CE: MM INSTITUTE OF PHYSICS PUBLISHING

MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 15 (2004) 1–9

PII: S0957-0233(04)79685-X

Measuring the shape of randomly arriving pulses shorter than the acquisition step Dimitar V Stoyanov, Tanja N Dreischuh, Orlin I Vankov and Ljuan L Gurdev Institute of Electronics, Bulgarian Academy of Sciences, 72 Tzarigradsko shosse, BG-1784 Sofia, Bulgaria E-mail: [email protected]

Received 15 April 2004, in final form 23 August 2004 Published DD MMM 2004 Online at stacks.iop.org/MST/15/1 doi:10.1088/0957-0233/15/0/000

Abstract In this paper we have developed and tested a novel method for measuring precisely the shape of pulses shorter than the acquisition step, which is effective for random delays of the input pulses with respect to the start pulse of the analog-to-digital converter (ADC). The method is based on conversion of the short pulses to be measured into longer damped oscillations and their correct acquisition (sampling) with saving the pulse information, rearranging of the sampled oscillations with respect to some reference time instant to form a finer-discretization high-precision oscillation, and retrieving the pulse shape by inverse algorithms. We demonstrated experimentally the good performance (5–7% rms error) of this method (by using 20 MHz/8 bits ADC) when measuring the shape of randomly arriving pulses, shorter than the ADC sampling step (50 ns), with an equivalent sampling frequency up to 2 GHz (0.5 ns equivalent sampling step). The resolving of shapes in a pulse pair with an inter-pulse delay shorter than the ADC sampling interval has also been demonstrated. The limiting equivalent sampling frequency is estimated to be up to 500 GHz. This method can be effectively applied for creation of some novel short-pulse measuring techniques, avoiding the problem of time synchronization to the start pulses in lidars and radars, nuclear experiments, tomography, communications, etc. Keywords: pulse shape measurement, randomly arriving pulses,

deconvolution

1. Introduction The shape of a pulse is a basic characteristic describing the time dependence of the pulse amplitude. The measurement of very short pulse shapes is one of the most important problems for many applications of pulsed electronic techniques in radars and lidars, communications, nuclear electronics, spectroscopy, etc [1–5]. This problem is closely related with the time and range resolution of corresponding systems, especially in their new areas of applications such as the tomography (medical and industrial), joint energy-time analysis of short-pulse sequences in multidetector systems (nuclear techniques), high-resolution absorption and dispersion analysis of transparent media. It should also be noted that the short pulse characterization 0957-0233/04/000001+09$30.00

is a typical problem in standard electronic measurements [6]. The ratio R of the pulse duration τp to the sampling step T imposes strong limitations on the correct acquisition of short pulses, due to the low number of pulse samples, if R = τp /T
1. For pulses randomly arriving with respect to the sampling instants, the probability for unsampling of pulses becomes very high. In spite of using fastest modern analog-todigital converters (ADC) of T ∼ 200 ps [7], the ratio R
1 remains typical for the communication and nuclear detectors and receivers as well as for some new types of high-resolution radar and lidar systems. There are some hardware approaches for short pulse sampling as, for instance, the boxcar (stroboscopic) techniques

© 2004 IOP Publishing Ltd Printed in the UK

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2. Formulation of the problem There are two basic problems to be solved for the successful measurement of the temporal shape of randomly arriving pulses shorter than the sampling step: (i) non-distorted, loss-free ADC-acquisition of the input pulse information; (ii) appropriate definition of some (specific for each pulse) time instant T0 , strictly linked to the pulse arrival. An effective solution of the first problem has been demonstrated in [12–14], where the input short pulses are converted (convolved) by a resonant circuit (resonator) into damped oscillations of a relatively long duration τr  T . In turn the damped oscillations are sampled precisely by the ADC. Further, by applying the inverse transformation (deconvolution) to the sampled (with a step T ) oscillations, one obtains output pulses having mass center positions determinately linked to the mass centers of the corresponding input short pulses (the true pulse shape is not recovered by this deconvolution). Then the calculation of the short pulse arrival times is straightforward. It was experimentally shown that using this technique one can measure the arrival times with an accuracy ∼0.3–0.5 ns for single pulse measurements using 50 ns ADC sampling step (and below 40 ps, if averaging over a set of pulses) [14]. The definition of a stable specific instant T0 could be realized using the same conversion of the input pulses, if the following conditions are fulfilled: ω0 τp  1,

βτp  1,

ω0 T  1,

(1)

where ω0 is the natural resonator frequency and β = 1/τr is the damping decrement. Certainly, in order to ensure an oscillatory resonator response, the inequality β  ω0 should also be satisfied. The first and the second inequalities in (1) outline in practice a nearly shock excitation of the resonance circuit leading to a clear-cut response (damped oscillation). In this case the initial transient period, being of the order of the 2

1.2 Tc

Amplitude (a.u.)

