HILBERT TRANSFORM ALGORITHM IN LABVIEW

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1 Scienific Bullein of he Peru Maior Universiy of îrgu Mureş Vol. 0 (XXVII) no., 03 ISSN-L (Prin, ISSN X (Online), ISSN (CD-ROM) HILBER RANSFORM ALGORIHM IN LABVIEW Oana MUNEAN echnical Universiy of Cluj-Napoca, Romania, Faculy of Elecrical Engineering, George Bariţiu s. no. 6-8, Cluj, Phone , Fax , oana.munean@ehm.ucluj.ro Absrac his paper presens an alernaive algorihm for Hilber ransform calculaion. his algorihm was implemened in LabVIEW sofware. I was esed for some differen elemenary signals. he resuls were compared wih he mehod proposed by LabVIEW programming environmen of Naional Insrumens Company. Keywords: Hilber ransform, Fourier ransform, analyical signal.. Inroducion I is known ha he Discree Fourier ransform is he mahemaical ool used in signal processing and analysis. Also, in programming environmens (such as Malab, Mahcad, LabView, C + +), he procedures for calculaing he Discree Fourier ransform are presened by various algorihms. his paper proposes an algorihm for compuing he Discree Hilber ransform, saring from known mehods for calculaing he discree Fourier ransform. Wih he mehod proposed in he paper, he user of a paricular programming environmen can quickly implemen he algorihm of Discree Hilber ransform. Hilber ransform is a linear operaor, inroduced in 905 by David Hilber. I is used in mahemaics and signal processing. Alhough i was discovered more han a cenury ago, i was lef over a while. Wih he developmen produced in he field of compuer echnology, i was possible o board wih success he Hilber ransform. Hilber ransform has shown is imporance no only as a mahemaical ool bu also in engineering. here are muliple applicaions of Hilber ransform in elecrical drives, auomoive, geophysical, mulimedia, radio and elevision. Hilber ransform is used in signal heory for analyical signals synhesis in he modulaion process. In Elecrical Engineering, Hilber ransform is well known because i manages o do he ± π/ phase shif exising among sinusoidal signals. An imporan aspec abou Hilber ransform is ha i keeps he original signal domain. his presen aricle proposes o develop a ime domain algorihm calculaion of Hilber ransform. I also proposed o sudy he algorihm behavior for differen inpu signals. In he beginning par of his paper, are presened heoreical consideraions of elemenary signals and he ransiion from ime o frequency domain. hen, i is se ou he Hilber ransform boh in ime domain and frequency domain. Hilber ransform of a real signal is inroduced by convoluion, as an imaginary par of he analyical signal and by ±π/ phase shif. Afer he Hilber ransform algorihm is developed, i is implemened in LabVIEW environmen. he procedure sars wih a harmonic signal. his signal is sampled and hen i is passed from ime domain o frequency domain. he signal suppors some mahemaical ransformaion before i reurns o ime domain. hus is obained an algorihm for compuing he Hilber ransform in ime domain. As well, his paper will highligh he Hilber ransform for differen inpu signals. ([3]-[7]). General presenaion here is informaion ha canno be highlighed in ime domain signals. For hose signals i is used frequency domain. Wih Fourier ransform i is possible o pass a signal from ime domain o frequency domain. Coninuous Fourier ransform: j X(j ) F{} e d () F X(j {X(j )e j )} d () A discree signal is given by a vecor of daa acquired from a ime coninuous signal. A ime coninuous signal is called analog. A discree signal is 49

2 called digial. Le here be x ( a ime coninuous signal. x [n] is obained by ime discreizaion of x ( wih. Where is he sampling sep. n, x [n ] x[n] k, X [jk ] X[k], Discree Fourier ransform: N jkn X [k] FD{x[n]} x[n]e N (3) k 0 N jkn X [k] FD {X[k]} X[k]e N (4) N n 0 Elemenary signals: his aricle uses he following signals: sine wave, square wave, riangle wave and sawooh wave. Hilber ransform. Definiions. Properies For x (, Hilber ransform produces a H {} funcion. In signal processing i is found like x ˆ(. H (} xˆ( [] (5) where x ˆ( is called he conjugae of x (. a. Hilber ransform defined by he convoluion produc Hilber ransform represens in fac he convoluion beween x ( and. H {} (6) xˆ ( v.p. ) d (7) xˆ( [lim ) d 0 (8) lim ) d ] 0 he negaive form of he original signal can be obained if Hilber ransform i is applied wice: [ x ˆ( ] H {ˆ( x } (9) he Hilber ransform used four imes on he same real signal gives us he original funcion back. b. Hilber ransform defined as imaginary par of an analyical signal A Hilber pair is formed by x ( signal ogeher wih he x ˆ( signal. An analyical signal is a complex signal. he real par is given by x ( and he imaginary par is given by x ˆ(. Le here be he analyical signal z (, hen: z ( jxˆ( (0) he module of he complex signal z ( is: z( x ( and i is called analyical signal envelope, and he phase of he complex signal is: xˆ ( () xˆ( arg z ( an () H {} Im{z(} (3) c. Hilber ransform defined by phase shif: By applying he Hilber ransform on a signal under harmonic form, each cosine harmonic becomes a specral componen wih he same magniude and wih ( ) phase shif. his mehod can be applied on periodic signals. For example i is given he signal cos( 0 ) and Hilber ransform is calculaed as follows: H{cos( 0 } cos( 0 ) sin( 0 (4) H{sin( 0 } sin( 0 ) cos( 0 (5) where 0 0 Hilber ransform in frequency domain: j,f 0 F { } 0,f 0 (6) j,f 0 In frequency domain, Hilber ransform is obained by muliplying he values of Fourier ransform a posiive frequencies wih j and a negaive frequencies wih j In fac, by muliplying by j he ±π/ phase shif i s realized. H{X(j )} Xˆ (j ) (7) jsgn( )X(j ) Inverse Hilber ransform in frequency domain: [Xˆ (j )] ( jsgn( )) X(j ) sgn ( )X(j ) X(j ) (8) F{[Xˆ (j )] } H{H{}} (9) Le here be z ( jxˆ( he Fourier ransform of z ( is: F{z(} Z(j ) X(j ) j( jsgn( )X(j )) (0) Z (j ) X(j ) sgn( )X(j ) () Where: Z (j ) X(j )[ sgn( )] () 0,if 0 sgn( ) (3),if 0 50

