The theory of optimum filtering, detection and estimation has a rather long history (see [5] [14] - [16] and references therein) and so, there is no need to repeat it hereafter. Applying as a criterion, for the optimal processing, the well-known criterion of the maximum a posteriori probability density function W_ps(∙), or its probabilistic functional, the whole theory of the optimum filtering, detection, estimation, etc. (and its quasi-optimum modifications, see [5] [14] - [16] etc.) was created and developed. This theory is based on the differential equation for the time evolution of W_ps(∙), applying the a priori characteristics of the Markovian model of the observed stochastic process.

This was already done, and it is called the Stratonovich-Kushner Equation (SKE). The interested reader might find numerous references related to it and its quasi-optimum modifications (see also the already mentioned References), for convenience it is introduced in (3.3). For the chaotic filtering based on the Markovian approach, the related modifications of the SKE (3.1.a) and its quasi-optimum solutions were also detailed in rather numerous sources [2] [5] [13] , etc. It is worth noticing that all this was done for the time domain.

The current material is dedicated to random fields (space-time) with application of the Markovian approach for their chaotic modeling as well. So, what are the specific features of this case? Actually, they are well known (see [7] [10] [11] ) and they will be briefly outlined hereafter. One must notice that when m → ∞, and the SDE (2.2) is transformed into SDE (2.5) the probability density function (PDF) of its solution is transformed as well to the probability density functional P(x(r, t)) - (PDF_n).

So, the way of stochastic description of random fields x(r, t) is drastically different from the random processes x(t): it is a “functional” description by means of probabilistic functionals i.e. “functions of functions x(r, t)”. Though, all the derivatives in terms of the Markovian approach: ordinary and partial derivatives, integrals, etc., start to be “functional (variation) derivatives”, “functional integrals”, etc.—components of the so-called well known long ago functional calculus (see, for example, [17] ) for the fields¹.

Note, that for the current material all the forthcoming equations for the a priori and a posteriori probability density functionals (PDF_n) have to be considered somehow in “symbolic” sense in order not to be involved with the “tiny” mathematical details which actually are not so important in the following.

This statement is true both for the scalar and the general vector random fields (for example MIMO case) $x\left(r,t\right)={\left[x_{\text{1}}\left(r,t\right),\cdots ,x_n\left(r,t\right)\right]}^{\text{T}}$ both stochastic and chaotic. Here “T” means transpose.

Returning to the Markovian approach, one has to remember, that a priori description of the corresponding PDF_n - P_pr(x, t), must be presented in the way similar to the Fokker-Plank-Kolmogorov Equation (FPKE) for m-dimensional Markovian processes. This equation, in the following, will also be called as Fokker-Plank-Kolmogorov Equation (FPKE) [12] [14] [15] but with “functional derivatives” and is given by:

$\begin{array}{l}\frac{\partial P_{pr}\left(x,t\right)}{\partial t}={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{D}{\int }\frac{\delta }{\delta x_i\left(r\right)}\left[f_i\left(t,r,x\right)\right]P_{pr}\left(x,t\right)\text{d}r}}\\ +\frac{1}{2}{\displaystyle \underset{i,j}{\overset{n}{\sum }}{\displaystyle \underset{D}{\iint }{\kappa }_{ij}\left(t,r_1,r_2\right)\frac{{\delta }^2{P}_{pr}\left(x,t\right)}{\delta x_i\left(r_1\right)\delta x_j\left(r_2\right)}\text{d}{r}_1\text{d}{r}_2}}\end{array}$,(3.1)

$P\left(x,t_0\right)=\delta \left[x\left(r\right)-x_0\left(r\right)\right]$,

where $\frac{\delta }{\delta x_i\left(F\right)}(\cdot )$ is a functional derivative, and ${\displaystyle \underset{D}{\int }(\cdot )}$ is a functional integral along the area D. The operator for P_pr(x, t) in (3.1) is linear though, the interpretation of (3.1) can be seen in the way:

$\frac{\partial P_{pr}\left(x,t\right)}{\partial t}=L\left\{P_{pr}\left(x,t\right)\right\}$,(3.2)

where L{} is a linear operator with functional derivatives. This form of the FPKE might be applied later, but notice once more, that all those formulas (for the moment!) have to be considered as “symbolic” and are just an illustration of the “idea” how the input signal must be treated. That is why, in the following, the way how to make concrete calculus is not discussed. Comparing FPK (3.1) with the FPK for the Markovian processes (see for example [12] [14] [15] ) and (3.3) one can easily see the complete formal similarity.