0.8

0.8 Amplitude (a.u.)

(see, e.g., [8, 9]). Here the set of pulses with a stable delay with respect to some start pulse are gated by pulses of varying delays (for each repetition period with respect to the start pulse), followed by low-pass filtering and arranging of acquired data. As a result, the pulse shape is transformed into a slow time scale. The resolution of boxcar techniques could be better than 50 ps [10]. Their essential disadvantage is the dependence of the resolution on the delay stability of the pulses with respect to the start. These methods are totally ineffective for the case of randomly arriving pulses because of the same reason. The pulse delay stability for very short pulses becomes an essential problem for any techniques, including streak-cameras [11], where resolutions below 1 ps are realized. The opportunity to measure the shapes of randomly arriving pulses with a resolution better than the ADC sampling step is of essential interest for a lot of applications. In this paper we present a novel technique effective for random delays of the input pulses with respect to the ADC start pulse and lossfree when τp /T < 1. The method can be effectively applied for creation of some novel short-pulse measuring techniques, avoiding the problem of time synchronization to the start pulses in lidars and radars, nuclear experiments, communications, tomography, etc.

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Figure 1. Input pulse shape (1) and the related damped oscillation (2) as functions of time. The damped oscillation sampled with T = 50 ns is given in the inset.

pulse duration and containing the essential information about the pulse effect, is much shorter than the period of oscillations. After the transient period is terminated we shall already have an entirely formed free oscillation with frequency ω0 and decrement β. Certainly, the amplitude and the phase shift of this oscillation will depend on the pulse shape. It is important to note as well that the relative phase shift between different free oscillations depends on the corresponding random pulse arrival times. The third inequality in (1) means that the sampling frequency should be sufficiently high to ensure an accurate acquisition of the free oscillations. The conversion of the signal pulses S(t) into decayed oscillations F (t) can be described analytically by their convolution with the resonator response function Rosc , i.e. F = S ⊗ Rosc . The input pulse shape and the related damped oscillation as functions of time are illustrated in figure 1. One can distinguish there both the characteristic time areas in the output damped oscillation: the area Ap (transient period), corresponding to the process of excitation of oscillations by the input pulse, and the area Aosc , corresponding to the process of free oscillations. Since both the areas are successive parts of the same process, they are strictly linked in time at constant oscillator parameters. Therefore, one can state that the difference Ts = Tosc − Tc between the time position Tosc of some of the characteristics of the damped oscillation in the area Aosc (say, the position of the first zero passing (see figure 1)) and the time position Tc of the input pulse center is constant (Ts = const). As long as the difference Ts has a fixed value, the position of Tosc could be used as the time instant T0 marking the arrival of the input pulses. Let us now consider the sampled oscillation (the inset in figure 1), when excited by the corresponding input pulse. The random pulse delay with respect to the ADC start will cause differences in the sampled values from pulse to pulse, including the signal (transient) area Ap ∼ τp . At the same time, because of the condition ω0 T  1, the free oscillation area Aosc will be sampled frequently enough to accurately determine the characteristic instant Tosc . After calculating Tosc , one could shift each sampled oscillation to some joint reference point TR after a preliminary transfer to a new finer time scale of step δT  T . By such transformations (see below), the amplitude samples of oscillations for different pulses will be arranged at different time positions on the finer

Measuring the shape of randomly arriving pulses

Pulse Source 1

Conversion block 2

Randomizing delay block 4

Computer

ADC 3

Controlling & processing software

triggering

Figure 2. General block scheme of the method for retrieving the shape of the randomly arriving pulses shorter than the acquisition step.

