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Locationaided uplink transmission for usercentric cellfree massive MIMO systems: a fairness priority perspective
EURASIP Journal on Wireless Communications and Networking volume 2022, Article number: 84 (2022)
Abstract
In this paper, we investigate the uplink transmission for usercentric cellfree massive multipleinput multipleoutput (MIMO) systems. The largestlargescalefadingbased access point (AP) selection method is adopted to achieve a usercentric operation. Under this usercentric framework, we propose a novel intercluster interferencebased (ICIB) pilot assignment scheme to alleviate pilot contamination. Considering the local characteristics of channel estimates and statistics, we propose a locationaided distributed uplink combining scheme to balance the relationship among the spectral efficiency (SE), user equipment (UE) fairness and complexity, in which local partial minimum meansquared error (LPMMSE) combining is adopted for some APs, while maximumratio (MR) combining is adopted for the remaining APs. A corresponding AP selection scheme based on a novel proposed metric representing interuser interference is proposed. We also propose a new fairness coefficient taking SE performance into account to indicate the UE fairness. Moreover, the performance of the proposed scheme is investigated under fractional power control and max–min fairness (MMF) power control. Simulation results demonstrate that the channel estimation accuracy of our proposed ICIB pilot assignment scheme outperforms that of the conventional pilot assignment schemes. It is also shown that compared with the benchmark LPMMSE combining, the proposed locationaided combining trades 13.45\(\%\) average SE loss for 26.61\(\%\) UE fairness improvement and 28.58\(\%\) complexity reduction when \(\gamma = 0.6\). And by adjusting the threshold \(\gamma\), a good tradeoff between the average SE, UE fairness and computational complexity can be provided by the proposed scheme. Furthermore, the proposed scheme with fractional power control can better demonstrate the advantages of tradeoff performance than MMF power control and full power transmission.
1 Introduction
Cellfree massive multipleinput multipleoutput (MIMO) is a promising technique for beyond5G and 6G networks due to its ability to provide a high spectral efficiency (SE) and tremendous macrodiversity with simple signal processing [1,2,3]. In cellfree massive MIMO systems, lots of access points (APs) jointly provide services to all user equipments (UEs) in the same timefrequency resource by utilizing the centralized coordination of the central processing unit (CPU), which is connected to all APs via fronthaul links [4].
The early studies on cellfree massive MIMO assumed that all APs serve all UEs in the network [5, 6]. Unfortunately, this assumption may lead to networkwide fronthaul signaling and data sharing, resulting in huge computational complexity, which is impractical for actual large networks [7]. To overcome this limitation, a usercentric framework has been introduced into cellfree massive MIMO; the main idea of this framework is that each UE is served by a subset of APs with the best channel conditions to improve the service efficiency of the system [8,9,10].
1.1 Motivation
In cellfree massive MIMO, there are two data transmission implementations consisting of the centralized and distributed signal processing, which are characterized by different degrees of cooperation among the APs [11]. Take the uplink as an example, in distributed signal processing, the channel estimation and receive combining are done at the APs, and data detection is done at the CPU, while in centralized processing, all operations mentioned above are done at the CPU, which leads to increased fronthaul load and deployment cost compared with the former. Nowadays, the distributed signal processing has been widely investigated due to its advantages of saving fronthaul overhead and flexible deployment [7, 11, 12]. A representative framework for scalable cellfree massive MIMO exploiting dynamic cooperation cluster concept was proposed in [7], and the uplink/downlink SE performance with maximumratio (MR) or local partial minimum meansquared error (LPMMSE) combining/precoding was studied, respectively.
There methods mentioned above are fully distributed and widely used in cellfree massive MIMO systems. However, the shortcomings of each scheme are also obvious. For example, the SE performance of MR combining/precoding is poor, although it has low complexity and good fairness among UEs. On the contrary, LPMMSE combining/precoding outperforms the standard MR method in terms of SE performance at the cost of increased computational complexity. Meanwhile, the UE fairness of these methods cannot be ensured effectively [7, 11]. These motivate us to explore a novel transmission scheme that can make an effective compromise between SE, UE fairness and complexity to improve comprehensive system performance.
Driven by the continuously increasing demands for high system throughput, low latency and improved fairness, locationaware communication for beyond5G and 6G networks has attracted lots of research interest since location information can be leveraged in wireless network design and optimization to complement existing technological developments [13]. Currently, lots of papers have focused on improving system performance or simplify system design for cellular massive MIMO systems by using location information obtained by means of global positioning system (GPS) or wireless positioning technologies [14], which can be classified into two categories: The first is to investigate pilot assignment schemes for alleviating pilot contamination [15, 16], and the second is devoted to the design of locationaided transmission schemes [17, 18]. Considering in cellfree massive MIMO, user location is closely related to its signal spatial structure, a natural thought is the following: Can we utilize location information to design and investigate the compromise transmission scheme mentioned above?
1.2 Related work
Currently, there are existing researches that have striven to design pilot assignment schemes for cellfree massive MIMO systems using location information [19, 20]. In [19], a structured pilot assignment scheme based on location information was proposed to maximize the minimum distance between UEs assigned to the same pilot by clustering all UEs. And the authors in [20] proposed a locationbased greedy (LBG) pilot assignment scheme; the difference between the proposed LBG pilot assignment scheme and the greedy pilot assignment scheme in [1] is that the initial pilot allocation is performed according to the location before applying the greedy algorithm.
On the other hand, there has been a broad amount of work on the design and study the transmission scheme of the cellfree massive MIMO [21,22,23,24,25,26,27], where most of them focus on the precoding in downlink. Specifically, in [21], the authors investigated the effect of channel hardening on system performance cellfree massive MIMO and compared the SE performance under the conjugate beamforming (CB) and normalized CB (NCB) schemes. Then, the work in [22, 23] proposed a novel enhanced normalized CB (ECB) precoding scheme for cellfree massive MIMO and studied the performance of NCB under different power allocation schemes. The above CB and its variants greatly boost the channel hardening enabling the UEs, but neglect the impact of interference between UEs. To further improve the SE performance, the authors in [24] proposed two distributed precoding schemes, referred to as local partial zeroforcing (PZF) and local protective PZF (PPZF), which can provide an effective tradeoff between suppressing interference and maintaining strong desired signal powers. In [25], a semidistributed precoding called joint maximumratio and zeroforcing (JMRZF) was proposed. The main idea is that part of APs are combined to perform centralized ZF (CZF), while other APs apply simple maximumratio transmission (MRT). In [26], the authors investigated the uplink transmission performance based on successive interference cancellation (SIC) for usercentric cellfree massive MIMO systems. The work in [27] was studied based on [24] but for uplink combining and proposed fullpilot ZF (FZF), partial FZF (PFZF), protective weak PFZF (PWPFZF) and local regularized ZF (LRZF) combining schemes, which can be performed in a fully distributed, coordinated and scalable fashion. However, the above schemes can only suppress the selfinterference of each AP, and its performance is limited by the interference between APs.
1.3 Contributions
Motivated by these observations, we introduce location information of UEs into the design of uplink transmission scheme for usercentric cellfree massive MIMO systems in this paper, which aims to design the transmission scheme to balance the SE, UE fairness and complexity for improving the comprehensive performance of the system. A twolayer decoding method is utilized to mitigate interuser interference, where each AP computes a local estimate of the data signals in the first layer, and then, the largescale fading decoding (LSFD) scheme is used to combine these estimates at the CPU in the second layer. In addition, to achieve the usercentric architecture, the largestlargescalefadingbased AP selection method is adopted. More specifically, the main contributions are listed as follows.

