America Joural of Egieerig Research (AJER) 5 America Joural of Egieerig Research (AJER) e-issn: 3-847 p-issn : 3-936 Volume-4, Issue-4, pp-3-4 www.ajer.org Research Paper Problem of calculatig time delay betwee pulse arrivals Ope Access Olugboji Oluwafemi Ayodeji., Jiya Joatha Yisa. Ajai Clemet Kehide 3 Mechaical Egieerig Departmet, Federal Uiversity of Techology, P.M. B. 65, Mia, Niger State. NIGERIA. ABSTRACT: This work compares the performace of four differet methods of estimatig the time delay betwee pulse arrivals at the sesors subjected to differet levels of atteuatio, distortio ad oise. The accuracy of the calculated time betwee the pulse arrivals at the sesors is determied ad aalysed for each of the methods based o the ideal atteuatio (o chage i shape), ideal atteuatio with added oise to the pulse sigal ad ideal atteuatio but with distortio. Based o the aalysis carried out, it is clear the cross correlatio method gives the best estimate of the delay i pulse arrival times irrespective of the sigal to oise ratio ad so is the preferred techique used i the remaider of this research. KEYWORDS: atteuatio, pulse, sigal, distortio, correlatio I. INTRODUCTION Whe calculatig the locatio of a evet, the ability to accurately determiethe time delay t ) betwee the ( t arrivals of a pulseat the sesors is very importat. Durig the propagatio of the pulse due to a evet as show i Figure, the shapes ad amplitudes of the pulse sigals at the sesors are differet due todistortio, frequecy depedet atteuatio ad oise. If sesor is closer to the evet, the pulse travellig to it suffers less propagatio loss tha does that travellig to sesor, ad therefore has a larger amplitude ad higher sigal to oise ratio, ad is also less distorted. This effect of distortio, frequecy depedet atteuatio ad oise o the pulse shape itroduces ucertaity i the measuremet of the time of arrivals ad hece i the estimatio of the time delay betwee arrivals, thus resultig i ucertaity i the locatio of the evet site. Figure Schematic represetatio of sesors o a pipelie w w w. a j e r. o r g Page 3
America Joural of Egieerig Research (AJER) 5 Atteuatio As the pulse propagates alog the pipelie, its itesity dimiishes with distace. The amplitude of the pulse is reduced by the spreadig of the pulse. Further weakeig of the pulse results from scatterig, which is the reflectio of the pulse i directios other tha the directio of propagatio. Atteuatio is the decay rate of the pulse as it propagates alog the pipelie ad is ormally expoetial. The amplitude chage of the decayig pulse ca be expressed as; A A e x () o I this expressio A is the uatteuated amplitude of the propagated pulse at some locatio. The amplitude A is the reduced amplitude after the pulse has travelled a distace x from that iitial locatio. The quatity is the atteuatio coefficiet of the pulse travellig i the x-directio. This value of the atteuatio is geerally proportioal to the square of frequecy [] ad it ca be obtaied experimetally.. Dispersio Dispersio is a pheomeo caused by the frequecy depedece of velocity of the pulse. As the pulse propagates, the phase relatio betwee the spectral compoets of the pulse varies with distace ad hece the pulse shape becomes progressively distorted, geerally wideig as the propagatio distace icreases. A pulse ormally comprises a rage of differet frequecies. The pulse as a whole propagates at the group velocity, whereas each frequecy compoet propagates withi the pulse at its ow phase velocity. This, coupled with the frequecy depedet atteuatio, ca cause the pulse shape to chage cosiderably with log distaces. II. METHODS OF CALCULATING TIME DELAY BETWEEN PULSE ARRIVALS The four methods of estimatig the time delay i arrivals betwee the pulse as cosidered by this research are: (i) peak detectio (ii) crossig (iii) cross correlatio (iv) pulse cetroid. Peak Detectio This process ivolves locatig the positio of the highest peak of the pulse sigals at the sesors. Cosider the pulse propagatig alog the pipelie as illustrated i Figure ; the geerated pulse propagates i both directios of the pipelie arrivig first at sesor the later after some time at sesor. Figure shows the arrival of a idealised pulse at sesors ad. The true arrival times are the poits where the sigals first leave the zero level ad the measured arrival times are the poits of highest magitude. If the pulses are the same shape, i.e. there is o distortio durig the propagatio, the the delay (d ad d) betwee the measured ad true time of arrivals of the pulse sigals at the sesors are the same ad the calculated time delay is correct but the distortio of the pulse will itroduce error i the calculatio..7.6.5.4.3 d - delay betw ee measured ad true arrival time of pulse at sesor d - delay betw ee measured ad true arrial time of pulse at sesor - true time delay.. -. -. d d 4 6 8 4 6 8 Figure Pulse arrival time measuremetsby peak detectio w w w. a j e r. o r g Page 33
