Phase Difference Measurement Based on Recursive DFT with Spectrum Leakage Considered

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1 Vol.8, o. (5), pp Phase Difference Measureent Based on Recursive DF ith Spectru Leaage Considered Huiyue Yang, Yaqing u, Haitao Zhang 3 and Ming Li 4 Departent of Inforation Engineering, Logistical Engineering University, Chongqing, China huiyue_yang@63.co bstract his paper investigates spectru leaage influence on the perforance of discrete Fourier transfor (DF) based phase difference easureent and proposes a novel ethod ith spectru leaage considered. In the proposed ethod, spectru is firstly corrected by interpolation algorith to reove the influence of short range leaage. o avoid negative frequency interference caused by the long range leaage, real-tie phase difference is calculated in succession by a recursive DF based algorith ith negative frequency contribution. nd then, precision of the proposed ethod is theoretically deonstrated in detail. Furtherore, the ethod is applied to Coriolis ass floeter (CMF). Siulation and experiental results sho that the accuracy of phase difference easureent has largely iproved, as copared ith the existing ethod based on the sliding Goertel algorith. Keyords: phase difference; spectru leaage; discrete Fourier transfor; negative frequency. Introduction Phase difference easureent of cosine signals ith the sae frequency is an iportant topic in any easureent and signal processing tass, such as radar, sonar, counication, electrical technology, poer systes and industrial autoation. In doain of high-precision flo easureent, Coriolis ass flo eter (CMF) operates properly by easuring the tie interval of to vibration signals hich is depended on the phase difference and frequency []. lot of ethods for phase difference easureent have been developed to fulfill the coprehensive applications requireents. In the existing ethods, ero crossing detection based ethod [] just requires sall aounts of calculation, but poor in anti-interference ability and deands excessively for hardare. Digital correlation ethod [3] is good at suppressing rando noise, but has difficult to eliinate haronic interference. What s ore, this ethod need to no the signal frequency in advance or required to aintain full period sapling hich is often hard to realie in practice. Methods based on DF [4] calculate phase difference by the subtraction of to DF phases at the axiu spectral line. his ethod has good resistance to haronic interference. Hoever, an unfeasible technique of integral period sapling is also particularly needed [5-6]. Otherise spectru leaage taes place inevitability hich has a passive effect for precision of phase difference easureent [7]. he Hilbert transfor based ethod [8-9] hich calculates phase difference by trigonoetry operation beteen the signal and its Hilbert transfor is proposed to detect variable phase difference. his ethod can figure out the phase difference at every sapling point in real-tie and perfors perfectly hen the phase difference changes rapidly. Hoever, it has feeble iune fro interference, hich liits practicalities of ISS: IJSIP Copyright c 5 SERSC

2 Vol.8, o. (5) the ethod. sliding Goertel algorith (SG) [] is also introduced to easure phase difference. But there is a slo convergence rate and a nuerical overflo. sliding DF is developed to easure the variable phase difference. o carry out the change of phase difference, a sall nuber of sapled data are alloed to be taen in the DF calculation. hen, spectru leaage becoes inevitable and rearable. Hoever, the current phase difference easureent ethods based on DF spectru analysis rarely consider the ipact of spectru leaage, especially the ipact of negative frequency coponents []. his is hy the results obtained either by the sliding DF or by the sliding Goertel algorith deviate fro the true values. o solve the aforeentioned probles, a novel recursive DF ethod ith spectru leaage considered is proposed in this paper, hich is expected to iprove the accuracy of variable phase difference easureent and shorten the convergence stage of calculation. his paper is organied into seven sections. In Section, principle of DF-based phase difference easureent is deduced. In Section 3, ho spectru leaage reduces the precision of phase difference estiation is deonstrated fro to aspects of short-range spectral leaage and long-range spectral leaage. Based on the recursive DF, a novel ethod ith spectru leaage considered is proposed in Section 4. Precision of the proposed ethod is analyed in Section 5. In Sections 6 and 7, results fro siulations and experients in CMF are given to validate the proposed ethod.. Phase Difference Detection based on DF he phase difference equals the subtraction of to DF phases at the signal frequency. For to sine signals ith the sae frequency, the sapling sequences can be expressed as follos: s ( n ) c o s [ f n / f ] ( n ) s s ( n ) c o s [ f n / f ] ( n ) s () Where n,,,, and are aplitudes, and are initial phases, f ( f f ) is sapling frequency and f s s is the signal frequency. ( n) is the discrete tie indo, hose length is. he DF of s ( n) at f can be expressed as: j f j j j [( f f ) / ] j [( f f ) / ] ( ) ( ) ( ) ( ) ( ) ( ) S e f f e f f W f e W f f e W f f e It is hard to ae f / f s ( Z () ) in sapling; accordingly spectru leaage becoes inevitably. Mar as frequency deviation and f as frequency resolution. We can get f ( ) f, is a positive integer. Ignoring the negative frequency coponents, then: j [( f f ) / ] j ( ) S ( ) W ( f f ) e W ( f ) e (3) he phase of s ( n) can be estiated as follos:. Siilarly, the phase of s ( n) can be expressed as. hen, phase difference can be obtained by the subtraction of the to phases, i.e., ˆ (4) 56 Copyright c 5 SERSC

