The Incoherent Approach Might Be Opportunistic for the 6G Wireless Dense Networks Design? Theoretic View

This material is aimed to attract attention to the “Incoherent approach for the information transmission in wireless channels”. Such kind of approach might be successfully applied in future (5G+, 6G, etc.) dense networks formed by high-speed-vehicles (HSV networks). Those scenarios take place in Doubly Selective communication channels typical for such kind of radio networks. To be concrete, the main part of the material is related to the Power MIMO-RIS-NOMA networks, which seems to be prospective for the future. The proposal for the incoherent view (“paradigm”) is based on several basic principles, thoroughly discussed in the main body of the article. First: rejection of the application of any type of Channel State Information (CSI, CSIT); Second: application of the modulation (demodulation) technique “invariant” to the communication channel's distortions; Third: Orthogonal Double Selective channel decomposition by means of “universal” eigen basis, as “artificial (virtual) trajectories” of wave propagation; and Finally: Chaos parameter settings for users UE's, as UE´s signatures together with multi-hypothesis sequential detection algorithms for users’ classification. The proposed approach seems rather opportunistic for effective utilization of radio resources and might simplify the implementation problems.

The challenges, provoked by the progress in the wireless network design: from 5G to 5G+, 6G, etc. certainly invoked many new trends (paradigms) for the network design.
Particularly, the last decades show, that the non-orthogonal multiple access (NOMA) attracts a considerable attention as a multiple access technique in dense networks 5G+ and beyond.
The hereafter material is dedicated to the Power NOMA, which still is rather practical.Recently, for the NOMA technology numerous schemes were proposed and thoroughly analyzed in the literature ( [1], [2], [3], [4], [5], [6]).It was shown, (see [1], [2], [3], [4]), that applying the correct parameter settings, the symbol level Power NOMA is able to strongly outperform the well-known orthogonal multiple access schemes (OMA).In dense networks which are expected to be in service for 5G+ and beyond (6G, etc.), it is natural to predict the "huge lack" of orthogonal radio-resources and though NOMA might be rather useful.
In this regard the application of MIMO, massive MIMO, etc. together with the intelligent channel reconfiguration (RIS) techniques seems to be very effective.Note that, for application in NOMA transmission, the channel reconfiguration is a complex problem for MIMO due to the very aggressive multi-user interferences (MAI) generated by intensive service demands from multiple User Equipment (UE) with always limited radio resources.
Note that for MIMO case this problem turns out to be rather complex and even provokes serious doubts regarding the future application of Power NOMA in MIMO channels (see [3], [5], [6]).The situation might be worse if the transmission system includes the utilization of RIS and relies on "coherent processing ideas" based on the Channel State Information (CSI, CSIT) which is always imperfect in real life scenarios, particularly in "High Velocity Channels" (HVC) (see [5], [7], [8], [9], [10]), generalized in the following through the term Doubly Selective Channels.One must notice, that the prospective scenarios for wireless networks related to 5G+, 6G, etc. might widely include applications of HSV, i.e., channels with large values for the parameters like Doppler Shifts, Frequency Offsets, time delays, etc.It was pointed out long ago in [7], [8] such kind of effects do not "allow" to effectively maintain the so-called "coherent" or "quasi coherent" paradigms for signal processing algorithms based on Channel State Information (CSI) estimation, because the time-frequency dynamic behavior of the channel simply does not support rather accurate estimation of CSI.
The latter is the reason why the present material suggests an "ideological shift" for the paradigm for MIMO-RIS-NOMA (see Fig. 1) system design from the coherent to the incoherent one with application of the modulation schemes named in [11], [12] as invariant to the distortions of the channel.
In [3] some doubts were recently expressed regarding the future implementation of Power NOMA, where it was stated that the "standard" Power NOMA approach might be useless and "impractical" for MIMO scenarios.The latter is explained (see below) due to application of SIC (Successive Interference Cancelation) for MAI mitigation and UE´s identification.That is why [3] suggested to concentrate on the Resource Splitting concept (RSMA) approach, but its application in Doubly Selective Channels was not illustrated [3].
Based on the "coherent" ideology the spread spectrum concept (the so-called WSMA NOMA) for NOMA-MIMO has been thoroughly analyzed [4], but its application in Double Selective Channels was not specified.
So, in short, how to manage multi-user interference in such kind of channels, together with Doppler Shift, frequency offset, etc. distortions affecting the desired information signals?
In this regard, hereafter is suggested, first to change the approach for the NOMA settings, i.e., refuse the idea of coherent ideology for NOMA settings for HVC and substitute them by the incoherent one, as "coherent tools" are anyway almost impossible to apply properly in real scenarios.
Secondly, apply the "invariant, robust," modulation technique to the above-mentioned distortions at the Doubly Selective Channels.
Third, it is reasonable to "decompose the rather complex" Doubly Selective Channel System Functions (in Bello sense) into the set of so-called "artificial trajectories" for wave propagation with its separated processing.
Fourth, based on the propagation phenomena in the Doubly Selective Channels, which transform the desired signals of each of UE's into the "almost" Gaussian Random Processes and the channel into the vector Gaussian Channel, it is suggested to apply for the UE's identification the "modified" idea for the SIC procedure in the form of "parallel filtering" approach, where the UE signatures are taken from the parameters of certain chaotic attractors which are used to model those UE's in the Gaussian Channel (see [13], [14], [15], [16], [17]) in order to precisely "separate" them and then classify (identify).
Those basic ideas applied to the NOMA-RIS-MIMO transmission design in the "incoherent fashion" might significantly simplify the system implementation, minimizing the SNR losses, etc. and providing realistic characteristics for the Noise Immunity and the Spectrum Efficiency of the system.
The article is organized as follows.Section II is totally dedicated to the so-called "Orthogonal channel decomposition" applied to the NOMA-MIMO-RIS.Section III describes the parallel filtering approach for MAI mitigation together with the sequential multi-hypothesis classification for the UE's.Section IV presents the sketch of the invariant DPSK-k modulation for the UE's and autocovariance demodulation algorithm together with their Noise Immunity characteristics evaluation.Section V is totally dedicated to Conclusions.

