Two-Dimensional Sampling-Recovery Algorithm of a Realization of Gaussian Processes on the Input and Output of Linear Systems

Based on the application of the conditional mean rule, a sampling-recovery algorithm is studied for a Gaussian two-dimensional process. The components of such a process are the input and output processes of an arbitrary linear system, which are characterized by their statistical relationships. Realizations are sampled in both processes, and the number and location of samples in the general case are arbitrary for each component. As a result, general expressions are found that determine the optimal structure of the recovery devices, as well as evaluate the quality of recovery of each component of the two-dimensional process. The main feature of the obtained algorithm is that the realizations of both components or one of them is recovered based on two sets of samples related to the input and output processes. This means that the recovery involves not only its own samples of the restored realization, but also the samples of the realization of another component, statistically related to the first one. This type of general algorithm is characterized by a significantly improved recovery quality, as evidenced by the results of six non-trivial examples with different versions of the algorithms. The research method used and the proposed general algorithm for the reconstruction of multidimensional Gaussian processes have not been discussed in the literature.


Introduction
The list of publications devoted to the study of sampling-recovery algorithms (SRA) for realization of random processes is huge and difficult to read. The problem, formulated in the title of the article, covers issues related to multidimensional SRA. Let us note two of the standard and most important of them: (1) In accordance with the selected criterion, it is necessary to determine the optimal structure of the device for restoring realizations of the selected random process for a given set of samples and (2) to assess the quality of restoration realizations. These two problems must be studied for many types of stochastic processes and for different types of sampling realizations. In the general case, the set of samples of realizations can be random and described by a stream of random points. Deterministic sampling can be periodic or non-periodic. When random jitter or gaps are present in the samples, the determinism of the samples disappears. In addition, the number of samples involved in recovery in all these cases can be arbitrary.
For each of the options mentioned, specific bibliographic lists of published works can be found. Here we will indicate only a few typical publications [1][2][3][4][5][6][7][8][9], in which SRAs of multidimensional In the mathematical literature, there is a result that is closely related to the problem formulated in here. Namely, in [22] (see also [23]), matrix expressions were obtained for the conditional mean vector and for the conditional covariance matrix of one vector for a fixed other vector. These relations have been derived for multidimensional Gaussian random variables. These formulas are given in the Appendix A and are designated by the letter "A". They cannot be used directly to solve the problem posed in the article. We generalize them to the case when two components are Gaussian processes with continuous time, and the other components (sets of samples) are random Gaussian variables with discrete time. For our purpose, we use different designations than those used in the Appendix A. Consider a column vector Z t, T (x) , T (y) that is analogous to the vector z (see Formula (A1)): Z t, T (x) , T (y) = Z 1 (t), Z 2 T (x) , T (y) T (1) where Y T (y) = y 1 T where N (x) , N (y) are the numbers of samples in both sets. The vector Z t, T (x) , T (y) is described by the mathematical expectation vector (see analogue the Formula (A2)): and covariance matrix where K 11 (t, t ),K 22 T (x) , T (y) are the covariance matrices of vectors Z 1 (t) and Z 2 T (x) , T (y) , respectively; K 12 t, T (x) , T (y) ,K 21 T (y) , T (x) , t -matrices of cross covariance between vectors Z 1 (t) and Z 2 T (x) , T (y) . Expression (9) is an analog of the matrix (A3) written in the new notation. We fix the vector Z 2 T (x) , T (y) , and the vectorZ 1 (t) remains random with its components conditional with respect to the vector Z 2 T (x) , T (y) . Then, the vectorZ 1 (t) is described by a Gaussian two-dimensional conditional probability density, which is characterized by a column vector of conditional mathematical expectations and a matrix of conditional covariances. The vector of conditional mathematical expectations instead of (A4) is written in the form: , T (y) Z 2 (T (x) , T (y) )-Z 2 (T (x) , T (y) ) (10) As in the one-dimensional case [13][14][15][16], based on (10), we introduce the definition of a multidimensional basic function b t, T (x) , T (y) = b (x) t, T (x) , T (y) , b (y) t, T (x) , T (y) T = = K 12 t, T (x) , T (y) K - 1 22 T (x) , T (y) (11) Relation (10) determines the optimal recovery structure for the sampled realizations of the two-dimensional process (see the Section 4). Recovery should be carried out sequentially at sampling intervals. The matrix of conditional covarianceK t, t T (x) , T (y) of the vector functionZ t T (x) , T (y) , when the vector Z 2 T (x) , T (y) is fixed, based on (A5), (9) takes the form: K t, t T (x) , T (y) =K 11 (t, t ) − K 12 t, T (x) , T (y) K - 1 22 T (x) , T (y) K 21 T (x) , T (y) , t Equating in (12) times t = t, it is possible to obtain relations that determine the functions of conditional variance, which characterize the quality of restoration of realizations of each component.
