Interference cancellation for non-orthogonal multiple access used in future wireless mobile networks

Abstract

Non-orthogonal multiple access (NOMA) is suggested as a radio access candidate for future wireless mobile networks. It utilizes the power domain for user multiplexing on the transmitter side and adopts a successive interference cancellation (SIC) as the baseline receiver scheme, considering the expected mobile device evolution in the near future. However, recent research focuses more on the performance evaluation of NOMA in context of assuming the perfect SIC at receiver side. In order to clarify the performance gap between the perfect and the practical SIC in NOMA schemes, and to examine the possibility of applying NOMA with practical SIC, this paper investigates the performance of NOMA applying multi-input multi-output (MIMO) technology with zero-forcing (ZF) and minimum mean square error (MMSE) SIC schemes. We propose an analysis on error effects of the practical SIC schemes for NOMA and in addition propose an interference-predicted minimum mean square error (IPMMSE) IC by modifying the MMSE weight factor using interference signals. According to the IPMMSE IC and analysis of IC error effect, we further suggest the remaining interference-predicted MMSE (RIPMMSE) IC to cancel the remaining interference. The simulation results show that by considering practical IC schemes, the bit error rate (BER) is degraded compared with conventional orthogonal multiple access (OMA). This validates that the proposed IC schemes, which can predict the interference signals, provide better performance compared to NOMA with conventional ZF and MMSE IC schemes.

1 Introduction

In the fourth-generation (4G) mobile communication systems, such as Long-Term Evolution (LTE), WiMAX, LTE-Advanced, and V2V networks [1–3], orthogonal access based on orthogonal frequency division multiple access (OFDMA) or single carrier-frequency division multiple access (SC-FDMA) was adopted. Orthogonal access is a reasonable choice for achieving good system throughput with a simplified receiver design. However, due to the vastly increased need for high-volume services, such as image transfer, video streaming, and cloud-based services, a new mobile communications system with further enhancement of system throughput is required for the next-generation (5G) mobile communication systems. In order to fulfill such requirements, non-orthogonal multiple access (NOMA) with a successive interference cancellation (SIC) receiver in downlink was presented as one of several promising candidate radio access technologies [4–10]. For downlink NOMA, non-orthogonality is achieved by introducing the power domain, either in time/frequency/code domains, for user multiplexing. User de-multiplexing is obtained through the allocation of a large power difference between the users on the transmitter side and the application of SIC on the receiver side. In this case, everyone can use the overall transmission bandwidth to get higher spectrum efficiency, and better user fairness can be achieved, compared with conventional orthogonal multiple access (OMA), by assigning greater power to the users under poor channel conditions. Furthermore, NOMA is suitable for the situations of massive connectivity, because it can support more simultaneous connections.

NOMA is a candidate technology for further performance enhancements of LTE and LTE-Advanced, and both the concept and system performance are discussed and analyzed, considering the perfect SIC on the receiver side [11–16]. Therefore, in this paper, we exploit the system performance based on the link-level simulation (LLS) of NOMA with practical SIC schemes (i.e., zero-forcing [ZF] and MMSE SIC).

We analyzed the effect of the error that is caused by the interference cancellation (IC) by considering practical SIC schemes for NOMA in. Based on the analysis, we propose a novel interference-predicted minimum mean square error (IPMMSE) IC scheme for NOMA downlink, which is based on MMSE criteria from prediction about interference signals. Moreover, based on the IPMMSE IC and the analysis of the IC error effect, we propose the remaining interference-predicted MMSE (RIPMMSE) IC to cancel the remaining interference, which can further improve the system performance. The link-level evaluation is provided, and the simulation results show that by using the proposed IC schemes, the bit error rate (BER) performance is enhanced, compared with conventional IC schemes for NOMA.

The rest of the paper is organized as follows. Section 2 introduces the basics of NOMA and describes a NOMA system model with ZF and MMSE SIC schemes by using multi-input multi-output (MIMO) technology [17]. In Section 3, we compare the conventional multiple access (MA) and NOMA with the practical IC scheme and analyze the IC error effect for NOMA. Based on IC error analysis, the proposed IPMMSE IC and RIPMMSE IC schemes are described in Section 4. Finally, we conclude this paper in Section 5.