1.
We propose a novel intercluster interferencebased (ICIB) pilot assignment scheme for reducing pilot contamination that takes into account the interference of UEs using the same pilot while being served by the same AP.

2.
We propose a locationaided distributed uplink combining scheme. In this scheme, a new metric based on location information and the AP intercluster interference to indicate the degree of interference between UEs is first proposed to select UEs with more severe interference. Second, the APs that provide service to the above UEs suffering from severe interference adopt LPMMSE combining, while MR combining is adopted for the remaining APs.

3.
We propose a new fairness coefficient that takes SE performance into consideration to compare the UE fairness of each combining scheme more reasonably.

4.
We investigate the performance of the proposed combining scheme under different power control schemes. The results demonstrate that the proposed scheme with the fractional power control can further improve the tradeoff performance compared with max–min fairness (MMF) and full power transmission.
The remainder of this paper is organized as follows. Section describes the system model. Section presents the proposed ICIB pilot assignment scheme and the locationaided uplink distributed combining scheme. Additionally, the complexity analysis and the power control schemes are also discussed in this section. Section provides results and discussion. Finally, the conclusions are summarized in Sect. .
Notation: Lowercase boldface symbols \({{\mathbf {x}}}\) and uppercase boldface symbols \({{\mathbf {X}}}\) denote vectors and matrices, respectively. \(\left( \cdot \right) ^{  1}\), \(\left( \cdot \right) ^{\mathrm{T}}\) and \(\left( \cdot \right) ^H\) denote the inverse, transpose and conjugate transpose, respectively. \(\left\lceil \cdot \right\rceil\) denotes the ceiling function. The expectation and variance operation are denoted as \({\mathbb {E}}\left\{ \cdot \right\}\) and \({\text {Var}}\left( \cdot \right)\), respectively. \(\left {{\mathbf {x}}} \right\) and \(\left\ {{\mathbf {x}}} \right\\) represent the norm and Euclidean norm, respectively, of vector \({{\mathbf {x}}}\). We also denote the cardinality and the nth element of the set \({\mathcal {B}}\) by \(\left {\mathcal {B}} \right\) and \({\mathcal {B}}\left( n \right)\), respectively.
2 System model
We consider an usercentric cellfree massive MIMO system with N APs and K singleantenna UEs randomly distributed in a large area, where each UE is served by a subset of APs, as shown in Fig. 1. The different colored regions represent AP clusters serving different UEs. Notably, the AP clusters partially overlap, which is a typical feature of a usercentric cellfree architecture. The largestlargescalefadingbased AP selection scheme in [28] is utilized to reduce the requirement for backhaul connection, where the subset of APs serving UE k and the predefined threshold for the UE to select APs are denoted by \({\mathcal {M}}_k\) and \(\varepsilon\), respectively. Moreover, each AP is equipped with a uniform linear array (ULA) containing M antenna elements and connected via an errorfree fronthaul link to a CPU, which facilitates data sharing and centralized operation for resource allocation tasks between all APs. The uplink transmission is considered in this paper, and each coherence block consists of \(\tau _{\mathrm{c}}\) samples, where \(\tau _{\mathrm{p}}\) is used for uplink pilots and \(\tau _{\mathrm{c}}  \tau _{\mathrm{p}}\) is used for uplink data transmission.
Then, we assume that there are numerous scatterers in the coverage area, and the propagation of a signal from an AP to a UE is composed of L paths. From the narrowband multipath transmission model, the \(M \times 1\) channel vector between the nth AP and the kth UE is given by [29, 30]
where \(\alpha _{nk}^l\) is the complex gain of the lth path distributed as \(\alpha _{nk}^l \sim {\mathcal {N}}_{\mathbb {C}} \left( {0,1} \right)\) and \({{\mathbf {a}}}\left( {\theta _{nk}^l } \right)\) is the array steering vector for path l, which can be expressed as
where \({\theta _{nk}^l }\) is the angle of arrival (AOA) of the lth path from the nth AP to the kth UE, \(\lambda\) is the signal wavelength, and d denotes the antenna spacing, which is usually assumed to be fixed. \(\beta _{nk}\) is the largescale fading coefficient.
The main system parameters are summarized in Table 1.
3 Methods
3.1 Intercluster interferencebased pilot assignment and channel estimation
In the uplink pilot training phase, we assume that a set of \(\tau _p\) mutually orthogonal uplink pilot sequences \(\varvec{\phi } _1 , \ldots ,\varvec{\phi } _{\tau _{\mathrm{p}} }\) are assigned to K UEs, where \(\varvec{\phi } _t \in {\mathbb {C}}^{\tau _{\mathrm{p}} \times 1}\) and \(\left\ \varvec{\phi }_t \right\ ^2 = \tau _{\mathrm{p}}\), for \(t \in \left\{ {1, \ldots ,\tau _{\mathrm{p}} } \right\}\), with \(\tau _{\mathrm{p}}\) being the number of uplink training samples in each coherence interval. Due to the limited length of the coherence interval, the practical scenario is a large network with \(K > \tau _{\mathrm{p}}\) so that a pilot sequence may be assigned to multiple different UEs. Such pilot reuse among different UEs may lead to a decrease in the channel estimation accuracy, known as pilot contamination. To mitigate pilot contamination effect, we propose a novel pilot assignment scheme based on intercluster interference. The main idea of this pilot assignment scheme is that orthogonal pilots are assigned to the UEs with large values according to the number of common APs serving UEs, where the number of common APs can indicate the degree of interference between AP cluster. Since the more common serving APs, the interference between UEs is more serious. Subsequently, pilots are assigned to the remaining UEs based on the largescale fading coefficient and the service relationship between the AP and UE. Specifically, the detailed steps of the proposed ICIB pilot assignment scheme are presented as follows:

1.
The CPU structures a matrix \({{\mathbf {S}}} \in {\mathbb {R}}^{K \times K}\) that represents the reuse of APs that provide service for different UEs based on \(\left\{ {{\mathcal {M}}_k } \right\}\), where the element of \({{\mathbf {S}}}\) is given by
$$\begin{aligned} {{\mathbf {S}}}_{ij} = \left\{ {\begin{array}{*{20}c} {\left {{\mathcal {M}}_i \cap {\mathcal {M}}_j } \right ,} &{} {{\text {if}}\;i \ne j} \\ {\infty ,} &{} {{\text {else}}} \\ \end{array} } \right. . \end{aligned}$$(3)The nondiagonal elements of \({{\mathbf {S}}}\) represent the number of common APs serving UEs, and \({{\mathbf {S}}}\) is a symmetric matrix; thus, we focus on the \(K\left( {K  1} \right) /2\) entries above the main diagonal.

2.
The CPU extracts the upper triangular elements of \({{\mathbf {S}}}\) and sorts them in descending order, denoted as \(\left\{ {{\tilde {{\mathbf{S}}}}_{ij}^r :i < j,r \in \left\{ {1, \ldots ,K\left( {K  1} \right) /2} \right\} } \right\}\). According to the sorting results, the corresponding index of UEs can be listed nonrepetitively. The first \(\tau _{\mathrm{p}}\) indexes are placed in index set \({\mathcal {F}}\), while the remaining indexes are placed in index set \({\mathcal {S}}\). Then, we assign \(\tau _{\mathrm{p}}\) orthogonal pilots to the UEs whose indices are elements in set \({\mathcal {F}}\).

