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

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

ISSN 0146-4116, Automatic Control and Computer Sciences, 2011, Vol. 45, No. 3, pp. 162–173. © Allerton 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.ong>Preciseong> ong>Measurementong> ong>ofong> ong>Eventong> ong>Flowong> ong>Timeong> ong>Coordinatesong> ong>Basedong>on the Digital Processing ong>ofong> a Triggered RelaxationOscillator Wave TrainA. S. Rybakov and V. Yu. VedinThe Institute ong>ofong> Electronics 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 ong>ofong> the interpolative measurement ong>ofong> event occurrencesbased on the digital processing ong>ofong> a relaxation oscillator generated during the event occurrence. It isshown that the use ong>ofong> linear algorithms for estimating the performance ong>ofong> a regression oscillation wavetrain model allows one to guarantee picosecond accuracy ong>ofong> the interpolative measurements. Thetechnique’s accuracy capabilities are illustrated by the results ong>ofong> actual measurements and computermodeling.Keywords: event timing, interpolator, oscillation wave train, triggered relaxation oscillator, regressionmodel, least-squares method, time interval, nonlinearityDOI: 10.3103/S01464116110300601. INTRODUCTIONThe time coordinates ong>ofong> an event flow are event occurrence time instants. The way these are measuredis referred to as event timing, and these measurements are made using discrete (rough) and interpolative(correction) techniques. The former type ong>ofong> measurement is confined to registering the actual conditionong>ofong> the meter ong>ofong> the increment interval (reference frequency) periods. The latter type ong>ofong> measurement isintended for refining the position ong>ofong> an event within the interval ong>ofong> the rough measurement discrete and isapplied with the help ong>ofong> an interpolator realized as a hardware and song>ofong>tware subsystem in a precision eventtimer.Interpolation based on digital signal processing is very promising in terms ong>ofong> the achievable accuracyand resolving power [1, 2]. These techniques have the following basic steps: the generation ong>ofong> a specificanalog signal during the event occurrence, the discretization and digitalization ong>ofong> its instant signal values,and the further digital processing ong>ofong> the computations, which results in the estimation ong>ofong> the signal timeposition within a discrete interval T s ong>ofong> the rough measurement.Works [3, 4] consider interpolative measurement using algorithms for estimating the location ong>ofong> thecenter ong>ofong> a unipolar analog signal according to its samples. It is shown that the estimations ong>ofong> the time positionong>ofong> the sample sequence center ong>ofong> an analog signal are nonlinearily dependent on the event occurrenceinstants. To compensate the nonlinearity ong>ofong> the interpolative measurement, additional correction ong>ofong> thetime position estimations is therefore necessary. A specific corrective function for estimating the signalcenter time position helps determine the interpolative estimation ong>ofong> the event occurrence instants. Thefunction may be derived after the obligatory and very complex calibration ong>ofong> the event timer and uponidentifying the transfer function ong>ofong> the interpolator.This work considers the results ong>ofong> the theoretical and experimental research ong>ofong> interpolative measurementthat does not require extra correction ong>ofong> the estimations ong>ofong> the analog signal time position and thereforemakes it unnecessary to preliminarily identify the transfer function ong>ofong> the interpolator. The studiedtechnique is based on estimating the parameters ong>ofong> the regression model ong>ofong> an analog signal according toa sequence ong>ofong> samples. The processing is finding a best fit ong>ofong> the regression function describing the waveformong>ofong> an analog signal to signal samples. The regression function coefficients chosen as informativeparameters ong>ofong> the modeled signal must depend on the level ong>ofong> the signal shift relative to a certain initialmoment ong>ofong> time.162


PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 163Cable delay1Imput signalD-Typeflip-flopRDC&PulsetrainRC Bit NCounter10-pole lowPass filterOutputwave trainto ADCFig. 1. Converter ong>ofong> input events into the output oscillation wave train ong>ofong> the triggered relaxation oscillator.The estimation γˆ ong>ofong> the parameter vector γ ong>ofong> the regression function g(t; γ) is determined by minimizingong>ofong> some error functionalˆγ= arg min Iγ ( ),(1)γwhere[ u g t ] 2Iγ ( ) = − ( ; γ) ,i.e., the estimation minimizes the sum ong>ofong> the squared signal sample { u ( )} Nk = u t k deviations measuredk=1during the instants { t } Nk = kT s from the values ong>ofong> the regression function where is the samplingk = 1gt ( k; γ),Tsfrequency period amounting to the interval ong>ofong> the rough measurement discrete. Such values ong>ofong> the regressioncoefficients are selected to estimate the parameters ong>ofong> the modeled signal, which allow the model toharmonize as best as possible with the signal samples in terms ong>ofong> the mean square deviation (least-squaresmethod). estimations ong>ofong> the regression coefficients are used to determine the time shift from the inceptionong>ofong> a relevant interval ong>ofong> rough measurement discrete.The algorithm ong>ofong> estimating the regression function coefficients with the help ong>ofong> the least-squaremethod was considered earlier in article [5] and was exemplified by the processing ong>ofong> exponentially fadingoscillations ong>ofong> an impact excitation loop. The use ong>ofong> computer modeling allowed revealing the possibilityong>ofong> picosecond accuracy without correcting the time shift values.This article considers algorithms for processing analog signal samples. The signal referred to is a wavetrain generated by a triggered relaxation oscillator when each event ong>ofong> 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 ong>ofong> any oscillations that are notnecessarily harmonic. To generate an oscillation wave train during an event occurrence, it is necessary touse a triggered relaxation oscillator. A relaxation oscillator is an oscillation generator where the active partoperates in the key “on/ong>ofong>f” mode. It should be noted that the device in question is a nonlinear dynamicsystem invariant to shifting: the time shift ong>ofong> an input signal (event) for a certain value generates a correspondingshift ong>ofong> the output signal ong>ofong> the same amount. The specified feature ensures that the oscillationsin a triggered relaxation oscillator are the same during each event occurrence.A triggered relaxation oscillator was used as the basis for developing a converter ong>ofong> input events into outputoscillation wave trains (see Fig. 1). The new device consists ong>ofong> a D-Type Flip-Flop logical element “AND” (gate&) with a cable delay in the delayed feedback, an pulse counter, and a 10-pole low pass filter.The converter operates as follows. The D-trigger switches on to the state ong>ofong> the logical “one” along theinput signal front, i.e., the event occurrence mark, and the logical element “AND” and the pulse counterare therefore unblocked. The “AND” element and the cable delay line constitute a triggered relaxationAUTOMATIC 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 ong>ofong> a series ong>ofong> 25 rectangular pulses with a repetition frequency ong>ofong> 27.9 MHz at the input ong>ofong> thelow pass filter (solid line) and the amplitude–frequency response ong>ofong> the low pass Chebyshev filter (dashed line).150200oscillator that starts to produce rectangular pulses with a period equal to a doubled sum ong>ofong> the cable andlogical element delays. After the set number ong>ofong> countings is reached, the counter generates a logical unitaccording to its output charge Bit N. The D-trigger and the counter are reset, and pulse generation stops.The series ong>ofong> pulses generated at the output ong>ofong> the relaxation oscillator goes through a