Overloaded Pilot Assignment with Pilot Decontamination for Cell-Free Systems

A large number of works in wireless communication theory is dedicated to the joint processing of spatially distributed antennas. This idea can be traced back to the work of Wyner [1], and has been “re-marketed” several times under different names with slight nuances, such as coordinate multipoint (CoMP), cloud radio access network (CRAN), and currently, it is promoted as cell-free massive MIMO (CF-mMIMO). As the name implies, one of the purposes of CF-mMIMO is eliminating cell boundaries and preventing user equipment’s (UE’s) performance degradation depending on their geographical positions.

The channel state information (CSI) acquisition with pilot signals at the infrastructure antenna side is important for taking advantage of CF-mMIMO’s spatial multiplexing gain. In this paper, we focus on uplink (UL) channel estimation in a time division duplex (TDD) system. Thanks to TDD operations and channel reciprocity, these estimates can also be used for the downlink precoding [2]. Generally, the number of orthogonal pilots is limited to reduce overhead and keep the training phase within coherence time. Therefore, pilot reuse is inevitable to increase the number of UEs to be accommodated, and it induces pilot contamination. In addition, CF-mMIMO system has no clear boundaries, so it is hard to apply a cell-based pilot reuse restriction.

As one of the pilot decontamination methods, in [3, 4], we have investigated a subspace projection (SP) based channel estimation, which uses receiver side discrete Fourier transform (DFT) processing. The channel subspace is spanned by dominant components of channel covariance matrices. Assuming uniform linear arrays (ULAs) or uniform planar arrays (UPAs), the channel covariance matrix is Toeplitz (for ULA) or Block-Toeplitz (for UPA), which both are approximately diagonalized by discrete Fourier transforms (DFT) on the columns and rows [5]. This means that the dominant channel subspaces are approximately spanned by subsets of the DFT columns. If the antennas have correlation due to a limited scattering angular spread, each radio unit (RU) and UE link has an individual channel subspace. Therefore, DFT-based projection onto the target UE’s subspace can reduce interference from co-pilot UEs if the subspaces of the co-pilot UEs are nearly orthogonal, which we call decontamination. We have shown that the effect of the pilot contamination can be reduced significantly with the SP-based decontamination, yielding a system performance close to the case of ideal CSI, where each RU has perfect knowledge of the channel vectors of its associated UEs [3].

The system model of CF in our previous works [3, 6] is similar to Björnson and Sanguinetti’s scalable CF system [7], where each user-centric cluster is formed by a finite number of RUs. In [3, 6], we have not allowed the RUs to assign the same pilot to different users, which we call non-overloaded pilot assignment (PA). However, the SP enhances RUs to estimate channels of co-pilot UEs under overloaded PA, where RUs can assign the same pilot to multiple UEs. Adopting the overloaded PA leads to an increased number of associated RUs per UE, which can increase the spatial multiplexing gain per UE and improve spectral efficiency (SE).

Most of related works, which aimed to address contamination by strategic PA methods, implicitly adopted overloaded PA because pilot selection is significantly constrained by the cluster formation, when the non-overloaded PA is adopted. In [8], a greedy PA algorithm was introduced as an early study in the CF literature. In [9], a PA scheme aiming at user throughput maximization is investigated, where the maximization problem is solved by using an iterative scheme based on the Hungarian algorithm. In [10], user-group PA was proposed which assigns the same pilot to UEs who share the fewest number of serving RUs. Furthermore, graphic framework based pilot assignment schemes are investigated in [11, 12]. Since these works do not adopt the SP-based decontamination, combining these overloaded PA methods and our decontamination method has potential to achieve better performance.

As mentioned above, to the best of our knowledge, combination of the SP decontamination and overloaded PA has not yet been investigated. Motivated by this, we propose two types of overloaded PA methods in this paper. We call these methods rough overloaded PA (R-OPA) and subspace information aided overloaded PA (SIA-OPA). The R-OPA forms clusters in advance without considering the contamination, and then assigns the pilots. We propose a specific implementation of R-OPA by adjusting a graphic framework based PA studied by Zeng et al. [12] to our SP based decontamination. The SIA-OPA has a restriction in the cluster formation phase that it adds an RU to a cluster of a UE only when the subspaces of co-pilot UEs are orthogonal.

The contributions of this work are summarized as follows.

• •

  We propose R-OPA and SIA-OPA for CF systems. In addition, a graphic framework-based PA is designed as a specific implementation of R-OPA.

• •

  We discuss the complexity of the proposed PA. Note that R-OPA with the graphic framework-based PA algorithm requires network-wide information exchange, whereas SIA-OPA does not.

• •

  We compare the sum SE and outage performance of the non-overloaded PA, R-OPA and SIA-OPA to reveal a region of parameters where the performance is best.

