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Sampling-Reconstruction Procedure with a Limited Number of Samples of Stochastic Processes at Random Times

VLADIMIR A. KAZAKOV

Department of Telecommunications of the School of Mechanical and Electrical Engineering National Polytechnical Institute of Mexico

U. Zacatenco, C.P. 07738, Mexico City MEXICO

Abstract: - The review of author’s publications is presented. All papers are based on the same methodological approach: the statistical description of the Sampling - Reconstruction Procedure (SRP) is given on the basis of the classical conditional mean rule which provides the minimum reconstruction error. There are two cases of non uniform sampling: the sample location is arbitrary and known and the location of samples is unknown and is described by probability density function (pdf). There are some principal distinctions between reviewed papers and known publications: the number of samples is limited and arbitrary, the pdf of sampled processes is taken into account, the sampled process can be stationary and non stationary, the samples with jitter are described by some new models: the jitter pdf is the Beta distribution, any samples can have their own jitter, other can have not jitter, the jitter effect can have stationary and non stationary character. New aspects of the SRP of Markovian processes with jumps are discussed also. Key-Words: - Random sampling, Optimal reconstruction, Stochastic processes. 1 Introduction The first wave of the scientific interest to the Sampling – Reconstruction Procedure (SRP) of stochastic processes sampled at random times was in the middle of the last century [1] - [3]. During the last years one can observe the second wave of such type of the interest [4] – [17]. All mentioned papers are characterized by some specific features: the number of samples is equal to infinity; the probability density function (pdf) of sampled process is not defined; the sampled process is stationary and is described by the covariance function or by the power spectrum; the jitter effect is characterized by the same pdf for all samples. The present paper is the review of author’s publications devoted to the statistical description of the SRP of different types stochastic processes sampled at random times. The well known conditional mean rule is used in these works. This rule provides the minimum reconstruction error [18]. As a result of the investigation the two principal SRP characteristics are obtained: the reconstruction function (this is the conditional mean function) and the error reconstruction function (this is the conditional variance function). The application of this rule to the SRP allows us to investigate some new aspects about this problem. For instance, it is possible to consider the following cases: 1) an arbitrary and limited number of samples; 2) the pdf of sampled

process is taken into account; 3) the stationary and non stationary sampled processes; 4) the reconstruction function and the error reconstruction functions are calculated in the time domain, this provides a possibility to demonstrate explicitly the influence of many different parameters of the sampled process and the sampling procedure on the principal SRP characteristics; 5) an arbitrary known location of samples; 6) the jitter effect of sampling can be described by some new models; 7) the SRP description of some Markovian processes with jumps. Below we concentrate our attention on the influence of features of sampled processes and of random time sampling on the main SRP characteristics: the reconstruction function and (especially) the error reconstruction function. 2 The SRP of Stochastic Processes with an Arbitrary Known Location of Samples Here we consider some different variants of sampled stochastic processes: 1) Gaussian processes, 2) non Gaussian continuous processes of common types, 3) non Gaussian processes on the output of non linear non inertial converters driven by Gaussian process, 4) the processes with jumps. Non Gaussian processes of common types and processes with jumps are considered in the Markovian interpretation.

Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp26-32)

2.1 Gaussian processes The multidimensional pdf is known for Gaussian processes. This pdf is determined by the mean ( )m t , the variance 2 ( )tσ and the covariance function

1 2( , )K t t . Let us fix an arbitrary set of samples of Gaussian process { }1 2, ( ), ( ),..., ( )NX T x T x T x T= . Then instead of the given process we have the conditional process ( )x t . All realizations of this new processes pass through all fixed points of the set ,X T . The main statistical characteristics of the conditional Gaussian process are known (see, for instance, [18, 19]). We write here the expressions for the conditional mathematical expectation

( , ) ( )m t X T m t= and the conditional variance 2 2( , ) ( )t X T tσ σ= . These formulas have the view:

( ) ( ) ( , ) ( ) ( )N N

i ij j ji j

m t m t K t T a x T m T⎡ ⎤= + −⎣ ⎦∑∑ , (1)

2 2( ) ( ) ( , ) ( , )N N

i ij ji j

t t K t T a K T tσ σ= − ∑∑ , (2)

where ija are an element of the inverse covariance

matrix 1

( , )i jK T T−

.