$\frac{\partial P_{pr}\left(x,t\right)}{\partial t}=-{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{\partial }{\partial x_i}\left[f_i\left(x,t\right)P_{pr}\left(x,t\right)\right]}+\frac{1}{2}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j=1}{\overset{n}{\sum }}\frac{{\partial }^2}{\partial x_i\partial x_j}}}\left[{\kappa }_{ij}{P}_{pr}\left(x,t\right)\right]$.(3.3)

In the previous section the complete similarity between FPKE for Markovian processes and fields was presented (in symbolic way for the case of fields). Absolutely the same issue takes place for the SKE for P_ps(x, t) which is nothing else, but an algorithm for optimum filtering (see [5] [14] - [16] , etc.) with the criterion of maximum a posteriori probabilistic functional. Omitting details about the development of the SKE (which almost coincide with those for the Markovian processes²) it finally yields:

$\frac{\partial P_{ps}\left(t,r,x\right)}{\partial t}=L\left\{P_{ps}\left(t,r,x\right)\right\}+P_{ps}\left(t,r,x\right)\left[F\left(t,x\right)-\langle F\left(t,x\right)\rangle \right]$,(3.4)

where

$F\left(t,x\right)={\displaystyle \underset{D}{\iint }{Q}_n\left(r,r^{\prime }\right)S\left(t,r^{\prime },x\right)\left[y\left(t,r\right)-\frac{1}{2}S\left(t,r,x\right)\right]\text{d}r\text{d}{r}^{\prime }}$,(3.5)

$y\left(t,r\right)=S\left(t,r,x\left(t,r\right)\right)+n\left(t,r\right)$,(3.6)

n(t, r)—Gaussian white noise field (δ-correlated in time) with covariation function:

$K_n\left(t,t^{\prime },r,r^{\prime }\right)=K_n\left(r,r^{\prime }\right)\delta \left(t-t^{\prime }\right)$,(3.7)

S(t, r, x(t, r))—Generally is assumed as a desired signal which depends on the desired field x(t, r)—object of filtering. Q_n(r, r') can be found from the integral equation:

${\displaystyle \underset{D}{\int }{Q}_n\left(r,r^{\prime }\right)K_n\left(r,r^{\prime }\right)\text{d}{r}^{\prime }}=\delta \left(r-r^{\prime }\right)$,(3.8)

$\langle F\left(t,x\right)\rangle$ means the averaging procedure for F(t, x) over P_ps(t, r, x).

For convenience let us present the SKE version for Markovian processes in the way

$\frac{\partial P_{ps}\left(x,t\right)}{\partial t}=-L_{pr}\left\{P_{ps}\left(x,t\right)\right\}+\frac{1}{2}\left[F\left(x,t\right)-{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}F\left(x,t\right)P_{ps}\left(x,t\right)\text{d}x}\right]P_{ps}\left(x,t\right)$, (3.9)

where P_ps is the a posteriori Probability Density Function which has to be found by solution of the SKE [12] [14] [15] .

The reader can compare the above presented sketch of the development of SKE, using P_ps(t, r, x), for Markovian random fields with the result of the standard procedure for Markov processes or chaos presented in [5] [14] - [16] and find out, that they are completely identical, for sure taking into account that the formulas for processes are “physical” and symbolic for fields. The latter offers really spectacular opportunities to obtain a powerful way to drive many results already obtained for processes, to fields, (with certain caution related to the above made comments regarding derivatives).