3 Vol.8, o. (5) It sees that the spectru leaage does not affect the phase difference estiation. But that is not the case; its reasons ill be deonstrated in the folloing. 3. Influence of Spectru Leaage 3.. Short-range Leaage: Spectral Lines Offset here are to types of spectral leaage: short-range leaage and long-range leaage []. Short-range leaage taes place in the ain lobe of spectru, hich ipact the phase difference estiation by. Under the bacground of the additive noise, sapling sequence can be represented as: x ( n ) s ( n ) ( n ), n,,, (5) Where s( n) is signal sapling sequence, and ( n) is hite Gaussian noise sequence hose variance is. he signal-to-noise ratio expressed as S R /( ). Short x( n) by a syetric indo ith sapling point, and then the sapling sequence can be expressed as: x ( n ) x ( n ) ( n ) ( n ) ( n ) (6) Ignore the negative frequency coponents of x ( ), that to say, only consider the first /part of spectru, there is: X ( ) S ( ) Z ( ),,, / (7) Generally, the axiu of aplitude spectru is obtained on line nuber or. Mar r as spectral line nuber corresponding to the axiu. On the condition of universal S R, it is satisfaction that Z R ( ) S ( ) and Z I ( ) S ( ) in. Set W ( f ) as continuous spectru of the syetric indo r ( n ), and then, e can get S ( ) W ( ) /. ccording to the forula (7), the phase spectru equals to: r X S ( ) Z ( ) ( ) a rc ta n S ( ) Z ( ) r r P r r r R R I I (8) ccording to the second-order aylor forula and ignore the higher order diensionless, forula (8) can be approxiately rerote as: P P (9) r r r X ( ) S ( ) Z P ( ) here R S ( ) S ( ) r I r R Z P ( ) Z ( ) Z ( ) r r r S ( ) S ( ) r r I P I P R c o s X ( ) Z ( ) sin X ( ) Z ( ) r r r r S ( ) r Mar P as average poer of the indo. s a result of the real and iaginary part of discrete spectru distribution of hite Gaussian noise sequence are subject to, P / and arbitrary to spectral lines of real part and iaginary part are independence ith each other, it s not difficult to no: R I R I () r r E ( Z ( )) E ( Z ( )), v a r( Z ( )) v a r( Z ( )) P / hus, taing the S R definition and syetric indo spectru into consideration, e can get: Copyright c 5 SERSC 57