Let us describe the general form of the submatrices included in expression (9). The two-dimensional Gaussian process x(t), y(t) T is described by the mathematical expectation vector (7) and the covariance matrix In (13), functions K x (t, t ), K y (t, t ) are covariance functions of processes x(t) and y(t), accordingly. The degree of statistical dependence between the processes is determined by the functions of cross covariance K xy (t, t ), K yx (t, t ). The remaining three sub-matrices are written this way: Here and below, the dots above the letters indicate centered random variables. .

Y T (y)
Using Formulas (13)-(16), we can specify the relations (10) and (12), which should be calculated sequentially in the intervals for interpolation T i−1 < t ≤ T i , i = 2, 3, . . . , N and for extrapolation at t ≥ T N . There is a retropolation option, when t ≤ T 1 . Here, the superscripts (x) and (y) are omitted.

Models of Used Gaussian Processes
Below, the use of the above general algorithm is illustrated with a series of examples in which two statistically related Gaussian processes appear. Covariance and cross-covariance functions of processes vary within wide limits. As indicated in Section 1, these processes are most simply described using a linear system with a given impulse response h(t). When the covariance function K x (τ) of the input process x(t) and characteristics h(t) change, the output process y(t) is described by various covariance functions. In this case, of course, the cross-covariance function between the input and output is also changed. There are general formulas [23], which can be used to determine the desired covariance functions for given K x (τ) and h(t). Let us write them out in relation to the stationary case, setting m x (t) = m y (t) = 0: Entropy 2020, 22, 1079 6 of 24 There are two cross-covariance functions K xy (τ), K yx (τ), that have the property K xy (τ) = K yx (−τ): For our purposes, when choosing linear systems, it is advisable to choose the simplest in structure and description. In this case, it can easily be demonstrated how the auto-and cross-covariance functions of the input and output processes of linear systems affect the main characteristics of the SRP: The structure of recovery devices (or basic functions) and the functions of recovery errors. As linear systems, it is appropriate to choose low-pass filters, consisting of series-connected integrating RC circuits, at the input of which there is white Gaussian noise. Such systems can have one or more connected integrating RC circuits separated by buffer cascades [23]. At the outputs of such systems, Gaussian processes with various statistical characteristics are formed. Below, this method will be used to describe both input and output processes. The simplest linear system is a single integrating RC circuit, at the input of which there is white noise. At the output of such a system, a Markov Gaussian process with an exponential covariance function is formed. At the outputs of two, three, and further circuits, the output processes will not be Markov.
Formulas (17) and (18) will be used below when considering examples.

