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Power minimization for GSICbased uplink cellfree massive MIMONOMA systems
EURASIP Journal on Advances in Signal Processing volume 2022, Article number: 40 (2022)
Abstract
Nonorthogonal multiple access (NOMA) and multipleinput multipleoutput (MIMO) are considered promising techniques to satisfy the demands for high spectrum efficiency and massive connectivity in future wireless communication. In this paper, a novel framework to realize transmission in cellfree massive MIMONOMA system with deep integration between MIMO and NOMA is proposed. A new method is developed to divide users into different groups according to their equivalent path loss, and then “grouplevel successive interference cancellation” (GSIC) is used to cancel the intergroup interference when demodulating the users. Based on the new framework, a rigorous closedform expression of the achievable sum rate in uplink cellfree massive MIMONOMA system is derived. In addition, a parallel iterative method is used to obtain the best power control scheme. Simulations show that the proposed scheme can effectively reduce the total power consumption and outperforms the orthogonal multiple access (OMA) transmission and traditional SICNOMA schemes based on user clustering.
1 Introduction
The Internet of things (IoT) is the “Internet of all things connected,” which is a huge network formed by combining various sensing devices with the network to interconnect people, machines, and things at any time, any space. However, the numbers of IoT terminals and big data services are rapidly growing, and wireless physical layers are facing new challenges [1]. There are the everincreasing demands for massive connectivity and high spectrum efficiency for the upcoming IoT using beyond the fifthgeneration (B5G) and the sixthgeneration (6G) networks; thus, nonorthogonal multiple access (NOMA) and massive multipleinput multipleoutput (MIMO) are considered key techniques [2]. NOMA mainly includes two categories: powerdomain NOMA and codedomain NOMA. This paper focuses on the former. In the powerdomain NOMA, multiple user messages are transmitted on the same resource blocks. At the receiver, the demodulation order of multiple users is determined by the channel state information (CSI), and the successive interference cancellation (SIC) technique is used to remove the interuser interference [3]; hence, the system capacity can be significantly increased. In [4, 5], NOMA was invoked in industrial IoT (IIoT), which satisfies the demands for large connectivity and high spectrum efficiency. According to the waterfilling principle, the maximal power was allocated to the node with the best channel in [5], which determined the demodulation order. However, in the system that combines NOMA with MIMO (MIMONOMA), user channels are in the form of matrices, which fails to determine the demodulation order for users during SIC based on CSI. Thus, research on massive MIMONOMA becomes more challenging.
In the existing studies for MIMONOMA, there are two main categories: user clustering methods and userspecific beamforming methods. In [6], the signal alignment method was used to optimize the MIMONOMA system. However, in the userclustered method based on signal alignment, the angle between equivalent channels after the signal alignment may be very small, for which some useful signals will be lost when the base station equalizer is used to suppress intercluster interference, which causes the problem of high power consumption. The userspecific beamforming method was used in [7]. Unlike the userclustered method, each user in this method had a different beamforming vector, so that the optimization degree of freedom was obviously greater than that of the userclustered method. However, this method failed to reflect the advantages of space division multiple access (SDMA) in MIMONOMA, which implies that SIC must be used among all users that occupy the same resource blocks. Once a user has error decoding, serious error propagation will occur, which greatly influences the performance of the system. Considering the limitations of these works, in [8], a precoding scheme based on the novel grouping was proposed in uplink MIMONOMA systems to minimize power consumption. Grouplevel successive interference cancellation (GSIC) was proposed in a reflecting intelligent surface (RIS)aided uplink NOMA system for the first time in [9]. Users were divided into different groups according to CSI, and the users in the same group were considered as a whole. The GSIC was used to remove intergroup interference by sequentially demodulating group users, while orthogonal multiple access (OMA)/SDMA was used among intragroup users. Wang et al. [8, 9] showed that the MIMONOMA system based on GSIC could reduce the total power consumption.
To the best of our knowledge, no prior literature has studied GSIC in a cellfree massive MIMONOMA system. Unlike traditional cellular networks, cellfree massive MIMO cancels the concept of cells, which enables it to avoid intercell interference [10]. Many distributed access points (APs) jointly provide uniform service to all users using the same resource blocks [11]. In addition, a central processing unit (CPU) is designed to connect all APs through a backhaul network. Then, simple signal processing is performed in each AP, and the CPU conducts complex processing. The distance between AP and CPU is close, which coincides with the demands for low power consumption and ultralow latency required by the IoT. In a cellfree massive MIMONOMA system, optimization based on SIC was deeply investigated in [12,13,14]. In [12], the authors took the first attempt to combine NOMA with cellfree massive MIMO and showed that NOMA could serve more users than OMA. In [13], a low complexity suboptimal userclustered method based on Jaccard coefficients was proposed to improve the sum rates of a cellfree massive MIMONOMA system. Nguyen et al. [14] formulated a max–min qualityofservice (QoS) power control problem in a cellfree massive MIMONOMA system and proved that NOMAbased cellfree massive MIMO could achieve higher power efficiency than OMA. These works were based on the userclustered method, and the users in the same cluster followed the principle of NOMA, while the users among different clusters followed the principle of SDMA. Similar to the singlecell MIMONOMA system, this clusteringbased scheme has the disadvantage of low power efficiency. Since the combination of GSIC and massive MIMONOMA is expected to significantly improve the power efficiency, this paper proposes to apply GSIC to uplink cellfree massive MIMONOMA systems by dividing users into different groups based on their differences in equivalent path loss. Through the transmission of NOMA among different groups and SDMA within each group, the transmission mechanism with deep integration between MIMO and NOMA is realized.
Green communication is the focal topic of future communication systems. With respect to renewable energy, Liu et al. [15] proposed simultaneous wireless information and power transfer (SWIPT). By using wireless information transfer (WIT) and wireless power transfer (WPT) in different time slots and power streams, SWIPT can simultaneously receive information and harvest energy. With respect to resource allocation, Li et al. [16] adopted a dynamic game model to achieve the best relationship between the intercell interference of multibeam satellite systems (MSS) and the power resources of users. In this paper, to optimize the power allocation scheme of a cellfree massive MIMONOMA system based on GSIC, a power optimization problem is formulated by jointly designing transmit power coefficients and equalizers, and a parallel iterative method is invoked to solve the optimization problem. Our contributions are listed as follows:

(1)
A novel framework of grouplevel optimization based on GSIC is proposed for the uplink cellfree massive MIMONOMA system, where users are divided into different groups based on their equivalent path loss for the first time. In the new framework, the intergroup multiple access follows the principle of NOMA, and GSIC is invoked to reduce the intergroup interference, while the intragroup multiple access follows the principle of SDMA. In addition, an achievable sum rate is derived considering the error propagation and intragroup interference.

(2)
A power minimization problem is formulated on the premise of ensuring the QoS and transmit power limit of each user. Using the relationship between the equalizers and the transmit power coefficients, the original joint optimization problem is transformed to a power optimization problem. In addition, the optimal power control scheme is obtained through a parallel iterative method. Simulations show that compared with OMA and SICNOMA based on user clustering, the proposed scheme can effectively reduce the total power consumption. The effect is significant when the number of users is large.
1.1 Notations
Throughout this paper, lowercase boldface letters denote vectors and matrices are represented by uppercase boldface letters. \({\mathbf {z}}^{H}\), \({\mathbf {z}}^{*}\), and \([{\mathbf {z}}]_{k}\) denote the conjugatetranspose, conjugate, and the kth element of vector \({\mathbf {z}}\), respectively. \(\mid \cdot \mid\) indicates the absolute operator. Finally, \({\mathbf {n}}\) \(\sim\) \(\mathcal {CN}(0,1)\) is a complex Gaussian random vector, whose mean is zero and variance is 1.
2 System model
In this paper, the uplink transmission of a cellfree massive MIMONOMA system is considered, where K users and L APs equipped with single antenna are randomly and uniformly distributed in a square area. All APs are connected to a CPU through a backhaul network. The users are divided into M groups based on their equivalent path loss to all APs, and the number of users in a group is \(\mathrm {U}_{\mathrm {k}}\). For convenience, the kth user in the mth group is defined as \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\). The channel gain between \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\) and all APs can be modeled as
with
where \(\varvec{\beta }_{m, k}\) is a diagonal matrix and \(\beta _{l, m, k}\) is the largescale channel gain which changes very slowly. In addition, each element in \({\mathbf {g}}_{m, k}\) independently follows a complex Gaussian distribution with a zero mean and unit variance.
3 Methods
3.1 New method of user grouping
The socalled GSIC implies that the users are divided into different groups following some principle and the users in a group are considered as a whole, which can also be considered a virtual user. After sorting the groups (virtual users), the virtual users can be sequentially demodulated and the intergroup interference can be removed.
To divide the users into multiple groups, this paper proposes a new method of user grouping based on the equivalent path loss of each user. The users in different groups follow the principle of NOMA, using GSIC to remove the intergroup interference, while the users in the same group follow the principle of SDMA. Therefore, two factors are considered during user grouping:

(i)
The number of users in each group: Since the users in the same group follow the principle of SDMA, there should be fewer users in each group than the number of APs, i.e., \(\mathrm {U}_{\mathrm {k}}<L\), which ensures that the users in each group can be demodulated in parallel.

(ii)
The equivalent pass loss of per user: In a singlecell massive MIMONOMA system, users can be divided into different groups according to their path loss to the base station. Motivated by this, we propose a new method of user grouping in a cellfree massive MIMONOMA system. Since users are served by all APs, user grouping can be implemented considering the equivalent path loss to all APs of each user. In other words, users with identical equivalent path loss are divided into a group. The equivalent path loss of \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\) to all APs is defined as follows:
where \(\beta _{l, m, k}\) is the largescale channel gain of \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\). The equivalent path loss of users in different groups should satisfy the following equation:
We present an example of equivalent path loss contours for five APs in Fig. 1, where users with equal equivalent pass loss are in the same contour.
3.2 Uplink data transmission
In the uplink transmission, user messages are simultaneously transmitted to all APs. The received signal at the lth AP can be expressed as
where \({\mathbf {h}}_{m, k}\) is the channel gain vector between \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\) and the lth AP, \(w_{m, k}\) and \(x_{m, k}\) are the transmit power coefficient and transmit signal of \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\), respectively. In addition, \({\mathbf {v}}\) is the noise vector with each element following \(\mathcal {CN}(0,\sigma _{m,k}^2 )\). To derive the uplink achievable sum rate formula, we rewrite the received signal at the lth AP in the form of a matrix:
with
where \({\mathbf {H}}_{m}\) and \({\mathbf {W}}_{m}\) are the channel gain matrix and transmit power coefficient matrix of the mth group, respectively. \({\mathbf {X}}_{m}\) is the transmit signal of users in the mth group.
3.3 Analysis of achievable sum rate
Considering the powerdomain NOMA, the order of demodulation is determined by the signal strength, and the users with higher power are prioritized. Therefore, in the cellfree massive MIMONOMA system based on GSIC, since the equivalent path loss in the first group is the smallest, we first detect the user signals in the first group. Before the user signals in the second group are decoded, all user signals in the first group will be removed, and so on, until the user signals in all groups are demodulated. The comparison between the new framework of GSICNOMA and the tradition SICNOMA based on user clustering is shown in Fig. 2.
Regarding different groups, in the new framework, most of the intergroup interference can be removed using GSIC. However, because of error propagation and the interference from undemodulated groups, some intergroup interference remains for the current demodulated group. Therefore, at the CPU, all user signals in the mth group can be expressed as
where \({\mathbf {E}}_{m}\) is the equalizer at the CPU for the users in the mth group. \(\varvec{\Sigma }_{j}\) is a diagonal matrix and represents the interference of the jth group due to error propagation, which can be written as
In the new framework, different users in each group are distinguished by their respective equalizer \({\mathbf {E}}_{m}\). In addition to the intergroup interference, interference among different users within a group remains, so the signal of \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\) can be expressed as
where \(\left[ {\mathbf {E}}_{m}\right] _{:, k}\) is the kth column of matrix \({\mathbf {E}}_{m}\), which is the equalizer of \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\).
To facilitate the power control constraints in the next subsection, using (12), an achievable sum rate for \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\) can be expressed as follows [10]:
where \({\text {SINR}}_{m, k}\) is the signal to interferenceplusnoise ratio (SINR) of \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\), which reflects the performance of users. \(P_{d}\) and \(P_{I_{i}}(\mathrm {i}=1,2,3)\) are the powers of the desired signal and interference signals, respectively, which can be expressed as follows:
By calculating the achievable sum rate of each user, the achievable sum rate of the uplink cellfree massive MIMONOMA system based on GSIC can be written as
3.4 Power control scheme
Much attention has been given to energyefficient communication in future wireless communication, so the total power consumption of the system is worth studying. In this paper, we optimize the power control scheme, which is subject to the QoS per user and maximum transmission power limit. Therefore, in the cellfree massive MIMONOMA system, the optimization formula of the total power consumption is
where \(R_{m, k}\) and \(P_{\max }\) are the minimum transmission rate and maximum transmission power of \(\mathrm {UE}_{\mathrm {m}, \mathrm {k}}\), respectively. Since the first constraint in the optimization is nonconvex, problem (\(\mathbf {P1}\)) is nonconvex.
To simplify the nonconvex optimization problem (\(\mathbf {P1}\)), we use the relationship between the equalizer and the transmit power. To maximize the SINR per user, in [8], the best linear equalizer is the minimum mean square error (MMSE) equalizer. Therefore, we use the MMSE equalizer to demodulate the user signals in the mth group, which can be expressed as
By substituting (20) into \({\text {SINR}}_{m, k}\) in (13), we can rewrite \({\text {SINR}}_{m, k}\) as
Therefore, the original nonconvex optimization problem (\(\mathbf {P1}\)) is transformed into
After the transformation, the original joint power control and equalizer optimization problem is transformed to a pure power control problem. To obtain the optimal power control scheme, by invoking the contradiction in Remark 1, the best transmit power per user can be expressed as (25).
