Structured Massive Access for Scalable Cell-Free Massive MIMO Systems

There is a large body of research on massive access in cellular massive MIMO [18, 19, 20, 21, 22, 23, 24, 25]. According to the user density in the network, massive access can be divided into structured access and random access. When the number of pilots is smaller than the number of UEs, but not dramatically like in Internet of Things (IoT) networks [26], structured access where each user is allocated a dedicated pilot resource is preferable [18]. In contrast, random access might outperform structured access in highly crowded scenarios. Structured access has been considered in [19, 20]. Specifically, the authors in [19] proposed a beam division multiple access to simultaneously serve multiple UEs via different beams in a multiuser massive MIMO network. From the perspective of array signal processing, the authors in [20] treated the multiuser massive MIMO as a type of non-orthogonal angle division multiple access to simultaneously serve multiple UEs. On the other hand, in [21], the authors improved the random access performance by averaging the pilot contamination across the transmission slots. In [22], the authors viewed the contaminated pilot signals as a graph code and analytically optimized performance by performing iterative belief propagation. The authors in [23] proposed a non-Bayesian algorithm to detect the activity of a large number of UEs for massive unsourced random access. Since the cell-free massive MIMO is widely used in indoor and hotspots scenarios, we focus on improving the structured access methods by suppressing the pilot contamination.

Cell-free massive MIMO was proposed in [7, 8], but builds on the heritage of coordinated multipoint [17, Sec. 7.4.3]. Four different ways to divide the signal processing between the APs and CPU are considered in [12]. The most promising distributed implementation uses minimum mean-squared-error (MMSE) combining along with large-scale fading decoding (LSFD) [27]. While all APs initially served all UEs, the user-centric approach has later become the leading way to achieve a practically implementable architecture [14, 15, 16, 28]. Several pilot assignment methods have been considered in the literature [9], including random assignment and brute-force optimization. A greedy algorithm was considered in [7] but it focused on limiting the coherent interference, which might not be the dominant part of pilot contamination and is also not scalable. Additionally, pilot assignment schemes based on tabu-search and K-means clustering were provided in [29] and [30], respectively. The former is also not scalable, while we will look into ways to improve the K-means approach in this paper. It was shown in [16] that pilot assignment can be made scalable by providing each accessing UE with the least bad pilot, but no optimization was carried out and the method is only evaluated for $L\gg K$ for which pilot assignment is fairly easy.

In this paper, we design a structured massive access uplink framework for scalable cell-free massive MIMO systems. Our main contributions are given as follows.

1. 1.

   We propose a scalable partial LSFD (P-LSFD) strategy for multi-antenna APs, which achieves roughly the same performance comparing to the optimal alternative.

2. 2.

   We propose a scalable algorithm based on a competitive mechanism which enables a large number of UEs to access the network and select the appropriate APs for service.

3. 3.

   We propose two pilot assignment schemes for structured massive access, namely User-Group scheme and interference-based K-means (IB-KM) scheme. Both of them are designed to suppress the mutual interference from the pilot sharing among UEs by partitioning the UEs in a proper manner, and shown to outperform the benchmarks.

4. 4.

   We propose a scalable fractional power control policy where a suitable tradeoff between fairness and average SE can be found by adjusting a parameter.

5. 5.

   We derive two novel closed-form SE expressions with maximum ratio (MR) combining, whereof one is suitable for arbitrarily fixed pilot assignment schemes and the other is dedicated to the random pilot switching scheme.

The remainder of this paper is organized as follows. Section II introduces the system model for scalable cell-free massive MIMO. The proposed P-LSFD strategy and its closed-form SE expression with MR combining are also provided in this section. Section III proposes a scalable algorithm for massive UEs to accessing the network and selecting APs for service. Another closed-form SE expression with MR combining and random pilot switching is provided in Section IV, and two novel pilot assignment schemes are proposed. The performance of the proposed structured massive access framework is numerically evaluated in Section VI. Finally, the major conclusions and implications are drawn in Section VII.

