a longr tim than th algorithms that us th discrt Fourir transform. This is xplaind by th fact that for DFT wr dvlopd fast calculus algorithms (FFT -Fast Fourir Transform). Obsrvation In litratur w mt th rlation: sn [ ] = TFD { Sk [ ] (3) And thn x [ is calculatd using this rlation: x [ = j m= x[ m] ~ s [ n m] = jx[ ~ s[ (3.a) 3.4. Windowing th positiv frquncis algorithm W now propos a nw algorithm for calculating th discrt Hilbrt transform. Similarly to rlation () th discrt complx analytic signal is dfind (also calld complx analytic squnc): z [ = x[ + j xˆ[ (3) Th discrt Fourir transform of th z[ signal is: Z [ k] = TFD{ z[ (3) Looking at (3 c) it can b writtn: Z[ k] = TFD{ x[ = X [ k], k =,, vn or: k =,, odd (3 a) Th window squnc R[k] is introducd: Or:, k = R[ k] =, k =,, vn (33 a), k =,, k = R [ k ] =, k =,, odd (33 b) +, k =, It can b obsrvd that: Rk [ ] = Sk [ ]( Sk [ ] + ) (33 c) In a mor compact form it can b writtn: Zk [ ] = RkXk [ ] [ ], k=, (34) This rlation rsults: zn [ ] = TFD Rk [ ] X[ k] (35) Whr DFT - has th maning of th invrs complx Fourir transform. It rsults that: x [ = R{ z[ (36) And: x ˆ[ = THD{ x[ = Im{ z[ (37) Rlations (3), (37) ar lading to an algorithm which allows calculating th discrt Hilbrt transform. Obsrvation: Th rlation (35) can also b writtn as:

6 49 ACTA ELECTROTEHICA M π jnk [ ] [ ] zn = Zk, M=, k = (38) for vn and M =, for odd. Th rlation (38) dosn t hav a practical us sinc th numric signal analysis nvironmnts alrady hav instrumnts for computing th DFT ( FFT ). As a rsult, it is prfrrd th algorithm xprssd by th rlations (3) (3). 4. IVERSE DISCRETE HILBERT TRASFORM Is dtrmind using th rlation: H H { x[ = { x[ (39) If, th squnc lngth is odd and th continuous componnt is missing w can writ: H { x [ = x[ (4) If is vn, whil th continuous componnt is diffrnt from zro or if is odd, that is w hav th yquist componnt, thn rlation (4) is not strictly tru any mor. Algorithms of th invrs discrt Hilbrt transform similar to thos prsntd abov for th dirct discrt Hilbrt transform rsult. 5. RESULTS AD COCLUSIOS This papr brifly prsnts known algorithms for calculating th Hilbrt transform and proposs and algorithm basd on th proprtis of th complx analytic signal. Th mthods of computation in tim and frquncy domains ar prsntd. For applications whr th hard/soft throughput as wll as tim ar important issus, this algorithm could rprsnt an advantag. ACKOWLEDGEMET This papr prsnts rsarch rsults of th CCSIS Program-o.556/7. Th scintific rsponsibility is assumd by th author. REFERECES. David G. Long, Ph.D. Commnts on Hilbrt Transform Basd Signal Analysis. MERS Tchnical Rport # MERS 4-.. Marpl, S.L., "Computing th discrt-tim analytic signal via FFT," IEEE Transactions on Signal Procssing, Vol. 47, o.9 (Sptmbr 999), pp Oppnhim, A.V., and R.W. Schafr, Discrt-Tim Signal Procssing, ndd., Prntic-Hall, Bracwll, R., Th Fourir Transform and Its Applications, McGraw-Hill, Fldman, M., "on-linar systm vibration analysis using Hilbrt Transform - I. Fr Vibration Analysis Mthod 'FREEVIB'", Mchanical Systms and Signal Procssing (994) 8(), Sanjit K. Mitra Digital Signal Procsing. A Computr-Basd Approach. McGraw-Hill Intrnational dition 6. ISB J.S. Bndat: Th Hilbrt Transform and Applications to Corrlation Masurmnts, Brul&Kjar, 985, BT Thran: Th Hilbrt Transform, Tchnical Rviw o , Brul&Kjar, BV M. Simon and G.R. Tomlinson. Us of th Hilbrt transform in modal analysis of linar and nonlinar structurs. Journal of Sound and Vibration (984) 96(4), pp Clarbout, J.F., Fundamntals of Gophysical Data Procssing, McGraw-Hill, 976, pp [] Mathias Johansson. Th Hilbrt Transform. Mastr Thsis. Mathmatics/Applid mathmatics. Vaxjo Univrsity.