Equation (25) shows that the optimal transmit powers of different users interact. Therefore, we use a parallel iteration method to achieve the best power control scheme. The socalled parallel iteration refers to obtaining the minimum power consumption of the system through multiple internal iterations and external iterations. Internal iteration refers to user iteration within a group. In the case of fixed intergroup interference, the internal user power within a group is updated according to (25). External iteration is the user iteration between all groups. After all groups complete an internal iteration, an external iteration is performed to determine whether the total power consumption of the system has converged. If it converges, the total power consumption of the system is considered the lowest at this time. The detailed procedure is summarized in Algorithm 1.
As long as the initial transmit power is feasible, the transmit power per user decreases when the number of iterations increases, which implies that the total power consumption of the system will converge to a stationary point [9]. The total computational complexity for Algorithm 1 is \(\sum _{m=1}^{M}(3 M+2 m4){\mathcal {O}} (L^{3})\).
Remark 1
First, a new function \({\mathfrak {R}}_{n, l}\left( w_{n, l}^{2}\right)\) is defined to indicate the effect of current user’s transmit power \(w_{n, l}^{2}\) on \({\text {SINR}}_{m, k}\). As the user’s transmit power \(w_{n, l}^{2}\) increases, the value of function \({\mathfrak {R}}_{n, l}\left( w_{n, l}^{2}\right)\) decreases.
Suppose \(\left\{ {\tilde{w}}_{m, k}^{2}\right\}\) is the optimal solution of (\(\mathbf {P2}\)), then any element \(\left\{ {\tilde{w}}_{m, e}^{2}\right\}\)in the set satisfies the following inequality:
Introducing a new user’s transmit power \({\widehat{w}}_{m, e}^{2}\) satisfies:
We can verify that \({\widehat{w}}_{m, k}^{2}\) also satisfies the constraints in (\(\mathbf {P2}\)):
In addition, as \({\mathfrak {R}}_{n, l}\left( w_{n, l}^{2}\right)\) is a monotonically decreasing function, \({\widehat{w}}_{m, k}^{2}\) also satisfies the QoS:
The above explanation contradicts with the fact that \(\left\{ {\tilde{w}}_{m, k}^{2}\right\}\) is the optimal solution of (\(\mathbf {P2}\)). Therefore, (25) is proved.
4 Results and discussion
In this section, we provide simulation results to assess the performance of cellfree massive MIMONOMA based on GSIC and compare it with OMA and SICNOMA based on user clustering. We consider that L APs and M users with singleantenna are uniformly and randomly distributed in a square of \(\mathrm {D} \times \mathrm {D} \mathrm {m}^{2}\). In this paper, the calculation method of largescale channel gain \(\beta _{l, m, k}\) is found in [13]. Other simulation parameters are presented in Table 1.
First, the effect of the minimum rate requirement for the users in a group on the total transmit power consumption is investigated in this paper. Figure 3 numerically depicts the user rate requirement versus the total transmit power consumption in the cellfree massive MIMONOMA system for L=50, M=2, and M=3. The total power consumption increases when the user rate requirement increases. Compared with OMA and the traditional SICNOMA based on user clustering, the proposed scheme has obvious advantages in total power consumption. The traditional SICNOMA has the worst performance because the userclustered method will make the equivalent channel gain between some users and AP quite small after beamforming, therefore, the system has high transmit power consumption. In the same time slot, compared with OMA, the proposed scheme in this paper can separate twice as many users as OMA. In addition, it removes the intergroup interference through GSIC, which effectively reduces the interuser interference and improves the SINR of the target user; therefore, the total power consumption of the system is reduced.
Next, the impact of the number of APs on the total transmit power consumption is evaluated. If both number of users and number of APs increase in proportion, Fig. 4 shows the total transmit power for different numbers of APs, with \(R_{1, k}\)=0.1 bit/s/Hz, M=2, and M=3. Figure 4 shows that when the GSICNOMA scheme and OMA scheme are used for transmission, when the number of APs increases, the total transmit power consumption of the system decreases. The reason is that when the number of APs increases, multiple users can be better distinguished during the demodulation period because the interference between users is effectively reduced. Therefore, the total transmit power consumption of the system is reduced. However, for the traditional SICNOMA based on user clustering, this benefit diminishes, and the interference between users sharply increases when the number of users increases, which drastically increases the total transmit power consumption of the system.
Figure 5 shows the total transmit power of the GSICNOMA and OMA schemes against the number of groups in the cellfree massive MIMONOMA system with L=50 and \(\mathrm {U}_{\mathrm {k}}\)=50. When the number of groups increases, the total transmit power increases. In addition, compared with OMA, more users are served by GSICNOMA and the proposed GSICNOMA scheme can significantly improve the power efficiency.
5 Conclusion
In this paper, the performance of an uplink cellfree massive MIMONOMA system based on GSIC is investigated. A new method of user grouping according to the equivalent path loss of each user is proposed to combine with GSIC. The intergroup users are distinguished based on the principle of NOMA, while the principle of SDMA is invoked during the intragroup transmission. This scheme realizes transmission with deep integration between MIMO and NOMA. Moreover, we use the relationship between the equalizer and the transmit power coefficient to transform the power optimization problem, and a parallel iterative method is used to optimize the power control of the system. The simulation results show that considering error propagation, compared with OMA and traditional SICNOMA based on user clustering, the proposed scheme can effectively reduce the total transmit power consumption of the system. The effect of GSICNOMA is more remarkable when the number of users is large.
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Abbreviations
 NOMA:

Nonorthogonal multiple access
 MIMO:

Multipleinput multipleoutput
 GSIC:

Grouplevel successive interference cancellation
 OMA:

Orthogonal multiple access
 IoT:

Internet of things
 CSI:

Channel state information
 SIC:

Successive interference cancellation
 SDMA:

Space division multiple access
 RIS:

Reflecting intelligent surface
 AP:

Access point
 SINR:

Signal to interferenceplusnoise ratio
 MMSE:

Minimum mean square error
 WIT:

Wireless information transfer
 WPT:

Wireless power transfer
 SWIPT:

Simultaneous wireless information and power transfer
 MSS:

Multibeam satellite systems
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The authors would like to thank NJUPT for their support and anyone who supported the publication of this paper.
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This work was supported by the National Natural Science Foundation of China (No. 62171235) and by the open research fund of National Mobile Communications Research Laboratory, Southeast University (No. 2021D10).
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Cui, T., Liu, C., Du, P. et al. Power minimization for GSICbased uplink cellfree massive MIMONOMA systems. EURASIP J. Adv. Signal Process. 2022, 40 (2022). https://doi.org/10.1186/s13634022008734
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DOI: https://doi.org/10.1186/s13634022008734
Keywords
 Cellfree massive MIMO
 Nonorthogonal multiple access
 Grouplevel successive interference cancellation
 Power control