Non Orthogonal Multiple Access (NOMA) Using a Nonlinear Energy Harvesting Model

Abstract

This paper derives and optimizes the throughput of NOMA with nonlinear energy harvesting. The Base Station (BS) harvests energy from the received Radio Frequency signal. Then, the BS uses the nonlinear harvested power to diffuse packets to N users. We suggest to optimize harvesting time and the powers of users to optimize the data rate. The optimization of NOMA powers and the harvesting process were not yet performed with a non linear energy harvesting model. The study is performed for Rayleigh channels in the presence of nonlinear energy harvesting model. Furthermore, the theoretical results were confirmed using MATLAB computer simulations.

1 Introduction and Literature Review

In Orthogonal Multiple Access (OMA), transmission to N users is performed over N different bands with width B/N where B is the total available band. Each user obtains a maximum throughput of $log_2(K)/N$ where K is the size of modulation. NOMA has been suggested to maximize the total throughput [1,2,3,4] since transmission to all users is performed over the available band B. Powers allocated to users should be optimized to reduce the effects of interference between users [5,6,7,8]. Weak user demodulates only its signal while strong users detects the weak user signal. In fact, the weak user signal is sent with a larger amplitude. Then, strong user subtracts the weak user signal to detect its signal [8]. NOMA has been deployed in Cognitive Radio Networks to transmit data to primary or secondary users [9,10,11,12]. Transmission to secondary users using NOMA can be performed when primary users are idle. Opportunistic spectrum access allows to benefit from unused bands [9,10,11,12]. Besides, NOMA transmission to primary or secondary users can be performed with adaptive power so that the interference is constrained to a given maximum value [9,10,11,12].

NOMA with power harvesting has been studied [13,14,15]. The base station harvests power from RF signals [13,14,15,16,17]. NOMA with a linear energy harvesting model was studied in [18]. Sum rate maximization of NOMA was suggested in [19]. Analysis of the maximum power harvesting capacity was derived in [20] assuming a linear energy harvesting model. The security of NOMA using multiple antennas was studied in [21]. A linear energy harvesting model has been considered in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The Furthermore, harvesting time has not been yet optimized. We study the throughput of NOMA with a non linear energy harvesting model. Powers of users and harvesting time are optimized to enhance the offered data rate this is the main contribution of the paper.

NOMA with two users is considered in Sect. 2. Section 3 generalize the model to N users. The results are shown in Sect. 4. Section 5 summarizes the results.

2 NOMA with Two Users

Figure 1 shows the network model with a Base Station and two users $U_1$ and $U_2$. $U_1$ is the weak user and $U_2$ is the strong user such that: $\lambda _1=E(|h_1|^2)<\lambda _2=E(|h_2|^2)$ where $h_i$ is channel from BS to $U_i$. In the first time duration, BS harvests power from the signal of A. Powee harvesting is done over $\beta F$ second where $0<\beta <1$ gives the harvesting time and F is frame length. The harvested energy is [17]

$$\begin{aligned} E=\alpha F P_A \beta |h_{ABS}|^2=\alpha L_0 E_A \beta |h_{ABS}|^2, \end{aligned}$$

(1)

where $E_A$ is the symbol energy of A, $P_A=\frac{E_A}{T_s}$, $T_s$ is the symbol duration, $\alpha$ is a proportional coefficient, $L_0=\frac{F}{T_s}$ and $h_{ABS}$ is channel from A to BS.

Fig. 1

NOMA with two users

The complexity of non linear energy harvesting is $\alpha L_0$ complex multiplications.

The non linear harvested energy is [17]

$$\begin{aligned} E_{nonlinear}=\Psi (E)=\Psi (\beta L_0 \alpha E_A |h_{ABS}|^2) \end{aligned}$$

(2)

where [27]

$$\begin{aligned} \Psi (x)=N\frac{1-e^{-Cx}}{1+e^{-C(x-B)}}, \end{aligned}$$

(3)

Typical values of parameters are $N=0.02$, $C=1500$ and $B=0.002$ [17].

