In this proposed method, two vehicles are allotted with an equal number of reflecting elements. Single antenna is used at RSU, near and far vehicles. More power is allocated to the far vehicle than the near vehicle to meet QoS criteria. The order in which the vehicles’ signals are decoded is fixed in FNOMA. The signal from the far vehicle is decoded first, with the signal from the near vehicle treated as interference. The far vehicle’s signal is decoded at the near vehicle, and its impact is removed from the received signal via SIC. The near vehicle’s signal is decoded from the residual signal.
A superposition-coded signal is generated at BS/RSU. It is given by
$$\begin{aligned} x_{s}=\sqrt{\beta _{\mathrm{nv}}P_{s}}\ x_{\mathrm{nv}}+\sqrt{\beta _{\mathrm{fv}}P_{s}\ }{\ x}_{\mathrm{fv}} \end{aligned}$$
(1)
where
\(\beta _{\mathrm{nv}}\) and
\(\beta _{\mathrm{fv}}\) are the fraction of the total transmit power
\(P_{s}\) allocated to near and far vehicles, respectively.
\(x_{\mathrm{nv}}\) and
\(x_{\mathrm{fv}}\) are the transmitted unit energy symbols corresponding to near and far vehicles, respectively. The power is allocated by BS/RSU to near and far vehicles such that the following constraint is met.
$$\begin{aligned} \beta _{\mathrm{nv}}+\beta _{\mathrm{fv}}=1, \beta _{\mathrm{fv}}>\beta _{\mathrm{nv}}>0 \end{aligned}$$
(2)
The channel between the BS/RSU and the
jth RIS element is
\(h_{j}=\alpha _{j}e^{-i\phi _{j}},\ h_{j}\sim C{\mathcal{N}}\left( 0,1\right)\), where
\(\alpha _{j}\) and
\({{\phi }}_{j}\) are the magnitude and phase of
\(h_{j}\). The channel between
jth RIS element and the vehicle is
\(g_{j}={\gamma }_{j}e^{-i\psi _{j}},g_{j}\sim C{\mathcal{N}} \left( 0,1\right)\), where
\({\gamma }_{j}\) and
\(\psi _{j}\) are the magnitude and phase of
\(g_{j}\). The number of reflecting elements assigned to the near and far vehicles are denoted by
\(\left( N_{\mathrm{nv}}\right)\) and
\(\left( N_{\mathrm{fv}}\right)\) , respectively. Assuming that they are equal, the received signal at the far vehicle is given by
$$\begin{aligned} y_{\mathrm{fv}}=\underbrace{\left( \sum ^{N}_{j=\frac{N}{2}+1}{h_{j}g_{j}}\right) }_{L}x_{s}+w_{\mathrm{fv}} \end{aligned}$$
(3)
where
\(w_{\mathrm{fv}}\) is the additive white Gaussian noise at the far vehicle, which follows
\(C{\mathcal{N}}\left( 0,N_o\right)\),
\(N_o\) is the noise variance. In Eq. (
3),
L is the dual-hop channel effect corresponding to a far vehicle. According to the CLT,
L follows a complex Gaussian distribution when the number of reflecting elements assigned to the far vehicle is greater than 16, i.e.,
\(L\sim C{\mathcal{N}}\left( 0,N_{\mathrm{fv}}\right)\) [
40]. By substituting (
1) in (
3), the received signal at far vehicle is rewritten as
$$\begin{aligned} y_{\mathrm{fv}}=L\ \sqrt{{\beta }_{\mathrm{nv}}P_{s}}\ x_{\mathrm{nv}}+L\ \sqrt{{{\beta }}_{\mathrm{fv}}P_{s}\ }{\ x}_{\mathrm{fv}}+w_{\mathrm{fv}} \end{aligned}$$
(4)
In FNOMA, the near vehicle receives less power, while the far vehicle receives more power. The first term in (
4) is not dominant over the second term because the cumulative dual-hop channel for the far vehicle is weaker and