II. CHANNEL ORTHOGONALIZATION APPROACH, ARTIFICIAL TRAJECTORIES AND RIS DESIGN A. HISTORIC COMMENTS
General ideas for channel characterization by means of the so-called Channel System Functions were presented in the fundamental works of P. A. Bello (see, for example, [18]).The Channel System Functions were applied first for orthogonalization purposes by R. Kennedy (1969) [19] in the design framework of the "West Ford" project and were further generalized and developed (see [20], [21], [22], [23] and the references therein).
In the following the concept of Channel Orthogonalization will be described and developed.

B. CHANNEL ORTHOGONALIZATION FOR MIMO CASE. ARTIFICIAL TRAJECTORIES AND UNIVERSAL EIGEN BASIS
The orthogonalization principle for MIMO channel representation will be considered as a representation of any Channel System Function (in Bello sense [18]) in terms of a reduced number of its eigen functions.Those functions in the following are denoted as virtual (artificial) trajectories (beams) for the propagation phenomena.
One can interpret the orthogonalization principle as a special case of the Generalized Fourier series approach, where orthogonalization basis is chosen in a specific way (see below).
Under the wide-sense stationary uncorrelated scattering (WSSUS) [18] assumption, in the following the focus is set on the so-called input impulse response time-delay-spread (FIR) matrix H(t, τ ) of the MIMO channel defined for "N" transmitting and "M" receiving antennas (N Tx , M Rx ) as: where t, τ and f are time, time delay and frequency; H(t, τ ) and H(t, f) are matrixes of size N Tx xM Rx .
It is well known, that the so-called KLE method is an optimum procedure (see [28]) (based on the so-called KLE integral equation) and is used to find a set of orthogonal eigen matrices (eigen functions), that can approximate the Gaussian H(t, τ ) with a minimum number of those eigen matrices and a predefined minimum mean-square error (MSE).Mathematical foundations for this can be found in [28] and are omitted hereafter.
In the general case for this matter the KLE integral equation must be applied [28]: where {λ l } denotes the set of eigen values,{ l ()} the set of eigen matrices obtained from covariance matrix R H of the Gaussian channel and the integration domain is angle beamwidth, T obs is the observation time for analysis, T max is the maximum delay excess time.Generally (2.2) is complex for its solution and for practical scenarios it might be simplified by several assumptions: separability for the covariance matrix R H (), WSSUS conditions, Comparing (2.2) and (2.3) it is obvious that the separability hypothesis for R H () is a "great" proposal for the KLE simplification.
One must notice the strong limitation of (2.2), (2.3): the dependence on their solutions i.e., eigen matrices (eigen functions) to the concrete form of R H ().
The latter encourages the authors of [20], [21] to propose the concept of "universal" eigen basis used under rather broad assumptions of the Channel System Functions (see the next paragraph).