3.
For the remaining UE \(k'\), \(k' \in {{\mathcal {S}}}\), when it uses the pilot t, \(t \in \left\{ {1, \ldots ,\tau _{\mathrm{p}} } \right\}\) one after another, we first calculate the sum of the average channel gains between the APs serving UE \(k'\) and the UEs that have already been assigned pilot t and served by these APs. Then, the pilot with the least interference at the APs in the cluster serving UE \(k'\) can be found and assigned to UE \(k'\).
The ICIB pilot assignment is summarized in Algorithm 1.
To further mitigate the interference among UEs caused by pilot reuse, we adopt the minimum meansquareerror (MMSE) channel estimation method. According to [31], the MMSE estimate \({{\hat{\mathbf {h}}}}_{nk}\) can be given by
where \(P_k\) denotes the transmit power of UE k, \({{\mathbf {R}}}_{nk} = {\mathbb {E}}\left\{ {{{\mathbf {h}}}_{nk} {{\mathbf {h}}}_{nk}^H } \right\}\) denotes the correlation matrix of channel \({{{\mathbf {h}}}_{nk} }\), \({{\mathbf {y}}}_{nk}^p\) is the inner product of the received signal of AP n with the pilot \(\varvec{\phi } _k\), and \({\mathbf {\Phi }}_{nk} = \left( {\sum \nolimits _{i \in {\mathcal {P}}_k } {P_i \tau _{\mathrm{p}} {{\mathbf {R}}}_{ni} } + {{\mathbf {I}}}_M \sigma ^2 } \right) ^{  1}\), in which \({{\mathcal {P}}_k }\) denotes the set of UEs that uses the same pilot as UE k, and \(\sigma ^2\) is the noise power. Moreover, the estimation error \({\tilde {\mathbf{h}}}_{nk} = {{\mathbf {h}}}_{nk}  {{\hat{\mathbf {h}}}}_{nk}\) has the correlation matrix \({{\mathbf {C}}}_{nk} = {\mathbb {E}}\left\{ {{\tilde{{\mathbf{h}}}}_{nk} {\tilde {{\mathbf{h}}}}_{nk}^H } \right\} = {{\mathbf {R}}}_{nk}  P_k \tau _{\mathrm{p}} {{\mathbf {R}}}_{nk} {\mathbf {\Phi }}_{nk} {{\mathbf {R}}}_{nk}\).
3.2 Locationaided uplink data transmission
In this subsection, we focus on uplink data transmission and propose a novel locationaided uplink combining scheme, which is described in detail in the following.
3.2.1 Distributed uplink transmission
During uplink data transmission, the received signal \({{\mathbf {y}}}_n^u \in {\mathbb {C}}^M\) at the nth AP is given by
where \({s_i }\) is the transmitted signal from UE i with power \(P_i\), and \({{\mathbf {n}}}_n \sim {\mathcal {N}}_{\mathbb {C}} \left( {{{\mathbf {0}}}_M ,{{\mathbf {I}}}_M \sigma ^2 } \right)\) is the received noise at the AP n. The distributed uplink transmission based on a twolayer decoding is considered in this paper. Specifically, in the first layer, AP n selects the combining vector \({{\mathbf {v}}}_{nk}\) and calculates the local data estimate \({\hat{s}}_{nk}\) by using the local channel estimates. Then, in the second layer, the local data estimates \({\hat{s}}_{nk}\) are sent to the CPU where they are fused into a final estimate of the UE data by using LSFD coefficients. Under the usercentric architecture, \({\hat{s}}_{nk}\) can be mathematically expressed as
where \({{\mathbf {D}}}_{nk} = \left\{ {\begin{array}{*{20}c} {{{\mathbf {I}}}_M ,} &{} {k \in {\mathcal {K}}_n } \\ {{{\mathbf {0}}}_M ,} &{} {k \notin {\mathcal {K}}_n } \\ \end{array} } \right.\) is the matrix denoting which UEs are served by which APs.
Next, the local data estimates of the APs serving UE k are collected at the CPU, after which the CPU performs the secondlayer decoding by computing the LSFD weighted signal as
where \(\mu _{nk}\) is the LSFD coefficient for AP n and UE k. For ease of presentation, we define \({{\mathbf {g}}}_{ki} = \left[ {{{\mathbf {v}}}_{1k}^H {{\mathbf {D}}}_{1k} {{\mathbf {h}}}_{1i} , \ldots ,{{\mathbf {v}}}_{Nk}^H {{\mathbf {D}}}_{Nk} {{\mathbf {h}}}_{Ni} } \right] ^{\mathrm{T}}\) as the Ndimensional vector with the receivecombined channels between UE k and all APs that serve UE k. Then, (7) can be expressed as
where \({\varvec{\mu }}_k = \left[ {\mu _{1k} , \ldots ,\mu _{Nk} } \right] ^{\mathrm{T}} \in {\mathbb {C}}^N\) is the LFSD weighting coefficient vector and \(\left\{ {{\varvec{\mu }}_k^H {{\mathbf {g}}}_{ki} :i = 1, \ldots ,K} \right\}\) denotes the effective channels.
By invoking the arguments as described in [11], the achievable uplink SE of UE k can be calculated as
with \({{\text {SINR}}_k }\) is given by
where \({{\mathbf {F}}}_k = \sigma ^2 {\text {diag}}\left( {{\mathbb {E}}\left\{ {\left\ {{{\mathbf {D}}}_{1k} {{\mathbf {v}}}_{1k} } \right\ ^2 } \right\} , \ldots ,{\mathbb {E}}\left\{ {\left\ {{{\mathbf {D}}}_{Nk} {{\mathbf {v}}}_{Nk} } \right\ ^2 } \right\} } \right) \in {\mathbb {R}}^{N \times N}\).
The effective SINR in (10) for UE k can be further maximized by
where \({{\tilde{\mathbf{D}}}}_k \in {\mathbb {R}}^{N \times N}\) is the diagonal matrix with the \(\left( {n,n} \right)\)th element being one if \(n \notin {\mathcal {M}}_k\) and zero otherwise. This leads to the maximum value
3.2.2 Proposed combining scheme
In this subsection, we propose a locationaided combining with an efficient AP selection scheme. The principle of locationaided combining is that only a subset of APs that serves UEs with severe interference performs LPMMSE combining, while the remaining APs perform MR combining, which can achieve an effective balance between the SE performance, UE fairness and complexity. The operation steps of the proposed combining scheme are as follows:

1.
The CPU first calculates the distances \(\left\{ {d_{ij} :i,j \in \left\{ {1, \ldots ,K} \right\} ,i < j} \right\}\) between the UEs according to the obtained location information.

2.
Next, we define a novel metric that indicates the degree of interference between UEs based on the distance and the number of common serving APs:
$$\begin{aligned} \Lambda _{ij} = \omega /d_{ij} + {{\mathbf {S}}}_{ij} , \end{aligned}$$(13)where \(\omega\) denotes the predefined coefficient that balances the relationship between the two metrics. The novel metric described above can be computed and sorted in descending order to obtain the sorted results \(\left\{ {{\tilde{\Lambda }} _{ij}^r :i < j,r \in \left\{ {1, \ldots ,K\left( {K  1} \right) /2} \right\} } \right\}\). Then, we define the set \({\mathcal {G}} = \emptyset\) in preparation for the following steps.

3.
For \({{\tilde{\Lambda }} _{ij}^r }\), \(r = 1, \ldots ,K\left( {K  1} \right) /2\), we can find the UE corresponding to \({\tilde{\Lambda }_{ij}^r }\) and the APs that provide service to these UEs. Then, the above AP index can be inserted into set \({\mathcal {G}}\) one after another until \(\left {\mathcal {G}} \right \geqslant \left\lceil {N\gamma } \right\rceil\) holds, where \(0 \leqslant \gamma \leqslant 1\) denotes the predefined threshold.