Now, it is time to make another main statement. The SKE as an optimum algorithm for filtering, estimation, identification, etc. (either for processes, fields, chaos, etc.) practically never (except for the Gaussian case) offers exact analytical solutions. Therefore, for application in Non-Gaussian cases quasi-optimum algorithms are always recommended.

The final comments here might help to understand why chaotic filtering (for both, processes and fields) offers high accuracy for low SNR. For this matter return once more to SKE (3.1), from where it can be seen that the SKE is made of two parts, the first is related to a priori data and the second to results of the filtering processes (a posteriori data). When the desired signal is a random process (field) then, the instantaneous value of the a priori part is certainly random (uncertain) and the analysis of the a posteriori part “compensates” this uncertainty of the a priori data and gives acceptable accuracy when $\text{SNR}\gg 1$.

When the model of the desired signal is chaotic (almost deterministic) then, the a priori part is somehow “predefined” or almost certain (known). Then, a posterior analysis is “practically” not needed, and SNR might be less than one (0 dB). Anyway, its influence on the filtered signal is practically negligible. This is in short words an illustration of the effect of “singularity” of chaos filtering. A more rigorous explanation can be found in [13] .

Note that in the framework of the statistical theory of communications, the singular scenarios are mainly considered to be non-practical when the accuracy does not depend on the SNR. But, if the models of the desired signals are somehow “shifted” from the random concept to the almost deterministic concept, like chaotic, then those scenarios seem to be rather opportunistic. This gives a “chance” to determine (identify, classify) many signals, images, etc. in “hard” conditions such as: low SNR, many targets, many users, etc. (see [18] [19] for example).

As a “last” thought, one might ask: what for these “symbolic equations” are presented if they are actually not real for calculus!? The answer is: to take advantage of the analogy of the real development of their analogues in time domain, which are absolutely real! (see the following) and apply them for concrete scenarios.

As it was mentioned in the previous subsection the SKE is a very important approach for optimum filtering however, it does not give any opportunity to get solutions for non-Gaussian scenarios, for which quasi-optimum (suboptimum) algorithms might be much more useful. The list of suboptimum solutions is rather long³ and the reader can find a rather “contemporaneous” version in [13] . Also, in [13] , an experimental study was presented, and showed that in the framework of the trade-off between computational complexity and filtering accuracy, the Extended Kalman Filter algorithm (EKF) seems to be a reasonable option. In the following, the EKF algorithm is presented for vector fields (both physical and chaotic) with the help of functional derivatives. It follows then:

$\begin{array}{l}\frac{\text{d}{\hat{x}}_k\left(\rho \right)}{\text{d}t}=-f_k\left(t,\rho ,{\hat{x}}_k\right)+{\kappa }_1^k\left(t,\rho \right)+{\displaystyle \underset{D}{\int }\frac{\delta F\left(t,x\right)}{\delta x_i\left(r\right)}{K}_{ik}\left(r,\rho \right)\text{d}r}\\ -\frac{1}{2}{\displaystyle \underset{D}{\iint }\frac{{\delta }^2{f}_k\left(t,\rho ,\hat{x}\right)}{\delta x_i\left(r_1\right)\delta x_j\left(r_2\right)}{K}_{ij}\left(r_1,r_2\right)\text{d}}{r}_1\text{d}{r}_2\\ \frac{\text{d}{K}_{l,m}\left({\rho }_1,{\rho }_2\right)}{\text{d}t}=-{\displaystyle \underset{D}{\int }\left[\frac{\delta f_i\left(t,{\rho }_1,\hat{x}\right)}{\delta x_i\left(r\right)}{K}_{lm}\left(r,{\rho }_2\right)+\frac{\delta f_m\left(t,{\rho }_2,\hat{x}\right)}{\delta x_i\left(r\right)}{K}_{il}\left(r,{\rho }_1\right)\right]\text{d}r}\\ +{\displaystyle \underset{D}{\iint }\frac{{\delta }^{2}F\left(t,\hat{x}\right)}{\delta x_i\left(r_1\right)\delta x_j\left(r_2\right)}{K}_{il}\left(r_1,{\rho }_1\right)K_{jm}\left(r_2,{\rho }_2\right)\text{d}}{r}_1\text{d}{r}_2\\ +{\kappa }_2^{lm}\left(t,{\rho }_1,{\rho }_2\right)\end{array}$ (3.10)

