Precise Measurement of Event Flow Time Coordinates Based on the ...

ISSN 0146-4116, Automatic C**on**trol and Computer Sciences, 2011, Vol. 45, No. 3, pp. 162–173. © Allert**on** Press, Inc., 2011.Original Russian Text © A.S. Rybakov, V.Yu. Vedin, 2011, published in Avtomatika i Vychislitel’naya Tekhnika, 2011, No. 3, pp. 56–69.**on** **the** Digital Processing **on**Oscillator Wave TrainA. S. Rybakov and V. Yu. VedinThe Institute **on**ics and Computer Scienceul. Dzérbenes 14, Riga, LV-1006 LatviaE-mail: alexryb@edi.lv, vedinv@edi.lvReceived March 1, 2011Abstract—This article is dedicated to **the** study **the** interpolative measurement **on** **the** digital processing **on** oscillator generated during **the** event occurrence. It isshown that **the** use **the** performance **on** oscillati**on** wavetrain model allows **on**e to guarantee picosec**on**d accuracy **the** interpolative measurements. Thetechnique’s accuracy capabilities are illustrated by **the** results **on** wave train, triggered relaxati**on** oscillator, regressi**on**model, least-squares method, time interval, n**on**linearityDOI: 10.3103/S01464116110300601. INTRODUCTIONThe time coordinates **the**se are measuredis referred to as event timing, and **the**se measurements are made using discrete (rough) and interpolative(correcti**on**) techniques. The former type **on**fined to registering **the** actual c**on**diti**on****the** meter **the** increment interval (reference frequency) periods. The latter type **the** positi**on** **the** interval **the** rough measurement discrete and isapplied with **the** help **on** eventtimer.Interpolati**on** based **on** digital signal processing is very promising in terms **the** achievable accuracyand resolving power [1, 2]. These techniques have **the** following basic steps: **the** generati**on** **the** event occurrence, **the** discretizati**on** and digitalizati**on** **the** fur**the**r digital processing **the** computati**on**s, which results in **the** estimati**on** **the** signal timepositi**on** within a discrete interval T s **the** rough measurement.Works [3, 4] c**on**sider interpolative measurement using algorithms for estimating **the** locati**on** **the**center **the** estimati**on**s **the** time positi**on****the** sample sequence center **on**linearily dependent **on** **the** event occurrenceinstants. To compensate **the** n**on**linearity **the** interpolative measurement, additi**on**al correcti**on** **the**time positi**on** estimati**on**s is **the**refore necessary. A specific corrective functi**on** for estimating **the** signalcenter time positi**on** helps determine **the** interpolative estimati**on** **the** event occurrence instants. Thefuncti**on** may be derived after **the** obligatory and very complex calibrati**on** **the** event timer and up**on**identifying **the** transfer functi**on** **the** interpolator.This work c**on**siders **the** results **the** **the**oretical and experimental research **on** **the** estimati**on**s **the** analog signal time positi**on** and **the**reforemakes it unnecessary to preliminarily identify **the** transfer functi**on** **the** interpolator. The studiedtechnique is based **on** estimating **the** parameters **the** regressi**on** model **the** regressi**on** functi**on** describing **the** waveform**on** functi**on** coefficients chosen as informativeparameters **the** modeled signal must depend **on** **the** level **the** signal shift relative to a certain initialmoment

PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 163Cable delay1Imput signalD-Typeflip-flopRDC&PulsetrainRC Bit NCounter10-pole lowPass filterOutputwave trainto ADCFig. 