4.
According to the obtained set \({\mathcal {G}}\) in the previous step, the APs in set \({\mathcal {G}}\) adopt LPMMSE combining scheme, while the remaining APs adopt MR combining scheme, which can be expressed as
$$\begin{aligned} {{\mathbf {v}}}_{nk}^{{\text {LA}}} = \left\{ {\begin{array}{*{20}c} {{{\mathbf {v}}}_{nk}^{{\text {LPMMSE}}} ,} &{} {n \in {\mathcal {M}}_k ,n \in {\mathcal {G}}} \\ {{{\mathbf {v}}}_{nk}^{{\text {MR}}} ,} &{} {n \in {\mathcal {M}}_k ,n \notin {\mathcal {G}}} \\ \end{array} } \right. , \end{aligned}$$(14)where \({{\mathbf {v}}}_{nk}^{{\text {LPMMSE}}} \! = \! P_k\! \left( {\sum \limits _{i \in {\mathcal {K}}_n } {\!P_i \left( {{{\hat{\mathbf {h}}}}_{ni} {{\hat{\mathbf {h}}}}_{ni}^H \!+\! {{\mathbf {C}}}_{ni} } \right) \!\!+\! {{\mathbf {I}}}_M \sigma ^2 } } \!\!\right) ^{  1} \!\!\!\!\!\!{{\mathbf {D}}}_{nk} {{\hat{\mathbf {h}}}}_{nk}\) and \({{\mathbf {v}}}_{nk}^{{\text {MR}}} = {{\mathbf {D}}}_{nk} {{\hat{\mathbf {h}}}}_{nk}\), respectively.
The locationaided combining is given in Algorithm 2.
For the proposed locationaided combining scheme, the APs that provide service to UEs are divided into two parts, one of which is detected by adopting LPMMSE combining, while the other by MR combining. Note that the closed form of the expectations in (10) and (12) cannot be computed when the APs using LPMMSE combining, but the expectations can be obtained by using Monte Carlo simulations easily. However, the expectations in (10) can be computed in closed form when the APs use MR combining, which is derived in the following.
Corollary
If the APs adopt MR combining, i.e., \({{\mathbf {v}}}_{nk} = {{\mathbf {D}}}_{nk} {{\hat{\mathbf {h}}}}_{nk}\) with \(n \in {\mathcal {M}}_k\), \(n \notin {\mathcal {G}}\), then the expectations in (10) can be given by
with \(\left[ {{\mathbb {E}}\left\{ {{{\mathbf {g}}}_{ki} {{\mathbf {g}}}_{ki}^H } \right\} } \right] _{nr} = \left[ {{\mathbb {E}}\left\{ {{{\mathbf {g}}}_{ki} } \right\} } \right] _n \left[ {{\mathbb {E}}\left\{ {{{\mathbf {g}}}_{ki}^* } \right\} } \right] _r\) for \(n \ne r\) and
while
Proof
It follows the similar approach as in [11, 21,22,23]. \(\square\)
3.3 Complexity analysis
In this subsection, we present complexity analysis for different distributed combining schemes. The complexity of a combining scheme originates mainly from the calculation of the combining vector. This is because the acquisition of the estimated channel and the secondlayer LSFD decoding are the same for each combining scheme, and the computational complexity in this process is not taken into consideration.
Next, we focus on the computational complexity of the combining vector for the proposed combining scheme.
More specifically, recalling that all APs are divided into two parts for the proposed combining scheme, hence the computational complexity of the proposed combining scheme should be analyzed with regard to these two aspects, respectively.
Proposition
The computational complexity of the locationaided uplink combining scheme is given by
Proof
Since the combining vector is computed by using elementary matrix operations, we can calculate the computational complexity with the help of the framework shown in [31]. Only complex multiplications and divisions are considered, while additions and subtractions can be neglected since these are much less complex. Specifically, for the APs \(n \in {\mathcal {G}}\), the complexity can be computed resorting to [31, Lemma B.1., B.2.]. On the other hand, for the remaining APs, when the AP serves at least one UE, MR combining is adopted. Since only the channel estimates are used for signal combining in this case, no additional calculations are required. Based on the above analysis, the computational complexity of the proposed scheme in proposition can be easily derived. \(\square\)
Remark
Note that (18) does not consider the calculation of distances between UEs. The main reasons for this are as follows. First, the calculation amount of distances between UEs are smaller than those of the combining vector. Second, the locations of UEs can be treated as unchanged within several or even tens of channel coherence blocks since the channel coherence time is short, whereas the position of a UE and its surrounding environment may not physically change in a comparable period of time. Hence, unlike combining vectors, it is not necessary to frequently calculate the distances between UEs. Based on this, the calculation of distance between UEs is negligible for the complexity of the proposed scheme.
Next, we summarize expressions for the computational complexity of other different combining schemes, obtained using the same methodology as the above analysis and shown in Table 2 at the top of the next page, where \({{\mathcal {S}}_n }\) and \({{\mathcal {W}}_n }\) represent the strong UEs set and weak UEs set in PWPFZF combining scheme, respectively, and \(\tau _{_{{\mathcal {S}} _n} }\) denotes the number of different pilots used by the UEs \(k \in {\mathcal {S}}_n\). Besides, \({\mathbb {I}}\left( {\text {E}} \right)\) is the indicator function, and it is equal to one if the logic statement \({\text {E}}\) is true.
3.4 Power control
Pragmatic power control is needed in practical implementations of cellfree massive MIMO systems. In the following, we introduce two power control schemes which are widely used in cellfree massive MIMO.
3.4.1 Max–min fairness power control
MMF power control consists in optimizing the SE achieved by the weakest UE to provide uniform service throughout the network [1, 5]. The MMF power control problem can be formulated as:
where \(P_{\max }\) denotes the maximum transmit power of UE. Without loss of generality, the problem given in (19) can be rewritten by introducing an auxiliary variable w:
Then, to improve the convergence rate, we equivalently replace the problem in (20) with a problem having the same constraints but where the total power is minimized, which is similar to [5], given by
Finally, if w is fixed, (21) is a convex program, which can be solved by using bisection search method with the help of the CVX toolbox [11].
3.4.2 Fractional power control
Fractional power control is a classical heuristic in the cellfree massive MIMO systems, which utilizes the local longterm channel statistics to balance performance for UEs with different channel gain [24]. In this paper, we also consider the distributed fractional power control scheme, which the power of UE k is given by [11]
where \(\chi\) denotes the power control behavior. Specifically, if \(\chi =\) 0, the fractional power control degenerates to full power transmission scheme, i.e., \(P_k = P_{\max }\), \(\forall k \in {\mathcal {K}}\).
4 Results and discussion
In this section, we evaluate the performance of the proposed ICIB pilot assignment scheme in comparison with that of the existing pilot assignment schemes, and we present simulation results to investigate the performance of the locationaided uplink combining scheme in terms of the SE performance, UE fairness and computational complexity, respectively. We consider a simulation scenario where all APs and UEs are randomly distributed in 300 \(\times\) 300 \(\hbox {m}^2\). The largescale fading coefficient \(\beta _{nk}\) models the path loss and shadow fading, as follows
where \({\text {PL}}_{nk}\) denotes the path loss, and \(10^{\frac{{\sigma _{sh} z_{nk} }}{{10}}}\) represents the lognormal shadow fading with standard deviation \(\sigma _{sh}\) and \(z_{nk} \sim {\mathcal {N}}_{\mathbb {C}} \left( {0,1} \right)\). The path loss follows the 3GPP Urban Microcell model in [24, 32], which assumes a 2 GHz carrier frequency, and is given by
where \({d_{nk} }\) denotes the distance between AP n and UE k including AP and UE’s heights. The shadow fading accounts for spatial correlations between APs and between UEs and follows
where \(a_n \sim {\mathcal {N}}_{\mathbb {C}} \left( {0,1} \right)\) and \(b_k \sim {\mathcal {N}}_{\mathbb {C}} \left( {0,1} \right)\) are independent random variables modeling the shadow fading impact on the channels from AP n to all the UEs and from UE k to all the APs, respectively, and \(\vartheta\) is the weighting parameter. The shadowing terms are correlated as
where \({d_{ni}^{{\text {AP}}} }\) is the distance between AP n and AP i, \({d_{kj}^{{\text {UE}}} }\) is the distance between UE k and UE j, and 9 meters is the decorrelation distance [24]. The standard deviation is set to \(\sigma _{sh} = 4\) dB, and the heights of AP and UE are set as 10 m and 1.6 m, respectively. The channel bandwidth \(B =20\) MHz. The ULA antenna spacing is \(d = \frac{\lambda }{2}\). Each coherence block contains \(\tau _{\mathrm{c}} =\) 100 samples. Unless specified, the fractional power control is utilized for transmission, where \(\chi =0.3\). Furthermore, the AP selection threshold and the predefined coefficient are set to \(\varepsilon =\) 90\(\%\) and \(\omega =\) 1000, respectively. The other specific parameters are summarized in Table 3.
Then, we use the normalized meansquared error (NMSE) to evaluate the performance of channel estimation. Considering the usercentric framework adopted in this paper, the expression of NMSE can be given by [33]
Figure 2 compares the NMSE of the channel estimation performance versus SNR with different pilot assignment schemes. Clearly, the proposed ICIB pilot assignment scheme is superior to the existing pilot assignment schemes such as random, greedy, LBG and basic pilot assignment (BPA) schemes. There are two reasons for this superiority. First, \(\tau _{\mathrm{p}}\) mutually orthogonal pilots are assigned to \(\tau _{\mathrm{p}}\) UEs with severe potential interference, rather than randomly. Second, for the UEs remaining after these orthogonal pilots are assigned, we consider the interference between the corresponding UE and the UEs served by the common APs that use the same pilot, and the interference relationship between UEs is used to assign pilots to the maximum extent under the usercentric framework. Furthermore, the NMSE performance is improved with the increase of SNR for all schemes and we can improve the channel estimation accuracy by increasing SNR appropriately.
Figure 3 depicts the comparison of perUE uplink SE, average SE and UE fairness of the proposed combining scheme with other combining schemes under different thresholds \(\gamma\), where the UE fairness is measured by the proposed fairness coefficient, which is inspired by [34] and taken the average SE into account, and defined as \({\text {Var}}\left( {R_k^u } \right) /\sqrt{{\mathbb {E}}\left\{ {R_k^u } \right\} }\). FZF, PWPFZF and LRZF adopt the same random pilot assignment as [27]. And MR and LPMMSE utilize the same BPA as [7]. While regarding the proposed locationaided combining scheme, we adopt the ICIB pilot assignment scheme. From Fig. 3a, we observe that the perUE uplink SE of our proposed locationaided combining is improved with the increase of \(\gamma\). This is intuitive since a larger threshold \(\gamma\) means that more APs use LPMMSE combining to suppress interuser interference. We can also observe in Fig. 3a that the locationaided combining gives a comparable 95\(\%\)likely SE to LPMMSE, and higher than that of MR, FZF, PWPFZF and LRZF. As shown in Fig. 3b, c, in terms of average SE, the proposed scheme is inferior to LPMMSE and LRZF, but better than MR, FZF and PWPFZF, while on the aspect of UE fairness, the above result is reversed. Take the LPMMSE among five benchmarks as an example, when \(\gamma = 0.6\), the proposed scheme trades 13.45\(\%\) average SE loss for 26.61\(\%\) UE fairness improvement compared with LPMMSE combining. These observations indicate that the proposed scheme can provide a good tradeoff between the UE fairness and average SE.