where ${\hat{x}}_k\left(\rho \right)$ is an averaged a posteriori estimation of the k-th component of the vector field at the “ρ” coordinate and the “t” time moment. In other words it is the result of the filtering process; ${\kappa }_{ij}\left(t,r_1,r_2\right)$ is a space covariation function for the external vector stochastic force (noise) delta-correlated in time, in the SDE (2.3) and (2.5); $K_{l,m}\left({\rho }_1,{\rho }_2\right)$ is a posteriori covariation matrix of the filtered field (which as usual is an error matrix or an evaluation of the filtering accuracy, for each component of the vector field!).

Once more, the above equations (3.10) are completely symbolic and are presented just to illustrate that they are identical to the EKF algorithm for random processes in continuous time (assuming, of course, the difference in notations for the derivatives, see [2] [13] and the references therein). Though, there is a clear opportunity to establish a “bridge” between the results for processes and fields (never mind random of chaotic) and apply the well-known results to new scenarios. In this context the next section will be dedicated to an application of the above results (certainly adapted) to MIMO terminals.

One must notice (as it was already stressed above) that all the presented material is considered in the framework of the Incoherent paradigm for 6G in NOMA-RIS-MIMO wireless communications, in the corresponding frequency bands [18] [20] [21] . The purpose of the filtering might be focused on the “identification” of the NOMA transmission users by means of the chaos filtering approach. Note that, the first attempt to do it for SISO case was presented in [18] with a rather high efficiency due to the “determinism” of the chaos modeling of the users’ received signals at the terminal. Tentatively, the same idea is behind the MIMO case but for the space time field scenario.

The necessary theoretical results were presented above, including the rational quasi-optimum chaos filtering algorithms as the Extended Kalman Filter. In order to properly adjust the previous theoretical results, it is reasonable to make some preliminary comments.

$y\left(t,r\right)=A\cos \left[{\omega }_{0}t+x\left(r\right)\right]+n\left(t,r\right)$,(4.1)

where n(t, r) is a noise, delta correlated in time and space; x(r) is a vector of random chaos field, modeled as SDE (2.5) with a “weak” processes noise. Approximately SDE (2.5) is:

$\frac{\text{d}x\left(r\right)}{\text{d}t}=0$.(4.2)

a) If the interval between antenna elements Δ_a for a linear planar antenna (PA) is much less than τ_f—a posteriori space correlation interval for the field x(r), then for x(r) the antenna aperture is somehow “solid” or “complete” and actually there is no MIMO, only a SISO scenario, and all the results related to chaos filtering (see [2] [13] , etc.) are valid for this case.

b) If Δ_a ~ τ_f, then chaos filtering is actually “space-time” and the latter must influence the filtering accuracy. The interested reader might have already figured out, that the Markovian concept of random field modeling might be applied only for each of the antenna elements of the aperture, if and only if, they are statistically independent. Otherwise, all the volume of the r_i coordinates have to be taken into account, i.e. admit field causality. Though, the algorithm (3.10) must be concretized for this case.

An additional assumption also must be taken into account, when Δ_a ≥ τ_f, then the MIMO antenna elements are decorrelated and the difference r_i - r_j is nothing else but a difference of the location of the antenna elements.

As processing is assumed to be held on each element of the MIMO antenna (i.e. {r_j} are known and fixed), then the processing algorithms have to be presented in digital way for the antenna elements and then summarized for all the area D. The latter gives a hope, that this “last stage”, can be considered as something like to the linear incoherent diversity addition that, might provide an additional increment on the filtering’s accuracy (diversity effect). This issue deserves to be detailed.