low pass ChebyshevFilter with a cutong>ofong>f frequency exceeding the pulse repetition in the series. A very smooth oscillation wavetrain is therefore generated at the output ong>ofong> the filter and fed to the input ong>ofong> a 12-digit analog-to-digitalconverter (ADC). The wave train sampling frequency is 100 MHz, which corresponds to a period ong>ofong>T s = 10 ns.When an event occurs, the output oscillation wave train is shifted by a certain value relative to the setframe ong>ofong> reference points, i.e., a position along the time axis ong>ofong> the sampling instants.The shape ong>ofong> an oscillation wave train at the output ong>ofong> the low frequency filter is generally not an idealsine wave. The high-frequency components ong>ofong> the spectrum ong>ofong> a series rectangular pulses cannot be fullysuppressed by the real low frequency filter (see Fig. 2) and must be taken into account in a modeled oscillationwave train by the corresponding selection ong>ofong> a regression function.The regression function approximating the oscillation wave train consists ong>ofong> a finite range ong>ofong> harmoniccomponents ong>ofong> frequencies divisible by the main oscillation frequency and includes odd as well as evenharmonics:gt (; γ ) = c + ( a cosnω t+ b sin nωt),0m∑n=1n c n cwhere 0 is a constant component ong>ofong> the oscillation wave train, γ= c0, a1, b1,..., an, bn,..., am,bmis thevector ong>ofong> the estimated regression coefficients, ω c is the basic frequency ong>ofong> the wave train oscillations (wavetrain fill frequency), and m is the number ong>ofong> frequency elements. In this study, the upper index “T” indicateshereafter the transposition ong>ofong> a vector or a matrix, while ω and Ω indicate angular frequencies. Introducingthe time shift parameter τ, a regression function 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 ong>ofong> the regressionfunction 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 ong>ofong> the “starting” values ong>ofong> the regression functioncoefficients. The dependence ong>ofong> the regression coefficients anand bnon 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 ong>ofong> the model (3) are the amplitudes { a , } m n b n n=1 ong>ofong> the quadratureharmonic components with the frequencies nωcand the wave train constant component c0. The regressionmodel (3) is linear in terms ong>ofong> all the estimated parameters. The parameters are estimated according to theminimum ong>ofong> functional:NNNwhere { uk} = { u( kT ) is a sequence ong>ofong> wave train samples at the sampling instants . Thek= 1 s }k=1{ kT s} k = 1parameters { a , } m n b n n=1 contain the data on the amplitude and the phase ong>ofong> the n frequency element ong>ofong> thewave train. A phase expressed in units ong>ofong> time is the start ong>ofong> the oscillation wave train shifted in time withinthe interval [0, Ts).The model parameter ωc(the main oscillation frequency) has no data on the wave traintime shift and is therefore not informative. The value ong>ofong> the main wave train oscillation frequency is presumedto be known before the regression coefficients are estimated.The relation ong>ofong> the regression coefficients with the frequency nω c amounts to the phase shift tangent ong>ofong>the n frequency element ong>ofong> the wave train(0)(0) ⎛bwheren⎞ (0)ψ n = arctan⎜is the constant 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 equation may be derived from relation (8) to describe the dependence ong>ofong> the time shift τ( nωc ) ong>ofong> theharmonic component with the frequency nω c on the regression 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< 0Considering the minimum ong>ofong> functional (7), the estimation ong>ofong> the regression coefficients presupposesthat the value ong>ofong> 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 regression coefficients in matrix notation is the following:v = Bγ +ε,where v is the column vector ong>ofong> dimension ( N × 1) ong>ofong> the oscillation 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 regression matrix ong>ofong> dimensionality ( 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 ong>ofong> dimension ((2m + 1) × 1) ong>ofong> regression coefficients,and ε is the column vector ong>ofong> dimension ( N × 1) ong>ofong> the errors in the signal model and in the quantizationong>ofong> the samples. The estimation ong>ofong> the parameters is confined to minimizing the square form consideringthe regression coefficient vector.