1. C**on**verter **the** output oscillati**on** wave train **the** triggered relaxati**on** oscillator.The estimati**on** γˆ **the** parameter vector γ **the** regressi**on** functi**on** g(t; γ) is determined by minimizing**on**alˆγ= arg min Iγ ( ),(1)γwhere[ u g t ] 2Iγ ( ) = − ( ; γ) ,i.e., **the** estimati**on** minimizes **the** sum **the** squared signal sample { u ( )} Nk = u t k deviati**on**s measuredk=1during **the** instants { t } Nk = kT s from **the** values **the** regressi**on** functi**on** where is **the** samplingk = 1gt ( k; γ),Tsfrequency period amounting to **the** interval **the** rough measurement discrete. Such values **the** regressi**on**coefficients are selected to estimate **the** parameters **the** modeled signal, which allow **the** model toharm**on**ize as best as possible with **the** signal samples in terms **the** mean square deviati**on** (least-squaresmethod). estimati**on**s **the** regressi**on** coefficients are used to determine **the** time shift from **the** incepti**on****the** regressi**on** functi**on** coefficients with **the** help **the** least-squaremethod was c**on**sidered earlier in article [5] and was exemplified by **the** processing **on**entially fadingoscillati**on**s **on** loop. The use **the** possibility**on**d accuracy without correcting **the** time shift values.This article c**on**siders algorithms for processing analog signal samples. The signal referred to is a wavetrain generated by a triggered relaxati**on** oscillator when each event **the** flows occurs.2. REGRESSION MODEL OF A TRIGGERED RELAXATIONOSCILLATOR WAVE TRAINIt is known that a wave train is a temporally or spatially limited sequence **on**s that are notnecessarily harm**on**ic. To generate an oscillati**on** wave train during an event occurrence, it is necessary touse a triggered relaxati**on** oscillator. A relaxati**on** oscillator is an oscillati**on** generator where **the** active partoperates in **the** key “**on**/**the** device in questi**on** is a n**on**linear dynamicsystem invariant to shifting: **the** time shift **on**dingshift **the** output signal **the** same amount. The specified feature ensures that **the** oscillati**on**sin a triggered relaxati**on** oscillator are **the** same during each event occurrence.A triggered relaxati**on** oscillator was used as **the** basis for developing a c**on**verter **on** wave trains (see Fig. 1). The new device c**on**sists **the** delayed feedback, an pulse counter, and a 10-pole low pass filter.The c**on**verter operates as follows. The D-trigger switches **on** to **the** state **the** logical “**on**e” al**on**g **the**input signal fr**on**t, i.e., **the** event occurrence mark, and **the** logical element “AND” and **the** pulse counterare **the**refore unblocked. The “AND” element and **the** cable delay line c**on**stitute a triggered relaxati**on**AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011N∑k=1kk(2)

164RYBAKOV, VEDINNomalized amplitude1.00.90.80.70.60.50.40.30.20.1050100Frequency, MHzFig. 2. Amplitude spectrum **on** frequency **the** input **the**low pass filter (solid line) and **the** amplitude–frequency resp**on**se **the** low pass Chebyshev filter (dashed line).150200oscillator that starts to produce rectangular pulses with a period equal to a doubled sum **the** cable andlogical element delays. After **the** set number **the** counter generates a logical unitaccording to its output charge Bit N. The D-trigger and **the** counter are reset, and pulse generati**on** stops.The series **the** output **the** relaxati**on** oscillator goes through a low pass ChebyshevFilter with a cut**the** pulse repetiti**on** in **the** series. A very smooth oscillati**on** wavetrain is **the**refore generated at **the** output **the** filter and fed to **the** input **on**verter (ADC). The wave train sampling frequency is 100 MHz, which corresp**on**ds to a period **the** output oscillati**on** wave train is shifted by a certain value relative to **the** setframe **on** al**on**g **the** time axis **the** sampling instants.The shape **on** wave train at **the** output **the** low frequency filter is generally not an idealsine wave. The high-frequency comp**on**ents **the** spectrum **the** real low frequency filter (see Fig. 2) and must be taken into account in a modeled oscillati**on**wave train by **the** corresp**on**ding selecti**on** **on** functi**on**.The regressi**on** functi**on** approximating **the** oscillati**on** wave train c**on**sists **on**iccomp**on**ents **the** main oscillati**on** frequency and includes odd as well as evenharm**on**ics:gt (; γ ) = c + ( a cosnω t+ b sin nωt),0m∑n=1n c n cwhere 