The CDF of $E_{nonlinear}=\Psi (E)$ is computed as

$$\begin{aligned} P_{E_{nonlinear}}(x)=F_{E}(\Psi ^{-1}(x))=1 -e^{-\frac{\Psi ^{-1}(x)}{\beta L_0 \lambda _{ABS}E_A\alpha }} \end{aligned}$$

(4)

where $\lambda _{ABS}=E(|h_{ABS}|^2)$

$$\begin{aligned} \Psi ^{-1}(x)=\frac{ln(y)+ln(N)+CB-ln(N-y)}{2C} \end{aligned}$$

(5)

Let

$$\begin{aligned} X_i=E_{nonlinear}|h_i|^2 \end{aligned}$$

(6)

The CDF of $X_i$ is

$$\begin{aligned} P_{X_{i}}(x)= & {} \int _0^{+\infty }P_{E_{nonlinear}} \left( \frac{x}{y}\right) \frac{e^{-\frac{y}{\lambda _i}}}{\lambda _i}dy,\nonumber \\= & {} 1-\int _0^{+\infty }e^{-\frac{\Psi ^{-1} \left( \frac{x}{y}\right) }{L_0 \beta \lambda _{ABS} E_A\alpha }}\frac{e^{-\frac{y}{\lambda _i}}}{\lambda _i}dy. \end{aligned}$$

(7)

where $\lambda _i=E(|h_i|^2)$.

In the second time duration, during $(1-\beta )F$ seconds, the BS transmits jointly $s_1$ and $s_2$ of users $U_1$ and $U_2$: $\sqrt{p_1}s_1+\sqrt{p_2}s_2$ where $1>p_1>p_2>0$ are the powers allocated to users such that $p_1>p_2$. The signal at $U_i$ is

$$\begin{aligned} r_i=\sqrt{E_{nonlinear}}[\sqrt{p_1}s_1+\sqrt{p_2}s_2]h_i+n_i \end{aligned}$$

(8)

where $n_i$ is Gaussian of variance $N_0$.

Weak user $U_1$ estimates $s_1$ with SINR

$$\begin{aligned} \gamma _1=\frac{p_1X_1}{p_2X_1+N_0} \end{aligned}$$

(9)

where $X_i$ is defined in (6).

The outage probability at $U_1$ is

$$\begin{aligned} P_{1,outage}(y)=P_{\gamma _1}(y)=P(\gamma _1 \le y)=P_{X_1} \left( \frac{N_0y}{p_1-p_2y}\right) \end{aligned}$$

(10)

where $p_{X_1}(y)$ is given in (7).

Strong user $U_2$ detects symbol $s_1$ since $p_1>p_2$ with SINR

$$\begin{aligned} \gamma _2^{2\rightarrow 1}=\frac{p_1X_2}{p_2X_2+N_0} \end{aligned}$$

(11)

Then $U_2$ subtracts $s_1$ and estimates $s_2$ with SNR

$$\begin{aligned} \gamma _2^{2\rightarrow 2}=\frac{p_2X_2}{N_0} \end{aligned}$$

(12)

There is no outage at $U_2$ if both SNR $\gamma _2^{2\rightarrow 2}$ and SINR $\gamma _2^{2\rightarrow 1}$ are greater than y:

$$\begin{aligned} P_{2,outage}(y)=1-P(\gamma _2^{2\rightarrow 2}>y, \gamma _2^{2\rightarrow 1}>y)=P_{X_2} \left( max\left[ \frac{N_0y}{p_1-p_2y},\frac{N_0y}{p_2}\right] \right) \end{aligned}$$

(13)

where $p_{X_2}(y)$ is given in (7).

The BLock Error Probability (BLEP) at $U_i$ is given by [22]

$$\begin{aligned} BLEP_i<P_{i,outage}(T_0) \end{aligned}$$

(14)

where

$$\begin{aligned} T_0=\int _0^{+\infty }blep(y)dy \end{aligned}$$

(15)

blep(y) is

$$\begin{aligned} blep(y)=1-[1-S(y)]^{Q} \end{aligned}$$

(16)

Q is packet size and S(y) is [23]

$$\begin{aligned} sep(y)=4\left( 1-\frac{1}{\sqrt{K}}\right) Q \left( \sqrt{\frac{3ylog_2(K)}{K-1}}\right) , \end{aligned}$$

(17)

and K is the constellation size of K Quadrature Amplitude Modulation (K-QAM).