General Optimal Structure of Restoration of Realizations of the Two-Dimensional Gaussian Process
Optimal recovery is understood to mean an algorithm that uses both sets of samples X T (x) , Y T (y) in the recovery of each of the components of a two-dimensional process Z 1 (t) = [x(t), y(t)] T . The structure of the optimal recovery device is determined by Formula (10) and is given in Figure 1.  Both inputs of the device receive sets of samples  (7) and (8)   Both inputs of the device receive sets of samples X T (x) , Y T (y) , which are stored in memory registers 1 and 2. The sets of samples are shifted in blocks 3 and 4 to obtain the best restoration quality (see, for example, Example 3). Then, information about the location of the samples along with the characteristics of the linear system is used to calculate the matrix elements K 12 t 1 , T (x) , T (y) ,K −1 22 T (y) , T (x) in blocks 5 and 6. In block 7, these matrices are multiplied. A priori information about the mathematical expectation functions (7) and (8) is stored in blocks 8, 9, and is used when subtracting average values m x T (x) , m y T (y) at the reference points in blocks 10, 11, and also when summing the functions m x (t), m y (t) in blocks 13, 14 In block 12, matrix multiplication is performed from the output of block 7 and elements of a centered a column vector of input samples. Recovered realizations m x (t), m y (t) are formed at outputs of blocks 13, 14.
We draw attention to the fact that the matrix of basic functions b t, T (x) , T (y) in the diagram in Figure 1 is not indicated. However, in accordance with (11), it is formed at the output of block 7. The elements of the matrix b t, T (x) , T (y) represent a set of an orthonormal system of functions. It means that superscripts are omitted here. The number of basic functions is the same as the total number of samples. To clarify the physical meaning of the elements of the matrix b t, T (x) , T (y) , consider a special case when N (x) = N (y) = 2. This option is explored in the Examples 1 and 2 in the next section. Let us concretize the matrices included in relation (11): As a result of multiplying (20) and (21), we obtain the matrix of basic functions whose elements are written in the form (we give only two of them): Let us change the notation: Using relations (22)-(25), we write expressions for basic functions in the form: in the general case is determined by the sum of a product of the autocovariance function with arguments t, T k , k = 1, 2, . . . , N (superscripts are omitted here) and elements of inverse covariance matrix. The difference is that in the case under consideration, we mean not only autocovariance functions K xx (·), K yy (·), but also cross-covariance functions K xy (·), K yx (·).
(2) It is clear for independent components, the sums with cross-covariances in (26) and (27) disappear. Then, formulas for the basic functions coincide with the expressions for the one-dimensional version, and the diagram in Figure 1 is split into two independent channels. Each example presented in the article is illustrated not only by the type of basic functions, but also by the corresponding graphs of recovery errors. Moreover, in the latter case, among many curves, a curve corresponding to the reconstruction algorithm is necessarily shown, in which only the own samples of the reconstructed realization are used. Comparison of the quality of restoration is performed for the same process models and selected parameters. Note that the always-proposed algorithm is characterized by an improvement in the quality of functioning.

Study Cases: Reconstruction of Realizations on One Sampling Interval
Shown in Figure 1, the general recovery scheme includes the option under consideration (one sampling interval) as a special case; therefore, a somewhat simplified scheme will not be discussed. Two of the most important characteristics of the SRA are detailed below: The basic functions for each sample involved in recovery and the error recovery functions. The purpose of considering a set of examples is to find out how the following parameters affect the specified characteristics: (1) The number and location of samples of input and output realizations, (2) input and output covariance functions, (3) their cross-covariance functions, and (4) the type of recovery procedure-on one interval or multiple intervals.
Further research requires specification of data on the number and location of samples. We note one important feature of the discussed algorithm, which will be considered when calculating the recovery errors in all the examples considered below. Formulas (10) and (12) are of a general nature, and their application for a large set of samples is associated with the complication of the device. Theoretically, each sample should participate in the formation of the output processes of the system shown in Figure 1. Actually, the samples of the realization of one component (say x(t)) affect the formation and the error of recovery of the other component y(t) only when the localization of samples of the first component is located near or inside the sampling interval of the recovered realization of the second component. The reason for this effect is that it is realized through the cross-covariance function: When the argument of this function is less than the covariance time τ (y) , then the value of the function K xy (τ) is close to the maximum and the influence of the corresponding sample on the quality of recovery is significant. In addition to the position of the maximum of the function K xy (τ), the discrepancy between the samples of the auxiliary and recovered realizations also affect the reduction of the recovery error. Such an effect occurs, for example, with unequal sampling periods Examples 5 and 6). In this case, the minimum of the recovery error will be in the interval close to the point ∆T j + t max (here t max is location of the point at which the function K xy (τ) reaches its maximum within the sampling interval T (y) ). The main characteristics of the SRA are also influenced by the elements of the inverse covariance matrix. However, it is difficult to establish at least some patterns of such influence.