As was shown in [18] , even for the SISO case, the normalized MSE (mean square error) for SNR ~ 0 dB is characterized with small percentage values. With the effect of diversity addition, it might diminish even to fractions of the percentage! Note that, an accurate analytical study of this, is hardly possible, but anyway, some evaluations might be done.

Taking into account the concretization scenario in the form (4.1), it is possible to rewrite the quasi-optimum algorithms (3.10) in the following way:

$\begin{array}{l}\frac{\text{d}\hat{x}\left(t,\rho \right)}{\text{d}t}=-\frac{A}{N_0}{\displaystyle \underset{D}{\int }y\left(t,r\right)B\left(r,\rho \right)\sin \left[{\omega }_{0}t+\hat{x}\left(t,r\right)\right]\text{d}r}\\ \frac{\text{d}B\left({\rho }_1,{\rho }_2\right)}{\text{d}t}=-\frac{A}{N_0}{\displaystyle \underset{D}{\int }y\left(t,r\right)B\left(r,{\rho }_1\right)B\left(r,{\rho }_2\right)\cos \left[{\omega }_{0}t+\hat{x}\left(t,r\right)\right]\text{d}r}\end{array}$(4.3)

The second equation in (4.3) can be additionally simplified for the band-pass equivalent signal processing approach, if all “high-frequency” components are neglected.

$\frac{\text{d}B\left({\rho }_1,{\rho }_2\right)}{\text{d}t}=-SNR{\displaystyle \underset{D}{\int }B\left(r,{\rho }_1\right)B\left(r,{\rho }_2\right)\text{d}r}$,(4.4)

here $SNR=-\frac{A^2}{2N_0}$ is signal to noise ratio on each element.

Then, for each element of the MIMO antenna, the digital version of (4.3) is

$\begin{array}{l}\frac{\text{d}{\hat{x}}_i\left(t\right)}{\text{d}t}=-\frac{A}{N_0}{\displaystyle \underset{j}{\sum }{y}_j\left(t\right)B\left(t,i-j\right)}\sin \left[{\omega }_{0}t+{\hat{x}}_j\left(t\right)\right]\\ \frac{\text{d}B\left(t,l\right)}{\text{d}t}=-SNR{\displaystyle \underset{m=-\infty }{\overset{\infty }{\sum }}B\left(t,m\right)B\left(t,l-m\right)}\end{array}$(4.5)

Please note, that the “m’s” here are simply the number of antenna elements, but “l” denotes actually, the “distances Δ_a” taken between the corresponding antenna elements and might be either, positive or negative. The “addition” operator in the second equation in (4.5) is concretely, the linear incoherent diversity addition operation and $B\left(t,i-j\right)=Q$ is the number of “effective” diversity “branches” taking part in the addition operation and can substitute “m” in (4.5).

If the “branches” are assumed to be homogeneous and statistically independent, then as a first approximation (roughly!) the resulting MSE can be considered as:

${\text{MSE}}_{\Sigma }~{\left({\text{MSE}}_i\right)}^Q$.(4.6)

If $Q\gg \text{1}$ it can be seen from (4.6), the resulting MSE might be negligible or very small.

In some sense, this statement might be seen beforehand, but one more important question arises: when is it reasonable to apply space-time processing? (see comments above as well). The answer is evaluated according to the degree of reduction of the MSE with the chaos space-time filtering on each antenna element compared with the MSE for the only time filtering case.

In order to make an analytical illustration of this, let us assume that the covariation of the fields, along the antenna elements, is exponential:

$K\left(m\right)~\text{exp}\left(-\beta \left|m\right|\right)$,(4.7)

where 1/β is the covariation interval.

As before, considering two limiting cases ${\beta }^{-\text{1}}\gg {\Delta }_a$ and ${\beta }^{-\text{1}}\ll {\Delta }_a$, it follows that, physically one is dealing (for the first case) with a complete (continuous) antenna aperture and there is no sense to use space-time processing (see above). But for the second case ( ${\beta }^{-\text{1}}\ll {\Delta }_a$) the situation is completely different.