4 Vol.8, o. (5) P P P P P E ( X ( )) S ( ), v a r( X ( )) r r r S ( ) S R W ( ) r () Using S R ( ) and S R ( ) as denotations of the to signal phase spectru respectively, r r e can obtain the variance of phase difference estiation, as follos: ˆ P var var S ( P ) var S ( ) P S R W r r ( ) () Obviously, the precision of DF-based phase difference estiation relates to S R, shape and length of the syetric indo and frequency deviation. 3.. Long-Range Leaage: egative Frequency Interference Long-range spectru leaage refers to the side-lobe spectru leaage, hich ay cause lines interference. For dense ulti-frequency signals, it is necessary to oo in spectru before phase difference estiation. For single frequency signal, long range spectru leaage ill cause negative spectru pea superposition to positive spectru, naely the negative frequency interference, hen the signal frequency is very lo or close to the yquist frequency. his phenoenon also taes place hen sapling point only a little hile. Here the ethod described by forula (-4) possesses a larger estiation error thans to the overlooing of negative frequency coponents. Denote and as the phases located the axiu and the sub-axiu spectru line respectively, and then, e coonly dee that there is interference spectru hen: (3) here is a little positive nuber. he phenoenon of negative spectru interference is shon in Figure. he real frequency is included in the range of (, ), naed positive frequency range. Range (, ) and range (, ) are syetrical spectru iage, called negative frequency range. When the signal frequency is lo or close to the yquist frequency, spectru pea closes to the ends of range (, ) and side-lobe interference increases rapidly, as shon in (b). When position of spectral pea is very close to or, and ill overlap and result in ain lobe interference, as shon in (c). Figure. Spectru ith egative Frequency Interference 58 Copyright c 5 SERSC

5 Vol.8, o. (5) 4. he Proposed Method 4.. Principle When a sall nuber of sapled data are taen in the DF calculation, the negative frequency interference becoes rearable, that brings about significant errors in phase difference estiation. herefore, both positive and negative frequency should be taen into consideration hen calculate the phase difference. In order to iprove the accuracy of phase difference easureent, a ethod ith spectru leaage considered is proposed. ae the negative frequencies into account, forula () equals to: n n j [ ( ) ] j n j [ ( ) ] j n S ( ) e e e e n n j n j n ( ) j j e e e e n n (4) If rectangular indo been used, e can deduce the folloing equation: tan ( ) c c tan c tan 3 (5) here is the phase of s ( ), c sin ( / ), c sin [ ( ) / ], c sin ( / ) sin [ ( ) / ]. 3 Siilarly, for s ( n ) tan ( ) c c tan c tan 3 (6) here is the DF phase of s ( n ) at. On the basis of (5) and (6), the forula for phase difference calculation is deduced as: cc (ta n ta n ) a rc ta n [ ] c c c (ta n ta n ) ( c c ) ta n ta n 3 3 If the Hanning indo is used, e can obtain another expression for phase difference calculation: (7) c (ta n ta n ) 4 a rc ta n [ ] c ta n ta n 4 (8) here, c 4 sin [ ( ) / ] D, sin ( / ) D D co s( / ) co s( / ) co s[ ( ) / ] co s( / ) D co s( / ) co s( / ) co s[ ( ) / ] co s[ ( ) / ] 4.. Recursive DF Recursive DF algorith introduced in this paper is based on the trigonoetric function. Supposing that e have obtained a sapling sequence x( n) at the point of, then x ( n ) x ( ), x (),, x ( ) and its DF is: Copyright c 5 SERSC 59

6 Vol.8, o. (5) j n X ( ) x ( n ) e x ( n ) co s n j sin n ( ) jb ( ) n n here (9) ( ) x ( n ) c o s n, n B ( ) x ( n ) sin n. n t the point of, the sapling sequence x( n) is updated by x( ) ith x ( ) discarded. Here, the DF of the updated x( n) are as follos: j ( n ) () n n X ( ) x ( n ) e x ( n ) c o s ( n ) j s in ( n ) ( ) jb ( ) here ( ) x ( n ) c o s ( n ) n n x ( n )[cos n cos sin n sin ] cos x ( ) cos x ( n ) cos n sin x ( ) sin x ( n ) sin n n n co s ( ) ( ) ( ) s in x x B ( ) Siilarly, B ( ) c o s B ( ) s in x ( ) ( ) x ( ) pparently, hen a ne point been taen into the sapling sequence, there is a recursive relationship beteen DFs of the to sequences Process ccording to the principle, process of the advanced ethod can be suaried as follos: Step: Saple to signals synchronously and intercept data ith the sae indo length. Step: Calculate the DFs of sapling signals s ( n) and ( ) s n. Step3: Correct the spectrus by interpolation algorith [3] and figure out. Step4: Calculate ( ) and B ( ). Step5: Calculate the phase difference by Eq. (7) or Eq. (8). 5. Precision DF of the signal defined by (5) can be express as: X ( ) S ( ) Z ( ) e b e,,,, j j () here S ( ) and Z ( ) are the DF transfors of s ( n ) and ( n ),, b and, are the j n aplitude and phase respectively, Z ( ) ( n ) e. s ( n ) is a Gaussian hite noise, n Z ( ) also obey Gaussian distribution, hose variance is v a r[ Z ( )]. For hite noise, there are E [ ( ) ( n )] hen n. herefore, autocorrelation of Z ( ) can be siplified as: 6 Copyright c 5 SERSC

