Method, transmitter device and receiver device for securing channel training on a power domain non-orthogonal multiple access system

Abstract

Secure channel training to enhance the confidentiality level of a power domain non-orthogonal multiple access (NOMA) communication system when impaired by eavesdropping attacks coming from inside and outside of the network. In a first scenario, a cooperative jammer available in the system defines an external source of entropy that is independent of the channel variation rate. While the jammer provides secrecy inside the network, the proposed invention is configured to secure the network from outside, encoding the system information, which is exchanged during the training phase, using only the channel state. In a second scenario, the cooperative jammer is not available; with the secrecy inside and outside of the network ensured through a different parameterization. That parameterization is done in a way that the required system information is encoded using not only the channels, but also a random phase defined in the data communication phase.

Claims

1. A transmitter device i for a secure channel training method applied to a power domain non-orthogonal multiple access system comprising: a plurality of transmitters and receivers, wherein the transmitter device i is configured to: estimate a channel between receiver device j and transmitter device i; estimate a channel between receiver device j+1 and transmitter device i; apply a channel processing block to encode the estimated channels between transmitter device i and receiver devices j and j+1; and transmit a signal of said encoded channels to receiver device j+1 containing information regarding the estimated channels between transmitter device i and receiver's devices j and j+1.

2. The transmitter device i according to claim 1, wherein the transmitter device i is further configured to apply said channel processing block to the estimated channels between transmitter device i and receiver devices j and j+1, by applying the following encoding operation: $\begin{array}{c}{x}_{F\left(i\right)}\left(k\right)=F_i\left[p_H{h}_{j+1,i}\left(k\right),h_{j,i}\left(k\right),r\left(k\right)\right]\\ =p_H.Math.h_{j+1,i}\left(k\right).Math.{.Math.h_{j,i}\left(k\right).Math.}^{-2}{e}^{j\left[{?}_{j,i}\left(k\right)-2{?}_{j+1,i}\left(k\right)-r\left(k\right)\frac{2?}{M}\right]}\end{array}$ where F.sub.i(.) defines the processing function to be applied, h.sub.j,i(k) is the small-scale channel fading coefficient between transmitter device i and receiver device j, h.sub.j+1,i(k) is the small-scale channel fading coefficient between transmitter device i and receiver device j+1, |h.sub.j,i(k)| is the magnitude of the channel between transmitter device i and receiver device j, |h.sub.j+1,i(k)| is the magnitude of the channel between transmitter device i and receiver device j+1, ?.sub.j,i(k) is the phase of the channel h.sub.j,i(k) between transmitter device i and receiver device j, ?.sub.j+1,i(k) is the phase of the channel h.sub.j+1,i(k) between transmitter device i and receiver device j+1, x.sub.F(i)(k) is the signal to be sent by transmitter device i containing the encoded channels between transmitter device i and receivers devices j and j+1, p.sub.H corresponds to a power allocation coefficient allocated to receiver j+1, r(k) is defined as 0 or as an integer random variable defined in the interval [1, M], wherein M is the order of a digital modulation for which the system is configured and k defines the system realization index.

3. A receiver device j+1 for a secure channel training method applied to a power domain non-orthogonal multiple access system comprising a plurality of transmitters and receivers, wherein the receiver device j+1 is configured to: receive a signal sent by transmitter device i containing the encoding of estimated channels between transmitter device i and receiver devices j and j+1; and apply a channel acquisition block to the signal containing the encoding of the estimated channels between transmitter device i and receiver devices j and j+1.

4. The receiver device j+1 according to claim 3, wherein the receiver device j+1 is further configured to apply said channel acquisition block to the signal sent by transmitter device i containing the encoding of the estimated channels between transmitter device i and receiver devices j and j+1, by applying the following decoding operation: $\begin{array}{c}{R}_i\left[y_{j+1,F\left(i\right)}\left(k\right),{?}_{j+1}{p}_H\right]={\left(\frac{.Math.y_{j+1,F\left(i\right)}\left(k\right).Math.}{{?}_{j+1}{p}_H}\right)}^{1/2}{e}^{-j{?}_{j+1,F\left(i\right)}\left(k\right)}\\ =h_{j+1,i}\left(k\right){h_{j,i}\left(k\right)}^{-1}{e}^{j\left[r\left(k\right)\frac{2?}{M}\right]},\end{array}\mathrm{where}\begin{array}{c}{y}_{j+1,F\left(i\right)}\left(k\right)={?}_{j+1}{x}_{F\left(i\right)}\left(k\right)h_{j+1,i}\left(k\right)\\ ={?}_{j+1}{p_H\left[.Math.h_{j+1,i}\left(k\right).Math.{.Math.h_{j,i}\left(k\right).Math.}^{-1}\right]}^2{e}^{j\left[{?}_{j,i}\left(k\right)-{?}_{j+1,i}\left(k\right)-r\left(k\right)\frac{2?}{M}\right]}\end{array}$ is the received signal at receiver device j+1 containing the encoding of the channels between transmitter device i and receiver devices j and j+1, |y.sub.j+1,F(i)(k)| is the magnitude of the received signal y.sub.j+1,F(i)(k), ?.sub.j+1,F(i)(k) is the phase of y.sub.j+1,F(i)(k), ?.sub.j+1 defines a constant path-loss coefficient for receiver j+1, h.sub.j,i(k) is the small-scale channel fading coefficient between transmitter device i and receiver device j, h.sub.j+1,i(k) is the small-scale channel fading coefficient between transmitter device i and receiver device j+1, p.sub.H is a power allocation coefficient allocated to receiver j+1, |h.sub.j,i(k)| is the magnitude of the channel between transmitter device i and receiver device j, |h.sub.j+1,i(k)| is the magnitude of the channel between transmitter device i and receiver device j+1, ?.sub.j,i(k) is the phase of the channel h.sub.j,i(k) between transmitter device i and receiver device j, ?.sub.j+1,i(k) is the phase of the channel h.sub.j+1,i(k) between transmitter device i and receiver device j+1, r(k) is defined as 0 or as an integer random variable defined in the interval [1, M], wherein M is the order of a digital modulation for which the system is configured, and k defines the system realization index.

5. The receiver device j+1 according to claim 3, wherein the receiver device j+1 is further configured to recover the ratio of the encoded estimated channels between transmitter device i and receiver devices j and j+1.

6. A system for a secure channel training method applied to a power domain, non-orthogonal, multiple access system, comprising: a plurality of transmitters according to claim 1 and a plurality of receivers according to claim 3.

7. The system according to claim 6, further comprising a jamming transmitter, wherein the jamming transmitter is configured for emitting a discrete jamming signal aligned with the transmissions of transmitter device i having the same order of a digital modulation for which the system is configured.

8. The system according to claim 7, wherein the jamming transmitter comprises a transmitter configured according to claim 2 and wherein r(k) is defined as 0.

9. The system according to claim 8, wherein the digital modulation for which the system is configured is a discrete square QAM constellation.

10. The system according to claim 6, wherein a digital modulation for which the system is configured is a discrete PSK constellation of order M with a constant phase shift ??=2?/M between any consecutive pair of points in the constellation and wherein r(k) is defined as an integer random variable distributed uniformly in the interval set [1, M].

11. The receiver device j+1 according to claim 4, wherein the receiver device j+1 is further configured to recover the ratio of the encoded estimated channels between transmitter device i and receiver devices j and j+1.

12. The system of claim 6, wherein the jamming transmitter comprises a transmitter according to claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The following figures provide preferred embodiments for illustrating the disclosure and should not be seen as limiting the scope of invention.