C. GENERALIZED KRONECKER CHANNEL MODEL (GKCM) FOR MIMO CASE
It must be stressed once more, that KLE method is an optimum procedure to approximate the Gaussian channel model with a minimum set of eigen matrices (eigen functions) and with a minimum value of the approximation error (MSE) (see [28]), but with a priori knowledge of R H (). The latter is a principal obstacle for a real implementation of KLE method for the real scenarios without channel sounding.
The ongoing idea was inspired by the fundamental characteristics of Prolate Spheroidal Wave Functions, PSWF, (see [20], [21], [23], [28] and references therein), which shows that PSWF might be suitable as a basis to propose a generic principle for channel modeling.It follows from their fundamental features, as they only require an a priori knowledge of a minimum set of parameters of the channel: channel bandwidth, F max , maximum delay spread, T max , and the angular bandwidth, θ .In this sense the PSWF basis can be considered as Universal Basis.
Once again, assuming the separability for R H () in spacefrequency-time domains it follows that the required number of PSWF for the approximation for each domain can be easily predicted in advance (see also [28]): r For time delay domain r For spatial domain (assuming ULA arrays for MIMO) where L is the number of elements of the ULA (for comments about it see the following); d is the separation between the ULA elements; λ is the working wavelength; ULA stands for Uniform Linear Antennas applied at MIMO.
Note that, for the ULA assumption in [21], it was shown that any Planar Antenna apertures (PA) can be successfully approximated (in covariance sense) by a non-uniform linear aperture (n-ULA).Then, the latter (also in covariance sense) can be approximated with a predefined error (MSE) by ULA. 1  That is why for the above presented expression of the number of PSWF the ULA approximation was finally chosen.
The approximation with PSWF obviously always requires more functions, than the KLE solution, because the latter is exact and optimum; at the same time the universal PSWF basis is introduced heuristically as a basis invariant to the statistics of the channel!Several examples of the possible increment of the required number of PSWF, comparing with the KLE approach, for some channel standards can be found in 1it must be mentioned that in [29] the same issue was proved from geometrical considerations.[30]; those examples show, that the increment of the number of PSWF is moderate.
Consider now, as it was proposed in [21], [30], the NOMA-RIS-MIMO as a channel with the dispersion and fading (including Reconfigurable Intelligent Surfaces, RIS) to be approximated by generic the Generalized Kronecker Channel Model (GKCM).
Then [21], [23], the H(t, τ ) of the GKCM might be represented in the following general way: where the superscript H indicates here the Hermitian transpose of the matrix; ˜ is the element wise square root of the Coupling Matrix (CM), which physically represents, how the eigen modes (eigen matrices) of the transmit and receive correlation matrixes (in our case PSWF) are connected through the scattering environment of the RIS; V's with certain indexes are "orthogonalization matrixes" G(t, τ ) is an N Tx xM Rx i. i. d. zero mean Gaussian CIR Matrix.
So the essence of the GKCM, including the MIMO-RIS-NOMA channel, is rather simple: the model contains a set of "independent" SISO channels, artificially created from the appropriate eigen basis (in our case-PSWF) accumulated in a model by the CM (Coupling Matrix).In this case RIS is nothing else, but a special case of CM.