The data rate at $U_i$ is computed as

$$\begin{aligned} Thr_i(\beta ,p_1,p_2)=log_2(K)(1-\beta )[1-BLEP_i] \end{aligned}$$

(18)

Therefore, the total data rate is

$$\begin{aligned} Thr(\beta ,p_1,p_2)=Thr_1(\beta ,p_1,p_2)+Thr_2(\beta ,p_1,p_2) \end{aligned}$$

(19)

Harvesting duration $\beta$ and powers $p_1$ and $p_2$ are optimized to maximize the total throughput

$$\begin{aligned} Thr^{optimal}=\underset{0<\beta <1,1>p_1>p_2>0}{max}Thr(\beta ,p_1,p_2) \end{aligned}$$

(20)

3 NOMA with N Users

Power harvesting is performed in the first time duration. In the second time duration, BS sends $\sum _{i=1}^N\sqrt{p_i}s_i$. The signal at $U_q$ is

$$\begin{aligned} r_q=h_q\sqrt{E_{nonlinear}}\sum _{i=1}^N\sqrt{p_i}s_i+n_q \end{aligned}$$

(21)

Fig. 2

NOMA for N users

As sown in Fig. 2, $U_q$ is the $(N-q+1)$ strongest user such that $\lambda _1<\lambda _2<...<\lambda _N$ where $\lambda _i=E(|h_i|^2)$. More power is allocated to $U_1$: $1>p_1>p_2>...>p_N>0$. User $U_q$ starts by detecting $s_1$ with SINR

$$\begin{aligned} \gamma _q^{q \rightarrow 1}=\frac{X_qp_1}{X_q\sum _{l=2}^Np_l+N_0}. \end{aligned}$$

(22)

Then $U_q$ performs subtracts the signal of $U_1$ and demodulates $s_2$ and the SINR is

$$\begin{aligned} \gamma _q^{q \rightarrow 2}=\frac{X_qp_2}{X_q\sum _{l=3}^Np_l+N_0}. \end{aligned}$$

(23)

Similarly, $U_q$ will detect $s_m$ for $m=1,2,...,q$ with SINR

$$\begin{aligned} \gamma _q^{q \rightarrow m} =\frac{X_qp_m}{X_q\sum _{l=m+1}^Np_l+N_0}. \end{aligned}$$

(24)

We deduce

$$\begin{aligned} P_{2,outage}(y)&=1-P(\gamma _q^{q \rightarrow 1}>y, \gamma _q^{q \rightarrow 2}>y,...,\gamma _q^{q \rightarrow q}>y)\nonumber \\&=P_{X_q}\left( \underset{1\le m\le q}{max} \left[ \frac{N_0}{p_m-y\sum _{l=m+1}^Np_l}\right] \right) \end{aligned}$$

(25)

The BLEP and throughput are computed using (14) and (18). The total throughput is computed as

$$\begin{aligned} Thr(\beta ,p_1,p_2,...,p_N)=\sum _{i=1}^NThr_i(\beta ,p_1,p_2,...,p_N) \end{aligned}$$

(26)

Harvesting duration $\beta$ and powers $p_1$ and $p_2$ are given by

$$\begin{aligned} Thr^{optimal}=\underset{0<\beta <1,1>p_1>p_2>...>p_N>0}{max} Thr(\beta ,p_1,p_2,...,p_N) \end{aligned}$$

(27)

4 Results and Discussion

Figures 3, 4, 5, 6 depict the BLEP and throughput at $U_1$ and $U_2$ for harvesting time $\beta =0.5$. There are two users located at 1 and 1.2 from the base station. We observe that the throughput increases at high SNR when 16QAM modulation is used with respect to Quadrature Phase Shift Keying (QPSK). Besides, the throughput at strong user is larger than that at weak user.

Fig. 3

BLEP at weak user

Fig. 4

BLEP at strong user

Fig. 5

$U_1$ Throughput

Fig. 6

$U_2$ throughput

In Fig. 7, the total throughput is shown for 16QAM and QPSK modulations. Optimal Power allocation (OP) increases the total throughput. It is crucial to optimize harvesting time $\beta$ to maximize data rates. The throughput is 8 bit/s/Hz at high SNR for 16QAM and when the harvesting process is optimized. However, the throughput is only 4 bit/s/Hz as obtained in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].

Fig. 7

Total throughput optimization

In Fig. 8, the throughput is shown for 16QAM modulations for 3 users at distances 1, 1.1 and 1.2 from the BS. We see that OP and optimal harvesting time allow to increase data rates. Optimal powers and optimal $\beta$ offer a throughput equal to 12 bit/s/Hz while the throughput is less than 6 bit/s/Hz when $\beta$ is not optimized as suggested in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Performance enhancement with respect to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] is about 22 dB.

Fig. 8

Total throughput optimization in the presence of three users

5 Conclusion

We have optimized the power harvesting process for NOMA systems. We used a realistic nonlinear power harvesting model and optimized the energy harvesting time to maximize the throughput. We obtained up to 8 dB gain by optimizing the power harvesting time and the results are valid for any network.