\({\beta }_{\mathrm{nv}}\) is smaller. As a result, the far vehicle’s signal is detected by considering the near vehicle’s signal as interference. The signal-to-interference plus noise ratio (SINR) for decoding a signal from a far vehicle while treating a signal from a near vehicle as interference is given by
$$\begin{aligned} {{\vartheta }}^{x_{\mathrm{fv}}}_{\mathrm{fv}}=\frac{{\left| L\right| }^2{{\beta }}_{\mathrm{fv}}{{\rho }}_{s}}{{\left| L\right| }^{2}{{\beta }}_{\mathrm{nv}}{{\rho }}_{s}+1} \end{aligned}$$
(5)
where
\(\rho _{s} =\frac{P_{s}}{N_0}\) is the transmit SNR. The outage occurs at the far vehicle when [
18],
$$\begin{aligned} {{{\log }}_2 \left( 1+{{\vartheta }}^{x_{\mathrm{fv}}}_{\mathrm{fv}}\right) \ }\le {\tilde{R}}_{\mathrm{fv}} \end{aligned}$$
(6)
where
\({\tilde{R}}_{\mathrm{fv}}\) is the desired rate demand of the far vehicle. Substituting (
5) in (
6), it is rewritten as
$$\begin{aligned} {{{\log }}_{2} \left( 1+\frac{{\left| L\right| }^{2}{{\beta }}_{\mathrm{fv}}{\rho }_{s}}{{\left| L\right| }^{2}{{\beta }}_{\mathrm{nv}}{\rho }_{s}+1}\right) \ }\le {\tilde{R}}_{\mathrm{fv}} \end{aligned}$$
(7)
Let
\(R_{\mathrm{fv}}=2^{{\tilde{R}}_{\mathrm{fv}}}-1\) and
\({\eta }_{\mathrm{fv}}={\left| L\right| }^{2}\), (
7) is simplified as
$$\begin{aligned} {\eta }_{\mathrm{fv}} \le \underbrace{\frac{R_{\mathrm{fv}}}{\left( {{\beta }}_{\mathrm{fv}}-{{\beta }}_{\mathrm{nv}}R_{\mathrm{fv}}\right) {{\rho }}_{s}}}_{r_{\mathrm{fv}}} \end{aligned}$$
(8)
The comprehensive derivation of (
8) is given in “Appendix 1.”
\({\eta }_{\mathrm{fv}}\) follows Chi-square distribution with two degrees of freedom. The corresponding mean of
\({\eta }_{\mathrm{fv}}\) is
\(E\left\{ {\eta }_{\mathrm{fv}}\right\} ={\delta }^{2}_{\mathrm{fv}}N_{\mathrm{fv}}\), where
\({\delta }^{2}_{\mathrm{fv}}\) is the average channel gain of far vehicle. The probability density function (pdf) of
\({\eta }_{\mathrm{fv}}\) is given by
$$\begin{aligned} f^{\left( {\eta }_{\mathrm{fv}}\right) }_{{\eta }_{\mathrm{fv}}}=\frac{1}{{\delta }^{2}_{\mathrm{fv}}N_{\mathrm{fv}}}e^{-\ \left( \frac{{\eta }_{\mathrm{fv}}}{{\delta }^{2}_{\mathrm{fv}}N_{\mathrm{fv}}}\right) },\; {\eta }_{\mathrm{fv}}\ge 0 \end{aligned}$$
(9)
The outage probability for the far vehicle is calculated using
$$\begin{aligned} P_{\mathrm{fv}}^o=\int _{0}^{r_{\mathrm{fv}}}{f_{\eta _{\mathrm{fv}}}^{\left( \eta _{\mathrm{fv}}\right) }\mathrm{d}\eta _{\mathrm{fv}}} \end{aligned}$$
(10)
Substituting (
9), the outage probability for the far vehilce is expressed as
$$\begin{aligned} P^{o}_{\mathrm{fv}}=1-\exp \left\{ -\left( \frac{R_{\mathrm{fv}}}{\left( {\beta }_{\mathrm{fv}}-{\beta }_{\mathrm{nv}}R_{\mathrm{fv}}\right) {{\delta }^{2}_{\mathrm{fv}}N_{\mathrm{fv}}\rho }_{s}}\right) \right\} \end{aligned}$$
(11)
The comprehensive derivation of (
11) is given in “Appendix 2.” The received signal at the near vehicle is given by
$$\begin{aligned} y_{\mathrm{nv}}=\underbrace{\left( \sum ^{\frac{N}{2}}_{j=1}{h_{j}g_{j}}\right) }_{M}x_{s}+w_{\mathrm{nv}} \end{aligned}$$
(12)
where
\(w_{\mathrm{nv}}\) is the additive white Gaussian noise at the near vehicle, which follows
\(C{\mathcal{N}} \left( 0,N_o\right)\).