(2) FIG. 1 is a schematic of the wireless communication scenario 1 where the present disclosure is implemented. In the respective schematic, five single antenna terminals are represented, being each one defined as: (1) is a legitimate transmitter; (2) is a cooperative jammer; (4) is the attacked legitimate receiver that is also the NOMA user with the weakest channel; (3) is the eavesdropper that attacks (4) from the inside of the network, being also the NOMA user with the strongest channel; (5) is a second eavesdropper that attacks (4) from outside of the network. Regarding the channel coefficients, (6) is the small-scale channel fading coefficient h.sub.E0A(k) between (3) and (1), (7) is the small-scale channel fading coefficient h.sub.BA(k) between (4) and (1), (8) is the small-scale channel fading coefficient h.sub.E1A(k) between (5) and (1), (9) is the small-scale channel fading coefficient h.sub.E0J(k) between (3) and (2), (10) is the small-scale channel fading coefficient h.sub.BJ(k) between (4) and (2), (11) is the small-scale channel fading coefficient h.sub.E1J(k) between (5) and (2).

(3) FIG. 2 represents a high-level block diagram of the channel training method proposed in the present disclosure for scenario 1. The diagram shows that terminal (3) transmits in resource element 1 (RE1) a RS that is used by (1) to estimate the channel h.sub.E0A(k) (6), and by (2) to estimate the channel h.sub.E0J(k) (9). Similarly, in resource element 2 (RE2) a RS is transmitted by (4) in order to allow (1) to estimate the channel h.sub.BA(k) (7), and (2) to estimate the channel h.sub.BJ(k) (10). Then, using as inputs the phases and magnitudes of the complex channels h.sub.E0A(k) (6) and h.sub.BA(k) (7), a channel processing block (19) defined by the function F.sub.A (.) is implemented at terminal (1). Similarly, using as inputs the phases and magnitudes of the complex channels h.sub.E0J(k) (9) and h.sub.BJ(k) (10), a channel processing block (18) defined by the function F.sub.J(.) is also implemented at terminal (2). Then, the output of F.sub.J(.) is transmitted from (2) to (4) in the resource element 4 (RE4), being the output of F.sub.A(.) transmitted from (1) to (4) in resource element 3 (RE3). Finally, in order to get the intended channel information, terminal (4) processes the signals received in RE3 and RE4 with the processing blocks (22) and (20), which are implemented by the functions R.sub.J(.) and R.sub.A(.), respectively.

(4) FIG. 3 represents the channel processing block (19) implemented by the proposed training method at terminal (1) for application scenario 1. The channel processing block (19) receives five inputs (24-28), processes those inputs according to the function F.sub.A(.), and outputs a complex signal (29) defined in equation (Eq. 7). Input (24) corresponds to the channel magnitude |h.sub.E0A(k)| between (3) and (1); input (25) corresponds to the signal p.sub.H|h.sub.BA(k)|, where |h.sub.BA(k)| defines the channel magnitude between (4) and (1), being p.sub.H a power allocation coefficient; input (26) corresponds to the phase ?.sub.E0A(k) of the channel h.sub.E0A(k) (6) between (3) and (1); input (27) corresponds to the phase ?.sub.BA(k) of the channel h.sub.BA(k) (7) between (4) and (1); and input (28) corresponds to an integer random variable defined as r(k), which for application scenario 1 is defined as 0.

(5) FIG. 4 represents the channel processing block (18) implemented by the proposed training method at terminal (2) for application scenario 1. The channel processing block (18) receives five inputs (30-34), processes those inputs according to the function F.sub.J(.) and outputs a complex signal (35) defined in equation (Eq. 8). Input (30) corresponds to the channel magnitude |h.sub.E0J(k)| between (3) and (2); input (31) corresponds to the signal p.sub.H|h.sub.BJ(k)|, where |h.sub.BJ(k)| defines the channel magnitude between (4) and (2), being p.sub.H a power allocation coefficient; input (32) corresponds to the phase ?.sub.E0J(k) of the channel h.sub.E0J(k) (9) between (3) and (2); input (33) corresponds to the phase ?.sub.BJ(k) of the channel h.sub.BJ(k) (10) between (4) and (2); and input (34) corresponds to an integer random variable defined as r(k), which for application scenario 1 is defined as 0.

(6) FIG. 5 represents the channel acquisition block (20) implemented by the proposed training method at terminal (4) for application scenario 1. The channel acquisition block (20) receives three inputs (36-38), processes those inputs according to the function R.sub.A (.) in equation (Eq. 11), and outputs a complex signal (39) defined in equation (Eq. 11). Input (36) corresponds to the magnitude |y.sub.B,F(A)(k)| of equation (Eq. 9) transmitted by (1) and received by (4) in RE3; input (37) corresponds to the phase ?.sub.B,F(A)(k) of equation (Eq. 9) transmitted by (1) and received by (4) in RE3; and input (38) is a constant defined by ?.sub.Bp.sub.H.

(7) FIG. 6 represents the channel acquisition block (22) implemented by the proposed training method at terminal (4) for application scenario 1. The channel acquisition block (22) receives three inputs (40-42), processes those inputs according to the function R.sub.J (.) in equation (Eq. 12), and outputs a complex signal (43) defined in equation (Eq. 12). Input (40) corresponds to the magnitude |y.sub.B,F(J)(k)| of equation (Eq. 10) transmitted by (2) and received by (4) in RE4; input (41) corresponds to the phase ?.sub.B,F(J)(k) of equation (Eq. 10) transmitted by (2) and received by (4) in RE4; and input (42) is a constant defined by ?.sub.Bp.sub.H.

(8) FIG. 7 is a schematic of the wireless communication scenario 2 where the present disclosure is implemented. In the respective schematic, four single antenna terminals are represented, being each one defined as: (44) is a legitimate transmitter; (46) is the attacked legitimate receiver that is also the NOMA user with the weakest channel; (45) is the eavesdropper that attacks (46) from the inside of the network, being also the NOMA user with the strongest channel; (47) is a second eavesdropper that attacks (46) from the outside of the network. Regarding the channel coefficients, (48) is the small-scale channel fading coefficient h.sub.E0A(k) between (45) and (44), (49) is the small-scale channel fading coefficient h.sub.BA(k) between (46) and (44), (50) is the small-scale channel fading coefficient h.sub.E1A(k) between (47) and (44).

(9) FIG. 8 represents a high-level block diagram of the channel training method proposed in the present disclosure for application scenario 2. The diagram shows that terminal (45) transmits in resource element 1 (RE1) a RS that is used by (44) to estimate the channel h.sub.E0A(k) (48). Similarly, in resource element 2 (RE2) a RS is transmitted by (46) in order to allow (44) to estimate the channel h.sub.BA(k) (49). Then, using as inputs the phases and magnitudes of the complex channels h.sub.E0A(k) (48), h.sub.BA(k) (49), and an integer random variable defined as r(k), a channel processing block (55) defined by the function F.sub.A (.) is implemented at terminal (44). Then, the output of F.sub.A (.) is transmitted from (44) to (46) in resource element 3 (RE3). Finally, in order to get the intended system information, terminal (46) processes the signal received in RE3 with the processing block (57), which is implemented by the function R.sub.A (.).

(10) FIG. 9 represents the channel processing block (55) implemented by the proposed training method at terminal (44) for application scenario 2. The channel processing block (44) receives five inputs (58-62), processes those inputs according to the function F.sub.A (.), and outputs a complex signal (63) defined in equation (Eq. 25). Input (58) corresponds to the channel magnitude |h.sub.E0A(k)| between (45) and (44); input (59) corresponds the signal p.sub.H|h.sub.BA(k)|, where |h.sub.BA(k)| defines the channel magnitude between (46) and (44), being p.sub.H a power allocation coefficient; input (60) corresponds to the phase ?.sub.E0A(k) of the channel h.sub.E0A(k) (48) between (45) and (44); input (61) corresponds to the phase ?.sub.BA(k) of the channel h.sub.BA(k) (49) between (46) and (44); and input (62) corresponds to an integer random variable defined as r(k) distributed uniformly in the integer set within [1, M], being M the order of the constellations applied for the information streams.