time scale. For a proper set of randomly arrived identical pulses (within the measurement time) at least one sampled oscillation will be positioned on each of the finer sampling steps δT . As a result, one can retrieve a novel oscillation with a step δT , where the signal pulse area Ap will be presented by the true signal samples as it is in boxcar techniques at fixed time-delay pulses. The random arrival of pulses will be here a necessary condition for the method performance. The method will also be effective for fixed-delay pulses, if randomizing the start pulses. The retrieved oscillation of shorter step δT is a convolution of the measured pulse with the response function of the resonator, excited by the pulses. In order to recover the pulse shape one can perform the corresponding inverse transformation. In summary, the problem for measuring with high precision the shape of pulses, shorter than the ADC sampling step could be solved by a set of successive procedures: (1) conversion of short pulses into longer damped oscillations to provide their correct acquisition and saving the pulse arrival information; (2) rearranging of sampled oscillations to some reference time instant and creation of high-resolved damped oscillation; (3) retrieving the pulse shape by deconvolutionbased algorithms.

3. Description of the method 3.1. Block scheme

each finer cell δT . The use of scheme 4 could be avoided, if ADC start pulses are fully independent of the ADC sampling generator. The conversion block 2 has been created by a resonance RC filter (1 MHz resonance frequency, relaxation time ∼5 µs) using LF357 operational amplifiers [15]. 3.2. Acquisition and rearranging of oscillations Let us consider a set of I sampled output signals {Fi (kT )}, created by a set of pulses, randomly arriving within the interval Trand > T ; i = 1, 2, . . . , I is the successive pulse number within the set, k = 0, 1, 2, . . . , KADC − 1 is the successive time sample number, KADC is the time series length. Then, the time argument t can be represented as t = kT + t, 0
t < T . The arrival time (ta )i of the ith pulse with respect to the ADC start can be represented in a similar way as (ta )i = (ka )i T + (ta )i , where (ka )i and (ta )i are the integer and non-integer parts of (ta )i /T , respectively. If we assume identical pulse shapes S(t), differing only in arrival times during the measurement procedure, the sampled values Fi [kT − (ta )i ] of the oscillations will depend (periodically, with a period equal to T ) on the time shift (ta )i of the arrival time with respect to the nearest sampling instant. Let us now divide the sampling interval T into Q  1 finer equivalent sampling cells δT = T /Q (Q is an integer). Then we can approximately write t ≈ kT + qδT = j δT , where q = 0, 1, 2, . . . , Q − 1 is defined by inequalities qδT
t < (q + 1)δT , j = 0, 1, 2, . . . , J − 1, and J = KADC Q. The arrival time (ta )i can also be presented by (ta )i ≈ (ka )i T + (qa )i δT . The following task to be solved is to form a new sampled oscillation of higher sampling step δT on the basis of the set of sampled output signals Fi (kT ). For this purpose we use a rearranging procedure requiring the knowledge of the reference characteristic point (Tosc )i of each sampled oscillation. The position (Tzer )i = (Tosc )i of the first zero passing (figure 1) easily provides a near real time performance for the same accuracy, if compared with the other possible reference points. The algorithm for calculation of (Tzer )i can be written by (Tzer )i = (jzer )i δT = (kzer )i T + (qzer )i δT = [(kzer )i Q (2a) + (qzer )i ]δT ,

It comprises (figure 2) a pulse source 1 (of defined pulse where (kzer )i is the integer part (in number of samples T ) shapes S(t) to be measured), a block 2 for conversion of of (Tzer )i defined by the inequalities F [(kzer )i T ]  0 and pulses into damped oscillations F (t) and an ADC-block 3 for F [[(kzer )i + 1]T ] < 0 and data acquisition and creation of triggering pulses. The entire 
Q|F [(kzer )i T ]| system performance is controlled by specialized software, (q ) = round . zer i |F [(kzer )i T ]| + |F [[(kzer )i + 1]T ]| combined with signal processing programs, according to the method. The computer-generated triggering pulses are passed (2b) through the scheme 4 for a time delay randomizing of pulses, driving the pulse source 1 with respect to the ADC start. This Here the value of Q = T /δT  1 is chosen by the is realized by attaching the triggering pulses to pulses of some processing program and could be arbitrary, in principle. The rearranging procedure starts with the definition of a second generator of a frequency ωrand
ωADC /2, where ωADC new time series U (j δT ) of a length J and a step δT . Then, is the ADC sampling frequency. The second generator is for each ith successive pulse one has to rearrange the samples nonsynchronized with the ADC sampling generator and thus, in F (kT ), according to the rule i its pulses will trigger the source 1 randomly within some  interval Trand larger than the sampling interval T . The Fi (kT ) for j = kQ + [jR − (jzer )i ] number of pulses per measuring time must be chosen to be Ui (j δT ) = 0 for j = kQ + [jR − (jzer )i ]. larger than the number T /δT of cells δT within the sampling interval in order to provide at least one sampled oscillation per (3) 3