America Joural of Egieerig Research (AJER) 5 Figure 3 shows the effect of oise o the estimate of the time delay by peak detectio. The additio of oise to the otherwise udistorted pulse sigals at the sesors results i the blurrig of the peaks which makes it ucertai where the highest istataeous peak value will occur. The horizotal lies withi the evelope boudig the oisy sigals give the rage of possible measured peaks sice the highest measured must lie o or above these lies. These are show expaded i Figures 4a ad 4b. If the oise level is the same at both sesors as show, the the amout of ucertaity at sesor is greater because the smaller sigal has a lower sigal to oise ratio. This ucertaity i the measured arrival times of the pulse sigals results i ucertaity i the estimate of the time delay ad hece i the estimate of the locatio of the evet site..7.6.5 A A, B - limit of ucertaity i measured arrivals time at sesor ad - true time delay - oise level.4.3. B. -. -. 4 6 8 4 6 8 Figure 3 Effect of oise o pulse arrival time by peak detectio.6 c c - ucertaity i arriv al time at sesor - oise lev el.6 c - ucertaity i arriv al time at sesor - oise lev el.5.5.4.4 c.3.3.... 4 6 8 4 6 8 (a) sesor (b) sesor Figure 4 Showig detailed sectio of limit of ucertaity (c ad c) i arrivals time by peak detectio at the sesors with the same level of oise () Figure 5 shows the effect of distortio o the estimate i time delay i arrivals betwee the pulse sigals by peak detectio. The shape of the pulse sigal at sesor idicates that the high frequecy compoets of the pulse resposible for the sharp pulse rise have atteuated i the course of pulse propagatio, resultig i the peak of the pulse arrivig late. With the pulse sigal at sesor distorted, the delay (d) betwee the measured ad true arrival times at this sesor is greater tha the delay (d) at sesor. The differece i estimate of the time delay of the measured ad true arrivals betwee the pulse sigals is large i this case, but it could be small if the distortio happeed to leave the maximum poit i the same place. This suggests that the error obtaied i the estimate of the time delay i arrivals will deped o the form ad amout of distortio ecoutered by the pulse durig propagatio. w w w. a j e r. o r g Page 34
America Joural of Egieerig Research (AJER) 5.7.6.5.4 d - delay betw ee measured ad true arrivals time of pulse at sesor d - delay betw ee measured ad true arrivals time of pulse at sesor - true time delay.3.. -. d d -. 4 6 8 4 6 8 Figure 5 Effect of distortio o pulse arrival time by peak detectio. Threshold Crossig The crossig method is aother simple techique for calculatig the time delay i arrivals usig estimates of the arrival times []. The time of arrival is take as the time whe the pulse first crosses a predetermied level as show i Figure 6. For this case where the pulse is atteuated but udistorted, the delay (d ad d) betwee the measured ad true arrival times of the pulse sigals are quite differet ad the estimate of the time delay is overestimated..7.6.5.4.3 d - delay betw ee measured ad true arrival time of pulse at sesor d - delay betw ee measured ad true arrial time of pulse at sesor - true time delay.. -. -. d d 4 6 8 4 6 8 Figure 6 Pulse arrival time measuremetsby crossig A practical limitatio i the use of the crossig method is show i Figure 6 where settig of the level too high (above.7, the highest peak of the pulse at sesor ) would result i o-detectio of the pulse at sesor, makig it impossible for the estimate i the time delay to be determied. This requires some advace kowledge of pulse height ad limits the smallest pulse that ca be detected i the presece of oise. To esure large sigals, sesors therefore eed to be spaced relatively closely alog the pipelie thus makig this process of damage detectio expesive. w w w. a j e r. o r g Page 35