The conditional mean ( )m t is the reconstruction function and the conditional variance 2 ( )tσ is the error reconstruction function. As one can see the expressions (1) and (2) describe the general case of a non stationary process ( )x t . The SRP description of some non stationary Gaussian process is given in [21, 22]. The expression (1) shows that the reconstruction function ( )m t is the linear function of samples. This is the specific feature of the reconstruction procedure of Gaussian processes. Below we restrict our consideration by the stationary sampled processes with the characteristics 2( ) 0, ( ) 1m t tσ= = and

1 2 1 2( , ) ( )K t t K t t= − . Then instead of (1) and (2) we have:

( ) ( ) ( )N N

i ij ji j

m t K t T a x T= −∑∑ , (3)

2 ( ) 1 ( ) ( )N N

i ij ji j

t K t T a K T tσ = − − −∑∑ . (4)

The expression (3) can be rewritten in the form

( ) ( ) ( )N

j jj

m t x T B t= ∑ , (5)

where the basic function ( )jB t is determined by the formula

( ) ( )N

j i iji

B t K t T a= −∑ . (6)

We can observe that the form of the basic functions ( )jB t depends on the current number j of the sample, the quantity of samples N, on the set of arbitrary sampling instants iT , on the covariance moments ( )i jK T T− which are involved into

elements ija , on the covariance function ( )iK t T− . Now let us introduce some typical variants of the covariance functions of Gaussian processes. It is convenient to consider the processes on the output of some linear systems driven by white noise [20].

Namely, on the output of the integrated RC circuit (with the parameter α ) we have the Gaussian Markov process with the normalized covariance function in the stationary regime: ( )( ) expK τ α τ= − . (7)

The output process on the two series integrated RC circuits with the equal parameters α has the next normalized covariance function: ( ) ( )( ) 1 expK τ α τ α τ= + − (8)

and the output process on the three series integrated RC circuits has the next normalized covariance function: ( ) ( )2 2( ) 1 / 3 expK τ α τ α τ α τ= + + − (9)

We can carry out the correct comparisons of the obtained results if all processes have the same covariance time 1cτ = . In order to do this we have to multiply the parameters α in (8) and (9) by the coefficients 2 and 8/3 correspondingly.

Fig. 1. The basic functions for the Markovian process

As examples the graphs of the basic function and the error reconstruction function are presented in Fig. 1 and Fig. 2. In Fig. 1 one can see the family of the basic functions ( )jB t for the Markovian process with the covariance function (7). The sampling intervals


are different between all pairs of samples. The forms of all basic functions are different as well. In Fig. 2 the error reconstruction function for the process with the covariance function (8) is presented. The sampling intervals are different and the maximum of the error curves are different as well. When the interval is bigger the error maximum is bigger also. The case of Markovian process is the simplest because the SRP of this process depends on only two nearest samples (on the left and on the right of the current reconstruction time). Particularly, it means that one can completely describe the SRP between any two samples with an arbitrary interval.

Fig. 2. The error reconstruction function for the process with the covariance function (8) The output process of the resonance filter has the covariance function 0( ) exp( )K Cosτ α τ ω τ= − . (10) The SRP of this process is described by the same manner, but there is one new parameter