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As seen, the values of Fi (kT ) are attached to some joint time position TR = jR δT by positions (jzer )i of their first zero passing (with accuracy δT ). This arranging just takes into account the information about the interval (ta )i ≈ (qa )i δT . The series Ui (j δT ) present the same sampled oscillations, where the samples of Fi (kT ) are disposed on the time axis through intervals of T = QδT and additionally shifted at (qa )i steps as they are on the ADC input. The cases of more than one sampled oscillation of the same values of (qa )i can be accounted, defining an additional integer time series  1 for j = kQ + [jR − (jzer )i ] (4) mi (j δT ) = 0 for j = kQ + [jR − (jzer )i ]. The output highly resolved oscillation V (j δT ) is created by averaging the entire set of rearranged oscillations Ui (j δT ) taking into account the number of oscillations of equal indices (qa )i along the finer time scale j = 1, . . . , J or  I I  Ui (j δT )  i=1 i=1 mi (j δT )  I V (j δT ) = (5) for i=1 mi (j δT ) > 0   I  0 for i=1 mi (j δT ) = 0. As a result we obtain a resultant oscillation V (j δT ) of step δT much shorter than the sampling interval T . The signal area Ap , containing the necessary information for retrieving the pulse shape S(j δT ) is also presented with a step δT . The noise n(t) in sampled oscillations can be of two main origins: (i) an ADC sampling noise; (ii) an input noise. The sampling noise is typically assumed as uncorrelated from sample to sample. The optimal bandwidth for input pulses is of the order of ωopt ≈ π/τp [16, 17] and thus, if τp < T the input noise on the conversion block 2 will be also uncorrelated. It should also be noted that SNRs ∼ 50–100 and more are typical for precise pulse measurements. The output noise in adjacent resolution cells δT will also be uncorrelated as the adjacent samples in V (j δT ) are created by different pulses. The √ noise within each cell δT can be reduced by a factor of ∼ M, where M ≈ I /Q is the mean number of averaged samples of equal indices q, if σa2
δT 2 (σa2 is the variance of the arrival time fluctuations). Moreover, the model of smooth short pulses without extremely short spikes (of the order of δT at Q  1) is a very good approximation here as the pulses are generated by (or passed through) bandwidth-limited electronic circuits. This offers the application of digital filtering to reduce additionally the noise influence. 3.3. Pulse shape retrieval algorithms According to the experimental realization of the method developed in this work, the output high-resolution oscillation V (t) can be expressed in the form of the following convolution:

t S(t  )Rosc (t − t  ) dt  V (t) = S ⊗ Rosc = t0

1/2  = A β 2 + ω02



t t0 

dt  S(t  ) cos[ω0 (t − t  ) + φ]

× exp[−β(t − t )],

(6)  2 where the resonator response function is Rosc (t) = A β + 1/2 ω02 exp(−βt) cos(ω0 t + φ), A is an amplitude factor, 4

1/2 

 1/2 

 = arcsin β β 2 + ω02 = φ = arccos ω0 β 2 + ω02 arctan(β/ω0 ); t0 is the initial moment of the oscillation V (t). The problem to be solved here is to obtain an estimate of the pulse shape S(t) on the basis of an experimental estimate of the high-resolution oscillation V (t). When V (t) is known (experimentally determined), equation (6) is the first kind of Volterra integral equation with respect to S(t). The kernel 1/2  K(t − t  ) = A β 2 + ω02 exp[−β(t − t  )] cos[ω0 (t − t  ) + φ] of this equation is obviously a continuous function of its arguments t and t  . Function V (t) is also continuous because of its integral physical nature. Then (6) has only one continuous solution. This unique solution, obtained in a general analytical form, is in fact an inverse algorithm for retrieving the pulse shape S(t). The solution of equation (6) can be obtained analytically in the following simple way. First, the differentiation of (6) with respect to t leads to the relation  1/2 V I (t) + βV (t) − Aω0 S(t) = − β 2 + ω02

t × dt  S(t  ) sin[ω0 (t − t  ) + φ] exp[−β(t − t  )]; (7) t0

the symbols such as ϕ I (x) and ϕ II (x) denote respectively the first and the second derivatives of the function ϕ(x) with respect to the variable x. By differentiating in turn equation (7) with respect to t and taking into account the expressions of (6) and (7) we obtain that 1/2  V (t) = Aω0 S I (t). V II (t) + 2βV I (t) + β 2 + ω02