America Joural of Egieerig Research (AJER) 5.7.6.5 A, B - limit of ucertaity i measured arrivals time at sesor ad - true time delay - oise level.4.3 A.. B -. -. 4 6 8 4 6 8 Figure 7 Effect of oise o pulse arrival time by crossig Figure 7 shows the effect of oise o the estimatio of the time delay i arrivals betwee the pulse sigals based o the crossig. The regio of the crossigs are show expaded i Figure 8. Figures 8a ad 8b shows how the ucertaity i the arrivals time measuremets of the pulse sigals at the sesors are determied. The slope of the true pulse sigals at the crossig level is take as the uderlyig slope of the oisy pulse sigals. The oise level is measured ear to the pulse ad is assumed to be approximately costat over the short period of the pulse. The ucertaity of the arrival time, obtaied from the first crossig of the pulse sigal at sesor is thus give by equatio ; c () ta Similarly, the ucertaity i the arrival time at sesor is give by equatio 3; c (3) ta The values of c ad c give the ucertaity i arrival times at sesors ad, respectively..6.5.4 c - ucertaity i time arriv al at sesor - oise lev el slope =/c c c.5.4 c - ucertaity i time arriv al at sesor - oise lev el slope of the true sigal at the lev el slope = /c.3.3 c c... true sigal. true sigal -. 5 5 -. 4 5 6 7 (a) sesor (b) sesor Figure 8 Showig detailed limit of ucertaity i the measured arrivals time by the crossig. w w w. a j e r. o r g Page 36
America Joural of Egieerig Research (AJER) 5 The calculated ucertaity of the pulse at sesor which has a steep slope is smaller tha that of the pulse at sesor havig a less steep slope at the crossig. This clearly shows that as the pulse propagates farther away from the evet site, the ucertaity i measuremets of arrival times of the pulse icreases. Thus, the estimate of the time delay betwee the measured ad true arrivals will be sigificatly large usig the crossig whe the sigal to oise ratio is small. Settig of the level is also a importat task i the use of this method with the presece of oise. Although, settig the level reduces ucertaity i the time of arrival measuremet, it presets the icoveiece of a high probability of false alarm. It causes the backgroud oise to cross the, resultig i a great deal of uwated data beig recorded. O the other had, a high level result i greater ucertaity but reduced probability of false alarms due to oise. Figure 9 shows the effect of distortio due to frequecy depedet atteuatio o time delay estimate by the crossig method. I this case, there is a large differece i the delay (d ad d) betwee the true ad measured arrival times; hece there will be a large error i the calculated time betwee arrivals. This also suggests that the magitude ad form of the distortio determies the size of the error i the calculated time delay usig this method..7.6.5.4 d - delay betw ee measured ad true arrivals time of pulse at sesor d - delay betw ee measured ad true arrivals time of pulse at sesor - true time delay.3.. -. d d Figure 9 Effect of distortio o pulse arrival time by crossig.3 Cross Correlatio Cross-correlatio is a method used to determie the similarities betwee two sigals as a fuctio of a time lag applied to oe of the sigals [3]. The cross correlatio A ( t ) CC ( ) of two sigals, x ( ) ad x ( ) is give by equatio 3; t CC x x ( ) x ( t ) x ( t ) dt (4) -. 4 6 8 4 6 8 Similarly, for discrete-time sigals, this equatio is writte as i equatio 5; N CC ( m ) x ( i ) x ( i m ) (5) X X N i The process of cross correlatio of the pulse sigals ivolves movemet of the propagatig pulse sigal at sesor alog the time axis by a small time icremet ad lookig at the similarities it has with the pulse sigal at sesor. By calculatig the cross correlatio fuctio of the sesor sigals a measure of the time delay is obtaied from the positio of the maximum peak of the cross correlatio fuctio as illustrated i Figures a ad b. Figure b shows the cross correlatio fuctio with a clear peak idicatig the locatio of the maximum best fit betwee the two pulses i Figure a. A advatage of usig this method is that it does ot require measuremets of the time of arrivals of the pulse sigals at the sesors i order to calculate the time delay, but rather gives the delay directly. xx x x t w w w. a j e r. o r g Page 37