M is the dual-hop channel effect corresponding to the near vehicle. According to the CLT,
M follows a complex Gaussian distribution when the number of reflecting elements assigned to the near vehicle is greater than 16, i.e.,
\(M\sim C{\mathcal{N}}\left( 0,N_{\mathrm{nv}}\right)\) [
40]. By substituting (
1) in (
12), the received signal at the near vehicle is rewritten as
$$\begin{aligned} y_{\mathrm{nv}}=M\ \sqrt{{\beta }_{\mathrm{nv}}{{P}}_{s}}\ x_{\mathrm{nv}}+M\ \sqrt{{{\beta }}_{\mathrm{fv}}{{P}}_{s}\ }{\ x}_{\mathrm{fv}}+w_{\mathrm{nv}} \end{aligned}$$
(13)
The second term in (
13) is dominant over the first term, because both the value of
M and
\({{\beta }}_{\mathrm{fv}}\) are higher. As a result, the signal from the far vehicle is detected first by the near vehicle. The SINR of decoding a signal from a far vehicle at a near vehicle is given by
$$\begin{aligned} {{\vartheta }}^{x_{\mathrm{fv}}}_{\mathrm{nv}}=\frac{{\left| M\right| }^{2}{{\beta }}_{\mathrm{fv}}{\rho }_{s}}{{\left| M\right| }^{2}{{\beta }}_{\mathrm{nv}}{\rho }_{s}+1} \end{aligned}$$
(14)
After eliminating the influence of
\({\ x}_{\mathrm{fv}}\) from
\(y_{\mathrm{nv}}\) using SIC, the received signal at the near vehicle is given by,
$$\begin{aligned} {\tilde{y}}_{\mathrm{nv}}\approx M\sqrt{{\beta }_{\mathrm{nv}}{{\rho }}_{s}}\ x_{\mathrm{nv}}+w_{\mathrm{nv}} \end{aligned}$$
(15)
The signal of a near vehicle is detected using (
15). The SNR for decoding
\(x_{\mathrm{nv}}\) at near vehicle is given by
$$\begin{aligned} {{\vartheta }}^{x_{\mathrm{nv}}}_{\mathrm{nv}}={\left| M\right| }^{2}{{\beta }}_{\mathrm{nv}}{\rho }_{s} \end{aligned}$$
(16)
The decoding of the far vehicle’s signal fails at the near vehicle when [
18]
$$\begin{aligned} {{{\log }}_{2} \left( 1+{{\vartheta }}^{x_{\mathrm{fv}}}_{\mathrm{nv}}\right) \ }\le {\tilde{R}}_{\mathrm{fv}} \end{aligned}$$
(17)
By substituting (
14) in (
17),
$$\begin{aligned} {{{\log }}_{2} \left( 1+\frac{{\left| M\right| }^{2}{{\beta }}_{\mathrm{fv}}{\rho }_{s}}{{\left| M\right| }^{2}{{\beta }}_{\mathrm{nv}}{\rho }_{s}+1}\right) \ }\le {\tilde{R}}_{\mathrm{fv}} \end{aligned}$$
(18)
After simplification, (
18) is written as,
$$\begin{aligned} {\left| M\right| }^{2}\le \frac{R_{\mathrm{fv}}}{\left( {\beta }_{\mathrm{fv}}-{\beta }_{\mathrm{nv}}R_{\mathrm{fv}}\right) {\rho }_{s}} \end{aligned}$$
(19)
The comprehensive derivation of (
19) is given in “Appendix 3.” The decoding of the near vehicle’s signal fails at the near vehicle when [
18],
$$\begin{aligned} {{{\log }}_{2} \left( 1+{{\vartheta }}^{x_{\mathrm{nv}}}_{\mathrm{nv}}\right) \ }\le {\tilde{R}}_{\mathrm{nv}} \end{aligned}$$
