Performance study of non-orthogonal multiple access (NOMA) with triangular successive interference cancellation

Abstract

The non-orthogonal multiple access (NOMA) allows allocating one carrier to more than one user at the same time in one cell. It is a promising technology to provide high throughput due to carrier reuse within a cell. In this paper, a novel interference cancellation (IC) technique is proposed for asynchronous NOMA systems. The proposed IC technique exploits a triangular pattern to perform the IC from all interfering users for the desired user. The bit error rate performance analysis of an uplink NOMA system with the proposed IC technique is presented, along with the comparison to conventional IC technique. The numerical and simulation results show that the NOMA with the proposed asynchronous IC technique outperforms the conventional IC. It is also shown that employing iterative IC provides significant performance gain for NOMA and the number of required iterations depends on the modulation level and the detection method. With hard-decision, two iterations are sufficient, however with soft-decision, two iterations are enough only for low modulation level, and more iterations are desirable for high modulation level.

Appendix: The mean square error of detection at the lth iteration

Given (10), $\left( {\text {D}}_{k}^{(\mathfrak {L})}[\varsigma ] \,\big \vert \, {\text {z}}_{k}^{(\mathfrak {L})}[\varsigma ] \right) = E[(X_{k}[\varsigma ] - \hat{X}_{k}^{(\mathfrak {L})}[\varsigma ])^{2}]$ represents the MSE of the detection at the $\mathfrak {L}$th iteration for the $\varsigma$th symbol of the kth user. For $\mathfrak {L}=0$, there was no a priori detection done for the symbol yet, so ${\text {D}}_{k}^{(0)}[\varsigma ] =1$. For $\mathfrak {L} \ge 1$, a priori detection was done for the symbol. With probability $(1-{\text {Pe}}_{k}^{(\mathfrak {L})}[\varsigma ])$ the detection was correct so $\left( {\text {D}}_{k}^{(\mathfrak {L})}[\varsigma ] \,\big \vert \,X_{k}[\varsigma ] = \hat{X}_{k}^{(\mathfrak {L})}[\varsigma ] \right) =0$; with probability ${\text {Pe}}_{k}^{(\mathfrak {L})}[\varsigma ]$ the detection was in error and $\left( {\text {D}}_{k}^{(\mathfrak {L})}[\varsigma ] \,\big \vert \, X_{k}[\varsigma ] \ne \hat{X}_{k}^{(\mathfrak {L})}[\varsigma ] \right)$ is obtained as follows. Let ${\text {P}}(m_{i} \,\big \vert \, m_{j})$ denote the probability of detecting constellation point $m_{i}$, given that constellation $m_{j}$ is transmitted. For M-QAM constellation $\left( {\text {D}}_{k}^{(\mathfrak {L})}[\varsigma ] \,\big \vert \, X_{k}[\varsigma ] \ne \hat{X}_{k}^{(\mathfrak {L})}[\varsigma ] \right)$ is given by

$$\begin{aligned} \left( {\text {D}}_{k}^{(\mathfrak {L})}[\varsigma ] \,\big \vert \, X_{k}[\varsigma ] \ne \hat{X}_{k}^{(\mathfrak {L})}[\varsigma ] \right) = \sum _{\begin{array}{c} i=1\\ i \ne j \end{array}}^{M} \left( m_{j} - m_{i} \right) ^{2} \cdot {\text {P}}(m_{i} \,\big \vert \, m_{j}). \end{aligned}$$

(19)

With hard-decision, the event with ${\text {P}}(m_{i} \,\big \vert \,m_{j})$ occurs when the power of residual interference plus noise exceeds half of the distance between nearest constellation points and the resultant ICed signal is closest to constellation point $m_{i}$. Since the power of residual interference plus noise is Gaussian distributed (as explained in Sect. 5) and Gaussian distribution has tail distribution with exponential falloff, the error probability for a constellation point for another point that is not one of its nearest neighbour is much less than that for a nearest neighbour. Thus nearest neighbour approximation to the detection error is used in this paper [12, Chap. 5.1]. Then, for Gray coded QAM constellation with average unit power,

$$\begin{aligned} \left( {\text {D}}_{k}^{(\mathfrak {L})}[\varsigma ] \,\big \vert \, X_{k}[\varsigma ] \ne \hat{X}_{k}^{(\mathfrak {L})}[\varsigma ] \right) = (m_{i} - m_{j})^{2} = (2 \cdot {\text {d}}_{{\text {unit}}})^{2} = (^{6}/_{(M-1)}), \end{aligned}$$

(20)

where ${\text {d}}_{{\text {unit}}} = \sqrt{^{3}/_{2(M-1)}}$ is half of the distance between two nearest neighbour constellation points at a constellation with average unit power. Then, the unified expression for all above cases is given by

$$\begin{aligned} &\big ({\text {D}}_{k}^{(\mathfrak {L})}[\varsigma ] \,\big \vert \, {\text {z}}_{k}^{(\mathfrak {L})}[\varsigma ] \big ) \\&\quad= {\left\{ \begin{array}{ll} 1, & {\text {for }}\, \mathfrak {L}=0, {\text {i.e. no priori detection was done for the symbol}} \\ ^{6}/_{(M-1)}, & {\text {for }}\, \mathfrak {L} \ge 1 {\text { and }} {\text {z}}_{k}^{(\mathfrak {L})}[\varsigma ] = {\text {e}}_{k}^{(\mathfrak {L})}[\varsigma ], {\text {i.e. priori detection was in error}} \\ 0, & {\text {for }}\, \mathfrak {L} \ge 1 {\text { and }} {\text {z}}_{k}^{(\mathfrak {L})}[\varsigma ] = {\text {c}}_{k}^{(\mathfrak {L})}[\varsigma ], {\text {i.e. priori detection was correct}}. \end{array}\right. } \end{aligned}$$

(21)