(11) FIG. 10 represents the channel acquisition block (57) implemented by the proposed training method at terminal (46). The channel acquisition block (57) receives three inputs (64-66), processes those inputs according to the function R.sub.A (.) in equation (Eq. 27), and outputs a complex signal (67) defined in equation (Eq. 27). Input (64) corresponds to the magnitude |y.sub.B,F(A)(k) of equation (Eq. 26) transmitted by (44) and received by (46) in RE3; input (65) corresponds to the phase ?.sub.B,F(A)(k) of equation (Eq. 26) transmitted by (44) and received by (46) in RE3; and input (66) is a constant defined by ?.sub.Bp.sub.H.

DETAILED DESCRIPTION

(12) The present disclosure relates to a physical layer security technique for wireless communications and comprises a secure channel training method designed to enhance the confidentiality level of a power domain non-orthogonal multiple access (NOMA) communication system when this is impaired by eavesdropping attacks coming from inside and outside of the network. Although not limited to, this disclosure targets two application scenarios. In the first scenario, a cooperative jammer available in the system defines an external source of entropy that is independent of the channel variation rate. While the jammer provides secrecy inside the network, the proposed disclosure is configured to secure the network from outside, encoding the system information, which should be exchanged during the training phase, using only the channel state. In the second scenario, the cooperative jammer is not available; being the secrecy inside and outside of the network ensured through a different parameterization of the proposed disclosure. That parameterization is done in a way that the required system information is encoded using not only the channels, but also a random phase defined in the data communication phase.

(13) In an embodiment, Table 1 contains simulated and theoretical results regarding the secrecy level obtained against the inside attacker (3) when the jamming signal d.sub.J and the information signal d.sub.B intended to (4) are square QAM constellations of the same order M and with similar magnitude scaling. The simulations results regarding the exact value of the mutual information I(d.sub.B; d.sub.B+d.sub.J) are located in the second column of this table. The third column contains a theoretical upper bound defined as

(14) $\stackrel{\_}{I\left(d_B;d_B+d_J\right).}$
All the mutual information results in this table were normalized to the entropy of d.sub.B, which is defined as log.sub.2 (M) for independent and uniformly distributed sources.

(15) TABLE-US-00001 TABLE 1 M $\frac{I\left(d_B\text{;}{d}_B+d_J\right)}{{\log }_2\left(M\right)}$ $\stackrel{\_}{\frac{I\left(d_B\text{;}{d}_B+d_J\right)}{{\log }_2\left(M\right)}}$ 4 0.5000 0.5850 16 0.3278 0.4037 64 0.2341 0.3023 256 0.1790 0.2385 1024 0.1440 0.1955 4096 0.1202 0.1648

(16) In an embodiment, Table 2 contains simulated and theoretical results regarding the secrecy level obtained against the outside attacker (5) when the jamming signal d.sub.J and the information signal d.sub.B are square QAM constellations of the same order M and with similar magnitude scaling. The results presented in this table were obtained considering a unitary scale parameter for the Rayleigh distributions that define the channel magnitudes |h.sub.E1A(k)| and |h.sub.BA(k)|, being the expected value of |d.sub.B|.sup.2 also unitary. In the second column of this table are defined the results of the mutual information I(|d.sub.B|; ?|d.sub.B?h.sub.E1A|, ?|h.sub.E1A?h.sub.BA|) obtained through simulation using Gaussian kernel functions for estimating probability densities. The third column contains the values of the theoretical upper bound

(17) $\stackrel{\_}{I\left(.Math.d_B.Math.;?.Math.d_B.Math..Math.h_{E1A}.Math.,?.Math.h_{E1A}.Math..Math.h_{\mathrm{BA}}.Math.\right)}$
derived in the secrecy evaluation of the proposed disclosure. Since ?=?.sub.E1p.sub.H is a fixed constant, has no impact in terms of mutual information for this case, which means that the results presented in this table are applicable for any value of ? greater than zero. All the mutual information results in this table were normalized to the entropy of d.sub.B, which is defined as log.sub.2 (M) for independent and uniformly distributed sources.

(18) TABLE-US-00002 TABLE 2 M $\frac{\begin{array}{c}I(.Math.d_B.Math.\text{;}?.Math.d_B.Math..Math.h_{E1A}.Math.,\\ ?.Math.h_{E1A}.Math..Math.h_{\mathrm{BA}}.Math.)\end{array}}{{\log }_2\left(M\right)}$ $\stackrel{\_}{\frac{\begin{array}{c}I(.Math.d_B.Math.\text{;}?.Math.d_B.Math..Math.h_{E1A}.Math.,\\ ?.Math.h_{E1A}.Math..Math.h_{\mathrm{BA}}.Math.)\end{array}}{{\log }_2\left(M\right)}}$ 4 0.0000 0.4953 16 0.1271 0.3513 64 0.1019 0.2457 256 0.0801 0.1863 1024 0.0655 0.1494 4096 0.0545 0.1245

(19) Example 1 represents 2 single antenna legitimate transmitters (1), (2); 2 single antenna legitimate receivers (3), (4); and 1 single antenna passive eavesdropper receiver (5).

(20) In an embodiment, FIG. 1 illustrates an example of an embodiment of the proposed disclosure in a wireless communication system for application Scenario 1. In the considered communication system, node (1) pretends to transmit in secrecy d.sub.B to node (4) and d.sub.E0 to (3), using for that a power domain non-orthogonal multiple access technique in which (3) is the strongest user and node (4) represents the weakest user. The embodiment illustrated in FIG. 1 considers that the distance between node (1) and (3) is lower than the distance between (1) and (4), therefore, as it is usually defined in the state of art techniques, (3) is the strongest user with node (4) being the weakest user. In power domain non-orthogonal multiple access systems, the strongest user (3) can only reach their intended information after decoding the interference generated by the weakest user (4), being this interference posteriorly cancelled from the overall received signal using a successive interference cancellation (SIC) equalizer. Alternatively, a joint decoding (JD) of interference and information can be also applied using a maximum likelihood (ML) equalizer. Because (3) will decode first the interference created by d.sub.B before accessing the information d.sub.E0 sent to him, this node (3) will be recognized by the network as a legitimate user that also has the capacity to eavesdrop d.sub.B attacking the network from inside. Additionally, in order to model the occurrence of an outside attack, the embodiment in FIG. 1 also assumes that a passive eavesdropper (5) not registered in the network, and not co-located with (4), tries to tap in an undetectable way the information d.sub.B sent to node (4). The assumption of a passive eavesdropper means that this node listens the communication and does not cause any intentional interference into the legitimate communication channel, staying therefore undetected to the network. Finally, to improve the secrecy level inside of the system, a jammer defined by node (2) cooperates with (1) generating a discrete jamming signal defined by d.sub.J. In the remaining of this document, the random variables defined by h.sub.RT with R?{E0, B, E1} and T?{A, J} represent small scale channel fading effects which are modelled by zero mean independent complex Gaussian distributions with variance ?.sub.h.sup.2. The path-loss attenuation between the transmitting nodes T and each one of the receiving nodes (3-5) in R is represented by the coefficients ?.sub.E0, ?.sub.B and ?.sub.E1, where it is assumed that ?.sub.B<?.sub.E0. Furthermore, in this embodiment, (1) and (2) are located at the same distance of each one of the other terminals (3-5), therefore, the same path-loss coefficient is applied to model the attenuation between these two terminals (1), (2) and each of the other receiving nodes (3-5). The noise impairment at the receiving node R is represented by n.sub.R and is modelled by zero mean complex Gaussian random variable with variance ?.sub.n.sup.2. Ideal RF up- and down-conversion is assumed, being the described baseband processing applied to an independent realization of the system denoted by k.