(8)

On the basis of equation (8), assuming a zero initial condition S(t = t0 ) = 0 (and respectively V (t = t0 ) = 0 and V I (t = t0 ) = 0, according to (6) and (7)), the solution of equation (6) is obtained in the following analytical form:  

t  2 −1 I 2 1/2   S(t) = (Aω0 ) V (t) + 2βV (t) + β + ω0 dt V (t ) . t0

(9) Equation (9) is in fact an algorithm for calculating (retrieving) the pulse shape S(t) on the basis of experimental estimates of the high-resolution oscillation V (t). Taking into account the sampling, one should imply that t = j δT . The resonator natural frequency ω0 and damping decrement β are determined on the basis of the experimentally-obtained shape of V (t). Let us note for completeness and clarity that an effective retrieval algorithm like (9) is easily derivable in the case of a LC circuit (eventually employed in the conversion block 2, figure 2) as well. Another way to retrieve S(j δT ) is to directly use the well-known Fourier-deconvolution procedure based on a form of the resonator response function Rosc (j δT ) that is derived from the oscillation V (j δT ). Both the above-described algorithms ensure in general a good retrieval accuracy. However, some time broadening appears when retrieving shorter pulses whose duration τp
T /2. This broadening can be considered as consequence of an additional convolution such that instead of V = S ⊗ Rosc we have V = S ⊗ R, where R = Rosc ⊗ Ra , and Ra is the corresponding additional response function. Such a

Measuring the shape of randomly arriving pulses

4. Simulations In this section we present some of the results from the computer simulations we have performed intended to analyse and illustrate the performance and the reliability of the proposed method. The corresponding computer model comprises the entire set of functions of the hardware blocks and the software procedures (algorithms for processing the sampled signals) used in experiments: generation of randomly arriving signal pulses, noise generation, conversion of the pulses into decayed oscillations, analog-to-digital conversion and sampling. We have used in the simulations pulses with different shapes and duration. The results presented here are for Gaussian-shaped and rectangular pulses. For optimal receiving performance the input noise is modeled with a bandwidth equal to the signal bandwidth; the signal-to-noise ratio (SNR) is close to this, estimated in the experiments (∼100–200), and the sampling step T is 50 ns. The finer sampling step used is δT = 0.5 ns. To avoid the necessity of enormously large number of scans, we have also used a smooth monotone sharp-cutoff digital filter with 7δT -wide window for smoothing the recovered pulse shape. Let us first consider the results from simulating the method performance for Gaussian-shaped pulses with different durations. The pulse shape retrieval is based on 4000 realizations (scans). The results show that for the pulses with full width at half maximum (FWHM)  10 ns the shapes retrieved by using the algorithm (9) practically coincide with the corresponding models. In figure 3 we have represented the corresponding shape profiles for the pulse with FWHM = 9.4 ns. The root-mean-square (rms) error in retrieving the pulse (averaged over the entire pulse) is ∼1%. The signal-tonoise ratio simulated here is SNR = 250. As mentioned above the values of the frequency ω0 and the decrement β, which the algorithm is based on are determined on the basis of the shape obtained of V (t). Similar results are also obtainable on the basis of a Fourier-deconvolution procedure. For shorter pulses (FWHM < 10 ns) the rms error increases with the decrease of the pulse width. The behavior of the rms error as a function of the pulse duration is given in figure 4, curve 1. When performing an additional deconvolution, as described in section 3.3, the rms error does not increase essentially for shorter pulse widths (see figure 4, curve 2). The results obtained for the pulses with FWHM = 4.7 ns

SNR ~ 250 ∆ T = 50 ns FWHM = 9.4 ns

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Figure 3. Model pulse shape (dashed curve) and the shape retrieved by use of algorithm (9) (full curve) in the case of 12 bits ADC for the Gaussian pulse with FWHM = 9.4 ns.