(14)The system ong>ofong> standard equations is the following:[ v −Bγ] T[ v −Bγ] → min.γ(15)whereas the solution ong>ofong> equation system (16) can be written as follows:where ˆ ˆ ˆ Tˆγ= ⎡ ⎣ cˆ0, aˆ1, b1,..., aˆn, bn,..., aˆm,b m⎤⎦is a column vector ong>ofong> dimension ((2m + 1) × 1) ong>ofong> the regressioncoefficients’ estimations.The time shift ˆ τ∈[0, T s ) ong>ofong> the oscillation wave train can be calculated using a relation similar to Eq. (10),only the regression coefficients must be replaced with their estimations. According to experimental2research, the amplitudes ong>ofong> the higher harmonies A 2 n = an + bn,n = 2, 3, ..., mwith the frequencies nωcare2 2nearly two hundred times lower than the amplitude A1 = a1 + b1ong>ofong> the main harmonic with the frequencyω c . It is therefore reasonable to determine the time shift only according to the wave train fillingfrequency (the main oscillation frequency is ω c ). This allows us to substantially decrease the effect ong>ofong> theadditive noise on the accuracy ong>ofong> the time shift measurements. The estimate ong>ofong> the time shift is determinedby the regression coefficients at the frequency according to the expression: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 ong>ofong> matching the regression function (3) with the helpong>ofong> the least-squares method to the samples ong>ofong> two oscillation wave trains with different time shifts relativeto the discretization moments.Numerical techniques were used to study the influence ong>ofong> the frequency inconsistency between thetheoretical model (3) and the actual oscillation wave train on the nonlinearity ong>ofong> the interpolative measurement.The dependence ong>ofong> the nonlinearity on the number ong>ofong> N wave train samples used in the estimationswas also considered. When the estimation ong>ofong> the regression 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 the2 πsampling frequency was 100 MHz (the sampling period was T s = 10 ns). The errors in the sample’s quantizationwere not taken into account; i.e., it was asserted that the values ong>ofong> the samples during the discretizationwere exactly equal to the instant values ong>ofong> the oscillation 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 140ong>Timeong>, nsFig. 3. Results ong>ofong> adjusting the regression function to samples ong>ofong> two wave trains with different time shifts relative to thesampling instants (the sampling frequency is 100 MHz).According to the results ong>ofong> the computer modeling, the mean degree ong>ofong> the error in estimating the timeshift Δτ = ˆτ − τ does not depend on the actual event’s occurrence period if there is no frequency mismatchbetween the modeled and the actual oscillation wave train:E[ Δτ ] = E[ ˆτ − τ ] = const, ∀τ ∈[0, T s ).(20)This means that a systematic error dependent ong>ofong> τ ong>ofong> the nonlinearity ong>ofong> the interpolative measurement isabsent and therefore we can adopt the estimation (18) ong>ofong> time shift τ as an interpolative estimation ong>ofong> theevent occurrence instant.The curves ong>ofong> the dependence ong>ofong> the error in the interpolative measurement Δτ = ˆτ − τ upon the eventoccurrence instant τ∈[0, T s ) shown in Fig. 4 allow us to conclude that, if there is a mismatch between thevalues ong>ofong> the frequencies ong>ofong> the regression model and the actual oscillation wave train, a systematic inaccuracyin the interpolative measurement will occur nonlinearily dependent on the event occurrenceinstant. In the case ong>ofong> a frequency mismatch, the nonlinearity error in the interpolative measurementΔτ = ˆτ − τ will also depend on the number ong>ofong> N samples used. In the curves in Fig. 4, the nonlinearity