D. RECONFIGURABLE INTELLIGENT SURFACE (RIS) DESIGN
The RIS as an element of the NOMA-MIMO transmission system was proposed rather recently to improve the characteristics of NOMA transmission against the traditional OMA setting with OFDMA (see [31], [32], [33]), see also Fig. 1.
As it was already mentioned (see below), the first attempt for the artificial "insertion" into the propagation media to improve the characteristics of the information transmission was first proposed long ago by R. Kennedy in the framework of the West Ford project [19].The GKCM idea follows from there.
Concretely, the possible scenario for NOMA-RIS-MIMO is presented in Fig. 1 and from it follows, that the UE's are located usually in small clusters surrounding each RIS in the system and the RIS is providing a beamforming for certain UE's.But considering the material above, this "beamforming" might be applied (certainly, in the artificial way!) through the "Coupling Matrix" (see GKCM), which might be used to provide the "propagation beams" for each UE's.
So, the RIS algorithm is the algorithm for "predefined, weighted by PAS, connections" between the eigen matrices (eigen functions), as artificial beams, at Tx and Rx.The aim of those connections is to provide the required SNR values for identification of UE's (user's decoding) at the Rx´s (see details in the next section).
Summarizing all this, one can consider that the RIS design is the synthesis of the CM by the RIS controller for required conditions of UE's, whose identification algorithms will be presented in the following.
Returning to the RIS design issue, one must assume, that (see Fig. 1) there are i = 1, n RIS located in different sites of the MIMO system; each site is assigned to several UE's ( j = 1, m).Then as it was shown in [1], [13] on each site (in Power NOMA), for successful identification at the Rx, the conditions for each UE must be significantly different and are assumed to form a variation series: where {h 2 j } are SNR's values required for each UE's to be successfully identified with a predefined precision.Assuming the diagonal form of , its "n-th" element is where "n" is a PSWF index associated with UE n , θ might be associated with PAS (Power Azimuth Spectrum) and denotes directions on RIS or mutual "losses" of the connected eigen modes, .< > denotes the statistical average operator.Then, if h 2 j for UE is a priori predefined, then: (2.7) Formula (2.7) is nothing else but a general form of RIS controller algorithm.Considering that, for the known characteristics and parameters of the MIMO system, eigen functions and {λ n } can be calculated a-priori, n,n can be easily calculated mainly a-priori and the algorithm for RIS is as easy as possible (see also some comments in [30]).
The latter gives an optimistic "hope" that such kind of RIS design, due to its simplicity, might be found as a prospective one for dense MIMO networks with large number of both UE's and sites.One of the reasons for such kind of optimism is based on the application of metamaterials for RIS implementation.

III. THE FILTER BANK APPROACH FOR MAI CANCELLATION WITH FURTHER UE'S IDENTIFICATION (CLASSIFICATION) A. BASICS OF CHAOS FILTERING
Keeping in mind that the current material is related to the scenarios corresponding to Incoherent Approach applied in Double Selective Channels, one must notice that the so-called SIC (Successive Interference Cancelation) method, proposed and implemented long ago for MAI mitigation (see, for example [34] and references therein.)won't be analyzed hereafter (see below and [13]).
The SIC design has been thoroughly investigated and developed (see [1], [35]), is mainly based on the utilization of the SIC, information, as it was already mentioned above (see section I); the low feasibility of an accurate user identification provoked therefore doubts regarding the application of the Power NOMA in MIMO transmission systems, particularly in Double Selective channels.
In this regard the Filter Bank approach was proposed for the UE's identification in [13] for HVC.Its application is based on

FIGURE. 2. Block diagram for the UE's Filtering Classification (identification).
specific chaos filtering methods for "parallel filtering" of all UE´s in a simultaneous fashion and then to pass the filtered results to the multiple-hypothesis sequential testing for UE's decoding (identification).This approach is characterized by high precision of the signal processing and by reduced processing times [13], [36].
It is reasonable to remind once more, but briefly, that due to the physical phenomena in Double Selective channels both in time and frequency domains, the transmitted signals are "destroyed" into the almost random processes with "double selective features", characterized by such distortions, as Doppler Shifts, Frequency Offsets, Frequency Shifts, etc.The latter gives an opportunity to apply the chaotic models (see [14], [15], [16], [17], [37], [38], [39] and references therein) as a UE's signatures, instead of random models, to take advantage of their "singular opportunities", i.e., practical "invariance" of its accuracy to SNR values of the corresponding signal processing algorithms.