2 System model

2.1 NOMA basics

The basic NOMA scheme with SIC for a 2-user equipment (UE) case in the cellular downlink is illustrated in Fig. 1 [1]. The transmit information for UE _i (i = 1, 2) at the base station (BS) is s _i , with transmission power p _i , so the transmit signal for UE _i is

Fig. 1

Basic NOMA scheme applying SIC in downlink

$$ {x}_i=\sqrt{p_i}{s}_i. $$

(1)

The sum of transmit power is restricted to p. So, the transmit signals are superposed as

$$ x={x}_1+{x}_2=\sqrt{p_1}{s}_1+\sqrt{p_2}{s}_2. $$

(2)

The received signal at UE _i is

$$ {y}_i={h}_ix+{n}_i, $$

(3)

where h _i is the complex channel coefficient between UE _i and the BS; n _i denotes the receiver’s Gaussian noise, including inter-cell interference. The power density of n _i is N _0,i . In the NOMA downlink, decoding is in the order of the increasing channel gain normalized by the noise and inter-cell interference power, |h _i | ² /N _0,i . For a 2-UE case, as shown in Fig. 1, we assume that |h ₁ | ² /N _0,1 > |h ₂ | ² /N _0,2, so UE₁ first decodes x ₂ and deletes its component from received signal y ₁. And UE₂ decodes x ₂ without interference cancellation, because it has the first decoding order. The throughput of UE _i , R _i is

$$ {R}_1={ \log}_2\left(1+\frac{P_1{\left|{h}_1\right|}^2}{N_{0,1}}\right),\kern1.1em {R}_2={ \log}_2\left(1+\frac{P_2{\left|{h}_2\right|}^2}{P_1{\left|{h}_2\right|}^2+{N}_{0,2}}\right). $$

(4)

2.2 NOMA-MIMO with practical SIC schemes

Sets of achievable rates for NOMA have been found by Cover [18], and the proof for the optimality of the sets of achievable rates for additive white Gaussian noise broadcast channels was given by Bergmans [19]. The capacity region of the uplink fading channel with receiver channel state information (CSI) was derived by Gallager [20], where he also showed that CDMA-type systems are inherently capable of higher rates than systems such as slow frequency hopping that maintain orthogonality between users. In [21], Tse gave a conclusion that NOMA is strictly better than OMA (except for the two corner points where only one user is being communicated to) in terms of sum rate, i.e., for any rate pair achieved by OMA, there is a power split for which NOMA can achieve rate pairs that are strictly larger. However, this conclusion is intended for single antenna systems and Tse did not give a proof for this conclusion. Here, one should be noted that the capacity gain of NOMA over OMA is achieved at the cost of more decoding complexity at the receivers for NOMA. The application of multiple-input multiple-output (MIMO) technologies to NOMA is important since the use of MIMO provides additional degrees of freedom for further performance improvement. The transceiver design for a special case of MIMO-NOMA downlink transmission, in which each user has a single antenna and the base station has multiple antennas, has been investigated in [22] and [23]. In [24], a multiple-antenna base station used the NOMA approach to serve two multiple-antenna users simultaneously, where the problem of throughput maximization was formulated and two algorithms were proposed to solve the optimization problem. In many practical scenarios, it is preferable to serve as many users as possible in order to reduce user latency and improve user fairness. Following this rationale, in [25], users were grouped into small-size clusters, where NOMA was implemented for the users within one cluster and MIMO detection was used to cancel inter-cluster interference. Different from conventional works, this paper focuses on the NOMA-MIMO with practical SIC instead of using perfect SIC assumption in conventional studies.

We assume orthogonal frequency division modulation (OFDM) signaling, although we consider non-orthogonal user multiplexing. Figure 2 illustrates the transmission scenario in the downlink NOMA-MIMO scheme for a 2-UE case, where UE₁ is a cell-center user (near user) and UE₂ is a cell-edge user (far user). Figure 3 shows the block diagram of transmitter and receiver for the downlink NOMA scheme. We assume there are two transmission antennas at the BS, and each antenna transmits signal to one UE. The transmit signal x _i for UE _i is

Fig. 2

Transmission scenario for the 2-UE downlink NOMA-MIMO scheme

Fig. 3

Block diagram of transmitter and receiver for the downlink NOMA-MIMO scheme

$$ {x}_i=\sqrt{p_i}{s}_i\kern0.4em ,\ \left(i=1,\kern0.5em 2\right)\kern0.1em , $$

(5)

where p _i is the allocated power, and s _i is the transmitted data for UE _i . After transmitting through a 2 × 2 channel H, as shown in Fig. 2, the 2-UE system is presented as

$$ Y=\left[\begin{array}{cc}\hfill {x}_1\hfill & \hfill {x}_2\hfill \end{array}\right]\left[\begin{array}{cc}\hfill {H}_{11}\hfill & \hfill {H}_{12}\hfill \\ {}\hfill {H}_{21}\hfill & \hfill {H}_{22}\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill {n}_1\hfill & \hfill {n}_2\hfill \end{array}\right], $$