We consider a CF-mMIMO system with $L$ RUs, each with $M$ antennas, and $K$ UEs. Both RUs and UEs are distributed on a squared region on the 2-dimensional plane. The set ${\cal C}_{k}\subseteq[L]=\{1,2,\dots,L\}$ denotes the cluster of RUs that serve UE $k$, and ${\cal U}_{\ell}\subseteq[K]$ denotes the set of UEs served by RU $\ell$, where the set of integers from 1 to $n$ is denoted by $[n]$. The RU-UE associations are described by a bipartite graph ${\cal G}$ whose vertices are the RUs and UEs, respectively. The set of edges accounting for associated RU-UE pairs is denoted by ${\cal E}$, i.e., ${\cal G}={\cal G}([L],[K],{\cal E})$. We also define ${\cal U}({\cal C}_{k})=\bigcup_{\ell\in{\cal C}_{k}}{\cal U}_{\ell}$ as the set of UEs served by at least one RU in ${\cal C}_{k}$.

In UL transmissions, the UEs transmit with the same energy per symbol $E_{\rm s}$, and we define the system parameter

where $N_{0}$ denotes noise power spectral density. The large-scale-fading-coefficient (LSFC) between RU $\ell$ and UE $k$ is denoted by $\beta_{\ell,k}$ and includes pathloss, blocking effects and shadowing, respectively. By assuming isotropic antennas, the maximum beamforming gain averaged over the small scale fading is $M$, therefore the maximum SNR at the receiver of RU $\ell$ from UE $k$ is $\beta_{\ell,k}M{\rm SNR}$.

We consider orthogonal frequency-division multiplexing (OFDM) modulation and channels following the standard block-fading model [2, 13, 14], such that they are random but constant over coherence blocks of $T$ signal dimensions in the time-frequency domain. The described methods are formulated for one resource block (RB), so the RB index is omitted for simplicity. We define channel matrices and its elements as follows. $\mbox{\bb H}\in\mbox{\bb C}^{LM\times K}$ denotes the overall channel matrix between all $LM$ RU antennas and all $K$ UE antennas on a given RB. Next, $\mathbbm{h}_{k}\in\mbox{\bb C}^{LM\times 1}$ denotes $k$-th column of $\mbox{\bb H}\in\mbox{\bb C}^{LM\times K}$. In addition, ${\bf h}_{\ell,k}\in\mbox{\bb C}^{M\times 1}$ denotes the channel vector between RU $\ell$ and UE $k$. Finally, $\mbox{\bb H}({\cal C}_{k})\in\mbox{\bb C}^{LM\times K}$ denotes the partial cluster-centric matrix for a cluster ${\cal C}_{k}$, whose $M\times 1$ blocks of RU-UE pairs $(\ell,k)\in{\cal E}$ are equal to ${\bf h}_{\ell,k}$, and equal to ${\bf 0}$ (the identically zero vector) otherwise.

For ease of analysis, we assume that the individual channels between RUs and UEs follow the single ring local scattering model [5] where the covariance matrix is perfectly diagonalized by DFT. Then ${\bf h}_{\ell,k}$ is given by

where ${\cal S}_{\ell,k}\subseteq\{0,\ldots,M-1\}$ and $\hbox{\boldmath$\nu$}_{\ell,k}$ are the angular support set according to [5], and an $|{\cal S}_{\ell,k}|\times 1$ i.i.d. Gaussian vector with components $\sim{\cal C}{\cal N}(0,1)$, respectively. The set ${\cal S}_{\ell,k}$ is constructed by the angular support $\Theta_{\ell,k}=[\theta_{\ell,k}-\Delta/2,\theta_{\ell,k}+\Delta/2]$ centered at angle $\theta_{\ell,k}$ of the LOS between RU $\ell$ and UE $k$, with angular spread $\Delta$. We let ${\bf F}$ denote the $M\times M$ unitary DFT matrix with $(m,n)$-elements $f_{m,n}=\frac{e^{-j\frac{2\pi}{M}mn}}{\sqrt{M}}$, and using a Matlab-like notation, ${\bf F}_{\ell,k}\stackrel{{\scriptstyle\Delta}}{{=}}{\bf F}(:,{\cal S}_{\ell,k})$ denotes the tall unitary matrix obtained by selecting the columns of ${\bf F}$ corresponding to the index set ${\cal S}_{\ell,k}$.

As mentioned in Sec. 1, the SP can enhance overloaded PA, and lead to improved SE compared to non-overloaded PA. This section introduces the non-overloaded PA, R-OPA and SIA-OPA schemes with their cluster formation rules.

Table I shows the basic parameter specifications of the simulations.

We consider a square coverage area of $A=2\times 2\ {\rm km}^{2}$ with a torus topology by using the wrapped around technique to avoid boundary effects. LSFCs are given according to the 3GPP urban microcell street canyon pathloss model from [15, Table 7.4.1-1], which differentiates between UEs in LOS and NLOS. The total number of receiving antennas is fixed as $LM=400$. For each set of parameters we generate 40 independent layouts, and small scale fading coefficients are varied 30 times for each layout. RUs and UEs are randomly distributed in each setup, and the sum SE is given by the sum of ${\rm SE}_{k}^{\rm ul}$ over all $k\in[K]$, and then averaging it with respect to the different layouts.