In Section 5.1 Example 1 and Section 5.2 Example 2, the numbers of samples are equal to two and the samples of both realizations are located at the same points. In Section 5.3 Example 3, the auxiliary sample is one, but its location varies within the sampling interval T

Example 1. System from One RC Chain with Markov Input Process
A Markov Gaussian process is formed at the output of an integrating RC circuit that is under the influence of white noise. Its normalized covariance function R(τ) = K(τ)/σ 2 in the stationary mode is determined by the formula where α = 1/RC is the constant parameter. We put m x (t) = m y (t) = 0. The linear system is also an integrating RC circuit with an impulse response Using expressions (18), (28) and (29) we determine the normalized covariance function of the output process y(t): as well as normalized cross-covariance functions (17) between the processes x(t) and y(t): Figure 2 shows the graphs of the cross covariance function R xy (τ) and R yx (τ) for various values of the parameters α and β. The curves are calculated for the following parameters: Curve 1-α = 2, β = 1; curve 2-α = 4, β = 1; curve 3-α = 4, β = 2 for R yx (τ) and curve 4-α = 2, β = 1; curve 5-α = 4, β = 1; curve 6-α = 4, β = 2 for R xy (τ). As can be seen, the cross-covariance functions are odd, and their maxima are shifted of the point τ = 0. Especially we note the curves 3 and 4 with their maxima in points τ = 0.25 and τ = −0.25 for R xy (τ) and R yx (τ), respectively. In general, when the value of parameters α, β of cross covariance functions R xy (τ), R yx (τ) increase, their maxima values decrease. This is explained, because the realizations of the input and output process are more chaotic when the bandwidth is increased, which is described by the value of parameters α, β.
Entropy 2020, 22, x FOR PEER REVIEW 10 of 26 In Figure 2 shows the graphs of the cross covariance function   The results of calculations of basic functions carried out according to formula (11) are shown in Figure 3. The values of the selected parameters are as follows: N (x) = N (y) = 2; the number of samples involved in the recovery of realizations is the same: The samples are located at the same points: Figure 3 shows the basic functions of the multidimensional algorithm b curves 5 and 6). These basic functions correspond to the restoration of realization of process x(t) at the input of the system. The samples of the realization of the output process y(t) are auxiliary samples here. The multivariate algorithm has four basic functions (for own and for auxiliary samples), while the unidimensional algorithm has two basic functions.
Curves 5 and 6 in Figure 3 refer to a one-dimensional algorithm. They are described by the first term in (26) and the covariance function (28). In accordance with (26), the multidimensional algorithm includes four basic functions, including two of them formed on the basis cross covariance functions (32). Moreover, these functions, elements of the inverse matrix, influence the calculation of the basic functions. It is obvious that the form of the multidimensional basic function changes radically in relation to the main functions in a one-dimensional algorithm.  Figure 3 refer to a one-dimensional algorithm. They are described by the first term in (26) and the covariance function (28). In accordance with (26), the multidimensional algorithm includes four basic functions, including two of them formed on the basis cross covariance functions (32). Moreover, these functions, elements of the inverse matrix, influence the calculation of the basic functions. It is obvious that the form of the multidimensional basic function changes radically in relation to the main functions in a one-dimensional algorithm. The results of calculations of recovery errors carried out according to formula (12) are shown in Figure 4. The values of the selected parameters are the same as in the comments to  The results of calculations of recovery errors carried out according to formula (12) are shown in Figure 4. The values of the selected parameters are the same as in the comments to Figure 3. Curve 1 describes the recovery error of realization of x(t) with multidimensional algorithm. It has a smoothed minimum close to the point τ = 0.25, because the function K yx (τ) has maximum at this point. The smoothness of the discussed extremum is influenced by the proximity of the control point, where the error is zero by the definition.