(6)

where n _i (i = 1, 2) is the receiver’s Gaussian noise for UE _i . The received signal y _i for UE _i is represented as

$$ {y}_i={H}_{ii}{x}_i+{\displaystyle \sum_{j\ne i}{H}_{ji}{x}_j+{n}_i}, $$

(7)

where H _ji denotes the channel between the jth antenna at the BS and the ith receiver, x _j is the transmit signal for UE _j , which is the interference for UE _i , and n _i the Gaussian noise. After reception, the signals are ranked in decreasing order by power. Channel estimation (CE) is performed for the interference signal y _ji with power p _j  > p _i (power of the desired signal). Then, SIC is employed until all interference signals are cancelled. The estimated received signal ${\tilde{s}}_i$ is obtained for UE _i . For the 2-UE case, from Eq. (6), the received signals for UE₁ and UE₂ are

$$ \begin{array}{c}\hfill {y}_1={H}_{11}{x}_1+{H}_{21}{x}_2+{n}_1\hfill \\ {}\hfill {y}_2={H}_{12}{x}_1+{H}_{22}{x}_2+{n}_2\hfill \end{array}, $$

(8)

respectively. Since NOMA allows UEs to share the same resources, and differentiates UE by power, IC is performed for UE with lower power to cancel the inter-user interference. Because UE₂ is a far user with greater power, the element H ₂₁ x ₂ is cancelled by the IC from y ₁, whereas y ₂ can be demodulated directly without IC.

In this paper, we consider practical SIC schemes based on the zero-forcing (ZF) and the MMSE criteria. As described in the system model, by assuming power p ₂ >> p ₁, the UE₂ can directly detect the signal without cancellation of interference. For UE₁, the received signal y ₁ is

$$ {y}_1={H}_{11}\sqrt{p_1}{s}_1+{H}_{21}\sqrt{p_2}{s}_2+{n}_1, $$

(9)

where interference is ${H}_{21}\sqrt{p_2}{s}_2$. We can get the weight factor of channel H ₂₁, i.e., Ĥ ₂₁|_ZF or Ĥ ₂₁|_MMSE, by ZF or MMSE criteria after getting the estimated channel ${\tilde{H}}_{21}$:

$$ {\left.{\widehat{H}}_{21}\right|}_{\mathrm{ZF}}={\left({\tilde{H}}_{21}^H{\tilde{H}}_{21}\right)}^{-1}{\tilde{H}}_{21}^H\kern0.7em ,\ {\left.{\widehat{H}}_{21}\right|}_{\mathrm{MMSE}}={\left({\tilde{H}}_{21}^H{\tilde{H}}_{21}+{\sigma}_{n_1}^2I\right)}^{-1}{\tilde{H}}_{21}^H, $$

(10)

where the sign H in the superscript in the equation represents the Hermitian transpose and ${\sigma}_{n_1}^2$ is the variance of noise, n ₁. Then, the estimated interference signal can be obtained as

$$ {\tilde{s}}_2\approx {\sqrt{p_2}}^{-1}{\widehat{H}}_{21}{y}_1={s}_2+{\sqrt{p_3}}^{-1}{\widehat{H}}_{21}\left({H}_{11}\sqrt{p_1}{s}_1+{n}_1\right), $$

(11)

The received signal is updated by subtracting the estimated interference signal:

$$ {y_1}^{\prime }={y}_1-{\tilde{H}}_{21}\sqrt{p_2}{\tilde{s}}_2\approx {H}_{11}\sqrt{p_1}{s}_1+{n}_1. $$

(12)

After the canceling the interference signal with high power, UE₁ can detect the desired information, s ₁, from the updated received signal.

3 IC error analysis

3.1 Comparison between conventional MA and NOMA with practical IC scheme

In this section, we compare the packet error rate (PER) and throughput performances of conventional MA and NOMA by LLS. We still use the system model mentioned above for 2-UE. In conventional MA (without NOMA), the power for UE₁ (near user), p ₁, is assigned as 0.8, and for UE₂ (far user), p ₂, is 0.2, which follows the water filling (WF) algorithm. For NOMA, e.g., p ₁ = 0.2, p ₂ = 0.8 [4], which is power control (PC). We imitate system-level simulation (SLS) by LLS, where power allocation is used instead of path loss for cases without and with NOMA (i.e., WF versus PC). According to the LTE specifications, simulation parameters are summarized in Table 1. Figure 4 shows the PER performance for cases with NOMA (case_1) and without NOMA (case_2), where the practical ZF IC is applied for UE₁ in case_2. According to the PER, the throughput C can be calculated by

Table 1 Simulation parameters

Fig. 4

PER performance with and without NOMA under AWGN channel

$$ C=\frac{\left(1-\mathrm{PER}\right)\times {n}_{\mathrm{bit}/\mathrm{packet}}}{T_{\mathrm{packet}}(s)}. $$

(13)

Therefore, the throughput performance for case_1 and case_2 is shown in Fig. 5. In the LLS, the path loss is not considered, and if we compare the performance between case_1 (p ₁ = 0.2, p ₂ = 0.8) and case_2 (p ₁ = 0.8, p ₂ = 0.2) under the scenario without NOMA, the performance of UE₁ in case_1 is same as UE₂ in case_2. And the performance of UE₂ in case_1 is the same as UE₁ in case_2. So, it is fair to compare the scenario of case_1 (with NOMA) and case_2 (without NOMA).