errordoes not exceed ±1.0 ps. If the frequencies mismatch, the nonlinearity error will not depend on the numberong>ofong> N wave train samples processed.4. ESTIMATION OF THE BASIC FREQUENCY OF THE OSCILLATION WAVE TRAINIf the regression model’s frequency is inconsistent with the main frequency ong>ofong> the actual wave train, aninaccuracy ong>ofong> the interpolative measurement nonlinearity dependent on the event occurrence instantappears. According to Fig. 4, to minimize the nonlinear inaccuracy, the value ong>ofong> the main wave train frequencyω c should be known precisely enough. This frequency may be calculated in a variety ong>ofong> ways, forexample, using modern digital spectral analyzers in the course ong>ofong> analyzing an oscillation wave train at theoutput ong>ofong> a low pass Chebyshev filter (see Fig. 1). Let us consider in brief the estimation ong>ofong> the wave trainfilling frequency ω c according to samples ong>ofong> a finite sequence ong>ofong> M wave trains.The number ong>ofong> N samples ong>ofong> a wave train does not suffice to evaluate the filling frequency accuratelyenough. Nonetheless, to evaluate this frequency, MN samples ong>ofong> a finite sequence ong>ofong> M wave trains generatedat the output ong>ofong> the converter (Fig. 1) during the timing ong>ofong> a periodic event flow may be used. A wavetrain u(t) is generated for each event at the output ong>ofong> the converter and then discreted and digitalized. Formore accurate estimation, the period ong>ofong> the event repetition should be aliquant to the sampling period.For a more detailed consideration ong>ofong> selecting an event repetition period, see study [4].A sequence ong>ofong> identically shaped M wave trains u(t) can be presented as follows:M −1∑yt () = ut ( −iT),i=0R(21)AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011


168RYBAKOV, VEDINong>Timeong> error, ps1.00.50–0.5+5+2.50–2.5–5N = 33Fc = 27.9 MHz–1.00 2 4 6 8 10ong>Timeong>, nsong>Timeong> error, ps1.00.50–0.5–1.00+5+2.50–2.5–5N = 35Fc = 27.9 MHz2 4 6 8 10ong>Timeong>, nsong>Timeong> error, ps0.30.20.10–0.1–0.2–5–2.50+2.5+5N = 38 N = 40Fc = 27.9 MHz0 2 4 6 8 10ong>Timeong>, ns0.50+5+2.50–2.5–5Fc = 27.9 MHz–0.50 2 4 6 8 10ong>Timeong>, nsFig . 4. Dependence ong>ofong> the interpolative measurement error Δτ = ˆτ − τ upon the event occurrence instant τ∈[0, Ts), thenumber ong>ofong> N samples, and the level ong>ofong> the frequency mismatch between the regression model and the oscillation wave train(the numbers above the relevant lines indicate the level ong>ofong> the frequency mismatch in kHz).ong>Timeong> error, pswhere T R is the period ong>ofong> the event repetition. If the period is not known beforehand, the spectral approach(a sufficiently detailed consideration ong>ofong> which is given in study [4]) can be used to evaluate this periodaccording to all the MN samples.The complex Fourier spectrum ong>ofong> a single oscillation wave train is determined according to the followingequation:∞−ω j tU( jω ) =∫u() t e dt,−∞where j = −1 is an imaginary unit. Using MN samples ong>ofong> a sequence ong>ofong> M wave trains (21), the estimationong>ofong> the complex Fourier spectrum ong>ofong> a single oscillation 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 ong>ofong> samples ong>ofong> M wave trains during the sampling instants and isk= 1{ tk} 1, Δωk=the step ong>ofong> the uniform frequency grid where the estimation ong>ofong> the Fourier spectrum ong>ofong> a single wave trainis reconstructed, Z is the number ong>ofong> nodes ong>ofong> the frequency grid. To avoid spectral peaks at null frequency, themean value ong>ofong> the sequence should be subtracted from the values ong>ofong> a sequence ong>ofong> samples before determiningthe spectrum estimation (23). The main frequency ong>ofong> the wave train oscillations (filling frequency) can be estimatedconsidering the position ong>ofong> the maximum ong>ofong> the amplitude spectrum on the frequency axis (see Fig. 5):ω ˆ = arg max U( jω) .c∀ω> 0The following sequence ong>ofong> operations can be underlined for the position determining the position ong>ofong> thepeak