Curves 3 and 4 describe the recovery errors σ 2 x (t), σ 2 y (t) for the one-dimensional algorithm, when the recovery is performed only on their own samples. The difference in the values of the curve maxima is explained by the difference in the time structure of the processes: The output process y(t) is smoother than the input process x(t). Curves 1 and 2 are obtained by a multidimensional algorithm, when both sets of samples participate in the restoration of each realization. The even form of curve 3 is explained, because this form is determined by the covariance functions R xy τ − T According to formula (27), the influential of these functions are weighed by the elements of the inverse matrix. A comparison of pairs of curves 2, 4, and 1, 3 indicates that the restoration using the multidimensional algorithm provides a higher quality of recovery than the similar procedure according to the one-dimensional algorithm.

Example 2. The Input Is Non-Markovian Process Formed by Two Sequential RC Chains. System Is One RC Chain
There is one difference between Example 1 and Example 2: Here, the input process is not Markovian. This circumstance changes all the covariance functions included in the expressions for the analysis of the studied algorithm.
The covariance function of the input process is determined by relation (30) with the change of index.
The results of calculations of basic functions and recovery errors are shown in Figures 6 and 7. The values of the selected parameters are as follows: The number of samples involved in the recovery of realizations is the same: N (x) = N (y) = 2; the samples are located at the same points: T    In Figure 6, the basic functions of the multidimensional algorithm (curves 1-4) and onedimensional algorithm (curves 5 and 6) are observed.
As in the previous example, covariance functions and elements of the inverse matrix influence the basic functions. The difference is explained by non-Markovian characteristics of the output process.
The results of calculations of recovery errors are shown in Figure 7. The Curves 1-4 are characterized by the same parameters as in Figure 6. When the basic functions change, the error recovery functions must also change. Comparison of the curves in Figures 4 and 7 shows that the maximum error values differ significantly. This fact is explained by the greater smoothness of the studied processes in this example compared to the processes in Section 5.1 Example 1 (see more about this effect in [13,14,16]). In addition, note that the curve 1 is asymmetric compared to curve 3. This is explained because the influence of the output process determines the reconstruction of the process at the input by means of the cross-covariance function. Meanwhile, curve 2 is a symmetric function, because the cross-covariance function ( ) yx R τ influences the reconstruction to a lesser extent.
In Section 5.1 Example 1 and Section 5.2 Example 2, the processes at the input and output of the linear system are different in the time structure: The process ( ) y t is more smoothed compared to the In Figure 6, the basic functions of the multidimensional algorithm (curves 1-4) and one-dimensional algorithm (curves 5 and 6) are observed.
As in the previous example, covariance functions and elements of the inverse matrix influence the basic functions. The difference is explained by non-Markovian characteristics of the output process.
The results of calculations of recovery errors are shown in Figure 7. The Curves 1-4 are characterized by the same parameters as in Figure 6. When the basic functions change, the error recovery functions must also change. Comparison of the curves in Figures 4 and 7 shows that the maximum error values differ significantly. This fact is explained by the greater smoothness of the studied processes in this example compared to the processes in Section 5.1 Example 1 (see more about this effect in [13,14,16]). In addition, note that the curve 1 is asymmetric compared to curve 3. This is explained because the influence of the output process determines the reconstruction of the process at the input by means of the cross-covariance function. Meanwhile, curve 2 is a symmetric function, because the cross-covariance function R yx (τ) influences the reconstruction to a lesser extent.
In Section 5.1 Example 1 and Section 5.2 Example 2, the processes at the input and output of the linear system are different in the time structure: The process y(t) is more smoothed compared to the process x(t). The results of restoration errors calculations in Section 5.1 Example 1 and Section 5.2 Example 2 show that the degree of influence of additional samples of one process on the restoration quality of another process is different. Specifically, when the process is more smoothed, then its positive influence on the restoration quality of another process is significantly higher than in the other situation. (see differences between curves 1 and 3, 2 and 4 in Figures 4 and 7).