(21) In an embodiment, the non-orthogonal multiple access communication process associated to the embodiment illustrated in FIG. 1 comprises two phases; a preliminary channel training phase (Phase I), and the main communication phase (Phase II). Although the main communication phase happens posteriorly to the training phase, the design of the secure training method associated to the present disclosure depends on the signal structure of the communication phase, hence, we will start by describing the communication phase.

(22) In an embodiment, in the communication phase, the information streams d.sub.B and d.sub.E0 are sent to the intended receivers, (4) and (3) respectively, in a way that the discrete jamming signal d.sub.J aligns with d.sub.B at node (3), protecting the network from inside attacks when discrete square QAM constellation structures of the same order M are applied for d.sub.B and d.sub.J. In order to align d.sub.B with d.sub.J at (3), and assuming that p.sub.L, p.sub.H are known power allocation coefficients computed according to the geographic distribution of (3) and (4), the signal transmitted by node (1) is formulated in equation (Eq. 1), being the one transmitted by node (2) defined in equation (Eq. 2).
x.sub.A(k)=h.sub.E0A.sup.?1(k)[p.sub.Ld.sub.E0(k)+p.sub.Hd.sub.B(k)](Eq. 1)
x.sub.J(k)=h.sub.E0J.sup.?1(k)p.sub.Hd.sub.J(k)(Eq. 2)

(23) In an embodiment, since the strongest user (3) is located closer to (1), the amount of power p.sub.L allocated to (3) should be lower than the power p.sub.H allocated to the weakest user (4). Taking into account equations (Eq. 1) and (Eq. 2), the signal observed by node (3) is given by equation (Eq. 3), with equations (Eq. 4) and (Eq. 5) defining the signals observed by (4) and (5), respectively.

(24) $\begin{array}{cc}\begin{array}{c}{y}_{E0}\left(k\right)={?}_{E0}\left[h_{E0A}\left(k\right)x_A\left(k\right)+h_{E0J}\left(k\right)x_J\left(k\right)\right]+n_{E0}\\ {?}_{E0}\left[p_L{d}_{E0}\left(k\right)+p_H\left(d_B\left(k\right)+d_J\left(k\right)\right)\right]+n_{E0}\end{array}& \left(\mathrm{Eq}.3\right)\\ \begin{array}{c}{y}_B\left(k\right)={?}_B\left[h_{\mathrm{BA}}\left(k\right)x_A\left(k\right)+h_{\mathrm{BJ}}\left(k\right)x_J\left(k\right)\right]+n_B\\ ={?}_B\left[\begin{array}{c}{h}_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)\left(p_L{d}_{E0}\left(k\right)+p_H{d}_B\left(k\right)\right)+\\ h_{\mathrm{BJ}}\left(k\right)h_{E0J}^{-1}\left(k\right)p_H{d}_J\left(k\right)\end{array}\right]+\\ n_B\end{array}& \left(\mathrm{Eq}.4\right)\\ \begin{array}{c}{y}_{E1}\left(k\right)={?}_{E1}\left[h_{E1A}\left(k\right)x_A\left(k\right)+h_{E1J}\left(k\right)x_J\left(k\right)\right]+n_{E1}\\ ={?}_{E1}\left[\begin{array}{c}{h}_{E1A}\left(k\right)h_{E0A}^{-1}\left(k\right)\left(p_L{d}_{E0}\left(k\right)+p_H{d}_B\left(k\right)\right)+\\ h_{E1J}\left(k\right)h_{E0J}^{-1}\left(k\right)p_H{d}_J\left(k\right)\end{array}\right]+\\ n_{E1}\end{array}& \left(\mathrm{Eq}.5\right)\end{array}$

(25) In an embodiment, in the case of (3), when ?.sub.E0p.sub.H is much larger than ?.sub.E0p.sub.L, a SIC equalizer can be applied to equation (Eq. 3) in order to get d.sub.E0. As it is typical of power-domain NOMA systems, first (3) decodes d.sub.B+d.sub.J treating ?.sub.E0p.sub.Ld.sub.E0 as noise, then, d.sub.B+d.sub.J is cancelled from equation (Eq. 3) in order to decode d.sub.E0 without interference. Alternatively, when the difference between ?.sub.E0p.sub.H and ?.sub.E0p.sub.L is small, a joint ML decoding of d.sub.E0 and d.sub.B+d.sub.J can be performed by (3). Note that for implementing the SIC processing described above it is required to use discrete constellations for signaling information, therefore, in the proposed scheme, pre-defined discrete constellations are considered for d.sub.E0, d.sub.B and d.sub.J. In terms of secrecy, the alignment of d.sub.B with d.sub.J in the same subspace forces (3) to tap the information related to d.sub.B only from the observation of d.sub.B+d.sub.J. Moreover, the approximation ?.sub.Bp.sub.L?0 is valid in the high signal-to-noise ratio (SNR) regime when p.sub.L<<p.sub.H. Therefore, the interference generated by d.sub.E0 at node B (4) can be neglected under such circumstance. Under this consideration, the signal observed by (4) can be reformulated as
?.sub.B(k)=?.sub.Bp.sub.H[h.sub.BA(k)h.sub.E0A.sup.?1(k)d.sub.B(k)+h.sub.BJ(k)h.sub.E0J.sup.?1(k)d.sub.J(k)]+n.sub.B(Eq. 6)
which means that by applying a maximum likelihood (ML) decoder, d.sub.B can be fully recovered in the high SNR regime when the equivalent channels h.sub.BAh.sub.E0A.sup.?1, and h.sub.BJh.sub.E0J.sup.?1 are known at node (4). When the difference between p.sub.H and p.sub.L is small, d.sub.B can be recovered directly from (Eq. 4) performing a joint decoding of d.sub.E0, d.sub.B and d.sub.J using an ML equalizer.

(26) In an embodiment, in the training phase, the design of the channel training method (FIG. 2) associated to the present disclosure is done taking into account three main goals that should be mutually achieved, which are: to provide information regarding h.sub.E0A (6) and h.sub.E0J (9) to nodes (1) and (2), respectively; to ensure that node (4) has access to the equivalent channels h.sub.BAh.sub.E0A.sup.?1, h.sub.BAh.sub.E0J.sup.?1; and to limit the amount of channel knowledge that the outside attacker (5) can obtain from this training process. While in the case of (3) the secrecy regarding d.sub.B is entirely ensured by the jamming component, in the case of (5) this protection will depend also on the design of the training phase. The secure training solution developed in the present disclosure is divided into four resource elements (RE) located within a coherence block of the channel, being the first two resource elements RE1,RE2 applied for channel estimation (14),(17), and the last two RE3,RE4 for channel state information feedback (21),(23). The purpose of the channel estimation process implemented in the first two resource elements RE1,RE2 of the training method is to provide the equivalent channels h.sub.BAh.sub.E0A.sup.?1 and h.sub.BAh.sub.E0J.sup.?1 to node (1) and (2), respectively. To achieve that, in resource element 1 (RE1) terminal (3) sends a reference signal (12) that provides h.sub.E0A (6) to (1) and h.sub.E0J (9) to node (2), while in resource element 2 (RE2) the RS (15) is sent by (4) giving h.sub.BA (7) to (1) and h.sub.BJ (10) to (2). In order to allow node (4) to extract d.sub.B from equation (Eq. 4) or (Eq. 6) using a joint ML decoding, the equivalent channels h.sub.BAh.sub.E0A.sup.?1 and h.sub.BAh.sub.E0J.sup.?1 acquired in the channel estimation phase (RE1,RE2) are fed back (21),(23) in secrecy to node (4) in the last two resource elements RE3,RE4 of the training method. In resource element 3 (RE3) node (1) feedbacks to (4) information regarding h.sub.BAh.sub.E0A.sup.?1 transmitting signal x.sub.F(A) in equation (Eq. 7), while in resource element 4 (RE4) node (2) exchanges h.sub.BAh.sub.E0J.sup.?1 with (4) sending signal x.sub.F(J) in equation (Eq. 8),