ong>ofong> discrete amplitude spectrum on the frequency axis:1) The discrete Fourier transformation (23) is calculated with a very small frequency interval Δω. Themean value ong>ofong> the sample sequence is provisionally 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. Reconstructed amplitude spectrum ong>ofong> a single wave train ong>ofong> 25 oscillation periods (upper curve) and its enlargedpart in the peak area (lower curve). The wave train filling frequency, i.e., 27.9 MHz, determines the position ong>ofong> the peakamplitude spectrum along the frequency axis.2) The peak discrete amplitude spectrum (the discrete Fourier transformation module) is searched foralong the frequency axis as the number ong>ofong> spectral samples ong>ofong> the highest amplitude:n5. EXPERIMENTAL RESULTS OF MEASURING EVENT FLOW TIME COORDINATESThe accuracy ong>ofong> the studied interpolation technique can be illustrated with experimental results ong>ofong>determining event occurrence instants. A sequence ong>ofong> valued instants was used to calculate the periods ong>ofong>time between two fellow events and to analyze the values’ statistical characteristics.The statistical characteristics ong>ofong> the estimations ong>ofong> the event flow time coordinates were studied usingthe method ong>ofong> the analysis ong>ofong> the fluctuations relative to the linear trend [6]. Unwanted fluctuations 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 ong>ofong> pulses generatedwas linked to the flow ong>ofong> periodic events. The interval between the events was about 204.794968 µs, while thetotal number ong>ofong> events was 160. When an event occurred, a wave train ong>ofong> 25 oscillations with the filling fremax= arg max Uˆ( nΔω) .n(25)3) The spectral peak position must be aligned by the parabolic interpolation for three spectral samples,and the samples ong>ofong> the highest amplitude should be put in the middle. This alignment is necessary sincethe peak spectrum may be actually positioned between two fellow spectral samples as the spectrum has adiscrete frequency. The necessary frequency correction ΔΩ for more precise determination ong>ofong> the spectralpeak is calculated according to the following equation: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 ong>ofong> which is added to thefrequency correction ΔΩ:ω ˆ = n Δω+ ΔΩ.cmax(27)AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011


170RYBAKOV, VEDINong>Timeong>, ps1050–5–100 20 40 60 80 100 120 140 160ong>Eventong> index1050–5–100 20 40 60 80 100 120 140 160Interval indexFig. 6. Results ong>ofong> the actual experiment. Jitter ong>ofong> the event occurrence instants (upper curve) and the sequence ong>ofong> errorsin measuring the time intervals between the events (lower curve).ωcquency f c = = 27.9 MHz was generated by the converter ong>ofong> the incoming events into wave trains (see2 πFig. 1). To digitize the instant values ong>ofong> the wave train, a 12-bit AD converter with a sampling frequencyong>ofong> 100 MHz was used. The number ong>ofong> wave train samples used in the processing was 35. However, the samplestaken at the initial, i.e., the unsettled, stage ong>ofong> the wave train were not considered, were discarded, andwere not used for the processing. The time position ong>ofong> the wave train was estimated using equations (17)and (18). The time interval between the events was estimated as the difference ong>ofong> the time instants whenfellow events occurred.As an example, the fluctuations ong>ofong> the estimations ong>ofong> the event flow time values relative to the lineartrend (event occurrence time jitters) are shown in Fig. 6. The estimation ong>ofong> the linear trend ong>ofong> the determinedtime coordinates ong>ofong> the event flow was displayed in the form ong>ofong> a straight line ong>ofong> regression, theparameters ong>ofong> which were determined using the least-squares method. The lower curve in Fig. 6 shows thesequences ong>ofong> errors in measuring the intervals between the events from the flow. Calculated using thesequence ong>ofong> intervals, the mean square error ong>ofong> the measurement was 2.885 ps.Figure 7 shows the diagrams ong>ofong> the autocorrelating functions calculated using the data sequences fromFig. 