Fig. 5

Throughput performance with and without NOMA under AWGN channel

From the simulation results shown in Fig. 4, we determine that in NOMA, UE₂ can detect information with greater power than interference from UE₁. Without NOMA, UE₂ cannot detect information owing to the lower power compared to the interference signal from UE₁. On the other hand, UE₁ with NOMA can benefit from IC, even though the power of the interference signal is much higher than the target signal. Without NOMA, UE₁ has performance similar to UE₂ with NOMA, owing to the existing interference from the lower power user. As the results shown in Fig. 5, NOMA can increase the sum throughput and improve the fairness between the near and far users, compared with the situations without NOMA.

In order to evaluate the performance of NOMA under simulation, we provide comparisons of the LLS for UE₁ under different modulation schemes, with the bound value for UE₁ that is obtained from Eq. (4), mentioned above. Figure 6 shows the simulation curves for UE₁ with NOMA under the perfect IC for different modulation and coding schemes. The coding scheme is convolutional code; the channel environment is single-path Rayleigh fading; and other simulation parameters are the same as in Table 1. From the comparison, all the simulation curves are lower than the bound and show a higher data rate with a higher order of modulation schemes. This verifies the LLS results and the analysis for the further research.

Fig. 6

Bound and simulation curves for UE₁ under NOMA with different modulation schemes

4 IC error under AWGN channel

For convenience, we use the same NOMA-MIMO transmission scenario as described in Section 2. At UE₁ (near user), IC is applied to cancel the interference from the UE₂ signal, as described in Fig. 7. Under the AWGN channel, the received signal for UE₁ is

Fig. 7

Block diagram for IC under AWGN channel

$$ {y}_1={x}_1+{x}_2+{n}_1, $$

(14)

and owing to the larger power of x ₂ compared to x ₁, x ₂ should be demodulated first. When demodulating x ₂ at UE₁, x ₁ becomes the interference component:

$$ \begin{array}{l}{y}_1={x}_2+\left({x}_1+{n}_1\right)\\ {}\kern0.5em ={x}_2+{i}_{1\_\mathrm{AWGN}}\kern0.1em ,\\ {}\kern0.5em ={\tilde{y}}_{x_2\left|\mathrm{U}\mathrm{E}1\right.}\end{array} $$

(15)

where i _{1_AWGN} is the interference for x ₂ plus noise, and we assume that the demodulated signal of x ₂ at UE₁ is ${\tilde{x}}_{2\left|\mathrm{U}\mathrm{E}1\right.}$. Then, the noise enhancement ratio when demodulating x ₂ at UE₁ is

$$ {e}_{\mathrm{AWGN}}=\frac{i_{1\_\mathrm{AWGN}}}{n_1}. $$

(16)

After that, ${\tilde{x}}_{2\left|\mathrm{U}\mathrm{E}1\right.}$ is re-modulated in order to cancel this signal from y ₁, and the received signal after IC for x ₂ is

$$ \begin{array}{l}{y}_1^{\prime }={y}_1-{\widehat{x}}_2\\ {}\kern1.3em ={x}_1+\left({x}_2-{\widehat{x}}_2\right)+{n}_1, \end{array} $$

(17)

where ${\widehat{x}}_2$ is the re-modulated signal and the component $\left({x}_2-{\widehat{x}}_2\right)$ is the remaining interference for x ₁ after IC. The power of the remaining interference for x ₁ is

$$ {p}_{\left({x}_2-{\widehat{x}}_2\right)}={p}_2\underset{\mathrm{ratio}\kern0.1em \mathrm{of}\kern0.1em \mathrm{remaining}\kern0.1em \mathrm{interference}}{\underbrace{\left(\frac{p_2}{i_{1\_\mathrm{AWGN}}}/\frac{p_2}{n_1}\right)}}=\frac{p_2}{e_{\mathrm{AWGN}}}\kern0.24em , $$

(18)

where $\frac{p_2}{i_{1\_\mathrm{AWGN}}}$ is the signal-to-interference-plus-noise ratio (SINR) and $\frac{p_2}{n_1}$ is the signal-to-noise ratio (SNR) of x ₂ from perfect IC. Therefore, the bound value for UE₁ after practical IC of x ₂ is

$$ {r}_{1\_IC\_\mathrm{AWGN}}={ \log}_2\left(1+\frac{p_1}{\frac{p_2}{e_{\mathrm{AWGN}}}+{n}_1}\right)\kern0.24em . $$

(19)

Figure 8 shows the loss due to IC error for UE₁ NOMA under practical IC compared with perfect IC (Fig. 6).