Boldface lowercase letters, $\bf x$, denote column vectors and boldface uppercase letters, $\bf X$, denote matrices. ${\bf X}_{ij}$ and ${{\bf{X}}_{\cdot j}}$ denote the entry $\left({i,j}\right)$ and the $j$th column of matrix $\bf X$, respectively. The superscripts ^T, ^∗, and ^H denote transpose, conjugate, and conjugate transpose, respectively. The $n\times n$ identity matrix is ${\bf I}_{n}$. We use $\buildrel\Delta\over{=}$ for definitions and ${\rm{diag}}\left({{\bf A}_{1},\ldots,{\bf A}_{n}}\right)$ for a block-diagonal matrix with the square matrices ${{\bf A}_{1},\ldots,{\bf A}_{n}}$ on the diagonal. The multi-variate circularly symmetric complex Gaussian distribution with correlation matrix $\bf R$ is denoted ${\cal N}_{\mathbb{C}}\left({{\bf 0},{\bf R}}\right)$. The expected value of $\bf x$ is denoted as ${\mathbb{E}}\left\{{\bf x}\right\}$. We denote by ${\left\|{\bf x}\right\|}_{2}$ the Euclidean norm of $\bf x$. We use $\left|{\cal A}\right|$ and ${\cal A}\left(n\right)$ to denote the cardinality and the $n$th element of the set $\cal A$, respectively.

We assume there are $\tau_{p}$ mutually orthogonal $\tau_{p}$-length pilot signals ${{\boldsymbol{\phi}}_{1}},\ldots,{{\boldsymbol{\phi}}_{\tau_{p}}}$ satisfying ${\left\|{{{\boldsymbol{\phi}}_{t}}}\right\|^{2}}=\tau_{p}$, with $\tau_{p}$ being a constant independent of $K$. Every UE is assigned to a pilot when it accesses the network. We consider a massive access scenario with a large number of UEs, in the sense that $K>\tau_{p}$. Hence, several UEs share the same pilot and these are referred to as pilot-sharing UEs. We denote by $t_{k}\in\left\{{1,\ldots,\tau_{p}}\right\}$ the index of the pilot assigned to UE $k$, and ${{\cal S}_{k}}$ the set of pilot-sharing UEs of UE $k$, including UE $k$ itself. When the UEs in ${\cal{S}}_{k}$ transmit pilot ${{\boldsymbol{\phi}}_{t_{k}}}$, AP $l$ receives the pilot signal ${\bf{y}}_{t_{k}l}^{\rm{p}}\in{{\mathbb{C}}^{N}}$ as [17, Sec. 3]

where $p_{i}$ denotes the pilot transmit power of UE $i$ and ${{\bf{n}}_{{t_{k}}l}}\sim{{\cal N}_{\mathbb{C}}}\left({{\bf{0}},{\sigma^{2}}{{\bf{I}}_{N}}}\right)$ is the thermal noise. The MMSE estimate of ${\bf h}_{kl}$ for $k\in{\cal S}_{k}$ is given by [17, Sec. 3]

where

is the correlation matrix of (1). The estimate ${\hat{\bf h}}_{kl}$ and estimation error ${\tilde{\bf h}}_{kl}={{\bf h}}_{kl}-{\hat{\bf h}}_{kl}$ are independent vectors distributed as ${\hat{\bf h}}_{kl}\sim{\cal N}_{\mathbb{C}}\left({{\bf 0},{\bf B}_{kl}}\right)$ and ${\tilde{\bf h}}_{kl}\sim{\cal N}_{\mathbb{C}}\left({{\bf 0},{\bf C}_{kl}}\right)$, where

Note that (1) indicates that sharing pilot ${\boldsymbol{\phi}}_{t_{k}}$ among the UEs in ${\cal S}_{k}$ generates mutual interference, and consequently degrades the system performance, which is the so-called pilot contamination.

During the uplink data transmission, AP $l$ receives the signal ${\bf y}_{l}\in{\mathbb{C}}^{N}$ from all UEs, as

where $s_{i}\sim{\cal N}_{\mathbb{C}}\left({0,p_{i}}\right)$ is the signal transmitted from UE $i$ with power $p_{i}$ and ${{\bf{n}}_{l}}\sim{{\cal N}_{\mathbb{C}}}\left({{\bf{0}},{\sigma^{2}}{{\bf{I}}_{N}}}\right)$ is the independent receiver noise.