rms error (%)

broadening is due to several factors. The first of them is the preliminary noise-damping filtration performed of V (j δT ). The second one is the finite time constant of the conversion block 2 of the measuring system (see figure 2). Thus, to compensate for the broadening one should obviously perform an additional deconvolution by using a properly identified shape of Ra (j δT ). It is natural an estimate of Ra (j δT ) to be determined by a testing experiment making use of a well-defined short pulse S0 (j δT ). Then, a (j δT ) can be considered as acceptable if the estimate R  the corresponding J retrieved pulse S0 (j δT )2 satisfies e.g. the −1  condition J j =1 [S0 (j δT ) − S0 (j δT )]
Dmin , where Dmin is some tolerable error. Here the estimation procedure a (j δT ) as a is based on searching for the true shape of R composition of smooth monotone filters.

18 16 14 12 10 8 6 4 2 0 0.0

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Figure 4. Root-mean-square retrieval error for the cases without (1) and with (2) additional deconvolution.

and FWHM = 3.6 ns are shown in figures 5(a) and (b), respectively. The corresponding rms errors decrease from 7.3% to 1.4% and from 16% to 4%. The effect of the number of bits of ADC is also investigated and the results for the pulse shape with FWHM = 18.8 ns are presented in figure 6. As can be seen the retrieved pulses at different number of bits do not differ essentially. The rms errors are 1% for 12 bits ADC, 1.2% for 10 bits ADC and 1.4% for 8 bits ADC. The dependence of the retrieval accuracy on SNR is very feeble, in general. For instance, for SNR = 1000 to 200, the retrieval error is ∼3% for a pulse shape with FWHM = 7 ns. Only when SNR < 200 the error increases slightly up to ∼6% for SNR = 100. It is pointed out in section 3.2 that the retrieval error should depend in general on the number of scans I so that the rms error ∝ I −1/2 . One can see in figure 7 that such a dependence is indeed consistent with the results from the simulations performed concerning this point. In this figure the rms error obtained by simulations at different number of scans (triangles), assuming Gaussian pulse shape with FWHM = 18.8 ns and SNR = 150, is compared with a curve describing the dependence ∝ I −1/2 . The starting point of the curve is obtained at I = 400 that is the minimum number 5

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Figure 5. Model pulse shape (dashed curve) and the profiles retrieved without (dotted curve) and with (full curve) additional deconvolution, for pulses with FWHM = 4.7 ns (a) and 3.6 ns (b).

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Figure 6. Retrieved pulse shape profiles by using algorithm (9) in the case of 12 bits- (full curve), 10 bits- (dotted curve) and 8 bits(chain curve) ADC. The input pulse shape model is given by a dashed curve.

of scans here ensuring at least one input pulse per each interval δT . We have considered above too short and sharp but still differentiable (and continuous) pulse shapes. At the same time it is interesting and useful to investigate by simulations the case of extremely sharp (discontinuous) shapes like the rectangular ones. The results from such simulations we have performed concerning rectangular pulses (with duration τp = T /3) are shown in figure 8; it is accepted that the SNR = 200, the 6

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Figure 8. Result from retrieving rectangular pulse with duration τp = T /3 at SNR = 200. The input rectangular shape model is given by a solid curve.

FWHM

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Figure 7. Root-mean-square retrieval error (triangles) versus number of scans I, for a Gaussian pulse shape with FWHM = 18.8 ns, compared with a curve (including circles) describing the dependence ∝ I −1/2 .

factor Q = 500 (δT = 0.1 ns), and the number of scans is 10 000. It is seen that, despite the unavoidable distortions, the recovered pulse shape is rectangular-like in character. The rise time of the retrieved rectangular-like pulses does not exceed some value of 0.05T depending on the SNR and the digital filtering procedures. These results show that the method, being suitable in general for too short and sharp but smooth pulse shapes, may be applicable to some discontinuous pulse shapes as well. As a whole, the results from the simulations confirm the expected good performance of the retrieval method developed here, under nearly the same conditions the experiment has been done.