0 is a c**on**stant comp**on**ent **the** oscillati**on** wave train, γ= c0, a1, b1,..., an, bn,..., am,bmis **the**vector **the** estimated regressi**on** coefficients, ω c is **the** basic frequency **the** wave train oscillati**on**s (wavetrain fill frequency), and m is **the** number **the** upper index “T” indicateshereafter **the** transpositi**on** **the** time shift parameter τ, a regressi**on** functi**on** can be written as follows:c [ ]m(0) (0) (0)0⎡n c n cn=1gt ( −τγ ; ) = c + ∑ ⎣a cos nω( t−τ ) + b sin nω( t−τ) ⎤ ⎦,where **the** upper index, i.e., **the** zero in round brackets, indicates **the** “starting” values **the** regressi**on**functi**on** coefficients at null shift; τ= 0; and(0) (0) (0) (0) (0) (0) (0) Tγ = ⎡ ⎣c0, a1 , b1,..., an , bn ,..., am , bm⎤ ⎦is **the** vector **the** “starting” values **the** regressi**on** functi**on**coefficients. The dependence **the** regressi**on** coefficients anand bn**on** **the** time shift τ is expressedas follows:AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011T(3)(4)

PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 165The estimated informative parameters **the** model (3) are **the** amplitudes { a , } m n b n n=1 **the** quadratureharm**on**ic comp**on**ents with **the** frequencies nωcand **the** wave train c**on**stant comp**on**ent c0. The regressi**on**model (3) is linear in terms **the** estimated parameters. The parameters are estimated according to **the**minimum **on**al:NNNwhere { uk} = { u( kT ) is a sequence **the** sampling instants . Thek= 1 s }k=1{ kT s} k = 1parameters { a , } m n b n n=1 c**on**tain **the** data **on** **the** amplitude and **the** phase **the** n frequency element **the**wave train. A phase expressed in units **the** start **the** oscillati**on** wave train shifted in time within**the** interval [0, Ts).The model parameter ωc(**the** main oscillati**on** frequency) has no data **on** **the** wave traintime shift and is **the**refore not informative. The value **the** main wave train oscillati**on** frequency is presumedto be known before **the** regressi**on** coefficients are estimated.The relati**on** **the** regressi**on** coefficients with **the** frequency nω c amounts to **the** phase shift tangent **the** n frequency element **the** wave train(0)(0) ⎛bwheren⎞ (0)ψ n = arctan⎜is **the** c**on**stant initial phase shift with **the** frequency ,(0) ⎟+ ηnnω c⎝an⎠(0)⎧ 0, if an≥ 0(0) ⎪(0) (0)η n = ⎨ π , if an < 0, bn≥ 0.⎪ (0) (0)⎩ −π , if an< 0, bn< 0An equati**on** may be derived from relati**on** (8) to describe **the** dependence **the** time shift τ( nωc ) **the**harm**on**ic comp**on**ent with **the** frequency nω c **on** **the** regressi**on** coefficients:where(0) (0)n n cos c n sin c(0) (0)n n c n ca = a nω τ−b nω τb = b cos nω τ+ a sin nω τ.Nm⎧⎪⎡ ⎤⎫ ⎪Iγ () = ∑⎨uk − ⎢c0+ ∑( ancosnω ckTs + bnsin nωckTs) ⎥⎬,k= 1⎪⎩ ⎣⎢n=1⎦⎭ ⎥⎪(0) (0)n n cos ωτ+ c n sin c(0) (0)n n cos ωτ− c n sin cb b n a nωτ(0)= = tan( nωcτ+ ψn),a a n b nωτ⎧ 0, if an≥ 0⎪η n = ⎨ π , if an < 0, bn≥ 0.(11)⎪⎩ −π , if an< 0, bn< 0C**on**sidering **the** minimum **on**al (7), **the** estimati**on** **the** regressi**on** coefficients presupposesthat **the** value **the** frequency is known.(0)n( )1 ⎡ ⎛bn⎞ ⎤ ψτ nω c = arctan + ηn− ,nω⎢ ⎜c a⎟ ⎥⎣ ⎝ n⎠ ⎦ nωcω c2(5)(6)(7)(8)(9)(10)3. ESTIMATION OF THE OSCILLATION WAVE TRAIN REGRESSIONMODEL COEFFICIENTSThe linear model for estimating regressi**on** coefficients in matrix notati**on** is **the** following:v = Bγ +ε,where v is **the** column vector **on** ( N × 1) **the** oscillati**on** wave train samples,[ ]v = u, u ,..., u N ,1 2T(12)(13)AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011