3 X(j ) ( ),if 0 Z (j ) (4) 0,if 0 I can be seen ha he analyical signal has always a null specral characerisic for negaive frequencies. ([]-[], [9], []-[4]) 3. Hilber ransform algorihm o complee he Hilber ransform algorihm, i is necessary o follow he nex seps: ) Discreize he signal ) Deermine he Discree Fourier ransform 3) Muliply he obained signal by ( j ) 4) Apply he Invers Fourier ransform ) he discree signal: x[n] {0;0.063;0.3;0.9;0.5;0.3; 0.37;0.43} ) he Fourier ransform: X[k] {.7 0.0i; i; 0. 0.i; 0. 0.i; 0. 00i; 0. 0.i; 0. 0.i; i} 3) he signal muliplied by ( j ) : Xˆ [k] { 0.7i; i;0. 0.i;0. 0.i; 0.i;0. 0.i;0. 0.i; i} 4) he Inverse Fourier ransform: xˆ( {0.3; 0.06; 0.06; 0.; 0.; 0.06; 0.06;0.4} For example, i is given x ( Fig. : Virual Insrumen Block Diagram 4. Device Descripion he implemened virual insrumen is a relaive simple one and i can be observed in Figure. o generae he signal, he program uses subrouine Simulae Signal. o deermine Hilber ransform i is uses he exising subvi Fas Hilber ransform.vi. o discreize he signal, he program uses subrouine Conver from Dynamic Daa o pass from ime o frequency domain, he program uses subvi FF.vi : o pass from frequency o ime domain, he program uses subvi Inverse FF.vi. o generae he discree signal sir[n]={,,...-,- }, i is creaed a subvi, Generae sir : Fig. : I is generaed a signal wih { } values. 5. Graphical Fig. 3: Virual Insrumen Block Diagram o deermine Hilber ransform by wo differen mehods For a sinusoidal signal inpu: As i can be seen in Figure 4 Hilber ransform is deermined wih wo differen mehods: ) he exising mehod in LabVIEW - Fas Hilber ransform.vi. ) he algorihm creaed and implemened in LabVIEW - Hilber algorim.vi. hen, i was chosen a sine wave signal. he Hilber ransform was deermined wih each mehod. he resuls were compared. 5

4 a) Fig. 6: Hilber ransform for a riangle wave inpu signal For a sawooh wave inpu signal: b) Fig. 4: Hilber ransform of he Sine wave signal inpu a) wih exising mehod b) wih implemened mehod For a square wave inpu signal: a H{ a (} ln (5) a Fig. 5: Hilber ransform for a square wave inpu signal For a riangle wave inpu signal: H { a, a} {ln a a ln } a a (6) Fig.7: Hilber ransform for a saw ooh wave inpu signal ([8], [0], [5]-[9]) 6. Conclusions ime domain calculaion of Hilber ransform was performed using polynomial approximaions. his algorihm is difficul o implemen. he numerical mehod presened in his paper allows he passage form frequency domain o ime domain, and vice versa, using he fas Fourier ransform ool ool which involves a small amoun of compuaion and high compuaion speed. his paper offers an alernaive mehod for Hilber ransform calculaion. he algorihm obained was implemened in LabVIEW sofware. he resuls were compared wih he mehod proposed by LabVIEW environmen. he Hilber ransform can be used o creae an analyic signal form a real signal. As we seen i is possible o use he muliple Hilber ransform o calculae he inverse Hilber ransform. Acknowledgmen his paper was suppored by he projec Docoral sudies in engineering science for developing he knowledge conrac no. 494/

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