7 Vol.8, o. (5) j ( l ) n j ( l ) n *, h e n l E [ Z ( ) Z ( l )] E [ ( n ) ( n )] e e n n, h e n l () lthough Z ( ) is a linear cobination of the sae set of rando variables, there is a lac of correlation in Z ( ) as a result of the orthogonal feature of DF basis function. hereby Z ( ) is coplex Gaussian hite noise, ith E [ b cos ] E [ b sin ] E [ Z ( )] and var[ b cos ] var[ b sin ] var[ Z ( )] / /. t the axiu, DF transfor of x ( n) can be represented as: R j f j ( ) S ( ) Z ( ) e be (3) here are the aplitude and phase of S ) respectively, and, f sin [sin ( ) co s( ) c / c ] sin sin ( ) c / c (4) f f 3 f f 3 ( hen, e rerite (3) as: j b f j ( f ) R ( ) e [ e ] j b b f e [ c o s ( ) j s in ( f f )] (5) b b ( ) c o s ( ) e f j f here is the phase of S ) f (,and b sin ( ) / arctan [ ] b co s( ) / (6) It is generally believed that b / hen S R is not particularly lo and is sufficient large. herefore: b arctan [ sin ( )] (7) f f f f here b and are stochastic hile and are invariable. b s in ( ) / f f Gaussian hite noise, and v a r[ ] /. ith Siilarly, e can obtain: f f is (8) here b sin( ) / is the phase of S ) and obey Gaussian distribution, f v a r[ ] / (. hen the phase difference can be get by: f 3 arctan [ ] (9) here f f c c c (tan tan ), ( c c ) tan tan, c c (tan tan ). f f f f 3 f f f f f 3 f f f 3 f f f f Set ] [ ] [ f f and E ] [ ] [, then f ( ) f f. t the point of E ] [ ] [ ] μ, the first order aylor series expansion of ( ) [ f f f is: Copyright c 5 SERSC 6

8 Vol.8, o. (5) f ( ) f f ( ) ( ) (3) hen c c (tan tan ) E[ ] f ( ) arctan[ ] f f f f μ (3) f f f f f v a r[ ] E f ( ) ( μ ) E [ ] C μ μ μ (3) here C is the covariance atrix of atrix. f c c sec [ c tan ( c tan c ) ] f f f f f f 3 f f f [ ] f f f 3 f c c sec [ c tan ( c tan c ) ] f f f f f f 3 f f f [ ] f f f 3 f μ f f ypically, Gauss hite noise is unrelated ith each other. hereby and relevant too. So is not / C (33) / hen, e can deduce that var[ f f ] ( ) (34) 6. Experient Results o verify the influence of spectral leaage and evaluate the perforance of the proposed ethod, coputer siulations are carried out, assuing that the signals are single-frequency real signals ith hite Gaussian noise. In siulations, the initialiation of phase difference is Influence of Frequency Deviation and Windos In siulations, the signal frequency equals 9 8 H, the sapling frequency equals H, S R db. 4. Rectangular indo, Hanning indo and Haing indo, are used respectively in siulations. Phase differences are calculated ties independently by the DF-based ethod ith ratio correction ethod as coparison. he root ean square error (RMSE) of phase difference estiation and the relationship beteen the results and deviation are shon in Figure. he theoretic values of RMSE are denoted by lines hile siulation results are shon by discrete points. s can be seen fro the Figure, the siulation results agreed ith the theoretical value. RMSE of the uncorrected DF-based ethod deduce ith the deviation closing to ero. We can also coe to the conclusion that RMSE ith the Hanning indo and Haing indo are siilar. When is sall, the RMSE ith a rectangular indo is saller copared ith Hanning indo or Haing indo. Hoever, the conclusion reverses hen is rearable. fter corrected, the RMSE is 6 Copyright c 5 SERSC