5. Experimental testing and analysis The experimental system for testing the method performance just corresponds to the block-schematic in figure 2. For precise control of the input pulse shapes we included additionally the LP142 LeCroy Digital 8-bits oscilloscope (boxcar type) of equivalent sampling rate up to 25 GHz [10]. In this case the randomizing delay scheme 4 was avoided to provide

Measuring the shape of randomly arriving pulses

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Figure 9. Highly resolved oscillation V (j δT ) and a sampled signal F (kT ) (inset) corresponding to the pulse pair shown in figure 12.

a stable (non-jitter, internal triggered) generation of pulses by the pulse generator 1 (Tektronix PG508 and TG502). The acquisition block 3 (figure 2) contains 8 bits/20 MHz ADC, T = 50 ns, with a sampling noise (uncertainty) of the order of one discrete (SNR ∼ 200). The choice of 50 ns sampling interval provides good opportunities to easily vary the pulse shapes by typical standard generators, to randomize the pulse arrival times, to measure the pulse shapes in boxcar regime by the oscilloscope LP142, and to observe them by a standard oscilloscope (Tektronix 2235). The testing includes the following basic procedures: (1) formation of the highly resolved oscillation V (j δT ); (2) determination of both reference functions Rosc (j δT ) and Ra (j δT ); (3) recovery of pulses of different shapes with a step δT  T ; (4) calculation of basic pulse shape parameters and estimation of restoration accuracy. We will discuss below the pulse recovery of randomly arriving pulses with considerably shorter (up to 100 times) step δT . 5.1. Formation of highly resolved oscillations The retrieved oscillations V (j δT ) are of very good quality as compared with their oscilloscope image. The oscillation V (j δT ) restored on the basis of algorithm (5) with step δT = 0.5 ns (equivalent sampling frequency 2 GHz) for a pair of short randomly arriving pulses (section 5.3) is shown in figure 9. An ADC-sampled oscillation F (kT ) is given in the inset in the same figure. As it is seen, the signal area Ap , carrying the shape information, is low-noise retrieved. Its specific form demonstrates the saving of the pulse information in the retrieved oscillation. This result is the key for the successful performance of the method. Moreover, the good quality of the retrieved oscillation within the free oscillation area Aosc offers the opportunity to derive the necessary oscillating reference function Rosc (j δT ).

200

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400

Figure 10. Pulse shape profiles recorded by the oscilloscope (dashed curve) and retrieving by using Fourier deconvolution (circles). The full curve is a fit of the retrieved profile.

The results obtained for pulses with different shapes and lengths are in accordance with the results from the simulations. The case of retrieving relatively wide pulses (FWHM ∼ T = 50 ns) is illustrated in figure 10 where the restored pulse shape (circles) is compared with that recorded by the boxcar oscilloscope (dashed curve). It is seen that in this case the pulse width is retrieved with a good accuracy. The retrieval rms error obtained (see section 4) is of the order of 7.5%. An additional filtering of the results leads to smoothing the restored pulse shape (see the full curve) and certainly to loosing its flat top part. As in the simulations, with decreasing the pulse length a bias error arises consisting in a broadening of the restored shape with respect to the true one. Because of the additional influence of the time constant of the measuring system (figure 2, block 2) the broadening obtained here appears at relatively wider pulses in comparison with the simulations. To compensate for this broadening one should determine the corresponding response function Ra (j δT ) by making use of a well-defined short pulse. Such a reference pulse we have used is shown in figure 11(a) by a dashed curve. In the same figure we have also shown the pulse shapes recovered with additional deconvolution (full curve). The rms retrieval error in this case (∼4.5%) is obviously considerably smaller than that obtained without additional deconvolution (∼18%). The resolution step achieved is δT = 4 ns. The additional deconvolution based on the response function Ra (j δT ) also essentially improves the accuracy of retrieving the asymmetric pulse shape given in figure 11(b) (FWHM = 26 ns); the error falls from 15% to 5%. The profile of the function Ra (j δT ) is shown in the inset in figure 11(a). It is a result of multiple convolution of a proper digital filter (having smooth frequency spectrum without zeros) [18] by itself. The number of convolutions is determined by the required upper limit Dmin of the retrieval error.

5.2. Recovery of pulse shapes shorter than the ADC sampling interval

5.3. Resolving a pair of short randomly arriving pulses with an inter-pulse delay shorter than the sampling step

Various experimental data were obtained and processed according to the procedures described in the preceding sections 3 and 4. Some of the results obtained are given below for illustrating the performance of the retrieval algorithms.