0 0/ / 2T T Tω π∆ = ∆ (here T∆ is the sampling interval). Besides this one can consider the model of Gaussian process with the restricted power spectrum ( )S ω . In this case it is necessary to use the Wiener - Khinchin transform and to determine the corresponding covariance function ( )K τ . Then applying the formulas (4) and (6) one can describe the SRP with limited number of samples. The basic function is not expressed by the function of type /Sinx x , and the error reconstruction is not equal to zero. The particular result of Balakrishnan‘s theorem [23] is valid only for the infinite number of samples [24, 25]. In the similar manner one can investigate the SRP of some multidimensional Gaussian processes [24, 26]. For instance, on the basis of corresponding matrix expressions we can describe the SRP of vector process on the outputs of many linear filters driven by Gaussian process. The minimum error will be in the case of non uniform sampling of output processes. The SRP of Gaussian fields can be analyzed in the same manner [24]. Here one can emphasize that the expressions (3) and (4) are known. But there are not their

applications in the SRP description of Gaussian processes (with the exception of some author’s papers). So, one can say that this material concerning the optimal SRP of Gaussian processes has the applied novelty. The examples play the very important role in this area because they are the basis of many actual practical recommendations about the choice of the sampling interval or the calculation of the error reconstruction function. There are two author’s patents with schemes of the interpolators of continuous stochastic processes with non uniform sampling intervals [28, 29]. 2.2 Non Gaussian continuous processes of

common types The complete description of the optimal SRP of non Gaussian processes is not possible because there are not acceptable general expressions of multidimensional pdf of any processes having a practical interest. This is the reason that we restrict our consideration by the SRP of non Gaussian Markovian processes of common types with two samples only. The main characteristic of any Markovian process is the transition pdf ( ( ) ( ))i jw x t x t . On the basis of

this pdf one can obtain the required conditional pdf [24]

1 2

1 2

2 1

( ( ) ( ), ( ))

( ( ) ( )) ( ( ) ( ))( ( ) ( ))

w x t x T x T

w x t x T w x T x tw x T x T

=

=. (11)

Knowing pdf (11) one can determine the reconstruction function as the conditional mean ( )m t and the error reconstruction function as the conditional variance 2 ( )tσ . So, in order to obtain the required expressions it is necessary to introduce into (10) the concrete formulas of the transition pdf of some chosen Markovian processes. There are some analytical formulas of the transition pdf for the processes with the Rayleigh, gamma etc. distributions in [19, 20, 27]. Here we do not give the final formulas because they are cumbersome. Some examples are given in [24]. It is quite possible to analyze the SRP of some non Markovian non Gaussian processes with two samples only. It is clear that this SRP description will be not optimal because here the reconstruction procedure depends on many samples around the current reconstruction time. We notice two general important points with respect of the SRP of non Gaussian processes: 1) the reconstruction function is non linear function of


samples, 2) the error reconstruction function depends on the values of samples. So, in order to eliminate this dependence it is necessary to average the conditional variance with the two dimensional pdf

1 2( ( ), ( ))w x T x T . 2.3 The processes at the output of non-linear

non-inertial converters For the sake of simplicity let us assume that the input process is Gaussian with known characteristics. Then changing the non linear performance of the converter we can have the different kinds of non Gaussian processes on its output. We consider three main types of the nonlinearity: 1) polynomial, 2) piecewise linear, 3) exponential. The unconditional characteristics of the output processes have been investigated in [19, 20]. Our problem is to describe the SRP of these output processes. In other words we have to investigate the conditional moment functions of the output processes, i.e. the conditional mean and the conditional variance. We restrict our consideration by the inverse non linear performances of simple types. The main idea is to recalculate the samples of the output process to the corresponding samples of the input Gaussian process and then to express the conditional output moment functions by the known conditional input moment functions. This method of the SRP description has been proposed in the first time by the author in [30]. The nonlinearities of the polynomial and piece lineal types have been analyzed in this work. As one example we give here the expression of the conditional moment functions on the output of the converter with the polynomial performance 2y ax= with one branch. Then the output reconstruction function 1( , )ym t N k− and the error reconstruction

function 2 ( , )y t N kσ − are expressed by the formulas:

1 2( , ) ( , )y xm t N k am t N k− = − ,

2 2 4 2 21( , ) 2 2( )x

y x xt N k a mσ σ σ⎡ ⎤− = +⎣ ⎦ ,

where (N-k) means the number of the reconstruction interval, 1 2,x xm m and 2 4,x xσ σ are the conditional moments of the input process ( )x t . The details of these investigations can be found in [30]. The case of the exponential nonlinearity has been investigated in [31]. The both variants of non Markovian and Markovian processes can be analyzed by this method. Once again, the reconstruction function is non linear function of samples and the error reconstruction function depends on the samples. In order to obtain the unconditional variance function it

is necessary to carried out the statistical average operation. Practically this operation can be realized only for Markovian input process. 2.4 The processes with jump First, it is necessary to note that the SRP of such types of the processes has been practically not investigated. We mark only two works [16, 32]. where the SRP of the binary Markovian process is considered within limits of the covariance theory, i.e. on the basis of Gaussian approximation. Here we mention some specific features of the SRP description of Markovian processes with jumps: 1) It is not necessary to determine the reconstruction function because the realization form of the sampled process is known: this is the straight horizontal lines on the right and on the left of the sample. 2) The problem is to determine the estimation of the jump point from the given state to another. So, this estimation point is the reconstruction point and the variance of this estimation is the reconstruction error. 3) Once again, we need to use the conditional mean rule when two different samples are known. The result of the solution is a number, but not a function. 4) There is a special type of the error here. This error can be occurred when the sampling interval is rather big with comparison the covariance time of sampled process. In fact, during this interval an undetected jump can be appeared: from the given state into another state and the regression into the first state. So, the duration of the sampling interval must be chosen on the basis of two demands: the variance of the estimation and the probability of the undetected jump must be given. If the duration of the sampling interval is rather small then the undetected jump can be neglected. In the case of the SRP of the binary Markovian process the conditional pdf of the jump moment between two different samples is the cut exponential pdf. The discussed problems have been investigated by the author and Y.A.Goritskiy (the unpublished results). There are two author’s patents with the reconstruction schemes of the processes with jumps on the basis of their samples [33, 34]. 3 The SRP with Jitter The SRP with jitter is related with the case of unknown location of the set of samples. Owing to some instability factors the real location of samples has not fixed positions but is characterized by a scatter around some deterministic points. The jitter effect is described by some probabilistic models.


There are quite a lot of publications devoted to the statistical analysis of the SRP with jitter. Beginning the Balakrishnan paper [1] nearly all other authors have carried out their investigations in the limits of Balakrishnan´s approach, i.e. the number of samples is equal to infinity, the pdf of the sampled process is not taken into account, all samples are characterized by the same pdf of jitter. Our approach provides a possibility to overcome these drawbacks and to investigate the SRP with jitter more deeply. 3.1 The classification of jitter models Here we mention all mathematical jitter models which we could find in the literature. The jitter effect can be described by discrete and continuous random variables. The jitters of some samples can be independent or independent. In the first case it is sufficiently to know one dimensional pdf, in the second case – the two or more dimensional pdf. On the basis of our approach one can investigate the variants when the samples can have the same or different jitter distributions. Particularly, some samples may have the jitter and some other samples can be without jitter. The jitter pdf can be changed in the time. In the discrete case this nonstationarity can be described by discrete Markovian chains. In the continuous case the nonstationarity can be characterized by the Fokker-Plank-Kolmogorov equation determined in some discrete times. In [35] the new convenient jitter model has been suggested: the jitter is described by beta distribution. 3.2 The SRP of Gaussian processes with jitter We use the formulas (6) and (4) for the basic function and for the conditional variance. There are many time arguments involved in these formulas. We pay the attention to the arguments of the covariance functions