Fig. 8

Bound for UE₁ NOMA with perfect and practical IC under AWGN channel

4.1 ZF IC error under single-path Rayleigh channel

Different from AWGN channel, various IC schemes can be used under the Rayleigh channel due to essentiality of channel equalization (CE). Firstly, we analyze ZF IC under the single-path Rayleigh channel. Similar to the AWGN case, the ZF IC is applied to cancel the interference from the UE₂ signal, as described in Fig. 9. The received signal for UE₁ is

Fig. 9

Block diagram for ZF IC under Rayleigh channel

$$ {y}_1={H}_{11}{x}_1+{H}_{21}{x}_2+{n}_1\kern0.24em , $$

(20)

where H ₁₁ and H ₂₁ are the channel coefficients for x ₁ and x ₂, respectively. Due to the larger power of x ₂, x ₂ should be demodulated first. When demodulating x ₂ at UE₁, H ₁₁ x ₁ becomes the interference. After (perfect) ZF CE for x ₂,

$$ \begin{array}{l}{H}_{21}^{-1}{y}_1={H}_{21}^{-1}{H}_{11}{x}_1+{x}_2+{H}_{21}^{-1}{n}_1\\ {}\kern2.8em ={x}_2+{H}_{21}^{-1}\left({H}_{11}{x}_1+{n}_1\right)\kern0.96em ,\\ {}\kern2.8em ={x}_2+{i}_{1\_\mathrm{Z}\mathrm{F}}\\ {}\kern2.8em ={\tilde{y}}_{x_2\left|\mathrm{U}\mathrm{E}1\right.}\kern0.1em \end{array} $$

(21)

where i _{1_ZF} is the interference for x ₂ plus noise, we assume the demodulated signal of x ₂ at UE₁ is ${\tilde{x}}_{2\left|\mathrm{U}\mathrm{E}1\right.}$. Then, the noise enhancement ratio when demodulating x ₂ at UE₁ is

$$ {e}_{\mathrm{ZF}}=\frac{i_{1\_\mathrm{Z}\mathrm{F}}}{n_1}\kern0.24em . $$

(22)

After that, ${\tilde{x}}_{2\left|\mathrm{U}\mathrm{E}1\right.}$ is re-modulated and then multiplied by the channel coefficient H ₂₁ to cancel this signal from y ₁. The received signal after ZF IC for x ₂ is

$$ \begin{array}{l}{y}_1^{\prime }={y}_1-{H}_{21}{\widehat{x}}_2\\ {}\kern1.3em ={H}_{11}{x}_1+{H}_{21}\left({x}_2-{\widehat{x}}_2\right)+{n}_1\kern0.36em ,\end{array} $$

(23)

where ${\widehat{x}}_2$ is the re-modulated signal and the component ${H}_{21}\left({x}_2-{\widehat{x}}_2\right)$ is the remaining interference for x ₁ after ZF IC. The power of the remaining interference for x ₁ is

$$ {p}_{\left({x}_2-{\widehat{x}}_2\right)}={p}_2\underset{\mathrm{ratio}\kern0.2em \mathrm{of}\kern0.1em \mathrm{remaining}\kern0.2em \mathrm{interference}}{\underbrace{\left(\frac{p_2{\left|{H}_{21}\right|}^2}{i_{1\_\mathrm{Z}\mathrm{F}}}/\frac{p_2{\left|{H}_{21}\right|}^2}{n_1}\right)}}=\frac{p_2{\left|{H}_{21}\right|}^2}{e_{\mathrm{ZF}}}\kern0.24em , $$

(24)

where $\frac{p_2{\left|{H}_{21}\right|}^2}{i_{1\_ZF}}$ is the SINR and $\frac{p_2{\left|{H}_{21}\right|}^2}{n_1}$ is the SNR of x ₂ from perfect IC. Therefore, the bound value for UE₁ after ZF IC of x ₂ is

$$ {r}_{1\_\mathrm{I}\mathrm{C}\_\mathrm{Z}\mathrm{F}}={ \log}_2\left(1+\frac{p_1{\left|{H}_{11}\right|}^2}{\frac{p_2{\left|{H}_{21}\right|}^2}{e_{\mathrm{ZF}}}+{n}_1}\right)\kern0.36em . $$

(25)

Figure 10 shows the bound and simulation curves for UE₁ NOMA with perfect and practical ZF IC under the single-path Rayleigh channel. From the figure, both bound and simulation curves show the loss due to the practical IC error, compared with perfect IC.