166RYBAKOV, VEDINB is **the** regressi**on** matrix **on**ality ( N × (2m+1)):[ ]⎡1 cosωcTs sin ωcTs ... cosmωcTs sinmωcTs⎤⎢1 cosωc2Ts sinωc2 Ts ... cosmωc2Ts sinmωc2T⎥sB = ⎢⎥,⎢ ... ... ... ... ... ... ⎥⎢⎣1 cosωcNTs sin ωcNTs ... cosmωcNTs sinmωcNT⎥s⎦Tγ= c0, a1, b1, ..., an, bn, ..., am,b m is **the** column vector **on** ((2m + 1) × 1) **on** coefficients,and ε is **the** column vector **on** ( N × 1) **the** errors in **the** signal model and in **the** quantizati**on****the** samples. The estimati**on** **the** parameters is c**on**fined to minimizing **the** square form c**on**sidering**the** regressi**on** coefficient vector.(14)The system **on**s is **the** following:[ v −Bγ] T[ v −Bγ] → min.γ(15)whereas **the** soluti**on** **on** system (16) can be written as follows:where ˆ ˆ ˆ Tˆγ= ⎡ ⎣ cˆ0, aˆ1, b1,..., aˆn, bn,..., aˆm,b m⎤⎦is a column vector **on** ((2m + 1) × 1) **the** regressi**on**coefficients’ estimati**on**s.The time shift ˆ τ∈[0, T s ) **the** oscillati**on** wave train can be calculated using a relati**on** similar to Eq. (10),**on**ly **the** regressi**on** coefficients must be replaced with **the**ir estimati**on**s. According to experimental2research, **the** amplitudes **the** higher harm**on**ies A 2 n = an + bn,n = 2, 3, ..., mwith **the** frequencies nωcare2 2nearly two hundred times lower than **the** amplitude A1 = a1 + b1**the** main harm**on**ic with **the** frequencyω c . It is **the**refore reas**on**able to determine **the** time shift **on**ly according to **the** wave train fillingfrequency (**the** main oscillati**on** frequency is ω c ). This allows us to substantially decrease **the** effect **the**additive noise **on** **the** accuracy **the** time shift measurements. The estimate **the** time shift is determinedby **the** regressi**on** coefficients at **the** frequency according to **the** expressi**on**:whereTT( B B) γ= B v,T −( ) 1Tˆ γ= B B B v,ω cbˆ11 ⎡ ⎛ ⎞ˆ arctan ˆ⎤τ= ⎢ ⎜ +η1,aˆ⎟ω ⎥⎣ ⎝ ⎠ ⎦c⎧0, if aˆ1 ≥ 0⎪η ˆ1 = ⎨ π , if aˆ1 < 0, bˆ1 ≥ 0 .(19)⎪⎩ −π , if aˆ1 < 0, bˆ1 < 0The example in Fig. 3 shows **the** graphic results **the** regressi**on** functi**on** (3) with **the** help**the** least-squares method to **the** samples **on** wave trains with different time shifts relativeto **the** discretizati**on** moments.Numerical techniques were used to study **the** influence **the** frequency inc**on**sistency between **the****the**oretical model (3) and **the** actual oscillati**on** wave train **on** **the** n**on**linearity **the** interpolative measurement.The dependence **the** n**on**linearity **on** **the** number **the** estimati**on**swas also c**on**sidered. When **the** estimati**on** **the** regressi**on** coefficients with different time instantsωcτ∈[0, T s ) was modeled **on** a computer, **the** wave train filling frequency was f c = = 27.9 MHz, and **the**2 πsampling frequency was 100 MHz (**the** sampling period was T s = 10 ns). The errors in **the** sample’s quantizati**on**were not taken into account; i.e., it was asserted that **the** values **the** samples during **the** discretizati**on**were exactly equal to **the** instant values **the** oscillati**on** wave train.1AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011(16)(17)(18)

PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 167Magnitude400035003000250020001500100050000 20 40 60 80 100120 140**the** regressi**on** functi**on** to samples **the**sampling instants (**the** sampling frequency is 100 MHz).According to **the** results **the** computer modeling, **the** mean degree **the** error in estimating **the** timeshift Δτ = ˆτ − τ does not depend **on** **the** actual event’s occurrence period if **the**re is no frequency mismatchbetween **the** modeled and **the** actual oscillati**on** wave train:E[ Δτ ] = E[ ˆτ − τ ] = c**on**st, ∀τ ∈[0, T s ).(20)This means that a systematic error dependent **the** n**on**linearity **the** interpolative measurement isabsent and **the**refore we can adopt **the** estimati**on** (18) **on** **the**event occurrence instant.The curves **the** dependence **the** error in **the** interpolative measurement Δτ = ˆτ − τ up**on** **the** eventoccurrence instant τ∈[0, T s ) shown in Fig. 4 allow us to c**on**clude that, if **the**re is a mismatch between **the**values **the** frequencies **the** regressi**on** model and **the** actual oscillati**on** wave train, a systematic inaccuracyin **the** interpolative measurement will occur n**on**linearily dependent **on** **the** event occurrenceinstant. In **the** case **the** n**on**linearity error in **the** interpolative measurementΔτ = ˆτ − τ will also depend **on** **the** number **the** curves in Fig. 4, **the** n**on**linearity