6. It is known that the autocorrelating function ong>ofong> realizing the noncorrelating process has a needlelikeshape while the argument value is null. The Dirac delta function, for example, is displayed in the curvein the form ong>ofong> a needle. The autocorrelating function shown in the upper diagram in Fig. 7 is not needlelike,and the jitter ong>ofong> the event occurrence instants is therefore correlated. The jitter Fractal Dimension [6](see the upper curve in Fig. 6) approximates 1.5, which corresponds to the fractal dimension ong>ofong> randomwalk actualizations. An assertion can be made that the event flow produced by the generator and used inthe experiment was not strictly periodical but was rather a case ong>ofong> Palm ong>Flowong>, which is also referred to as arecurrent stream ong>ofong> limited residual effect or as a stream with accumulating dispersion ong>ofong> event occurrences.To analyze the effect ong>ofong> the generator frequency fluctuations on the interval measurement errors, a caseong>ofong> event timing was modeled on a computer. The results ong>ofong> the computer modeled measurement with thesource data identical to the actual experiment are given in Figs. 8 and 9, although not a Palm flow but aquasi-periodical stream ong>ofong> events with jittering time values was imitated. The mean square error ong>ofong> the jitteringinstant was set as 1.0 ps.The mean square measurement error calculated according to a sequence ong>ofong> intervals amounts to 1.559 ps.Figure 9 shows curves ong>ofong> autocorrelating functions calculated according to the data sequences in Fig. 8. Theneedlelike shape ong>ofong> the autocorrelating function in the upper curve ong>ofong> Fig. 9 indicates that the event occurrencejitter is not correlated (see the upper diagram in Fig. 8). The jitter’s fractal dimension (see the upperdiagram in Fig. 8) is 1.995, which corresponds to the fractal dimension ong>ofong> “white noise” actualizations.ong>Timeong>, psAUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011


PRECISE MEASUREMENT OF EVENT FLOW TIME COORDINATES 171NormalizedNormalizedauto-correlation auto-correlation1.00.50–0.5–150 –100 –50 0 50 100 150ong>Timeong> delay in number ong>ofong> the repetition period ong>ofong> events1.00.50–0.5–1.0–150 –100 –50 0 50 100 150ong>Timeong> delay in number ong>ofong> the repetition period ong>ofong> eventsFig. 7. Results ong>ofong> the actual experiment. The autocorrelation function ong>ofong> the jitter ong>ofong> the event occurrence instants (uppercurve) and the autocorrelation function ong>ofong> the sequence ong>ofong> errors in measuring the time intervals between the events (lowercurve).10ong>Timeong>, ps50–5–100 20 40 60 80100120 140 160ong>Eventong> index10ong>Timeong>, ps50–5–100 20 40 60 80100120 140 160Interval indexFig. 8. Computer modeling results. Jitter ong>ofong> the event occurrence instants (upper curve) and the sequence ong>ofong> errors in measuringthe time intervals between the events (lower curve).Comparing the experimental results with the results ong>ofong> the computer modeling, the following conclusioncan be reached: the mean square error obtained in the experiment during the measurement ong>ofong> thetime intervals by the event timing was 2.885 ps. Nevertheless, this value does not allow one to make an ultimatejudgement about the accuracy ong>ofong> the interpolation. The given value is overestimated since it containsnot only errors in the interpolation measurement but also errors caused by the unsteady frequency ong>ofong> thegenerator producing event streams similar to Palm flows. The results ong>ofong> the modeling allows one to assertthat, if real measurements ong>ofong> time intervals between events are made, the type ong>ofong> interpolation in questionAUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011