The option considered in the first two examples of Section 5, in addition to theoretical, is of practical interest. We repeat that the proposed method refers to the case when the transmitted messages must have a statistical relationship. In telemetry systems, such messages are transmitted over different channels. It is quite possible that a message described by the simplest covariance function (in our model this is an input process) must be reconstructed with greater accuracy. Then, naturally, the message samples with a more complex covariance function (this is an output process) will play an auxiliary role.

Example 3. Displacement of the Auxiliary Sample within the Sampling Interval of the Main Component
Again, consider the system studied in Section 5.1 Example 1. That means there is a system of one RC circuit with a parameter β = 1. A Markov process x(t) with a parameter α = 2 acts at its input. There are three important differences: (1) The input x(t) is an auxiliary process, (2) the set of samples For this simple variant, we specify the relations (14)-(16): Elements of matrices (37) and (38) show that cross-covariance functions have an important role in calculating reconstruction error.
In this example, the auxiliary sample x 1 T  It should be noted again that a realization to be restored belongs to the output process, which is characterized by the cross-covariance function in Figure 1, curve 4 in contrast to Section 5.1 Example 1 and Section 5.2 Example 2.
In Figure 8, the basic functions of the multidimensional algorithm b  The influence of the auxiliary sampling moments T (x) 1 on the reconstruction depends on the location of the maximum of the cross-covariance function K xy (τ) in the interpolation region. It should be noted that the maximum of the K xy (τ) covariance function (Figure 2 curve 4) is located at t = −0.45; that is, the maximum of the cross covariance function is located at t < T (y) 1 = 0. As a consequence, K xy (τ) this, the lobe of the basic function of the auxiliary sampling instant b    3 (t) have the same shape as the basic functions of the one-dimensional algorithm b 1 (t), b 2 (t).
In the proposed method, with a limited number of counts, each of the counts has its own basic function. This is true even for a one-dimensional algorithm. In the multidimensional version, the situation is more complicated, since the form of the basic function is influenced by both its own samples and the samples of the auxiliary realization. Moreover, the first of them affect the form of the basic function through their own covariance function, and the second through the cross-covariance function. In addition, in both cases, the elements of the inverse covariance matrix and the temporal  In Figure 8, the basic functions of the multidimensional algorithm  (Figure 9). For example, when the cross-covariance function K xy (τ) is at T (x) 1 = 0, curve 1 is tilted to the right (Figure 9). This is explained by the influence of the maximum of the cross-covariance function K xy (τ) manifesting itself in the region close to the sampling instant T  In Figure 8, the basic functions of the multidimensional algorithm

Study Cases: Reconstruction of Realizations on Several Sampling Intervals
There are three examples here with multiple sampling intervals SRA. The input process realizations are auxiliar. The realization of the output process should be restored. Each example has its own peculiarity. Section 6.1 Example 4 and Section 6.2 Example 5 are described by the same input process and system as in Section 5.1 Example 1. Section 6.1 Example 4 differs in sampling procedures: The sampling of the input realization is non-periodic; the sample of the output realization is periodic. The number of samples is equal N (x) = N (y) = 3. In Section 6.2 Example 5, the sampling of the realizations of both processes is periodic, but the instance points are offsets. The number of samples is equal N (x) = N (y) = 6. Section 6.3 Example 6 examines the SRA when the input process is not Markov. The number of samples is equal N (x) = N (y) = 7.

Example 4. SRA Algorithm with Non-Periodic Sampling of Auxiliary Input Process
There is a system of one RC circuit with a parameter β = 1. A Markov process x(t) with a parameter α = 2 acts at its input. The option of recovering the output process at several intervals, when the procedures for sampling the processes x(t) and y(t) are different, is considered. The numbers of samples are the same, i.e., N (x) = N (y) = 3. Sampling intervals of the process y(t) are periodic: Samples of the realization of the process x(t) are non-periodic: 5 . This is a non-trivial case, which, however, is easily studied by the applied methodology.