errordoes not exceed ±1.0 ps. If **the** frequencies mismatch, **the** n**on**linearity error will not depend **on** **the** number**the** regressi**on** model’s frequency is inc**on**sistent with **the** main frequency **the** actual wave train, aninaccuracy **the** interpolative measurement n**on**linearity dependent **on** **the** event occurrence instantappears. According to Fig. 4, to minimize **the** n**on**linear inaccuracy, **the** value **the** main wave train frequencyω c should be known precisely enough. This frequency may be calculated in a variety **the** course **on** wave train at **the**output **on**sider in brief **the** estimati**on** **the** wave trainfilling frequency ω c according to samples **the** filling frequency accuratelyenough. N**on**e**the**less, to evaluate this frequency, MN samples **the** output **the** c**on**verter (Fig. 1) during **the** timing **the** output **the** c**on**verter and **the**n discreted and digitalized. Formore accurate estimati**on**, **the** period **the** event repetiti**on** should be aliquant to **the** sampling period.For a more detailed c**on**siderati**on** **on** period, see study [4].A sequence

168RYBAKOV, VEDIN**the** interpolative measurement error Δτ = ˆτ − τ up**on** **the** event occurrence instant τ∈[0, Ts), **the**number **the** level **the** frequency mismatch between **the** regressi**on** model and **the** oscillati**on** wave train(**the** numbers above **the** relevant lines indicate **the** level **the** frequency mismatch in kHz).**the** period **the** event repetiti**on**. If **the** period is not known beforehand, **the** spectral approach(a sufficiently detailed c**on**siderati**on** **the** MN samples.The complex Fourier spectrum **on** wave train is determined according to **the** followingequati**on**:∞−ω j tU( jω ) =∫u() t e dt,−∞where j = −1 is an imaginary unit. Using MN samples **the** estimati**on****the** complex Fourier spectrum **on** wave train is expressed as follows:MNˆ( Δω ) = ( )−jnΔω( tkmod TR)s∑k, = 0, − 1,k=1Un T yt e n Zwhere { ( )} MNMNyt k is a sequence **the** sampling instants and isk= 1{ tk} 1, Δωk=**the** step **the** uniform frequency grid where **the** estimati**on** **the** Fourier spectrum **on**structed, Z is **the** number **the** frequency grid. To avoid spectral peaks at null frequency, **the**mean value **the** sequence should be subtracted from **the** values **the** spectrum estimati**on** (23). The main frequency **the** wave train oscillati**on**s (filling frequency) can be estimatedc**on**sidering **the** positi**on** **the** maximum **the** amplitude spectrum **on** **the** frequency axis (see Fig. 5):ω ˆ = arg max U( jω) .c∀ω> 0The following sequence **on**s can be underlined for **the** positi**on** determining **the** positi**on** **the**peak **on** **the** frequency axis:1) The discrete Fourier transformati**on** (23) is calculated with a very small frequency interval Δω. Themean value **the** sample sequence is provisi**on**ally subtracted from **the** sequence itself.AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011(22)(23)(24)

PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 1691.0NormalizedamplitudeNormalizedamplitude0.51.00.50002420 4025 26 27 2860 80 100Frequency, MHz29 30 31 32Frequency, MHzFig. 5. Rec**on**structed amplitude spectrum **on** periods (upper curve) and its enlargedpart in **the** peak area (lower curve). The wave train filling frequency, i.e., 27.9 MHz, determines **the** positi**on** **the** peakamplitude spectrum al**on**g **the** frequency axis.2) The peak discrete amplitude spectrum (**the** discrete Fourier transformati**on** module) is searched foral**on**g **the** frequency axis as **the** number **the** highest amplitude:n5. EXPERIMENTAL RESULTS OF MEASURING EVENT FLOW TIME COORDINATESThe accuracy **the** studied interpolati**on** technique can be illustrated with experimental results **the** periods **the** values’ statistical characteristics.The statistical characteristics **the** estimati**on**s **the** event flow time coordinates were studied using**the** method **the** analysis **the** fluctuati**on**s relative to **the** linear trend [6]. Unwanted fluctuati**on**s relativeto **the** linear trend are referred to in **the** scientific literature as jitters.To generate an event flow, a high stability pulse generator was used. The sequence **the** flow **the** events was about 204.794968 µs, while **the**total number **on**s with **the** filling fremax= arg max Uˆ( nΔω) .n(25)3) The spectral peak positi**on** must be aligned by **the** parabolic interpolati**on** for three spectral samples,and **the** samples **the** highest amplitude should be put in **the** middle. This alignment is necessary since**the** peak spectrum may be actually positi**on**ed between two fellow spectral samples as **the** spectrum has adiscrete frequency. The necessary frequency correcti**on** ΔΩ for more precise determinati**on** **the** spectralpeak is calculated according to **the** following equati**on**:Un ˆ[ ˆmax −1] − Un [ max + 1]ΔΩ =Δω.2 [ 1] 2 [ ] [ 1]( Un ˆmax − − Un ˆ ˆmax + Unmax+ )(26)4) The spectral sample number n max is translated into **the** frequency, **the** value **the**frequency correcti**on** ΔΩ:ω ˆ = n Δω+ ΔΩ.cmax(27)AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011

170RYBAKOV, VEDIN**the** actual experiment. Jitter **the** event occurrence instants (upper curve) and **the** sequence **the** time intervals between **the** events (lower curve).ωcquency f c = = 27.9 MHz was generated by **the** c**on**verter **the** incoming events into wave trains (see2 πFig. 1). To digitize **the** instant values **the** wave train, a 12-bit AD c**on**verter with a sampling frequency**the** processing was 35. However, **the** samplestaken at **the** initial, i.e., **the** unsettled, stage **the** wave train were not c**on**sidered, were discarded, andwere not used for **the** processing. The time positi**on** **the** wave train was estimated using equati**on**s (17)and (18). The time interval between **the** events was estimated as **the** difference **the** time instants whenfellow events occurred.As an example, **the** fluctuati**on**s **the** estimati**on**s **the** event flow time values relative to **the** lineartrend (event occurrence time jitters) are shown in Fig. 6. The estimati**on** **the** linear trend **the** determinedtime coordinates **the** event flow was displayed in **the** form **on**, **the**parameters **the** least-squares method. The lower curve in Fig. 6 shows **the**sequences **the** intervals between **the** events from **the** flow. Calculated using **the**sequence **the** mean square error **the** measurement was 2.885 ps.Figure 7 shows **the** diagrams **the** autocorrelating functi**on**s calculated using **the** data sequences fromFig. 6. It is known that **the** autocorrelating functi**on** **the** n**on**correlating process has a needlelikeshape while **the** argument value is null. The Dirac delta functi**on**, for example, is displayed in **the** curvein **the** form **on** shown in **the** upper diagram in Fig. 7 is not needlelike,and **the** jitter **the** event occurrence instants is **the**refore correlated. The jitter Fractal Dimensi**on** [6](see **the** upper curve in Fig. 6) approximates 1.5, which corresp**on**ds to **the** fractal dimensi**on** **on**s. An asserti**on** can be made that **the** event flow produced by **the** generator and used in**the** experiment was not strictly periodical but was ra**the**r a case **on** **the** effect **the** generator frequency fluctuati**on**s **on** **the** interval measurement errors, a case**on** a computer. The results **the** computer modeled measurement with **the**source data identical to **the** actual experiment are given in Figs. 8 and 9, although not a Palm flow but aquasi-periodical stream **the** jitteringinstant was set as 1.0 ps.The mean square measurement error calculated according to a sequence **on**s calculated according to **the** data sequences in Fig. 8. Theneedlelike shape **the** autocorrelating functi**on** in **the** upper curve **the** event occurrencejitter is not correlated (see **the** upper diagram in Fig. 8). The jitter’s fractal dimensi**on** (see **the** upperdiagram in Fig. 8) is 1.995, which corresp**on**ds to **the** fractal dimensi**on** **on**s.

PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 171NormalizedNormalizedauto-correlati**on** auto-correlati**on**1.00.50–0.5–150 –100 –50 0 50 100 150**the** repetiti**on** period **the** repetiti**on** period **the** actual experiment. The autocorrelati**on** functi**on** **the** jitter **the** event occurrence instants (uppercurve) and **the** autocorrelati**on** functi**on** **the** sequence **the** time intervals between **the** events (lowercurve).10**the** event occurrence instants (upper curve) and **the** sequence **the** time intervals between **the** events (lower curve).Comparing **the** experimental results with **the** results **the** computer modeling, **the** following c**on**clusi**on**can be reached: **the** mean square error obtained in **the** experiment during **the** measurement **the**time intervals by **the** event timing was 2.885 ps. Never**the**less, this value does not allow **on**e to make an ultimatejudgement about **the** accuracy **the** interpolati**on**. The given value is overestimated since it c**on**tainsnot **on**ly errors in **the** interpolati**on** measurement but also errors caused by **the** unsteady frequency **the**generator producing event streams similar to Palm flows. The results **the** modeling allows **on**e to assertthat, if real measurements **the** type **on** in questi**on**AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011

172RYBAKOV, VEDINNormalizedauto-correlati**on**Normalizedauto-correlati**on**1.00.50–0.5–150 –100 –50 0 50 100 150**the** repetiti**on** period **the** repetiti**on** period **on** functi**on** **the** jitter **the** event occurrence instants (upper curve)and **the** autocorrelati**on** functi**on** **the** sequence **the** time intervals between **the** events (lowercurve).(using a 12-bit AD c**on**verter) will ensure that **the** mean square error in **the** separate measurements doesnot exceed 2.5 picosec**on**ds.6. CONCLUSIONSThe interpolati**on** measurement **on** digital processing **on**wave train samples is studied. A wave train is generated with each event occurrence. The processing **the**samples is **the** estimati**on** **the** coefficients **the** regressi**on** functi**on**s approximating **the** wave train.The processing is realized using **the** regressi**on** model **on** wave train with linear informati**on**-carryingparameters. The model is a finite sequence **on**ic comp**on**ents with frequencies divisibleby **the** basic oscillati**on** frequency. The time shift **the** beginning **the** corresp**on**dingdiscrete interval is determined using **the** estimated amplitudes **the** quaternary harm**on**iccomp**on**ents.It is shown that **the** use **the** parameters **the** regressi**on** model **the**oscillati**on** wave train allows **on**e to ensure **the** picosec**on**d accuracy **the** interpolati**on** measurements.It is shown that **the** c**on**sidered type **on** does not require extra correcti**on** **the** obtainedestimati**on**s **the** oscillati**on** wave train time positi**on**. The latter estimati**on**s can **the**refore be used directlyas interpolati**on** estimati**on**s **on**e to avoid **the** provisi**on**al identificati**on****the** interpolator’s transfer functi**on**.Numerical approaches were used to study **the** effect **the** frequency mismatch between **the** **the**oreticaland **the** actual oscillati**on** wave train **on** **the** n**on**linearity **the** interpolati**on** measurement.The accuracy **the** interpolati**on** is proved with **the** specified experimental results **the**time intervals between events **the** flow with an inc**on**sistency **on**ds.REFERENCES1. Artyukh, Yu., Vedin, V., and Rybakov, A., A New Approach to High Performance C**on**tinuous **on** **on**trol and **on**f.),Warsaw, Poland, 1998, pp. 139–143.2. Artyukh, Yu., Bespal’ko, V., and Boole, E., Potential **the** DSP-**on**ics and Electrical Engineering, Kaunas: Technologija, 2009.AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011

PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 1733. Rybakov, A.S., Estimating **the** **on** **the** Pulse Signal Midpoint by a Small Number **on** **on**t. Comp. Sci. (Engl.Transl.), 2009, vol. 43, no. 1, pp. 9–16].4. Rybakov, A.S., Rec**on**structi**on** **the** Corrective Comp**on**ent **the** Transfer Functi**on** **the** Interpolator in **the**Process **on** **the** Precisi**on** **on**t.Comp. Sci. (Engl. Transl.), 2010, vol. 44, no. 1, pp. 11–21].5. Rybakov, A.S. and Vedin, V.Yu., **on** Digital Processing **the** Resp**on**se **on**icOscillator Avtomat. Vychisl. Tekhn., 2010, No. 6, pp. 43–56. [Aut. C**on**t. Comp. Sci. (Engl. Transl.), 2010,vol. 44, no. 6, pp. 337–346].6. Pavlov, A.N. and Anishchenko, V.S., Multifractal Analysis