Fig. 10

Bound and simulation curves for UE₁ under NOMA with perfect and practical IC under single-path Rayleigh channel

4.2 MMSE IC error under single-path Rayleigh channel

Similar to ZF IC analysis, we analyze MMSE IC under the single-path Rayleigh channel. At UE₁, the MMSE IC is applied to cancel the interference from the UE₂ signal, as described in Fig. 11. The received signal for UE₁ is

Fig. 11

Block diagram for MMSE IC under Rayleigh channel

$$ {y}_1={H}_{11}{x}_1+{H}_{21}{x}_2+{n}_1\kern0.24em , $$

(26)

where H ₁₁ and H ₂₁ are the channel coefficients for x ₁ and x ₂, respectively. Due to the greater power of x ₂, x ₂ should be demodulated first. When demodulating x ₂ at UE₁, H ₁₁ x ₁ becomes the interference. We assume perfect channel estimation, so the MMSE weight factor for channel H ₂₁ is

$$ {W}_{\mathrm{MMSE}\_{H}_{21}}={\left({H}_{21}^H{H}_{21}+{\sigma}_{n_1}^2I\right)}^{-1}{H}_{21}^H, $$

(27)

After (perfect) MMSE CE for x ₂

$$ \begin{array}{l}{W}_{\mathrm{MMSE}\_{H}_{21}}{y}_1={W}_{\mathrm{MMSE}\_{H}_{21}}{H}_{11}{x}_1+{x}_2+{W}_{\mathrm{MMSE}\_{H}_{21}}{n}_1\\ {}\kern5.4em ={x}_2+{W}_{\mathrm{MMSE}\_{H}_{21}}\left({H}_{11}{x}_1+{n}_1\right)\kern1em ,\\ {}\kern5.4em ={x}_2+{i}_{1\_\mathrm{MMSE}}\\ {}\kern5.4em ={\tilde{y}}_{x_2\left|\mathrm{U}\mathrm{E}1\right.}\kern0.1em \end{array} $$

(28)

where i _{1_MMSE} is the interference for x ₂ plus noise, we assume the demodulated signal of x ₂ at UE₁ is ${\tilde{x}}_{2\left|\mathrm{U}\mathrm{E}1\right.}$. Then, the noise enhancement ratio when demodulating x ₂ at UE₁ is

$$ {e}_{\mathrm{MMSE}}=\frac{i_{1\_\mathrm{MMSE}}}{n_1}\kern0.24em , $$

(29)

After that, ${\tilde{x}}_{2\left|\mathrm{U}\mathrm{E}1\right.}$ is re-modulated, then multiplied by the channel coefficient H ₂₁ to cancel this signal from y ₁. The received signal after MMSE IC for x ₂ is

$$ \begin{array}{l}{y}_1^{\prime }={y}_1-{H}_{21}{\widehat{x}}_2\\ {}\kern1.3em ={H}_{11}{x}_1+{H}_{21}\left({x}_2-{\widehat{x}}_2\right)+{n}_1\kern0.24em ,\end{array} $$

(30)

where ${\widehat{x}}_2$ is the re-modulated signal and the component ${H}_{21}\left({x}_2-{\widehat{x}}_2\right)$ is the remaining interference for x ₁ after MMSE IC. The power of the remaining interference for x ₁ is

$$ {p}_{\left({x}_2-{\widehat{x}}_2\right)}={p}_2\underset{\mathrm{ratio}\kern0.2em \mathrm{of}\kern0.2em \mathrm{remaining}\kern0.2em \mathrm{interference}}{\underbrace{\left(\frac{p_2{\left|{H}_{21}\right|}^2}{i_{1\_\mathrm{MMSE}}}/\frac{p_2{\left|{H}_{21}\right|}^2}{n_1}\right)}}=\frac{p_2{\left|{H}_{21}\right|}^2}{e_{\mathrm{MMSE}}}\kern0.36em , $$

(31)

where $\frac{p_2{\left|{H}_{21}\right|}^2}{i_{1\_\mathrm{MMSE}}}$ is the SINR and $\frac{p_2{\left|{H}_{21}\right|}^2}{n_1}$ is the SNR of x ₂ from perfect IC. Therefore, the bound value for UE₁ after MMSE IC of x ₂ is

$$ {r}_{1\_\mathrm{I}\mathrm{C}\_\mathrm{MMSE}}={ \log}_2\left(1+\frac{p_1{\left|{H}_{11}\right|}^2}{\frac{p_2{\left|{H}_{21}\right|}^2}{e_{\mathrm{MMSE}}}+{n}_1}\right)\kern0.36em . $$

(32)

Figure 12 shows the bound and simulation curves for UE₁ NOMA with practical ZF and MMSE IC under the single-path Rayleigh channel. From the simulation results, both bound and simulation curves show the gain from MMSE IC compared with ZF IC.