Here we demonstrate the capabilities of the method to resolve randomly arriving pulses of complicated shapes shorter than the acquisition interval. A traditional experiment we have performed here is to resolve two close pulses. Both the pulses 7

D V Stoyanov et al FWHM = 16 ns ∆T = 50 ns δT = 4 ns

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recorded by the boxcar oscilloscope are given in figure 12 by a dashed curve. Figure 12(a) displays the pulse pair recovered with step δT = 1 ns (equivalent sampling frequencies 1 GHz) and rms accuracy ∼5%. Figure 12(b) corresponds to 0.5 ns sampling step, or 100 times finer than the ADC acquisition step (2 GHz equivalent sampling frequency); the rms retrieval accuracy is 5% again. In the cases of δT = 4 ns and 2 ns we have also achieved good retrieval accuracy of the order of 8% and 6.5%, respectively. The increase of the equivalent sampling frequency (or the ratio T /δT ) is limited by the noise. The minimum tolerable sampling step δT can be estimated by the condition δT ≈ σa , where σa is the rms deviation of the arrival times for pulses without randomized delays (avoiding block 4 in figure 2). In our measurements we obtained σa
0.5 ns for single measurements. The recovered pulse pair in figure 12(c) displays the same case as in figure 12(b), but for the lowest number of single measurements Imin ∼ 400 , when at least one pulse was arriving per each interval δT . As seen, the retrieved pulses are also of good quality. The tolerable number Imin is estimated to be of the order of Imin ∼ (2 − 4)T /δT . As evident, by performing larger number of single measurements I  Imin one can additionally reduce the rms variations, and thus increase the tolerable equivalent sampling frequency ωsam = 2π/δT . The increase of ωsam can also be achieved by using high capacity (say 12 bits) ADC.

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Figure 11. Pulse shape profiles with FWHM = 16 ns (a) and 26 ns (b) recorded by the boxcar oscilloscope (dashed curve) and retrieved with additional deconvolution (full curve). The inset of (a) represents the corresponding additional response function Ra (j δT ).

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Figure 12. Pulse pair profiles recorded by the boxcar oscilloscope (dashed curve) and retrieved with a step δT = 1 ns (a) and 0.5 ns (b), (c), on the basis of 4000 (a), (b) and 400 scans (c).

6. Conclusion We developed and tested a novel method for measuring the shape of pulses shorter than the acquisition step, which is effective for random delays of the input pulses with respect to the start pulse. It is based on conversion of the short pulses to be measured into longer damped oscillations and their correct acquisition (sampling) with saving the pulse information, rearranging of the sampled oscillations with respect to some reference time instant to form a finer discretizing high-precision oscillation, and retrieving the pulse shape by inverse algorithms. We demonstrated experimentally the good performance (5–7% rms error) of this method (by using 20 MHz/8 bits ADC) to measure the shape of randomly

Measuring the shape of randomly arriving pulses

arriving pulses, shorter than the sampling interval (50 ns), with an equivalent sampling frequency up to 2 GHz (0.5 ns equivalent sampling step). The maximum tolerable equivalent sampling frequency can be estimated by accepting the realistic value of the ratio T /δT = 100. If using the fastest (5 GHz, T = 200 ps) commercial ADC [7] one could obtain a value of (ωsam )max ∼ 500 GHz for the maximum equivalent sampling frequency (δT = 2 ps). Thus, using this method one could analyse very short, randomly arriving pulses. It is also of interest to compare approximately the minimum number of pulse repetitions Imin required by this method with the repetitions (Imin )bc , required by the boxcar technique. As mentioned above, for T /δT = 100 one needs of the order of Imin ∼ 200–400 pulse repetitions. In boxcar regime (assuming pulse duration τp
T , the same equivalent sampling step δT = T /100 and the same repetition number) it is required that (Imin )bc ∼ Imin in order to scan (without averaging) the entire pulse shape, including some time intervals around both the pulse sides. The more precise estimates for Imin and (Imin )bc could be obtained by taking into account the effect of noise. In general, one could state that the two methods require approximately the same number of pulse repetitions. The method can be effectively applied for creation of some novel short-pulse measuring techniques, avoiding the problem of time synchronization to the start pulses in lidars and radars, nuclear experiments, tomography, communications, etc, where the useful information is carried by the time resolved profiles of short pulse processes.

Acknowledgments This work was supported in part by the Bulgarian National Science Fund under grant F-907.

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