America Joural of Egieerig Research (AJER) 5.7.6 - true time delay 4 35.5 3.4.3.. 5 5 -. 5 -. 5 5 5 5 (a) pulse sigals (b) cross correlatio fuctio Figure Pulse time delay estimatio by cross correlatio Figure shows the cross correlatio of the pulse sigals with the additio of oise. It ca be see that eve with the additio of oise to the pulse, performig a correlatio betwee the pulse sigals still gives a good estimate of the measured time delay i arrivals of the pulse sigals. This is attributable to the cross correlatio method actig as a itegrator (equatio 5) which averages the radom oise preset i the pulse sigals..7.6 - true time delay - oise level 4 35.5 3.4.3.. 5 5 -. 5 -. 5 5 5 5 (a) oisy pulse sigals (b) cross correlatio fuctio Figure Effect of oise o pulse time delay estimatio by cross correlatio Figure shows the effect of distortio o the time delay measuremet by cross correlatio. There is some error usig this method, though ot as much as the ucertaity i the peak detectio ad crossig methods. The magitude of the cross correlated peak is lower due to the poorer fit ad the size of the error will deped the size ad form of the distortio i each applicatio. w w w. a j e r. o r g Page 38
America Joural of Egieerig Research (AJER) 5.7.6.5.4.3 35 - true time delay 3 5. 5. -. 5 -. 5 5 5 5 (a) distorted ad udistorted sigals (b) cross correlatio fuctio Figure Effect of distortio o pulse time delay estimatio by cross correlatio Figure shows the ufiltered ad filter pulse sigals at the sesors ad Figure shows their cross correlatio..4 Pulse Cetroid Aother way to measure betwee pulse arrivals is by the use of a cetroid timig techique. The cetroid of a pulse is its geometrical cetre. For complicated shapes such as experimetally measured pulses the cetroid ca be foud by umerical itegratio..7.6.5.4 t t d - delay betw ee measured ad true arrival time of pulse at sesor d - delay betw ee measured ad true arrial time of pulse at sesor - true time delay.3.. -. -. d d t t - arrival time at sesor - arrival time at sesor 4 6 8 4 6 8 Figure 3 Pulse arrival time by use of cetroid Figure 3 shows how the cetroid of the pulse sigals is used i estimatig the time delay betwee the pulse sigals. The cetroid of each pulse sigal is determied by equatio 6 usig umerical itegratio; t L L t q ( t ) dt i q ( t ) dt i i (6) where L is the legth of the pulse, t ad i t are both measured from the same datum. The delay is differece betwee the calculated values of t. w w w. a j e r. o r g Page 39
America Joural of Egieerig Research (AJER) 5 Cosiderig Figure 3, the two pulse sigals show oly differ i magitude with o oise or distortio o them. The positios of the cetroids from the start of the pulses (d ad d) are the same ad so there is o error i the calculated delay. Figure 4 shows the pulses with added oise o which the method of cetroids was tried. The oise o the pulse sigals makes it impossible to determie the exact limits of itegratio to be take ad this affects computatio of the cetroid. Hece, it is ot possible to determie the delay by the cetroid method..7.6 - true time delay - oise level.5.4.3.. -. -. 4 6 8 4 6 8 Figure 4 Effect of oise o pulse arrival time by cetroid Figure 5 shows the effect of distortio o the estimate of the time delay i arrivals by the use of cetroid. It ca be see that the differece i delay (d ad d) betwee the measured ad true arrivals time are differet. This idicates that whe pulse sigals are distorted there will be errors i the estimate of the time delay..7.6.5.4 d - delay betw ee measured ad true arrivals time of pulse at sesor d - delay betw ee measured ad true arrivals time of pulse at sesor - true time delay.3.. -. d d -. 4 6 8 4 6 8 Figure 5 Effect of distortio o pulse arrival time by cetroid III. RESULTS Test for Effectiveess of the Methods usig Experimetal Pressure Pulse Sigals To test for the effectiveess of these methods i estimatig the time delay betwee arrivals of the pulse sigals, sesor sigals from a true pulse obtaied i a experimetal work were used. These sigals, show i Figure 6, were aalysed usig a program writte i Matlab programmig laguage to measure the time delay usig the peak detectio, crossig ad cross correlatio methods. The cetroid is ot used for the reasos give above. w w w. a j e r. o r g Page 4