Note that the basic functions of the multidimensional algorithm b   (Figure 10). This influence is weighted by the elements a ij of the inverse covariance matrix. This influence is most clearly seen in the basic function b  Attention should be paid to Figure 11 that shows the auxiliary main functions (curves 1-3) have a variable shape. This is explained because the sampling intervals ∆T (x) between the sampling instants T decreases with increasing sampling interval. This means that, as the sampling interval ∆T (x) increases, the influence between the mutual covariance functions K xy τ − T (x) i , i = 1, 2, . . . , N (x) decreases; this is manifested in the coefficients a ij in the inverse covariance matrix.  The results of calculations of recovery errors are presented in Figure 12. Curve 1 describes the recovery error using the multidimensional algorithm, and curve 2 refers to the one-dimensional version. As you can see, the character of curve 1 is different on both sampling intervals due to nonperiodicity of auxiliary samples. The results of calculations of recovery errors are presented in Figure 12. Curve 1 describes the recovery error using the multidimensional algorithm, and curve 2 refers to the one-dimensional version. As you can see, the character of curve 1 is different on both sampling intervals due to non-periodicity of auxiliary samples. The results of calculations of recovery errors are presented in Figure 12. Curve 1 describes the recovery error using the multidimensional algorithm, and curve 2 refers to the one-dimensional version. As you can see, the character of curve 1 is different on both sampling intervals due to nonperiodicity of auxiliary samples.   In this example, the question of using the proposed algorithm when restoring the realization of the output process at sampling intervals 5 and 6 is considered.
The description of the system and the input process coincides with the data of Section 5.1 Example 1. All covariance functions are characterized by expressions (28), (30)-(32). The example is considered when the numbers of samples are equal N (x) = N (y) = 6, and the sampling of the input x(t) and output y(t) processes occurs with different periods. So, the set of input and output processes is described by such data: Sample sets X T (x) , Y T (y) are used to reconstruct the realization of the output process y(t).
The basic functions of the multidimensional algorithm b (y) 12 (t) (odd curves) and basic functions of the one-dimensional algorithm b 1 (t) − b 6 (t) (even curves) are shown in Figure 13. Note that the shape of the basic functions of the multidimensional algorithm differs from the functions of the one-dimensional algorithm close to the moments of the auxiliary samples . This means that this difference is caused by the functions of cross-covariance . . , N (x) (as one can see in Figure 2). As can be seen in Figure 14, the form in the interpolation region of the auxiliary basic function b         This means that this difference is caused by the functions of cross-covariance  As can be seen in Figure 14, the form in the interpolation region of the auxiliary basic function  The recovery error of the output process ( ) y t by multidimensional and one-dimensional algorithms is illustrated in Figure 15. Curve 1 characterizes the recovery using a multidimensional algorithm. Curve 2 relates to a one-dimensional algorithm.
There is a small smoothed minimum at the highs of curve 2 in the middle of the total interval. This effect for non-Markov processes is described in the analysis of a onedimensional algorithm [13,14,16]. In this example, the difference in the maxima of the one-dimensional curves is insignificant. On curve 1, this effect is seen by the dependence among their samples. The recovery error of the output process y(t) by multidimensional and one-dimensional algorithms is illustrated in Figure 15. Curve 1 characterizes the recovery using a multidimensional algorithm. Curve 2 relates to a one-dimensional algorithm. The influence of the displacement of the auxiliary samples with respect to the own samples T (y) i , i = 1, 2, . . . , N (y) is observed. This means that the maximums of the cross-covariance functions are located the interpolation region. This location corresponds to the minimum of the reconstruction error function, that is There is a small smoothed minimum at the highs of curve 2 in the middle of the total interval. This effect for non-Markov processes is described in the analysis of a one-dimensional algorithm [13,14,16]. In this example, the difference in the maxima of the one-dimensional curves is insignificant. On curve 1, this effect is seen by the dependence among their samples.