B. PROPERTIES OF CHAOS FILTERING AND UE'S IDENTIFICATION (CLASSIFICATION)
The user's identification utilizing their simultaneous filtering or estimation seems to be a rather opportunistic idea for the Doubly Selective Channels (see Fig. 2).
This "hope" is based on the statement that the Filter Bank Solutions depend on the high precision of the identification of the many different users by applying only their chaotic "signatures" in Power NOMA transmission systems.
In [13] the SNR and SINR conditions, for effective user decoding (identification), were experimentally proposed (see the following).
One must notice, that the corresponding background for chaos filtering was exhaustively presented in the already mentioned references, though there is no sense on repeating it!The most important extractions from those references are presented in the following: r As an "optimum" option hereafter the Extended Kalman Filter (EKF) is recommended, implemented in the socalled "one moment" (1MMEKF) or "two moments" fashion (2MMEKF) [14], [15], [16].This option offers a good balance between filtering accuracy and computational complexity.
r The 1MMEKF and 2MMEKF filtering algorithms show some slight differences in terms of filtering accuracy, but in terms of computational complexity and processing time the differences are practically negligible; both are "simple" and "fast."r The accuracy of the 1MMEKF and 2MMEKF obviously do not demonstrate "singularity", as they are heuristic and not the optimum ones [28], [37]; anyway those algorithms are rather accurate, efficient and universal.The extractions (and some relevant comments) of the simulations results are presented in the following.
Under the influence of additive white noise, the simulations illustrate the efficiency of the Filter Bank approach for identification of OMA, UE´s and their further (final) classification by means of multiple-hypothesis testing with sequential analysis algorithms [36], [40].
Fig. 3 illustrates the efficiency of the 1MMEKF and 2MMEKF algorithms for filtering OFDMA signals as OMA in presence of NOMA (MAI) interferences and additive white noise.
For simulations, the OMA and NOMA signals were modeled considering 50 carriers (see [13] for details and related current Standard settings).From the corresponding figures it follows, that for SNR = 13-15 dB (usually applied in wireless communications) and SIR of the same order, the normalized (regarding the signal power) Mean Square Error (NMSE) of the filtering is less than 5%.This NMSE value shows, that the filtered OFDMA (OMA signal) practically preserves its average power after filtering, while the SNR losses are less than 1dB.Though, the noise immunity of OMA is almost preserved.
From the Fig. 3 one might find, that the concept of sufficiently different channel conditions for the identification of OMA and NOMA users (see [1]) might be considered as around 9-12 dB, i.e., it is almost like SNR.So, the OMA signal can be successfully eliminated from the aggregate input signal to classify the NOMA users.The final efficiency of filtering for the individual NOMA users UE´s is illustrated in Table I of [30], which shows that it is also less than 5%.After the separation of all UEs (OMA, NOMA) by means of the Filter Bank (see Fig. 2), a classification procedure is required.It might be achieved through the application of sequential m-hypothesis testing algorithms.As it was known for a long time, they are characterized by reduced processing times (see [35], [36] and references therein).The latter is detailed in [13], [35] and is omitted in the following.Finally it might be stressed that the real-time implementation of the proposed algorithms is based on standard DSP (Digital Signal Processing) methods and it does not represent any problems.
It was also mentioned in [13], [35] that, for evaluation of the influence of the classification errors on the final noise immunity of the NOMA transmission system, the methodology of the Chernoff Upper bounds might be successfully applied together with the so-called minimum Kullback-Leibler distance between Gaussian hypothesis in Double Selective channels for the scenarios of significant difference between them.Finally, it was shown, that the classification error is much less than the Error Probability-P err of the system and it might be neglected [13].
This issue can be interpreted also as the essence for the explanation of the application of the "significant difference requirement" for the UE's classification (see above).

IV. INCOHERENT AND INVARIANT DPSK DEMODULATION TECHNIQUE A. PREFACE
One must notice that the "incoherent concept" for application in Doubly Selective Channels has been discussed in [7], [8], [24], [25], [26], where the utilization of DPSK was considered, but the "invariant" (robust) aspects were not properly presented there.
This topic is provided hereafter, based on the material of [11], with application of the differential phase shift keying of high order, i.e., DPSK-k for MIMO case.!!
The modulation DPSK-k is a generalization of the DPSK, that introduces k higher orders for the phase differences and it was thoroughly investigated in [12].Their opportunistic practical applications to achieve invariant properties to Doppler Shift, Frequency Offset, and other channel dispersions were stressed in [12] as well.
The ideas proposed hereafter were inspired by [12], as an attempt to generalize DPSK-k for MIMO transmission over Doubly Selective Channels applying the channel orthogonalization tools together with incoherent demodulation for MIMO reception.
So, how all those issues might be useful for HSV channels, dense networks, etc. at 5G+, 6G and beyond, is to be seen hereafter.