Introducing for each antenna element the parameter $g^{-1}=\frac{{\text{MSE}}_{\text{T}}}{{\text{MSE}}_{\text{S/T}}}$, where

MSE_T is the mean square error of the filtering for pure time processing and MSE_S/T is the same, but for the space-time case, one can see, that g^-1 is nothing else but a metric of how much the MSE is reduced due to the space-time processing. So, when g takes its minimum value, the gain of the space-time processing is maximum.

It might be shown that, when it is possible to neglect the time fluctuations of the field (fadings), one has:

$g_{\min }=\sqrt{1-\exp \left(-2\beta \right)}$.(4.8)

From (4.8) it follows that, when $\beta \ll \text{1}$ (for example β ~ 0.1) then, the gain of the space-time filtering (i.e. the MSE) might be more than twice, which is really outstanding taking into account that the real gain is ~ 2^Q.

As it follows from the earlier presented data, even a rather “modest” (no more than twice) improvement for the MSE_i of the filtering on each antenna element might be a significant improvement on the resulting MSE_Q due to the diversity effect.

The latter might be encouraging for intentions to, even lightly, improve the filtering accuracy of the space-time processing on the antenna elements. Tentatively, one of the possible trends might be the “multi-moment” approach presented in [13] , etc., which showed very good results for “pure time” processing.

Remind that the essence of the multi-moment approach is the introduction of the time instants [13] : $t_1<t_2<\cdots <t_n$, and the consideration of $x=x_i\left(t_i\right)$, $i=\overline{1,n}$ which forms a new vector $x\left(t\right)={\left[x\left(t_{\text{1}}\right),\cdots ,x\left(t_n\right)\right]}^{\text{T}}$, as a new “input” to be processed in an “aggregate” way.

The possible “improvement” of the filtering accuracy might follow from the aggregate processing of the field x(t, r), (r fixed), of “n” instants (some kind of “time diversity”) (see [2] [13] ).

For this matter, it is reasonable to invoke Equations (3.1)-(3.4). The first two are dedicated to the description of the a priori Probability Functional, P_PR(t, x), of the input chaotic field and the last one to the generalized SRE processing of chaotic space-time filtering.

Applying then the straightforward analogy between the generalized SRE for filtering in partial derivatives and the variation (functional) derivatives (see section 3 above) one can write the multi moment generalized SRE in the way

$\frac{\partial P_{ps}\left(t,r,x\right)}{\partial t_1\cdots \partial t_n}=L_{pr}\left\{P_{ps}\left(t,r,x\right)\right\}+P_{ps}\left(t,r,x\right)\left[F\left(t,x\right)-\langle F\left(t,x\right)\rangle \right]$,(4.9)

with $t={\left[t_{\text{1}},t_{\text{2}},\cdots ,t_n\right]}^{\text{T}}$.

It is obvious (and there is no need for additional comments), that the real multi-time space-time filtering (4.9), even in the quasi-optimum fashion, might be rather complex in computational sense and therefore impractical. But for each antenna element and applying the simplest EKF algorithm (for two-moment scenario) the possible improvement for the MSE_i (and some illustrations about it) can be seen from the following tables [22] .

In order to interpret correctly the data from the following tables, some comments are necessary:

−10 dB; AWGN denotes the white Gaussian noise of the channel; the interferences were considered as the total interference signal of all carriers, which do not depend on the concrete selected signal for filtering.

As it can be seen from the tables, for the simple EKF algorithm with low SINR (less than 0 dB), the chaotic filtering operates with an MSE_i of less than 10% (on each antenna element), i.e. it gives rather good results for extremely tough scenarios. Next, the difference in the filtering efficiency between 1 MM and 2 MM approaches is not significant (less than 10%), but taking into account the diversity effect for MSE_Q, it might be seen as an important factor (see above) for the MIMO space-time processing with $Q\gg \text{1}$.