( ), ( )i jK t T K T t− − and the covariance moment

( )i jK T T− . The inverse covariance matrix with the

elements ija contains many covariance moments

( )i jK T T− . All time arguments iT in the formulas (3) – (6) are deterministic values. Now we consider the situation when the instant time iT of any sample ( )ix T can have a jitter. First we study the case of the independent jitter of samples times, i.e. i i iT T ε= + , where iε is the random variable with known pdf ( )iw ε . In result the

functions ( )jB t and 2 ( )tσ become random

functions with a complex dependence of many random time arguments ,i jT T There must be the restrictions on the type of

( )iw ε : i) the tails of the pdf with infinite limits must tend to zero very quickly, i.e.: 1 1i i i i iT T Tσ σ − −+ ∆ = − (12) where 1,i iσ σ − are the standard deviations of

( )iw ε , 1( )iw ε − ; ii) if the pdf ( )iw ε is defined by the

restricted limits [ ], ( )i i i ia b b a≥ then

1 1( ) ( )i i i i ib T T a T− −− + − ≤ ∆ (13) By the way, the beta - distribution of jitter is satisfied to the equation (13) and provides a wide possibility for approximation of many real jitters by the convenient analytical expression. The main idea of the SRP with jitter investigation is connected with the statistical average operations of the expressions of the basic functions and the error reconstruction function: 2 ( ) 1 ( ) ( )

N N

i ij ji j

t K t T a K T tσ = − − −∑∑ (14)

( ) ( )N

j i iji

B t K t T a= −∑ (15)

where the angle parentheses mean the statistical average with respect of all time random variables iT and jT . The calculations of the average functions (14) and (15) can be fulfilled numerically. On the basis of the algorithm (14) and (15) many types of the SRP with jitter have been analyzed. To be exact, we have considered the SRP of Gaussian processes with the covariance functions (7) – (10), with the different number of samples N, with all above mentioned models of jitter, including the case with the system with two sources of jitter [36 - 39]. Finally, one can notice two main effects: 1) the basic functions change their forms and become smoother; 2) the error reconstruction functions are not equal to zero in the points iT . In Fig. 3 one can see the typical graph of of the error reconstruction function. There are 4 samples. The samples with number 1 – 3 have jitter with the beta distribution and with some different parameters. The first and the second samples have the symmetric law; the third is characterized by the asymmetric beta distribution. There are three curves in Fig. 3 for the Gaussian processes with the covariance functions (7) – (9). The corresponding numeration of the curves is 1 – 3. Curve 3 is the average error reconstruction function for the


covariance function (9) and it is smaller in comparison with curves 1 and 2, because the realizations of this process are very smooth. The Markovian process, as the chaotic process, has the biggest errors (curve 1). It is natural that all curves have their minimum on the points of the samples, with the exception of the third sample because the jitter is asymmetric here. The error on the point of the fourth sample is equal to zero because there is not any jitter here.

Fig. 3. Average error reconstruction functions given by three different types of processes and considering

four samples with different jitter.. The multidimensional algorithm of the SRP with jitter with respect of a vector Gaussian process and Gaussian fields can be carried out in the same manner. 3.3 The SRP of non-Gaussian processes with

jitter The same idea must be applied in order to analyze the SRP with jitter of above mentioned non Gaussian processes: the continuous processes of common type and the processes on the output on the non lineal non inertial converters. In these cases the basic functions can not be used. So, it is necessary to average the reconstruction function ( )m t and the conditional variance 2 ( )tσ with respect of the values of samples and jitters. It is obviously the numerical calculations become cumbersome. 4 Conclusion The review of author’s papers devoted to the SRP with random sampling of many stochastic processes is given. The analysis is carried out on the basis of the one method – the conditional mean rule. Taking into account the pdf of sampled process and the limited number of samples we have a possibility to