B. GENERALIZED DPSK-K MODULATION AND MIMO RECEPTION
The virtual trajectories for the MIMO channels (see section 2) were already broadly applied for multi-carrier system design in Doubly Selective Channels (see [24], [25], [26] and references therein).
The following material is completely dedicated to the invariant MIMO reception in Doubly Selective Channels applying the DPSK-k modulation-incoherent demodulation technique [11].
The DPSK-k generalizations from the SISO (see [11]) to the MIMO scenarios require some mathematical work, whose result yields rather cumbersome expressions [11].Here, only a sketch with the simplified explanation is presented.
For the N T × N R MIMO system with the application of Space Time Block Codes (STBC) or Orthogonal SBTC (OS-TBC), the same logic as in [12], is valid with the difference, that "scalar" description in [12] must be "substituted" by matrixes (see [11]).
Though, the constellation symbols form the matrix b (for the time instant "b") and the transmitted difference modulation matrix is: Then, for demodulation, the estimated date b is calculated in the way: where k is the received space modulated matrix.The structures of these matrixes are based on the binomial coefficients (see analogy to scalar case in [12]) and show the necessary multiplications of 1 .The same logic is for f z ( k ) and the necessary multiplication for k (see [11]), if the autocovariance demodulator is considered.
Though, it follows from (4.1) and (4.2) that the autocovariance demodulator in the MIMO case is a representation of the autocovariance demodulation principle for the estimated transmitted matrix over a "b" time interval (instant) and the q-th artificial trajectory.
Note, that the autocovariance demodulator is really easy for implementation and is also computationally the simplest option, but for sure at the same time is the solution with the lowest noise immunity properties.
The adequate "balance" between noise immunity requirements and simplicity for the implementation is presented in the next paragraph.

C. NOISE IMMUNITY EVALUATIONS AND SIMULATIONS
It must be noticed that the exact calculus of the noise immunity for the DPSK-k autocovariance demodulation, particularly for the MIMO case to the best of our knowledge, is impossible to obtain, as even for SISO scenarios they are unknown [12].
For this matter in the following the well-known Chernoff bounds are applied (see [41]), as an upper bound for the BER in the binary case, as it was done in [11] with its further simulation validation.The latter provides the expression for the Chernoff bound of the BER for the autocovariance demodulation in the form: where h 2 is the SNR at the demodulator input and the α k can be found for any DPSK-k.Consequently: for the DPSK-1 α 1 = 1 4 (see [11]) and for DPSK-2 α 2 = 1 8 and for the DPSK-3 α 3 = 1 16 , etc.For the MIMO reception with the quadratic incoherent addition for homogeneous conditions at all "diversity" branches (together with the artificial trajectories), finally, one can get the final expression for BER [41]: It might be noticed that the artificial trajectories reduce the number of diversity branches to achieve the channel hardening (see [23] for details), which obviously reduce the fading strength and "converts" the Doubly Selective Channel into the "constant" channel, but with aggregate SNR from all diversity branches!The latter drastically improves the noise immunity of the MIMO system and requires less complex channel error correcting codes (see also the simulation results at [11]).
The simulation results presented in [11] serve to verify the "robustness" of the upper bounds of the expressions (4.1), (4.4).Particularly, they show that, such upper bounds offer no more than 3 dB difference between the simulation data and the upper bound for the SNR around P err ∼ 10 −4 , which is acceptable.
As it was pointed out before, due to the noise immunity losses from application of the autocovariance demodulators, the error correcting codes (BCH, for simple example) might be a plausible solution for "cancelling" those losses for MIMO systems over Doubly Selective Channels, which together with the channel hardening can reduce the negative impact of the noise increment for the autocovariance demodulators for DPSK-k.!!

V. CONCLUSION
The conclusions must start with the statement that the material presented below doesn't have anything to do with the attempt to create doubts for the practical value of the broadly applied coherent paradigm.It is totally wrong and even absurd!
The goal of the article is different and dedicated to attracting attention to the features of "incoherent paradigm", which is possible to apply, when the coherent methods might be hard to implement!For Power NOMA-MIMO-RIS transmission system design in Doubly Selective Channels, the presented material illustrates different aspects of the incoherent approach certainly without any relation to the CSI, CSIT estimation of the channel.
r Artificial (virtual) trajectories approach for the decom- position of the Channel System Functions to achieve a channel hardening at diversity combining in RX terminal, as fast as possible.r Filtering method together with an m-hypothesis sequen- tial testing for effective classification of the UE's.
r Invariant DPSK-k modulation together with autoco- variance demodulation to achieve an incoherent UE's demodulation processing, as simple as possible.It is possible to characterize each of these approaches (see above) as somehow fast and simple, which place them, as a good option for operation in the Doubly Selective Channels for the 5G+, 6G networks.

FIGURE. 3 .
FIGURE. 3. Illustration of the efficiency for OMA filtering in presence of MAI and noise.

r
Application of the artificial trajectories for RIS design by means of the Coupling Matrix of the GKCM as a model of MIMO-RIS-NOMA channel.