(27) 0$\begin{array}{cc}\begin{array}{c}{x}_{F\left(A\right)}\left(k\right)=F_A\left[p_H{h}_{\mathrm{BA}}\left(k\right),h_{E0A}\left(k\right),r\left(k\right)=0\right]\\ =p_H.Math.h_{\mathrm{BA}}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-2}{e}^{j\left[{?}_{E0A}\left(k\right)-2{?}_{\mathrm{BA}}\left(k\right)-r\left(k\right)\frac{2?}{M}\right]}\\ =p_H.Math.h_{\mathrm{BA}}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-2}{e}^{j\left[{?}_{E0A}\left(k\right)-2{?}_{\mathrm{BA}}\left(k\right)\right]}\end{array}& \left(\mathrm{Eq}.7\right)\\ \begin{array}{c}{x}_{F\left(J\right)}\left(k\right)=F_J\left[p_H{h}_{\mathrm{BJ}}\left(k\right),h_{E0J}\left(k\right),r\left(k\right)=0\right]\\ =p_H.Math.h_{\mathrm{BJ}}\left(k\right).Math.{.Math.h_{E0J}\left(k\right).Math.}^{-2}{e}^{j\left[{?}_{E0J}\left(k\right)-2{?}_{\mathrm{BJ}}\left(k\right)-r\left(k\right)\frac{2?}{M}\right]}\\ =p_H.Math.h_{\mathrm{BJ}}\left(k\right).Math.{.Math.h_{E0J}\left(k\right).Math.}^{-2}{e}^{j\left[{?}_{E0J}\left(k\right)-2{?}_{\mathrm{BJ}}\left(k\right)\right]}\end{array}& \left(\mathrm{Eq}.8\right)\end{array}$
where |h.sub.RT| and ?.sub.RT defines the magnitude and phase of the channel h.sub.RT for R?{E0, B, E1} and T?{A, J}. As can be seen from equations (Eq. 7) and (Eq. 8), in order to configure the proposed disclosure for this application scenario, the parameter r(k) is set to zero. Taking into account equations (Eq. 7) and (Eq. 8), the signal received by (4) in RE3 is formulated in equation (Eq. 9), being the one received in RE4 defined by equation (Eq. 10).

(28) $\begin{array}{cc}\begin{array}{c}{y}_{B,F\left(A\right)}\left(k\right)={?}_B{x}_{F\left(A\right)}\left(k\right)h_{\mathrm{BA}}\left(k\right)\\ ={?}_B{p_H\left[.Math.h_{\mathrm{BA}}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-1}\right]}^{-2}{e}^{j\left[{?}_{E0A}\left(k\right)-{?}_{\mathrm{BA}}\left(k\right)\right]}\end{array}& \left(\mathrm{Eq}.9\right)\\ \begin{array}{c}{y}_{B,F\left(J\right)}\left(k\right)={?}_B{x}_{F\left(J\right)}\left(k\right)h_{\mathrm{BJ}}\left(k\right)\\ ={?}_B{p_H\left[.Math.h_{\mathrm{BJ}}\left(k\right).Math.{.Math.h_{E0J}\left(k\right).Math.}^{-1}\right]}^2{e}^{j\left[{?}_{E0J}\left(k\right)-{?}_{\mathrm{BJ}}\left(k\right)\right]}\end{array}& \left(\mathrm{Eq}.10\right)\end{array}$

(29) In an embodiment, considering that ?.sub.Bp.sub.H is a large scale coefficient known at (4), the extraction of h.sub.BAh.sub.E0A.sup.?1 and h.sub.BAh.sub.E0J.sup.?1 at node (4) is done through the implementation of the processing operations defined in equations (Eq. 11) and (Eq. 12), being ?.sub.B,F(A)(k) and ?.sub.B,F(J)(k) defined as the phases of equations (Eq. 9) and (Eq. 10), respectively.

(30) $\begin{array}{cc}\begin{array}{c}{R}_A\left[y_{B,F\left(A\right)}\left(k\right),{?}_B{p}_H\right]={\left(\frac{.Math.y_{B,F\left(A\right)}\left(k\right).Math.}{{?}_B{p}_H}\right)}^{1/2}{e}^{-j{?}_{B,F\left(A\right)}\left(k\right)}\\ =h_{\mathrm{BA}}\left(k\right){h_{E0A}\left(k\right)}^{-1}\end{array}& \left(\mathrm{Eq}.11\right)\\ \begin{array}{c}{R}_J\left[y_{B,F\left(J\right)}\left(k\right),{?}_B{p}_H\right]={\left(\frac{.Math.y_{B,F\left(J\right)}\left(k\right).Math.}{{?}_B{p}_H}\right)}^{1/2}{e}^{j{?}_{B,F\left(J\right)}\left(k\right)}\\ =h_{\mathrm{BJ}}\left(k\right){h_{E0J}\left(k\right)}^{-1}\end{array}& \left(\mathrm{Eq}.12\right)\end{array}$

(31) In an embodiment, although the overhead of four resource elements, the training solution proposed in the present disclosure allows to protect the channels that (5) needs to know for extracting the information d.sub.B through the observation of equation (Eq. 5). In the mathematical formulations presented above no channel estimation errors are considered and a noiseless channel is assumed for the feedback phase, i.e. the channel training process is ideal. Furthermore, it is also considered that the length of the channel coherent interval is wide enough to perform training and transmission in the considered resource elements RE1-RE4.

(32) In an embodiment, to quantify the secrecy level provided by the present disclosure according to physical layer security principles, the mutual information will be used as the assessment metric. Although the present embodiment is focused in securing the system against the outside attacker (5), the developed training method was designed to be integrated in a jamming scenario, therefore, the secrecy level provided by the jammer (2) against the inside attacker (3) is also evaluated in the secrecy analysis presented next.

(33) In an embodiment, in the case of the inside attacker (3), the alignment of d.sub.B with d.sub.J in (Eq. 3) ensures that (3) is not capable to obtain more information regarding d.sub.B than the one he can extract through the observation of d.sub.B+d.sub.J. In order to protect d.sub.B, the integer addition d.sub.B+d.sub.J has to generate equivocation in (3), being this equivocation dependent on the structure of the discrete constellations selected for d.sub.B and d.sub.J. It is known that the use of complex square QAM signals is efficient in generating this equivocation, therefore, this type of constellation is used for d.sub.B, d.sub.E0 and d.sub.J. Considering independent and uniformly distributed square QAM signals of order M=L.sup.2 for d.sub.B and d.sub.J, an upper bound on the normalized mutual information I(d.sub.B; d.sub.B+d.sub.J) is defined in equation (Eq. 13).

(34) $\begin{array}{cc}\frac{I\left(d_B;d_B+d_J\right)}{{\log }_2\left(M\right)}?\frac{\stackrel{\_}{I\left(d_B;d_B+d_J\right)}}{{\log }_2\left(M\right)}={\log }_2\left(2-\frac{1}{L}\right)?\frac{1}{{\log }_2\left(L\right)}& \left(\mathrm{Eq}.13\right)\\ \underset{L.fwdarw.?}{\lim }\left[{\log }_2\left(2-\frac{1}{L}\right)?\frac{1}{{\log }_2\left(L\right)}\right]=0& \left(\mathrm{Eq}.14\right)\end{array}$

(35) In an embodiment, the result in equation (Eq. 14) shows that asymptotically when L.fwdarw.? the amount of information related to d.sub.B obtained by the inside attacker (3) is zero, which means that full secrecy is reached when the order M of d.sub.B and d.sub.J grows to infinity.