There is a small smoothed minimum at the highs of curve 2 in the middle of the total interval. This effect for non-Markov processes is described in the analysis of a onedimensional algorithm [13,14,16]. In this example, the difference in the maxima of the one-dimensional curves is insignificant. On curve 1, this effect is seen by the dependence among their samples. Figure 15. Recovery errors in Section 6.2 Example 5. Figure 15. Recovery errors in Section 6.2 Example 5.

Example 6. SRA When the Input Process Is Non-Markovian
Consider another example, which is an analogue of Section 5.2 Example 2. Here, the system is an RC circuit with a parameter γ, and the input process is formed from white noise by two consecutive RC circuits with parameters α, β. Covariance functions are defined by expressions (34)-(36). The input process x(t) here is non-Markovian. Input and output processes are sampled as follows: As can be seen, the number of samples is different and equal to 7. Input samples are delayed for a while t = 0.6. A realization of the output process y(t) is reconstructed.
In Figure 16, the basic functions of the multidimensional b (y) 12 (t) (odd curves) and one-dimensional algorithm b 1 (t) − b 6 (t) (even curves) are compared. Note that the maximum of the basic functions of the multidimensional and one-dimensional algorithm corresponds to the sampling instant T 12 (t) are narrower than the functions of the one-dimensional algorithm b 1 (t) − b 6 (t). This is because all the cross-covariance functions K xy τ − T     Figure 17 is greater. This is related to the cross-covariance function K xy τ − T  6 (t) of the last instant of the auxiliary sample T (x) 6 = 5.6 concentrates the influence in an additive form (that is, the coefficients of the inverse matrix a ij are positive in this last auxiliary sampling instant) of the covariance function K y (τ) and the mutual covariance function K xy (τ).  In Figure 17, the auxiliary basic functions   The form in Figure 18 of the curves shows an analogy with Figure 15 in Section 6.2 Example 5. The main differences in (18) (with a comparison of Figure 15) are associated with a significant decrease in the values of the errors n and the asymmetric nature of the curves related to multivariate recovery. The reasons are obvious: (1) The output process ( ) y t is smoother and (2)   The form in Figure 18 of the curves shows an analogy with Figure 15 in Section 6.2 Example 5. The main differences in (18) (with a comparison of Figure 15) are associated with a significant decrease in the values of the errors n and the asymmetric nature of the curves related to multivariate recovery. The reasons are obvious: (1) The output process y(t) is smoother and (2) the shift of the samples of the set T (x) as compared with the samples of the set T (y) , and with the size of the sampling interval, is insignificant (0.15). Curve 1 shows the effect of reducing the error in the center between the extreme samples. Obviously, this is a reflection the greater statistical relationship between samples in the considered non-Markov process.
The increase in the quality of restoration (in Figure 18) is physically explained by the fact that in the known method when restoring realizations, only its own samples are used. In the proposed method, the number of samples participating in the reconstruction is increased due to samples from another, statistically related realization. Moreover, the number of additional samples can be arbitrary. It is obvious that the restoration of realizations from a larger number of samples leads to an increase in the quality of restoration.

Conclusions
The problem investigated in the article work to the problem of sampling-recovery of two-dimensional Gaussian processes. The dimensionality of the problem is not limited by the presence of two random processes at the input and output of the linear system, since, in addition to them, the problem includes two sets of samples fixed in the realizations of these processes. The algorithm developed differs in that the reconstruction of the realizations of both components, or one of them, is carried out on the basis of two sets of samples. This means that the recovery occurs not only with the participation of its own realization samples, but with the realization samples of another component.
The considered examples illustrate some applications of the proposed algorithm. They studied the options when the following changes: (1) The type and input of the system, (2) the number of intervals on which the restoration is performed, (3) as well as the number of auxiliary samples involved in the functioning of the multidimensional algorithm.
In all cases, there are basic functions and error recovery functions. These functions are optimal and characterize the estimation of yields using the recovery algorithm studied. These reconstruction characteristics allow us to demonstrate the advantage of using the algorithm based on the quality of the reconstruction. This result will be used as long as the random processes have a statistical dependency.