form some practical recommendations about the choice of the reconstruction functions and sampling intervals when the error reconstruction is given. There is another variant: if the sampling intervals and the jitter characteristics are given then one can calculate the quality of the reconstruction procedures. References: [1] A. V. Balakrishnan, On the problem of time jitter in sampling”, IRE Trans. on Information Theory, vol. IT-8, 1962, pp. 226-236. [2] W. M. Brown, “Sampling with random jitter”, J. SIAM, vol. 11, 1963, pp. 460-473. [3] B. Liu and T. P. Stanley, “Error bound for jittered sampling”, IEEE Trans. Automat. Contr., vol. AC-10, No. 4, 1965. [4] W. L. Gans, “The measurement and deconvolution of time jitter in equivalent-time waveform samplers”, IEEE Trans. Instrum. Meas., vol. IM.32, No. 1, 1983, pp. 126-133. [5] E. I. Plotkin, L. M Roytman and M. N. S. Swamy, “Reconstruction of nonuniformly sampled band-limited signals and jitter error reduction” Signal Processing, vol. 7, 1984, pp. 151-160. [6] Y. C. Yenq, “Digital spectra of nonuniformly sampled signals: fundamentals and high-speed waveform digitizers”, IEEE Trans. Instrum. Meas., vol. 37, No 2, 1998, pp. 245-251. {7} Y. C. Yenq, “Digital spectra of nonuniformly sampled signals: digital look-up tunable sinusoidal oscillators”, IEEE Trans. Instrum. Meas., vol. 37, No 3, 1998, pp. 358-362. [8] E. Van der Onderaa, J. Renneboog, “Some formulas and applications of nonuniform sampling of bandwidth-limited signals”, IEEE Trans. Instrum. Meas., vol. 37, No 3, 1998, pp. 353-357. [9] T. M. Souders, D. R. Flach, C. Hagwood and G. L. Yang, “The effects of timing jitter in sampling systems”, IEEE Trans. Instrum. Meas., vol. 39, No 1, 1990, pp. 80 - 85. [10] M. Shinagawa, Y. Akazava and T. Wakimoto, “Jitter analysis of high-speed sampling systems”, IEEE J. Solid-State Circuits, vol. 25, No. 1, 1990, pp. 220-224. [11] G. Tong and T. M. Souders, “Compensations of Markov estimator errors in time-jittered sampling of nonmonotonic signals”, IEEE Trans. Instrum. Meas., vol. 42, No 5, 1993, pp. 931-935. [12] J. Schoukens, R. Pintelon and G. Vandersteen, “A sinewave fitting procedure for characterizing data acquisition channels in the presence of time base distortion and time jitter”, Trans. Instrum. Meas., vol. 46, No. 4, 1997, pp. 1005-1010.


[13] G. Vandersteen and R. Pintelon, “Maximum likelihood estimator for jitter noise models”, IEEE Trans. Instrum. Meas., vol. 49, No 6, 2000, pp.1282-1284. [14] N. Da Dalt, M. Harteneck, C. Sander and A. Wiesbauer, “On the jitter requirements of the sampling clock for analog-to-digital converters”, IEEE Trans. Circuits Syst. 1, vol. 49, No 9, 2002, pp. 1354-1360. [15] J. Walker, “An improved jitter time-stamp method for source timing recovery in AAL2”, in Proc. of Fourth International Symposium in Communication Systems, Networks and Digital Signal Processing, Newcastle, UK, July 20-22, 2004, pp. 318 – 321. [16] B. Lacaze, “Stationary clock changes on stationary processes”, Signal Processing, vol. 55, 1996, pp. 191-205. [17] B. Lacaze, “Reconstruction of stationary processes sampled at random times”, in the book: F. Marvasti, Editor. “Nonuniform Sampling: Theory and Practice”, New York, Kluwer Academic / Plenum Publishers, p. 924, 2001. [18] G. Cramer, “Mathematic Methods of Statistic”, Princeton, N. J.: Princeton University Press, 1946. [19] R.L.Stratonovich, Topics in the Theory of Random Noise, Vol. 1, Gordon and Breach, N.Y. 1963. [20] V. I. Tikhonov, Statistical Radio engineering, Moscow, “Radio i Swiaz”, 1982. (In Russian.) [21]. V.A.Kazakov, M.A.Beliaev. Sample-reconstruction procedure of some nonstationary processes. Radioelectronics and Communication Systems, vol.40, No 9, pp. 43-49. 1997 [22] V.A. Kazakov and M. A. Belyaev, “Sampling reconstruction procedure for Non-Stationary Gaussian processes based on conditional expectation rule”, Sampling Theory in Signal and Image Processing, vol. 1, No. 2, pp. 135–153, May 2002. [23] A. V. Balakrishnan, “A note on the Sampling Principle for Continuous Signals”, IRE Trans. on IT, IT-3, pp. 143-146, 1957. [24] V. Kazakov. “The sampling-reconstruction Procedure with a Limited Number of Samples of Stochastic Processes and Fields on the Basis of the Conditional Mean Rule”, Electromagnetic Waves and Electronic Systems. Vol. 10, No 1, 2005, pp. 98-116. [25] V.A.Kazakov, S.A.Afrikanov, M.A.Beliaev. Comparison of two algorithms of the realization restorations using random numbers of counts.. Radioelectronics and Communication Systems . vol. 37. pp. 43-45. No 4. 1994. [26].V.A.Kazakov, M.A.Beliaev. Reconstruction of realizations of Gaussian processes from readings of