Figure 4 compares the perUE uplink SE, average SE and UE fairness of the proposed locationaided combining scheme with another benchmark for different thresholds \(\gamma\), where the difference between the benchmark and the proposed scheme lies in the metric used to select APs. Specifically, similar to (13), we define a metric \(\Omega _{ij,n} = {\mathbb {E}}\left\{ {{{\mathbf {h}}}_{ni}^H {{\mathbf {h}}}_{nj} } \right\} ,i \ne j\) based on the channel correlation, which represents the degree of interference between different UEs relative to AP n. The subsequent AP selection process is the same as the proposed scheme except for the selection metric. For convenience of expression, this benchmark can be called the channel correlationbased combining scheme. As shown in Fig. 4a, the proposed combining scheme can provide a comparable perUE uplink SE to the channel correlationbased combining regardless of \(\gamma = 0.5\) or \(\gamma = 0.7\). From Fig. 4b, c, the proposed scheme is also comparable to the channel correlationbased combining in terms of average SE, while in terms of UE fairness, the channel correlationbased combining is even slightly inferior to the proposed scheme. The AP selection metric proposed in (13) not only considers the distance between UEs, but also considers the common AP service situation determined by the largestlargescalefadingbased AP selection scheme. From this point of view, the proposed locationaided scheme also takes the impact of largescale fading into account. Moreover, the metric of AP selection for the channel correlationbased combining needs to be obtained through longterm observation and calculation, which requires more time and computational overhead compared to the acquisition of location information and common AP situation in the proposed scheme. Based on the above analysis, the proposed locationaided combining scheme is reasonable and meaningful.
Figure 5 shows the average SE and the number of complex multiplications of different combining schemes versus \(\gamma\). Similarly, the random pilot assignment is adopted for FZF, PWPFZF and LRZF, the BPA is adopted for MR and LPMMSE, while the ICIB scheme is used for the proposed locationaided combining. As shown in Fig. 5a, b, both average SE and the number of complex multiplications of proposed scheme grow linearly with the increase of \(\gamma\), whereas those of the five other schemes remain unchanged. Specifically, compared with LPMMSE combining scheme, the proposed scheme can reduce the complexity by 28.58\(\%\) at the cost of average SE only losing 13.45\(\%\) when \(\gamma = 0.6\), while \(\gamma = 0.5\), the SE loss is 18.2\(\%\), and the complexity can be greatly reduced by 42.28\(\%\). Therefore, the proposed combining scheme can select an appropriate \(\gamma\) to balance the uplink SE and computational complexity according to the actual requirements of the network.
Figure 6 compares perUE uplink SE, average SE and UE fairness with different power controls for proposed combining scheme. We use the proposed ICIB pilot assignment as before. In Fig. 6a, we can see that when \(\gamma = 0.4\) or \(\gamma =0.6\), the SE distribution with fractional power control both outperforms the MMF power control and full power transmission in terms of 95\(\%\)likely SE. This is because that fractional power control allows UEs with worse channel conditions to transmit with higher power when \(\chi =0.3\). In addition, the MMF power control can provide the best uniformly service than the other two schemes and thus can achieve best UE fairness, as shown in Fig. 6c. However, we can observe in Fig. 6b that the average SE of the proposed scheme with MMF power control is much less than the other schemes. Moreover, regardless of the value of \(\gamma\), the fractional power control outperforms the full power transmission in both average SE and UE fairness, which verifies the advantages of fractional power control for the proposed locationaided combining scheme.
5 Conclusion
This paper investigates the uplink transmission of cellfree massive MIMO systems based on a usercentric topology. The largestlargescalefadingbased AP selection method is adopted, and a novel pilot assignment scheme based on the intercluster interference is proposed to reduce pilot contamination. Based on the location information of UEs and the service relationship of APUE pairs, we design a new metric to represent the degree of interference between UEs. Then, we propose a locationaided distributed uplink combining scheme that includes a novel proposed AP selection to select the APs served by UEs with large interuser interference to adopt LPMMSE combining, while MR combining is adopted for the remaining APs. A new fairness coefficient that takes SE performance into account is proposed to characterize UE fairness. Furthermore, the performance of the proposed combining scheme is investigated under different power control schemes. Simulation results show that the proposed ICIB pilot assignment scheme achieves better channel estimation performance than the existing pilot assignment schemes. Moreover, our proposed locationaided uplink combining scheme effectively compromises between the average SE, UE fairness and complexity by adjusting the threshold \(\gamma\), which has obvious advantages when considering the comprehensive performance of the above three metrics in comparison with the existing combining schemes. Finally, applying fractional power control can further improve the tradeoff performance compared with MMF power control and full power transmission.
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Abbreviations
 MIMO:

Massive multipleinput multipleoutput
 AP:

Access point
 ICIB:

Intercluster interferencebased
 SE:

Spectral efficiency
 UE:

User equipment
 LPMMSE:

Local partial minimum meansquared error
 MR:

Maximum ratio
 MMF:

Max–min fairness
 CPU:

Central processing unit
 GPS:

Global positioning system
 LOS:

Line of sight
 LBG:

Locationbased greedy
 CB:

Conjugate beamforming
 NCB:

Normalized CB
 ECB:

Enhanced normalized CB
 PZF:

Partial zeroforcing
 PPZF:

Protective PZF
 JMRZF:

Joint maximumratio and zeroforcing
 CZF:

Centralized ZF
 MRT:

Maximumratio transmission
 SIC:

Successive interference cancellation
 FZF:

Fullpilot ZF
 PFZF:

Partial FZF
 PWPFZF:

Protective weak PFZF
 LRZF:

Local regularized ZF
 LSFD:

Largescale fading decoding
 ULA:

Uniform linear array
 AOA:

Angle of arrival
 MMSE:

Minimum mean square error
 NMSE:

Normalized meansquared error
 BPA:

Basic pilot assignment
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This work was supported in part by the National Natural Science Foundation of China under Grant 62071485, Grant 61901519, Grant 62001513 and Grant 62171119, in part by the Basic Research Project of Jiangsu Province under Grant BK 20192002 and the Natural Science Foundation of Jiangsu Province under Grant BK 20201334 and BK 20200579 and in part by the National Key Research and Development Program of China under Grant 2020YFB1807201 and Key Research and Development Plan of Jiangsu Province under Grant BE20210133.
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Wei, C., Xu, K., Shen, Z. et al. Locationaided uplink transmission for usercentric cellfree massive MIMO systems: a fairness priority perspective. J Wireless Com Network 2022, 84 (2022). https://doi.org/10.1186/s1363802202171x
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DOI: https://doi.org/10.1186/s1363802202171x