For the large-scale network deployment, we prefer to offload most of the computational tasks to the APs to avoid overloading the CPU. More specifically, every AP preprocesses its signal by computing local estimates of the data and then passes them to the CPU for final decoding, which is the so-called LSFD. Although all APs can physically receive the signal from all UEs, only the APs in the set ${\cal M}_{k}$ take part in the signal detection for UE $k$ due to the AP selection. We denote by ${\bf a}_{kl}\in{\mathbb{C}}^{N}$ the combining vector selected by AP $l$ for UE $k$, where $k\in{\cal D}_{l}$. Then, the local estimate of $s_{k}$ is given by

Any combining vector can be adopted in the above expression. MR combining with ${\bf a}^{\rm{MR}}_{kl}={\hat{\bf h}}_{kl}$ was considered in [27], while [16] has recently advocated for using the local partial MMSE (LP-MMSE) combining

Then the local estimates $\left\{{{{\tilde{s}}_{kl}}}\right\}$ are sent to the CPU, where they are linearly combined by using the weights $\left\{{{{w}_{kl}}}\right\}$ to obtain ${{\hat{s}}_{k}}=\sum\limits_{l=1}^{L}{w_{kl}^{*}{{\tilde{s}}_{kl}}}$, which is eventually used to decode $s_{k}$. From (7), we have the final estimate of $s_{k}$, as

where ${{\bf{W}}_{k}}={\rm{diag}}\left({{w_{k1}}{{\bf{I}}_{N}},\ldots,{w_{kL}}{{\bf{I}}_{N}}}\right)\in{{\mathbb{C}}^{\left({LN}\right)\times\left({LN}\right)}}$.

Since the CPU does not have the knowledge of channel estimates, we utilize the so-called use-and-then-forget (UatF) bound [17, Th. 4.4] to obtain the achievable SE.

The structure of (1) is a generalized Rayleigh quotient with respect to ${\bf w}_{k}$. As a consequence, the maximum value of $\rm{SINR}_{k}$ is achieved as [17, Lem. B.10]

with the optimal LSFD weight

The fronthaul load required to gather all the statistical matrices for computing the LSFD vector in (17) and the related computational complexity are summarized in Table I. Clearly, they grow very fast with the size of the network, making the implementation of the optimal LSFD unscalable.

To achieve the implementation, we propose to use the alternative P-LSFD vector as

where ${{\cal P}_{k}}=\left\{{i:{{\bf{A}}_{kl}}{{\bf{A}}_{il}}\neq 0,l\in\left\{{1,\ldots,L}\right\}}\right\}$ is the index set of the UEs which are served by partially the same APs as UE $k$. Only those UEs in ${{\cal P}_{k}}$ might cause substantial interference to UE $k$. Note that $\left|{{\cal P}_{k}}\right|\leq\left({{\tau_{p}-1}}\right)\left|{{\cal M}_{k}}\right|+1$, where the upper bound is achieved in the unlikely case that all the APs in ${\cal M}_{k}$ serve UE $k$ but otherwise serve entirely different sets of UEs. Importantly, the upper bound is independent of $K$. The fronthaul load related to the statistical parameters and the total number of complex multiplications required by P-LSFD is given in Table I. It is important to note that the proposed P-LSFD is a scalable strategy whose complexity does not grow with $K$.

The expectations in (1) cannot be computed in closed-form when using LP-MMSE, but can be easily computed using Monte-Carlo simulations. Similar to [17, Cor. 4.5], we can obtain the following closed-form expression as a simple baseline when using MR combining.

When the random pilot assignment scheme is applied, every UE in the network is assigned a pilot at random from $\tau_{p}$ orthogonal pilots and uses it in all blocks. Random pilot switching is another approach to assign pilots, in which each UE does not pick one pilot at random, but switches between pilots in a random fashion over blocks to average over the pilot contamination [31]. When random pilot switching is applied, the pilot-sharing UEs for UE $k$ will vary. We use a binary random variable

instead of ${\cal P}_{k}$ to indicate whether a UE $i$ is a pilot-sharing UE of UE $k$ or not, since it is easier to define the statistics of ${\chi_{ik}}$ than ${\cal P}_{k}$. The probability of ${\chi_{ik}}=1$ is $\frac{1}{{{\tau_{p}}}}$ and $1-\frac{1}{{{\tau_{p}}}}$ otherwise. With this notation, the despreaded pilot signal received at AP $l$ in (1) can be rewritten as

As a consequence, we have

where ${\mathbb{E}}_{\left\{{\bf h}\right\}}\left\{\cdot\right\}$ denotes the expectation with respect to the channel and noise realizations. Since no randomness appears in $\chi_{kk}=1$, we rewrite the SINR expression in (1) as

for the following derivation. A closed-form expression of SINR when using MR combining and random pilot switching is obtained as follows.