(36) In an embodiment, in the case of the outside attacker (5), the secrecy level provided by the proposed invention depends not only on the signal y.sub.E1 defined in equation (Eq. 5) observed during the main communication phase, but also depends on the signals received during the training phase. Although d.sub.B is not present in any of the signals defined by equations (Eq. 7) and (Eq. 8), some information about the communication channels, which are required at (5) to extract d.sub.B from equation (Eq. 5) can be acquired during this phase. Therefore, in the following secrecy analysis both phases are considered to evaluate the level of protection against node (5). Assuming a noiseless scenario, the signals observed during the training phase at node (5) are formulated in equations (Eq. 15) and (Eq. 16).

(37) $\begin{array}{cc}\begin{array}{c}{y}_{E1,F\left(A\right)}\left(k\right)={?}_{E1}{x}_{F\left(A\right)}\left(k\right)h_{E1A}\left(k\right)\\ ={?}_{E1}{p}_H.Math.h_{\mathrm{BA}}\left(k\right).Math..Math.h_{E1A}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-2}\\ e^{j\left[{?}_{E1A}\left(k\right)+{?}_{E0A}\left(k\right)-2{?}_{\mathrm{BA}}\left(k\right)\right]}\end{array}& \left(\mathrm{Eq}.15\right)\\ \begin{array}{c}{y}_{E1,F\left(J\right)}\left(k\right)={?}_{E1}{x}_{F\left(J\right)}\left(k\right)h_{E1J}\left(k\right)\\ ={?}_{E1}{p}_H.Math.h_{\mathrm{BJ}}\left(k\right).Math..Math.h_{E1J}\left(k\right).Math.{.Math.h_{E0J}\left(k\right).Math.}^{-2}\\ e^{j\left[{?}_{E1J}\left(k\right)+{?}_{E0J}\left(k\right)-2{?}_{\mathrm{BJ}}\left(k\right)\right]}\end{array}& \left(\mathrm{Eq}.16\right)\end{array}$

(38) In an embodiment, in equation (Eq. 15) y.sub.E1,F(A) defines the signal observed at (5) during the feedback of equation (Eq. 7) from (1) to (4) in RE3, being equation (Eq. 16) the signal acquired during the transmission of equation (Eq. 8) from (2) to (4) in RE4. In the case of the main communication phase, the signal obtained by (5) was formulated in equation (Eq. 5). Since the target of (5) is to tap information regarding d.sub.B, the interference generated by d.sub.E0 and d.sub.J in equation (Eq. 5) reduces always the capacity of (5) to get the intended information, which means that an upper bound on the mutual information I(d.sub.B; y.sub.E1) is obtained when these two interference components are eliminated. In order to simplify the secrecy evaluation related to the outside attacker (5), in the following analysis d.sub.E0 and d.sub.J will be defined as zero. Therefore, under the non-interference scenario, the signal observed at (5) can be reformulated as defined in equation (Eq. 17).

(39) $\begin{array}{cc}\begin{array}{c}{\hat{y}}_{E1}\left(k\right)={?}_{E1}{p}_H{h}_{E1A}\left(k\right){h_{E0A}\left(k\right)}^{-1}{d}_B\left(k\right)\\ ={?}_{E1}{p}_H.Math.h_{E1A}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-1}.Math.d_B\left(k\right).Math.\\ e^{j\left[{?}_{E1A}\left(k\right)-{?}_{E0A}\left(k\right)+{?}_B\left(k\right)\right]}\end{array}& \left(\mathrm{Eq}.17\right)\end{array}$

(40) In an embodiment, because the interference caused by the jamming signal d.sub.J was removed in equation (Eq. 17), the observation of the signal in equation (Eq. 16) at (5) does not add any information useful to extract d.sub.B, which means that an upper bound on the mutual information at (5) can be calculated using only equations (Eq. 17) and (Eq. 15). Considering |h.sub.E0A| known at (5), and for a constant ?=?.sub.E1p.sub.H, a theoretical upper bound on the amount of information obtained by the outside attacker (5) is formulated in equation (Eq. 18).

(41) $\begin{array}{cc}I\left(d_B;{\hat{y}}_{E1},y_{E1,F\left(A\right)}\right)?I\left(.Math.d_B.Math.;?.Math.d_B.Math..Math.h_{E1A}.Math.,?.Math.h_{E1A}.Math..Math.h_{\mathrm{BA}}.Math.\right)?\stackrel{\_}{I\left(.Math.d_B.Math.;?.Math.d_B.Math..Math.h_{E1A}.Math.,?.Math.h_{E1A}.Math..Math.h_{\mathrm{BA}}.Math.\right)}=\frac{1}{2}{\log }_2\left(2?e{?}_A^2\right)+\frac{1}{2}{\log }_2\left(2?e{?}_B^2\right)-\left[E\left({\log }_2.Math.d_B.Math.\right)+h\left(.Math.h_{E1A}.Math.\right)+E\left({\log }_2.Math.h_{E1A}.Math.\right)+h\left(.Math.h_{\mathrm{BA}}.Math.\right)\right]& \left(\mathrm{Eq}.18\right)\end{array}$
In equation (Eq. 18) ?.sub.A.sup.2 defines the variance of the product |d.sub.B?h.sub.E1A|, ?.sub.B.sup.2 the variance of |h.sub.E1A?h.sub.BA|, being h(x) and E(x) the differential entropy and the expected value of the random variable x, respectively.

(42) $\begin{array}{cc}\underset{M.fwdarw.?}{\lim }\frac{\stackrel{\_}{I\left(.Math.d_B.Math.;?.Math.d_B.Math..Math.h_{E1A}.Math.,?.Math.h_{E1A}.Math..Math.h_{\mathrm{BA}}.Math.\right)}}{{\log }_2\left(M\right)}=0.& \left(\mathrm{Eq}.19\right)\end{array}$

(43) In an embodiment, as defined in (Eq. 19), the asymptotic behaviour of the theoretical upper bound formulated in equation (Eq. 18) makes the percentage of information acquired by (5) goes to zero when the cardinality M of d.sub.B increases. In an embodiment, although the application example associated to the embodiment presented in FIG. 1 only considers two single antenna transmitters (1-2) and two single antenna receivers (3-4), the proposed disclosure can be integrated in an up-scaled version of the system illustrated in FIG. 1, up-scaling not only the number of transmitters (1-2) and receivers (3-4), but also the number of antennas applied by these nodes (1-4).

(44) Example 2 represents 1 single antenna legitimate transmitter (44); 2 single antenna legitimate receivers (45), (46); and 1 single antenna passive eavesdropper receiver (47).

(45) In an embodiment, FIG. 7 illustrates an example of an embodiment of the proposed disclosure in a wireless communication system for application Scenario 2. In the considered communication system, node (44) pretends to transmit in secrecy d.sub.B to node (46) and d.sub.E0 to (45), using for that a power domain non-orthogonal multiple access system in which (45) is the strongest user and node (46) represents the weakest user. The embodiment illustrated in FIG. 7 considers that the distance between node (44) and (45) is lower than the distance between (44) and (46), therefore, as it is usually defined in the state of art techniques, (45) is the strongest user with node (46) being the weakest user. In power domain non-orthogonal multiple access systems, the strongest user (45) can only reach their intended information after decoding the interference generated by the weakest user (46), being this interference posteriorly cancelled from the overall received signal using a successive interference cancellation (SIC) equalizer. Because (45) will decode first the interference created by dg before accessing the information d.sub.E0 sent to him, this node (45) will be recognized by the network as a legitimate user that also has the capacity to eavesdrop d.sub.B attacking the network from inside. Additionally, in order to model the occurrence of an outside attack, the embodiment in FIG. 7 also assumes that a passive eavesdropper (47) not registered in the network, and not co-located with (46), tries to tap in an undetectable way the information d.sub.B sent to node (46).