spread of estimate i time delay (s) (arbritary uits) America Joural of Egieerig Research (AJER) 5..8.6.4..8.85.9.95..5..5. time (s) Figure 6 True pulse sigal from experimet Cosiderig Figure 6, the true time delay betwee the startig poits of the two pulses is.4s. The delay calculated usig the peak detectio method was.5 s; both the cross correlatio ad crossig methods gave.3s. The effect of oise was ivestigated by addig a radom umber to each data poit of the experimetal sigals. Te repetitios at various oise levels were carried out for each of the three methods ad the spread of values of the calculated delay determied ad plotted agaist oise level i Figure 7. The oise level is measured i the same arbitrary uits as the pulse, so that the maximum oise level (.3 uits) is approximately oe third of the magitude of the smaller pulse. 6 x -4 5 4 peak detectio 3 b crossig cross correlatio.5..5..5.3.35 oise level Figure 7 Graph of spread i time delay agaist icreasig oise level At low oise levels below.5 uits (5% of the magitude of the smaller pulse) the three techiques give very similar results with very small spread (<.ms). At higher oise levels the peak detectio method became progressively less reliable tha the crossig ad cross correlatio methods, though it should be oted that eve at the highest oise level ivestigated the spread of results was oly.6ms, or <6% of the true value. Up to a oise level of.8 the crossig ad cross correlatio methods gave similar results, with spread of results below.6ms (.5% of the true value). Above a oise level of.8 (positio b i Figure 7) the crossig method could ot be used reliably, but the cross correlatio method cotiued to give similarly reliable results up to the maximum oise level ivestigated. IV. DISCUSSION OF RESULTS Four methods have bee preseted to measure the time delay i arrivals betwee pulse sigals. The peak detectio method uses the measured time betwee pulse arrivals, defied as the time of the highest peak values. Whe oise ad distortio are ot preset, this method works well ad is accurate. Noise causes blurrig of the peaks of the pulse ad therefore itroduces ucertaity. w w w. a j e r. o r g Page 4
America Joural of Egieerig Research (AJER) 5 The crossig method uses arrival times i the same way, but i this case defied as the time where the sigal first crosses a level. This method, though simple, resulted i overestimatio of the time delay. The method is effective at high sigal to oise ratios but becomes ureliable i higher oise ad becomes uusable whe the oise level approaches the size of the smaller pulse. The cross correlatio method estimated the time delay directly without the eed for measurig the arrival times. The method is effective eve i the presece of high oise. The oly disadvatage is that it requires much more computatio time tha the other methods, but this is ot sigificat with today s moder computers. The use of the pulse cetroid estimated the time delay effectively for pulse sigals of similarly shapes but results i errors whe the pulse sigals are distorted ad could ot be used i the presece of sigificat oise because of the difficulty of determiig the limits of the pulses. V. CONCLUSION Based o the foregoig aalysis, it is clear the cross correlatio method gives the best estimate of the delay i pulse arrival times irrespective of the sigal to oise ratio ad so is the preferred techique used i the remaider of this research. REFERENCES [] Cao, C.F., Baik, S., Choi, J.B. ad Kim, Y.J., Protectio of Udergroud Gas Pipelies from Third Party Damage by O-Lie Moitorig Usig Piezoelectric Accelerometers, Proc. of the Istitutio of Mechaical Egieers, Part E, J. Process Mechaical Egieerig,, pp. 6 67. [] Walker, J. F., 97, Performace Data for a Double-Threshold Detectio Radar, IEEE Trasactios o Aerospace ad Electroic System, 7(), pp. 4-46. [3] Sug, P. J., Cha, Y.C., Ji, H.N., Si-Hyug, L., Doghoo, S. ad Tae-Yog, C.,, Numerical Study of Leakage Detectio ad Locatio i a Simple Gas pipelie brach usig a Array of Pressure Sesors,. J. Mechaical Sci. ad Tech., 4(4), pp. 983 99. [4] Steve, W. S., 997, The Scietific ad Egieer s Guide to Digital Sigal Processig, Califoria Techical Publishig. w w w. a j e r. o r g Page 4