9 RMSE / RMSE / RMSE / RMSE / International Journal of Signal Processing, Iage Processing and Pattern Recognition Vol.8, o. (5) alays in a near constant value and has nothing to do ith. he constant value (about. 5 3 ) is deterined by SR and (a)rectangular indo heoretic value DF calculation Corrected DF calculation.45.4 (b)hanning indo heoretic value DF calculation Corrected DF calculation.45.4 (c)haing indo heoretic value DF calculation Corrected DF calculation δ δ δ Figure. Relationship beteen RMSE and the Deviation 6.. Influence of the SR and Sapling Length Under the condition of 4, and.4, the relationship beteen the RMSE and SR is sho in Figure 3. he phenoenon of RMSE changes ith sapling length under the condition of 3,.4 and S R db is also denoted in Figure 3. It can be seen that the siulation results and forula calculation results are consistent. What s ore, the higher S R, the saller the RMSE of phase difference estiation heoretic value =64 =8 =56 =5 =4 = SR /db Figure 3. Relationship of RMSE ith SR and Sapling Length 6.3. Precision of the Proposed Method In siulations, the nuber of sapled points equals 4, the sapling frequency equals H and the frequency resolution is f f / H. he indos of s rectangular and Hanning are used respectively. In order to visually reflect the inherent la of signal spectru, signal frequency is selected ith frequency resolution as a basic unit. Signal frequency range fro.5 f to.5 f and f to f, step length equals. 5 f. he phase differences are calculated by the proposed ethod and the relative errors of calculation are shon in Figure 4. pparently, the proposed ethod has higher precision than universal DF-based ethod. he error of DF ethod fell sharply close to the loer liit of double precision arithetic hen the relative frequency f / f d is an integer. his is because side lobe of the negative frequency in positive spectru is exactly ero and doesn t have any ipact on positive spectru. Copyright c 5 SERSC 63

10 Relative error /% International Journal of Signal Processing, Iage Processing and Pattern Recognition Vol.8, o. (5) -5 DF ith rectangular indo DF ith Hanning indo he proposed ethod ith rectrangular indo he proposed ethod ith Hanning indo f / f d Figure 4. Relative Error of Phase Difference Measureent 6.4. pplication in CMF o experientally validate the proposed ethod, a syste for CMF signal processing has been developed, as shon in Figure 5. We select a ind of CMF(FS) ith a 7R transitter. 4-channal dynaic signal acquisition I 934 hich operates ith sapling rate of H is used to saple the CMF s oscillation signals. he signal frequency of CMF sensors equals about 98H. Mass flo easured by the scale is deeed as the actual value. D5 Pup Valve CMF(FS) CMF(Q-884) Water tan Valve scale Valve3 Figure 5. Bloc of the Experiental Syste ccording to CMF s principle, the ass flo rate is calculated by the tie interval t, hich depends on the frequency and the phase difference. he SG is taen as coparisons. able. Experiental Results Mass flo (g/in) heoretic value of tie interval(g/in) he SG (g/in) he proposed ethod (g/in) s shon in able, the results of the proposed ethod are ore close to the theoretic values copared ith the SG. hen, e coe to the conclusion that the proposed ethod is effective and practical. 7. Conclusion For phase difference easureent ethod based on DF, there is a proble unable to slide over that spectru leaage has a disadvantage ipact on precision. In this paper, e deonstrate ho spectru leaage influence the perforance of the DF based ethod and put forard a novel ethod based on recursive DF ith short range leaage and long range leaage both been considered. Siulation and experiental results sho that the accuracy of phase difference easureent has largely iproved copared ith the SG. Precision of the proposed ethod can be iproved father by selected appropriate 64 Copyright c 5 SERSC