the process and linear transformation of it. Telecommunications and Radioengineering. vol. 49. pp. 97-102. No 9. 1995. [27] V.I. Tikhonov, M.A.Mironov, Markovian Processes. M., “Sov. Radio”, 1977. (In Russian.) [28].V.A.Kazakov, A.N.Gubin. Linear Interpolator. Patent of USSR No 894740, 1980. (In Russian.) [29]V.A.Kazakov, A.N.Gubin. Interpolator. Patent of USSR No 885975, 1980. (In Russian.) [30] V. A. Kazakov, “Regeneration of samples of random processes following nonlinear inertialess convertions”, Telecommunication and Radioengineering, vol. 43, No. 10, 1988, pp. 94-96. [31] V.A. Kazakov and S. Sánchez, “Sampling-reconstruction procedure of random processes at the output of exponential Non-Linear converters”, Electromagnetic Waves and Electronic Systems, vol. 8, No. 7-8, 2003, pp.77-82. [32] B.Lacaze, Periodic bi-sampling of stationary processes. Signal Processisng, vol. 68, 1998, pp. 283-293. [33]V.A.Kazakov. Interpolator of the binary processes, Patent of USSR No 756425, 1980. (In Russian.) [34] V.A.Kazakov. Device for the interpolation of the jump processes, Patent of USSR No 1023350, 1981. (In Russian.) [35] V.Kazakov, D.Rodriguez. “Sampling-Reconstruction Procedure of Gaussian processes with a Finite Number of Samples with Jitter”, Proceedings of the Fourth International Symposium 20-22 July 2004 “Communication Systems, Networks and Digital Signal Processing”- CSNDSP-04. Newcastle, UK., pp. 557-560. [36] V.Kazakov, D.Rodriguez. “Sampling-Reconstruction Procedure of Gaussian Processes with an Arbitrary Number of Samples and Uniform Jitter”, Electromagnetic Waves and Electronic System,. vol. 9, No 3-4, 2004, pp. 108-114. [37] V. Kazakov, D. Rodriguez. “Sampling Jitter and Reconstruction Procedure of Gaussian Processes with a Limited Number of Samples”. WSEAS Trans. On information Science & Applications. Issue 3, vol. 2, March 2005, pp. 287-294. [38] V. Kazakov, D. Rodriguez. “Sampling-Reconstruction Procedure of Gaussian Processes with two Jitter Sources”. WSEAS Transactions On Electronics. Issue 2, vol. 2, April 2005, pp. 53-57. [39] V. Kazakov, D. Rodriguez. “Reconstruction of Gaussian processes with an arbitrary number of samples and discrete jitter”. IETE Journal of Research, vol. 51, No 5, September- October, 2005, pp. 361- 370.


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