Table II shows the largest sum SE achieved by the different PA schemes and the corresponding system parameters, where R-OPA Random denotes rough overloaded PA using random pilot assignment and R-OPA WGF denotes rough overloaded PA using WGF-based pilot assignment. Firstly, we confirm that $\tau_{p}=30$ outperforms $\tau_{p}=20$ for the R-OPA and the SIA-OPA even taking into account the overhead in eq. (14), while $\tau_{p}=20$ is the best for the non-overloaded PA. As a comparison with existing studies, the results of R-OPA with the WGF algorithm and PM channel estimation represent [12]. From Table II, the SIA-OPA with SP achieves the highest sum SE. In the following, we focus on the setup $\tau_{p}=30,L=25,M=16$ to analyze the behavior around the largest sum SE.

Fig. 1 shows the mean cluster size and outage probability vs. $K$, and Fig. 2 illustrates the sum SE vs. $K$. Note that outage occurs in the region of more than 800 UEs for the SIA-OPA and the non-overloaded PA. From Fig. 1, the cluster size with rough overloaded PA (R-OPA) is almost equal to $Q=10$ for all evaluated $K$. From this observation we deduce for the given simulation parameters that if there is no limitation on cluster formation imposed by the pilot assignment such as non-overloaded, $Q$ RUs with the largest channel gain will form a user-centric cluster for most UEs. Note that the cluster size with R-OPA is independent of the pilot assignment scheme, since the clusters are formed before assigning pilots. The cluster size of the non-overloaded PA scheme is significantly reduced by the increase of $K$, while that of the SIA-OPA scheme decreases more gradual. For small channel estimation error, UEs with larger clusters can be given higher SE by obtaining more spatial diversity. Thus, the cluster formation of R-OPA and SIA-OPA have potential to outperform the non-overloaded PA in terms of SE.

Then let us focus on Fig. 2 where Ideal CSI lines indicate no channel estimation error. The SE degradation factor can be divided into two: channel estimation error due to pilot contamination, and deviation from the optimal multiplexing gain of massive MIMO. The fact that the sum SE saturates in the ideal CSI case indicates the latter factor.

The R-OPA WGF scheme with PM channel estimation is not tolerant of an increase in UE density. This is an interesting effect when compared to the non-overloaded PA with PM, where the sum SE first grows with the number of UEs, and then decreases again. The degradation of the R-OPA WGF scheme with PM is mainly due to an increased channel estimation error, while the non-overloaded PM is specifically designed to avoid assigning the same pilot to UEs that can potentially cause large mutual pilot contamination. The R-OPA WGF scheme allows co-pilot UEs with large potential interference if their subspaces are mutually orthogonal. Since the PM channel estimation only eliminates pilot contamination from UEs with a different UL pilot (and does not take into account the channel subspaces), the channel estimate is potentially contaminated to a large extent by co-pilot users with orthogonal subspaces, which leads to a strong degradation of the sum SE for growing $K$.

The non-overloaded PA with the SP achieves almost the same sum SE as the ideal case, which means that the SP effectively eliminates the contamination. For the R-OPA, the performance of random assignment significantly degrades compared to ideal CSI as $K$ increases due to that randomly assigned pilots generate many co-pilot UEs within the same subspace. The WGF-based R-OPA also degrades compared to the ideal case, however, it achieves the highest sum SE at $K\leq 400$ among all PA methods. At last, SIA-OPA slightly outperforms the R-OPA at $K\geq 600$ and achieves the highest sum SE at $K=600$. In addition, the gap between the SP and the ideal CSI stays very small, thus the SIA-OPA works as expected. In the region of $K\geq 800$, the sum SE saturates and slightly degrades, which can be considered the effect of increased outage as seen in Fig. 1.

The WGF R-OPA has the highest sum SE at a certain UE density ($K\leq 400$), where the per UE SE (sum SE$/K$) is the largest. On the other hand, the SIA-OPA achieves a competitive sum SE at $K=400$ and is easier to implement in terms of scalability than WGF R-OPA.

This paper has investigated overloaded PA and cluster formation methods that outperform the non-overloaded PA in CF-mMIMO systems. The WGF R-OPA achieved the highest sum SE in the region of a certain UE density ($K\leq 400$), while it is challenging for practical implementation due to the unscalable metric calculation and information exchange. The SIA-OPA, which assigns pilots to UEs with orthogonal subspaces, achieves the highest sum SE in the region of $K\geq 600$. Furthermore, it also achieves a competitive sum SE at $K\geq 400$ with a more relaxed PA procedure. The appropriate method in a practical system can be chosen according to the trade-off between their performance and feasibility.