Note that the normalization of ${{{\bf{\hat{h}}}}_{kl}}$ with ${\bf{B}}_{kl}^{{\rm{-}}1}$ in Corollary 1 makes the expected channel gain equal to ${\bf{w}}_{k}^{\rm{H}}{{{\bf{\bar{u}}}}_{kk}}$ as in (33), and thereby enables us to derive the closed-form expressions in Corollary 1.

We treat the closed-form SE expression obtained in Corollary 1 as a “worst” case, since all UEs in the network are possibly suffering from strong pilot contamination in random pilot switch. Therefore, we mainly consider the random pilot assignment, which is widely considered in previous works, as a benchmark. The reason is that two UEs that are close to each other will occasionally share the same pilot and then create strong mutual interference. This can be avoided by a structured pilot assignment.

A K-means-type pilot assignment scheme was proposed in [30] and we call it geography-based K-means (GB-KM) pilot assignment since the geographic location of the UEs is utilized. Inspired by this scheme, we propose another K-means-type pilot assignment scheme where instead the distances between all UEs and APs are considered. Note that no extra processing is needed for this distance information since it is intermediate when the APs and CPU obtain $\left\{{\bf R}_{kl}\right\}$. Since this scheme aims to suppress the interference generated by the pilot-sharing UEs, we refer it to as IB-KM pilot assignment scheme. Before we elaborate the scheme, we first make the following key assumption.

The rationale behind Assumption 2 is that the inter-user interference occurs when the pilot-sharing UEs communicate with the same AP. The strength of the interference depends on the signal power of the pilot-sharing UEs, which is mainly determined by the distances between the pilot-sharing UEs and the same AP when the channel distribution and the transmit power are roughly the same. A simple example with $3$ UEs and $4$ APs is provided in Fig. 2 to explain Assumption 2. The distances between UE $k$ and its serving APs (i.e., ${\rm{diag}}\left({{\bf{d}}_{k}}\right){{\bf A}_{\cdot k}}$) are marked with “Serving” in Fig. 2. In the example, we can see that UE $2$ and UE $3$ are located in the similar positions but served by different subsets of APs (${\cal M}_{2}=\left\{1,3\right\}$ and ${\cal M}_{3}=\left\{2,3,4\right\}$). When comparing the cases of these $3$ UEs in Fig. 2, we can conclude that UE $1$ and UE $2$ will generate less inter-user interference if they share the same pilot than UE $1$ and UE $3$. The reason is that UE $1$ and UE $2$ are served by disjoint subsets of APs while UE $1$ and UE $3$ have a common serving AP, i.e., AP $3$. Then we back to (41) and find that ${\rm{Dis}}_{12}=9150>{\rm{Dis}}_{13}=315$, since ${\rm{Dis}}_{ik}$ indicates the difference between service quality of UE $i$ and UE $k$ from the APs of their corresponding ${\cal M}_{i}$ and ${\cal M}_{k}$.

Based on Assumption 2, the basic idea of the IB-KM pilot assignment scheme is that the $K$ UEs are separated into $\left\lceil{K/{\tau_{p}}}\right\rceil$ disjoint clusters centering on $\left\lceil{K/{\tau_{p}}}\right\rceil$ centroids, whose minimum ${\rm{Dis}}$ with each other is as large as possible. When the APs are deployed, the location of these centroids can be trained with a large number of points randomly locating in the coverage area ,which could be generated by the MATLAB function “rand” [30]. Every such cluster comprises at most $\tau_{p}$ UEs, which have the smallest values of the ${\rm{Dis}}$ with the corresponding centroid. UEs in the same cluster are assigned mutually orthogonal pilots, as shown in Fig. 3. The algorithm initiates with $\left\{{\cal C}_{m}=\emptyset:m=1,\ldots,\left\lceil{K/{\tau_{p}}}\right\rceil\right\}$ and $\varepsilon=0.001$. Note that the distance between APs and UEs $\left\{{{{\bf{d}}_{k}}}\right\}$ is generated when the spatial correlation matrices $\{{\bf R}_{kl}\}$ are generated, which depends on the simulation setup.

Our proposed IB-KM pilot assignment scheme operates through the following steps.

1. 1.