Fig. 12

Bound and simulation curves for UE₁ under NOMA with practical ZF and MMSE IC under single-path Rayleigh channel

5 Proposed interference-predicted MMSE IC schemes for NOMA

5.1 Proposed interference-predicted MMSE IC scheme

Based on the analysis of MMSE IC in the previous section, we propose modifying the MMSE weight factor by introducing the information on interference signals to improve the link-level performance of NOMA. From Eq. (28), we can determine that the interference for x ₂ plus noise after MMSE CE for x ₂ is

$$ {i}_{1\_\mathrm{MMSE}}={W}_{\mathrm{MMSE}\_{H}_{21}}\left({H}_{11}{x}_1+{n}_1\right)\kern0.24em . $$

(33)

We assume that i _{1_MMSE} is the background noise for demodulating x ₂, so then, the IPMMSE weight factor is

$$ {W}_{\mathrm{IPMMSE}\_{H}_{21}}={\left({H}_{21}^H{H}_{21}+{\sigma}_{i_{1\_\mathrm{MMSE}}}^2I\right)}^{-1}{H}_{21}^H\kern0.24em , $$

(34)

where ${\sigma}_{i_{1\_\mathrm{MMSE}}}^2$ is obtained from the variance of i _{1_MMSE}:

$$ \begin{array}{l}\operatorname{var}\left({i}_{1\_\mathrm{MMSE}}\right)=E\left({i}_{1\_\mathrm{MMSE}}{i}_{1\_\mathrm{MMSE}}^H\right)\\ {}\kern5.8em =E\left(\left({W}_{\mathrm{MMSE}\_{H}_{21}}\left({H}_{11}{x}_1+{n}_1\right)\right){\left({W}_{\mathrm{MMSE}\_{H}_{21}}\left({H}_{11}{x}_1+{n}_1\right)\right)}^H\right)\\ {}\kern5.8em ={W}_{\mathrm{MMSE}\_{H}_{21}}{H}_{11}E\left({x}_1{x}_1^H\right){H}_{11}^H{W}_{\mathrm{MMSE}\_{H}_{21}}^H+\\ {}\kern.5em {W}_{\mathrm{MMSE}\_{H}_{21}}E\left({n}_1{n}_1^H\right){W}_{\mathrm{MMSE}\_{H}_{21}}^H\kern8.5em ,\\ {}\kern5.8em ={W}_{\mathrm{MMSE}\_{H}_{21}}{H}_{11}{\sigma}_{x_1}^2{H}_{11}^H{W}_{\mathrm{MMSE}\_{H}_{21}}^H+\\ {}\kern0.5em {W}_{\mathrm{MMSE}\_{H}_{21}}{\sigma}_{n_1}^2{W}_{\mathrm{MMSE}\_{H}_{21}}^H\end{array} $$

(35)

where the notation E represents estimation. After deriving Eq. (35), we find that the MMSE weight factor can be modified by considering interference, which makes this algorithm practical. Equation (28) can be replaced by

$$ \begin{array}{l}{W}_{\mathrm{IPMMSE}\_{H}_{21}}{y}_1={W}_{\mathrm{IPMMSE}\_{H}_{21}}{H}_{11}{x}_1+{x}_2+{W}_{\mathrm{IPMMSE}\_{H}_{21}}{n}_1\\ {}\kern6.3em ={x}_2+{W}_{\mathrm{IPMMSE}\_{H}_{21}}\left({H}_{11}{x}_1+{n}_1\right)\kern3em .\\ {}\kern1em ={x}_2+{i}_{1\_\mathrm{IPMMSE}}\\ {}\kern1em ={\tilde{y}}_{x_2\left|\mathrm{U}\mathrm{E}1\right.}\kern0.1em \end{array} $$

(36)

Figure 13 shows the BER curves for UE₁ with conventional OMA and NOMA for different IC schemes under single-path Rayleigh fading channel. The modulation and coding scheme used in the simulation is quadrature phase-shift keying (QPSK) with 1/2 convolutional code. From the simulation results, we find that OMA has the best BER performance, because there is no interference for OMA signaling. As for NOMA, ZF IC gives the worst BER, but it has the lowest complexity. And MMSE IC shows the better BER performance than ZF IC, which is in accordance with our previous analysis in Section 3. Finally, the proposed IPMMSE IC gives the best BER performance among the NOMA IC schemes, because it considers the effect of the interference signals. When the target BER is 10⁻³, the IPMMSE IC scheme outperforms the ZF IC scheme by around 1 dB and by 0.2 dB over the MMSE IC scheme.

Fig. 13

BER for conventional OMA and NOMA with ZF, MMSE, and IPMMSE IC under single-path Rayleigh channel

5.2 Proposed remaining interference-predicted MMSE IC scheme

Based on the analysis of the IC error effect in the previous section, after IPMMSE CE, the noise enhancement ratio when demodulating x ₂ at UE₁ is

$$ {e}_{\mathrm{IPMMSE}}=\frac{i_{1\_\mathrm{IPMMSE}}}{n_1}\kern0.36em . $$

(37)