172RYBAKOV, VEDINNormalizedauto-correlationNormalizedauto-correlation1.00.50–0.5–150 –100 –50 0 50 100 150ong>Timeong> delay in number ong>ofong> the repetition period ong>ofong> events1.00.50–0.5–150 –100 –50 0 50 100 150ong>Timeong> delay in number ong>ofong> the repetition period ong>ofong> eventsFig. 9. Computer modeling results. Autocorrelation function ong>ofong> the jitter ong>ofong> the event occurrence instants (upper curve)and the autocorrelation function ong>ofong> the sequence ong>ofong> errors in measuring the time intervals between the events (lowercurve).(using a 12-bit AD converter) will ensure that the mean square error in the separate measurements doesnot exceed 2.5 picoseconds.6. CONCLUSIONSThe interpolation measurement ong>ofong> event occurrence instants based on digital processing ong>ofong> oscillationwave train samples is studied. A wave train is generated with each event occurrence. The processing ong>ofong> thesamples is the estimation ong>ofong> the coefficients ong>ofong> the regression functions approximating the wave train.The processing is realized using the regression model ong>ofong> an oscillation wave train with linear information-carryingparameters. The model is a finite sequence ong>ofong> harmonic components with frequencies divisibleby the basic oscillation frequency. The time shift ong>ofong> an event occurrence from the beginning ong>ofong> the correspondingdiscrete interval is determined using the estimated amplitudes ong>ofong> the quaternary harmoniccomponents.It is shown that the use ong>ofong> linear algorithms ong>ofong> estimating the parameters ong>ofong> the regression model ong>ofong> theoscillation wave train allows one to ensure the picosecond accuracy ong>ofong> the interpolation measurements.It is shown that the considered type ong>ofong> interpolation does not require extra correction ong>ofong> the obtainedestimations ong>ofong> the oscillation wave train time position. The latter estimations can therefore be used directlyas interpolation estimations ong>ofong> event occurrence instants, which allows one to avoid the provisional identificationong>ofong> the interpolator’s transfer function.Numerical approaches were used to study the effect ong>ofong> the frequency mismatch between the theoreticaland the actual oscillation wave train on the nonlinearity ong>ofong> the interpolation measurement.The accuracy ong>ofong> the interpolation is proved with the specified experimental results ong>ofong> measuring thetime intervals between events ong>ofong> the flow with an inconsistency ong>ofong> about several picoseconds.REFERENCES1. Artyukh, Yu., Vedin, V., and Rybakov, A., A New Approach to High Performance Continuous ong>Timeong> IntervalCounting, Application ong>ofong> Microprocessors in Automatic Control and ong>Measurementong>s (Proc. XI Polish Nat. Conf.),Warsaw, Poland, 1998, pp. 139–143.2. Artyukh, Yu., Bespal’ko, V., and Boole, E., Potential ong>ofong> the DSP-ong>Basedong> Method for Fast ong>Preciseong> ong>Eventong> Timing,in Electronics 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 ong>Timeong> Position ong>ofong> the Pulse Signal Midpoint by a Small Number ong>ofong> Samples inHigh–Precision ong>Eventong> Timing, Avtomat. Vychisl. Tekhn., 2009, No. 1, pp. 15–25 [Aut. Cont. Comp. Sci. (Engl.Transl.), 2009, vol. 43, no. 1, pp. 9–16].4. Rybakov, A.S., Reconstruction ong>ofong> the Corrective Component ong>ofong> the Transfer Function ong>ofong> the Interpolator in theProcess ong>ofong> Calibration ong>ofong> the Precision ong>Eventong> ong>Timeong>r, Avtomat. Vychisl. Tekhn., 2010, No. 1, pp. 17–31 [Aut. Cont.Comp. Sci. (Engl. Transl.), 2010, vol. 44, no. 1, pp. 11–21].5. Rybakov, A.S. and Vedin, V.Yu., ong>Preciseong> ong>Eventong>s Timing ong>Basedong> on Digital Processing ong>ofong> the Response ong>ofong> a HarmonicOscillator Avtomat. Vychisl. Tekhn., 2010, No. 6, pp. 43–56. [Aut. Cont. Comp. Sci. (Engl. Transl.), 2010,vol. 44, no. 6, pp. 337–346].6. Pavlov, A.N. and Anishchenko, V.S., Multifractal Analysis ong>ofong> Complex Signals, Usp. Fiz. Nauk, 2007, vol. 177,no. 8, pp. 859–876 [Phys.-Uspekhi, 2007, vol. 50, no. 8, pp. 819–834].AUTOMATIC CONTROL AND COMPUTER SCIENCES Vol. 45 No. 3 2011

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