(46) In an embodiment, in the following, the random variables defined by h.sub.RT with R?{E0, B, E1} and T?{A} represent small scale channel fading effects which are modelled by zero mean independent complex Gaussian distributions with variance ?.sub.h.sup.2. The path-loss attenuation between the transmitting nodes T and each one of the receiving nodes (45-47) in R is represented by the coefficients ?.sub.E0, ?.sub.B and ?.sub.E1, where it is assumed that ?.sub.B<?.sub.E0. Ideal RF up- and down-conversion is assumed, being the described baseband processing applied to an independent realization of the system denoted by k.

(47) In an embodiment, the non-orthogonal multiple access communication process associated to the embodiment illustrated in FIG. 7 comprises two phases; a preliminary channel training phase (Phase I), and the main communication phase (Phase II). Although the main communication phase happens posteriorly to the training phase, the design of the secure training method associated to the present disclosure depends on the signal structure of the communication phase, hence, we will start by describing the communication phase.

(48) In an embodiment, in the communication phase, the information signals d.sub.B and d.sub.E0 are sent to the intended receivers, (46) and (45) respectively, using for both streams a discrete PSK constellation of order M featuring a structure with a constant phase shift ??=2?/M between any consecutive pair of points in the constellation set. Assuming that p.sub.L and p.sub.H are power allocation coefficients computed according to the geographic distribution of (45) and (46), the signal transmitted by node (44) is formulated in equation (Eq. 20).

(49) $\begin{array}{cc}\begin{array}{c}{x}_A\left(k\right)=h_{E0A}^{-1}\left(k\right)\left[p_L{d}_{E0}\left(k\right)+p_H{d}_B\left(k\right)e^{j{?}_K\left(k\right)}\right]\\ =h_{E0A}^{-1}\left(k\right)\left[p_L{d}_{E0}\left(k\right)+p_H{d}_B\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}\right]\\ =h_{E0A}^{-1}\left(k\right)\left[p_L{e}^{j{?}_{d_{E0}}\left(k\right)}+p_H{e}^{j\left[{?}_{d_B}\left(k\right)+r\left(k\right)\frac{2?}{M}\right]}\right]\\ =h_{E0A}^{-1}\left(k\right)\left[p_L{d}_{E0}\left(k\right)+p_H{d}_{B,K}\left(k\right)\right]\end{array}& \left(\mathrm{Eq}.20\right)\end{array}$

(50) In an embodiment, as defined in equation (Eq. 20), the information associated to d.sub.B(k) is coded in the phase ?.sub.d.sub.B(k), while the information of d.sub.E0(k) is carried in the phase ?.sub.d.sub.E0(k), being ?.sub.d.sub.B(k) and ?.sub.d.sub.E0(k) independent uniform random variables selected from the phases of a PSK constellation of order M. Additionally, a random variable ?.sub.K(k) is formulated as ?.sub.K(k)=r(k)[2?/M], being r defined as an independent random variable distributed uniformly in the integer set within the interval r?[1, M]. Then, as shown in equation (Eq. 20), d.sub.B(k) is processed with ?.sub.K(k) resulting in the signal

(51) $d_{B,K}\left(k\right)=e^{j\left[{?}_{d_B}\left(k\right)+r\left(k\right)\frac{2?}{M}\right]}.$

(52) In an embodiment, because the strongest user (45) is located closer to (44), the amount of power p.sub.L allocated to (45) should be lower than the power p.sub.H allocated to the weakest user (46). Taking into account equation (Eq. 20), the signal observed by node (45) is given by equation (Eq. 21), with equations (Eq. 22) and (Eq. 23) defining the signals observed by (46) and (47), respectively.

(53) 0$\begin{array}{cc}\begin{array}{c}{y}_{E0}\left(k\right)={?}_{E0}{h}_{E0A}\left(k\right)x_A\left(k\right)+n_{E0}\\ ={?}_{E0}\left[p_L{d}_{E0}\left(k\right)+p_H{d}_{B,K}\left(k\right)\right]+n_{E0}\\ ={?}_{E0}\left[p_L{e}^{j{?}_{d_{E0}}\left(k\right)}+p_H{e}^{j\left[{?}_{d_B}\left(k\right)+r\left(k\right)\frac{2?}{M}\right]}\right]+n_{E0}\end{array}& \left(\mathrm{Eq}.21\right)\\ \begin{array}{c}{y}_B\left(k\right)={?}_B{h}_{\mathrm{BA}}\left(k\right)x_A\left(k\right)+n_B\\ ={?}_B{h}_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)\left[p_L{d}_{E0}\left(k\right)+p_H{d}_{B,K}\left(k\right)\right]+n_B\\ ={?}_B{h}_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)\left[\begin{array}{c}{p}_L{e}^{j{?}_{d_{E0}}\left(k\right)}+\\ p_H{e}^{j\left[{?}_{d_B}\left(k\right)+r\left(k\right)\frac{2?}{M}\right]}\end{array}\right]+\\ n_B\end{array}& \left(\mathrm{Eq}.22\right)\\ \begin{array}{c}{y}_{E1}\left(k\right)={?}_{E1}{h}_{E1A}\left(k\right)x_A\left(k\right)+n_{E1}\\ ={?}_{E1}{h}_{E1A}\left(k\right)h_{E0A}^{-1}\left[p_L{d}_{E0}\left(k\right)+p_H{d}_{B,K}\left(k\right)\right]+n_{E1}\\ ={?}_{E1}{h}_{E1A}\left(k\right)h_{E0A}^{-1}\left(k\right)\left[\begin{array}{c}{p}_L{e}^{j{?}_{d_{E0}}\left(k\right)}+\\ p_H{e}^{j\left[{?}_{d_B}\left(k\right)+r\left(k\right)\frac{2?}{M}\right]}\end{array}\right]+\\ n_{E1}\end{array}& \left(\mathrm{Eq}.23\right)\end{array}$

(54) In an embodiment, in the case of (45), when ?.sub.E0p.sub.H is much larger than ?.sub.E0p.sub.L, a SIC equalizer can be applied to equation (Eq. 21) in order to get d.sub.E0. As it is typical of power-domain NOMA systems, first (45) decodes d.sub.B,K(k) treating ?.sub.E0p.sub.Ld.sub.E0(k) as noise, then, d.sub.B,K(k) is cancelled from equation (Eq. 21) in order to decode d.sub.E0(k) without interference. Alternatively, a joint ML decoding of d.sub.E0(k) and d.sub.B,K(k) can also be applied to (Eq. 21). In terms of secrecy, d.sub.B(k) is internally protected against (45) if the effect of ?.sub.K(k) can be entirely resolved in (46) ensuring at the same time that ?.sub.K(k) is not accessible at (45).

(55) In an embodiment, the approximation ?.sub.Bp.sub.L?0 is valid in the high SNR regime when p.sub.L<<p.sub.H, therefore, the interference generated by d.sub.E0(k) at node B can be neglected for simplification purposes. Under this consideration, the signal observed by (46) can be reformulated as

(56) $\begin{array}{cc}\begin{array}{c}{\hat{y}}_B\left(k\right)={?}_B{h}_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)p_H{d}_{B,K}\left(k\right)+n_B\\ ={?}_B{h}_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)p_H{e}^{j\left[{?}_{d_B}\left(k\right)+r\left(k\right)\frac{2?}{M}\right]}+n_B\\ ={?}_B{p}_H{h}_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}{d}_B\left(k\right)+n_B\end{array}& \left(\mathrm{Eq}.24\right)\end{array}$
which means that by applying a maximum likelihood (ML) decoder, d.sub.B(k) can be fully recovered in the high signal to noise ratio (SNR) regime when the equivalent channel

(57) $h_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}$
is known at node (46). Alternatively, using again

(58) $h_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]},$
a joint ML decoding of

(59) $d_{E0}\left(k\right)e^{-j\left[r\left(k\right)\frac{2?}{M}\right]}$
and d.sub.B(k) can be applied directly to expression (Eq. 22) in order to get d.sub.B(k).