   Arbitrarily generate $K_{\rm p}$ points and $\left\lceil{K/{\tau_{p}}}\right\rceil$ centroids in the coverage area, where $K_{\rm p}$ is a large number. Each point and centroid measures its distance with all APs, generates distance vector ${{\bf{d}}^{\prime}_{p}}={\left[{{{d}^{\prime}_{p1}},\ldots,{{d}^{\prime}_{pL}}}\right]^{\rm{T}}}$, $p=1,\ldots,{K_{\rm{p}}}$ and ${{\boldsymbol{\mu}}_{m}}={\left[{{\mu_{m1}},\ldots,{\mu_{mL}}}\right]^{\rm{T}}}$, $m=1,\ldots,\left\lceil{K/{\tau_{p}}}\right\rceil$, respectively.

2. 2.

   Each point selects the centroid

   $$m^{*}=\arg{\min_{1\leq m\leq\left\lceil{K/{\tau_{p}}}\right\rceil}}{\left\|{{{\bf{d}}^{\prime}_{p}}-{{\boldsymbol{\mu}}_{m}}}\right\|^{2}_{2}},\ p=1,\ldots,K_{\rm p},$$

   (42)

   and join the corresponding cluster ${\cal C}_{m^{*}}$.

3. 3.

   Each centroid updates its distance vector as

   $${{\boldsymbol{\mu}}^{\prime}_{m}}=\frac{1}{{\left|{{{\cal C}_{m}}}\right|}}\sum\limits_{{p}\in{{\cal C}_{m}}}{{{\bf{d}}^{\prime}_{p}}},\ m=1,\ldots,\left\lceil{K/{\tau_{p}}}\right\rceil,$$

   (43)

   and go back to step 1), until

   $${\max_{1\leq m\leq\left\lceil{K/{\tau_{p}}}\right\rceil}}{\left\|{{{\boldsymbol{\mu}}^{\prime}_{m}}-{{\boldsymbol{\mu}}_{m}}}\right\|^{2}_{2}}<\varepsilon,$$

   (44)

   where $\varepsilon$ is a small number.

4. 4.

   Each UE generates its distance vector ${{\bf{d}}_{i}}={\left[{{d_{i1}},\ldots,{d_{iL}}}\right]^{\rm{T}}}$, $i=1,\ldots,{K}$.

5. 5.

   Each UE selects the centroid

   $$m^{*}=\arg{\min_{1\leq m\leq\left\lceil{K/{\tau_{p}}}\right\rceil}}{\left\|{{\rm{diag}}\left({{\bf{d}}_{i}}\right){{\bf A}_{\cdot i}}-{{\boldsymbol{\mu}}_{m}}}\right\|^{2}_{2}},\ i=1,\ldots,K,$$

   (45)

   and join in the corresponding cluster ${\cal C}_{m^{*}}$. A competition mechanism similar to the one in Algorithm 1 could be applied if a generic centroid $m$ is selected by more than $\tau_{p}$ UEs. Or more succinctly, each cluster chooses $\tau_{p}$ UEs with the smallest values of $\rm{Dis}$ with the corresponding centroid in sequence, until all UEs are allocated into $\left\lceil{K/{\tau_{p}}}\right\rceil$ disjoint clusters; a UE only can be chosen by one cluster.

6. 6.

   Find a cluster with $\tau_{p}$ UEs and arbitrarily assign the UEs $\tau_{p}$ mutually orthogonal pilots. Without loss of generality, we assume $\left|{\cal C}_{1}\right|=\tau_{p}$ and assign the UEs in ${\cal C}_{1}$ pilots $\left\{{{\boldsymbol{\phi}}_{1}},\ldots,{{\boldsymbol{\phi}}_{\tau_{p}}}\right\}$.

7. 7.

   Each UE in ${\cal C}_{1}$ finds UE

   $$i^{*}=\arg{\max_{i\in{\cal C}_{m}}}{\left\|{{\rm{diag}}\left({{\bf{d}}_{i}}\right){{\bf A}_{\cdot i}}-{\rm{diag}}\left({{\bf{d}}_{k}}\right){{\bf A}_{\cdot k}}}\right\|^{2}_{2}},\ k\in{\cal C}_{1},$$

   (46)

   in ${\cal C}_{m}$, $m=2,\ldots,\left\lceil{K/{\tau_{p}}}\right\rceil$, and shares pilot with this UE. If a UE $i^{*}$ in ${\cal C}_{m}$ is selected by multiple UEs in ${\cal C}_{1}$, then only the UE, whose value of $\rm{Dis}$ with UE $i^{*}$ is the largest, shares pilot with UE $i^{*}$; the rest UEs find another UE based on (46), until each UE in the network are assigned a pilot.

The pseudo code of this algorithm is given Algorithm 2.