11 Vol.8, o. (5) indos. What s ore, the proposed ethod can also be used to trac dynaic phase difference. Further research is under discussion. cnoledgent he authors greatly appreciate the financial supports fro the ational atural Science Foundation of China (67449, 6375) and the Chongqing atural Science Foundation of CSC (CSCjj46, CSC3jcyj43). References []. Martin, D. Wolfgang and R. lfred, Coriolis ass floeters: Overvie of the current state of the art and latest research, Flo Measureent and Instruentation, vol. 7, (6), pp [] J. J. Zheng and S. B. Yu, Research of real-tie detection technology of phase, Electrical Measureent & Instruentation, vol. 4, no. 456, (3), pp [3] Y. Q. u,.. Shen, M. Li and H.. Zhang, Research on phase difference easureent algorith for non-integer period sapling signal based on ulti-layer correlation, Chinese Journal of Scientific Instruent, vol. 35, no. 7, (4), pp [4] Y. Q. Jiang and Y. G. He, e algorith for high-accuracy phase difference easureent based on indoed DF, Journal of Circuits and Systes, vol., no., (5), pp. -6. [5] J. Li and Y. F. Wang, DF phase estiation algorith and noise sensitive frequency region Journal of Electronics & Inforation echnology, vol. 3 no. 9, (9), pp [6] S. Schuster, S. Scheiblhofer and S. ndreas, he influence of indoing on bias and variance of DF-based frequency and phase estiation [J], IEEE ransactions on Instruentation and Measureent, vol. 58, no.6, (9), pp [7] S.S. Young, K.J. Ho, B.W. Weui, Y. Su and J.. Se, study on the leaage error in the spectru of acoustic intensity, JSME International Journal Series, vol. 47, no., (4), pp [8]. M. Vuˇcija and L. V. Saranovac, siple algorith for the estiation of phase difference beteen to sinusoidal voltages, IEEE ransactions on Instruentation and Measureent, vol. 59, no., (), pp [9] Y. Q. u, H.Y. Yang, H.. Zhang and X.Y. Liu, CMF signal processing ethod based on feedbac corrected F and Hilbert transforation Measureent Science Revie, vol. 4, no., (4), pp [] K. J. Xu and W. F Xu, signal processing ethod based on FF and SG for coriolis ass floeters C Metrologica SIIC, vol. 8, no., (7), pp [] Y.Q. u, H.. Zhang, Y.W. Mao,.. Shen and H.Y. Yang, Unbiased phase delay estiator ith negative frequency contribution for real sinusoids, Journal of pplied Sciences, vol. 3, no. 8, (3), pp [] K. F. Chen, J.. Jiang and S. Crosen, gainst the long-range spectral leaage of the cosine indo faily Coputer Physics Counications, vol. 8, (9), pp [3] K. F. Chen, J. L. Yang and S. W. Zhang, Spectru correction based on the coplex ratio of discrete spectru around the ain-lobe, Journal of Vibration Engineering, vol., no. 3,(8), pp uthors Huiyue Yang, he received his B.S. degree in utoatic fro Logistical Engineering University, China, in 9 and his M.S. degree in Detection echnology and utoation Devices in. o, he is a Ph.D. candidate in Logistical Engineering University. His ain research interests include signal processing, intelligent detection and instruentation. Yaqing u, he received his B.S. degree in 984 fro autoatic control engineering in Chengdu university of Science and echnology, China, M.S. degree in 99 fro autoatic control theory and application in Chongqing University, China, and Ph.D. degree in 994 fro precise instruent and achinery in Chongqing University. o he is a professor and Ph.D. supervisor in Logistical Engineering University. His ain research interests include intelligent detection and instruentation, intelligent autoation syste. Copyright c 5 SERSC 65

12 Vol.8, o. (5) 66 Copyright c 5 SERSC