(60) In an embodiment, the design of the channel training method (FIG. 7) associated to the present disclosure is done taking into account three main goals that should be mutually achieved, which are: to provide information regarding h.sub.E0A (48) to node (44); to ensure that node (46) has access to the equivalent channel

(61) $h_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}$
in a way that the random phase ?.sub.K(k)=r(k)[2?/M] cannot be resolved at (45); and to limit the amount of channel and encoding knowledge that the outside attacker (47) can obtain from this training process.

(62) In an embodiment, the secure training solution (FIG. 8) developed in the present disclosure applied to application scenario 2 is divided into three resource elements (RE) located within a coherence block of the channel, being the first two resource elements RE1,RE2 applied for channel estimation (52),(54), and the last one RE3 for channel and encoding information feedback (56). The main objective of the channel estimation process implemented in the first two resource elements RE1, RE2 of the training method is to provide the equivalent channels h.sub.BA(k) and h.sub.E0A(k) to node (44). To achieve that, in RE1 terminal (45) sends a reference signal (51) that provides h.sub.E0A(k) (48) to (44), while in RE2 the RS (53) is sent by (46) giving h.sub.BA(k) (49) to (44). In order to allow node (46) to decode d.sub.B(k) from equation (Eq. 22) or (Eq. 24) using a ML equalizer, the equivalent channel

(63) $h_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}$
is fed back (56) to node (46) in the last resource element RE3 of the training method. Therefore, in RE3 node (44) feedbacks to (46) information regarding

(64) $h_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}$
transmitting signal x.sub.F(A) in equation (Eq. 25).

(65) $\begin{array}{cc}\begin{array}{c}{x}_{F\left(A\right)}\left(k\right)=F_A\left[p_H{h}_{\mathrm{BA}}\left(k\right),h_{E0A}\left(k\right),r\left(k\right)\right]\\ =p_H.Math.h_{\mathrm{BA}}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-2}{e}^{j\left[{?}_{E0A}\left(k\right)-2{?}_{\mathrm{BA}}\left(k\right)-r\left(k\right)\frac{2?}{M}\right]}\end{array}& \left(\mathrm{Eq}.25\right)\end{array}$
where |h.sub.RT| and ?.sub.RT defines the magnitude and phase of the channel h.sub.RT for R?{E0, B, E1} and T?{A}. Taking into account equations (Eq. 25), the signal received by (46) in RE3 is formulated in equation (Eq. 26).

(66) $\begin{array}{cc}\begin{array}{c}{y}_{B,F\left(A\right)}\left(k\right)={?}_B{x}_{F\left(A\right)}\left(k\right)h_{\mathrm{BA}}\left(k\right)\\ ={?}_B{p_H\left[.Math.h_{\mathrm{BA}}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-1}\right]}^2{e}^{j\left[{?}_{E0A}\left(k\right)-{?}_{\mathrm{BA}}\left(k\right)-r\left(k\right)\frac{2?}{M}\right]}\end{array}& \left(\mathrm{Eq}.26\right)\end{array}$

(67) In an embodiment, considering that ?.sub.Bp.sub.H is a large scale coefficient known at (46), the extraction of

(68) 0$h_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}$
at node (46) is done through the implementation of the processing operations defined in equation (Eq. 27), being ?.sub.B,F(A)(k) the phase of y.sub.B,F(A)(k).

(69) $\begin{array}{cc}\begin{array}{c}{R}_A\left[y_{B,F\left(A\right)}\left(k\right),{?}_B{p}_H\right]={\left(\frac{.Math.y_{B,F\left(A\right)}\left(k\right).Math.}{{?}_B{p}_H}\right)}^{1/2}{e}^{-j{?}_{B,F\left(A\right)}\left(k\right)}\\ =h_{\mathrm{BA}}\left(k\right)h_{E0A}^{-1}\left(k\right)e^{j\left[r\left(k\right)\frac{2?}{M}\right]}\end{array}& \left(\mathrm{Eq}.27\right)\end{array}$

(70) In an embodiment, although the overhead of three resource elements, the training solution proposed in this disclosure allows to protect the channels and the encoding information that (47) needs to know for extracting the information d.sub.B(k) through the observation of equation (Eq. 23). In the mathematical formulations presented above no channel estimation errors are considered and a noiseless channel is assumed for the feedback phase, i.e. the channel training process is ideal. Furthermore, it is also considered that the length of the channel coherent interval is wide enough to perform training and transmission in the considered resource elements RE1-RE3.

(71) In an embodiment, in order to show that the secure training method proposed in the present disclosure ensures full secrecy of d.sub.B(k) against the inside (45) and outside (47) eavesdropping attack, the signals observed during the training and communication phase at nodes (45) and (47) should be jointly analysed using the mutual information I[.] as the assessment metric. Since the signals observed by (45) and (47) in the communication phase were already defined in equations (Eq. 21) and (Eq. 23), respectively, the next step required to quantify the achieved secrecy, is the formulation of the signals observed during the channel training phase at (45) and (47), which are defined in equations (Eq. 28) and (Eq. 30), respectively.

(72) $\begin{array}{cc}\begin{array}{c}{y}_{E0,F\left(A\right)}\left(k\right)={?}_{E0}{x}_{F\left(A\right)}\left(k\right)h_{E0A}\left(k\right)\\ ={?}_{E0}{p}_H.Math.h_{\mathrm{BA}}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-1}{e}^{j{?}_{E0,F\left(A\right)}\left(k\right)}\end{array}& \left(\mathrm{Eq}.28\right)\end{array}$$\begin{array}{cc}{?}_{E0,F\left(A\right)}\left(k\right)=2{?}_{E0A}\left(k\right)-2{?}_{\mathrm{BA}}\left(k\right)-r\left(k\right)\frac{2?}{M}& \left(\mathrm{Eq}.29\right)\end{array}$$\begin{array}{cc}\begin{array}{c}{y}_{E1,F\left(A\right)}\left(k\right)={?}_{E1}{x}_{F\left(A\right)}\left(k\right)h_{E1A}\left(k\right)\\ ={?}_{E1}{p}_H.Math.h_{E1A}\left(k\right).Math..Math.h_{\mathrm{BA}}\left(k\right).Math.{.Math.h_{E0A}\left(k\right).Math.}^{-2}{e}^{j{?}_{E1,F\left(A\right)}\left(k\right)}\end{array}& \left(\mathrm{Eq}.30\right)\end{array}$$\begin{array}{cc}{?}_{E1,F\left(A\right)}\left(k\right)={?}_{E1A}\left(k\right)+{?}_{E0A}\left(k\right)-2{?}_{\mathrm{BA}}\left(k\right)-r\left(k\right)\frac{2?}{M}& \left(\mathrm{Eq}.31\right)\end{array}$

(73) In an embodiment, taking into account the observed signals in equations (Eq. 21) and (Eq. 28), the amount of information that (45) can obtain about d.sub.B can be quantified as

(74) $\begin{array}{cc}\begin{array}{c}I[d_B;y_{E0}.Math.y_{E0,F\left(A\right)}]=H\left(d_B\right)-H(d_B.Math.y_{E0},y_{E0,F\left(A\right)})\\ ?H\left(d_B\right)-H(d_B.Math.d_{B,K},y_{E0,F\left(A\right)})\\ =H\left({?}_{d_B}\right)-H({?}_{d_B}.Math.{?}_{d_B}+{?}_K,{?}_{E0,F\left(A\right)})\\ =H\left({?}_{d_B}\right)-H({?}_{d_B}.Math.{?}_{d_B}+r\frac{2?}{M},{?}_{E0,F\left(A\right)})\\ =0\end{array}& \left(\mathrm{Eq}.32\right)\end{array}$