Then, the power of the remaining interference for x ₁ is

$$ {p}_{H_{21}\left({x}_2-{\widehat{x}}_2\right)\_\mathrm{IPMMSE}}={p}_2{\left|{H}_{21}\right|}^2\underset{\mathrm{ratio}\kern0.2em \mathrm{of}\kern0.2em \mathrm{remaining}\kern0.2em \mathrm{interference}}{\underbrace{\left(\frac{p_2{\left|{H}_{21}\right|}^2}{i_{1\_\mathrm{IPMMSE}}}/\frac{p_2{\left|{H}_{21}\right|}^2}{n_1}\right)}}=\frac{p_2{\left|{H}_{21}\right|}^2}{e_{\mathrm{IPMMSE}}}\kern0.36em , $$

(38)

where $\frac{p_2{\left|{H}_{21}\right|}^2}{i_{1\_\mathrm{IPMMSE}}}$ is the SINR and $\frac{p_2{\left|{H}_{21}\right|}^2}{n_1}$ is the SNR of x ₂ from perfect IC. The next step is to demodulate x ₁, so we propose the remaining interference-predicted MMSE (RIPMMSE) IC for x ₁ to cancel the remaining interference. The RIPMMSE weight factor for channel H ₁₁ is

$$ {W}_{\mathrm{RIPMMSE}\_{H}_{11}}={\left({H}_{11}^H{H}_{11}+{\sigma}_{i_{1\_\mathrm{IPMMSE}}+n}^2I\right)}^{-1}{H}_{11}^H\kern0.36em , $$

(39)

where ${\sigma}_{i_{1\_\mathrm{IPMMSE}}+n}^2$ is obtained from the variance of the remaining interference plus noise

$$ \begin{array}{l}\operatorname{var}\left({H}_{21}\left({x}_2-{\widehat{x}}_2\right)+{n}_1\right)=E\left[\left({H}_{21}\left({x}_2-{\widehat{x}}_2\right)+{n}_1\right){\left({H}_{21}\left({x}_2-{\widehat{x}}_2\right)+{n}_1\right)}^H\right]\\ {}\kern10.50001em =E\left[\left({H}_{21}\left({x}_2-{\widehat{x}}_2\right)+{n}_1\left)\right({\left({x}_2-{\widehat{x}}_2\right)}^H{H}_{21}^H+{n}_1^H\right)\right]\\ {}\kern10.50001em =E\left[{H}_{21}\left({x}_2-{\widehat{x}}_2\right){\left({x}_2-{\widehat{x}}_2\right)}^H{H}_{21}^H\right]+E\left({n}_1{n}_1^H\right)\kern1.44em .\\ {}\kern10.50001em ={H}_{21}E\left[\left({x}_2-{\widehat{x}}_2\right){\left({x}_2-{\widehat{x}}_2\right)}^H\right]{H}_{21}^H+{\sigma}_{n_1}^2\\ {}\kern10.50001em ={H}_{21}{p}_{H_{21}\left({x}_2-{\widehat{x}}_2\right)\_\mathrm{IPMMSE}}{H}_{21}^H+{\sigma}_{n_1}^2\end{array} $$

(40)

After deriving Eq. (40), we find that the MMSE weight factor can be updated by considering the power of the remaining interference, which makes this algorithm practical. Figure 14 shows the BER performances for UE₁ under conventional OMA and NOMA with different IC schemes under single-path Rayleigh fading channel. The modulation and coding scheme used in the simulation is QPSK with 1/3 convolutional code. The simulation results show that RIPMMSE IC can further improve the BER performance compared to IPMMSE IC for NOMA. When the target BER is 10⁻³, the RIPMMSE IC scheme can provide a 1.5-dB gain, compared to the ZF IC scheme, and a 0.5 dB gain over the MMSE IC scheme. Under the same principle, for 3-UE NOMA, the MMSE weight factor can be updated for a third time to further improve the BER performance.

Fig. 14

BER for conventional OMA and NOMA with ZF, MMSE, IPMMSE, and RIPMMSE IC under single-path Rayleigh channel

6 Conclusions

In this paper, we exploit the performance of a NOMA-MIMO system for the perfect and the practical SIC schemes, which clarifies the necessity for the investigation into IC schemes for NOMA. Because many previous works focused on NOMA, some of the research topics, such as employment of the practical SIC and the error effect due to IC, are still in the early stages or not fully developed. In this paper, we perform the error analysis considering ZF and MMSE SIC schemes for NOMA; and based on the analysis, we propose a novel IPMMSE IC scheme by predicting the MMSE weight factor using information about interference signals. In addition, based on the IPMMSE IC scheme and the analysis of the IC error effect, we propose RIPMMSE IC to further boosts the system performance. We provide the link-level evaluations for a 2-UE scenario under NOMA-MIMO system on the single-path Rayleigh channel, and the simulation results show that the RIPMMSE IC scheme outperforms ZF and MMSE IC schemes by around 1.5 and 0.5 dB, respectively, at a target BER of 10⁻³. In the future work, a more general case, i.e., larger number of users will be examined with our proposed schemes.