(20)
where
\({\tilde{R}}_{\mathrm{nv}}\) is the desired rate demand of the near vehicle. Substituting (
16) in (
20) gives
$$\begin{aligned} {{{\log }}_{2} \left( 1+{\left| M\right| }^{2}{{\beta }}_{\mathrm{nv}}{\rho }_{s}\right) \ }\le {\tilde{R}}_{\mathrm{nv}} \end{aligned}$$
(21)
After simplification, (
21) is written as,
$$\begin{aligned} {\left| M\right| }^{2}\le \frac{R_{\mathrm{nv}}}{{{\beta }}_{\mathrm{nv}}{\rho }_{s}} \end{aligned}$$
(22)
where
\(R_{\mathrm{nv}}=2^{{\tilde{R}}_{\mathrm{nv}}}-1\). “Appendix 4” illustrates the comprehensive derivation of (
22). As a result, the condition for which decoding of a near vehicle’s signal fails at a near vehicle is provided by
$$\begin{aligned} {{\eta }_{\mathrm{nv}}=\left| M\right| }^{2}\le \underbrace{\max \left\{ \frac{R_{\mathrm{fv}}}{\left( {{\beta }}_{\mathrm{fv}}-{{\beta }}_{\mathrm{nv}}R_{\mathrm{fv}}\right) {{\rho }}_{s}},\ \frac{R_{\mathrm{nv}}}{{{\beta }}_{\mathrm{nv}}{\rho }_{s}}\right\} }_{r_{\mathrm{nv}}} \end{aligned}$$
(23)
\({\eta }_{\mathrm{nv}}\) follows Chi-square distribution with two degrees of freedom. The corresponding mean is
\(E\left\{ {\eta }_{\mathrm{nv}}\right\} ={\delta }^{2}_{\mathrm{nv}}N_{\mathrm{nv}}\), where
\({\delta }^{2}_{\mathrm{nv}}\) is the average channel gain of near vehicle. The pdf of
\({\eta }_{\mathrm{nv}}\) is given by
$$\begin{aligned} f^{\left( {\eta }_{\mathrm{nv}}\right) }_{{\eta }_{\mathrm{nv}}}=\frac{1}{{\delta }^{2}_{\mathrm{nv}}N_{\mathrm{nv}}}e^{-\ \left( \frac{{\eta }_{\mathrm{nv}}}{{\delta }^{2}_{\mathrm{nv}}N_{\mathrm{nv}}}\right) },\;\; {\eta }_{\mathrm{nv}}\ge 0 \end{aligned}$$
(24)
The outage probability of near vehicle is calculated using
$$\begin{aligned} P_{\mathrm{nv}}^{o}=\int _{0}^{r_{\mathrm{nv}}}{f_{\eta _{\mathrm{nv}}}^{\left( \eta _{\mathrm{nv}}\right) }{\mathrm{d}}\eta _{\mathrm{nv}}} \end{aligned}$$
(25)
By substituting (
24) in (
25) and simplifying,
$$\begin{aligned} P^{o}_{\mathrm{nv}}=1-\exp \left\{ -\frac{1}{{\delta }^{2}_{\mathrm{nv}}N_{\mathrm{nv}}}\left( \max \left\{ \frac{R_{\mathrm{fv}}}{\left( {\beta }_{\mathrm{fv}}-{\beta }_{\mathrm{nv}}R_{\mathrm{fv}}\right) {\rho }_{s}},\ \frac{R_{\mathrm{nv}}}{{{\beta }}_{\mathrm{nv}}{\rho }_{s}}\right\} \right) \right\} \end{aligned}$$
(26)
The comprehensive derivation of (
26) is given in “Appendix 5.” The throughput of blind RIS-IR-FNOMA is given by [
41],
$$\begin{aligned} T={\tilde{R}}_{\mathrm{nv}}\left( 1-P^o_{\mathrm{nv}}\right) +{\tilde{R}}_{\mathrm{fv}}\left( 1-P^o_{\mathrm{fv}}\right) \end{aligned}$$
(27)
where
\(P^o_{\mathrm{nv}}\) and
\(P^o_{\mathrm{fv}}\) are given in (
26) and (
11), respectively.