Tomlinson-Harashima Cluster-Based Precoders for Cell-Free MU-MIMO Networks

Coordinated base stations (BSs) have been deployed worldwide to establish cellular network services. However, wireless applications are evolving constantly with an increasing demand for more resources [1, 2, 3, 4]. For high throughput and quality-of-service required for future networks, it is desired to further densify BSs. However, this approach is impractical. As an alternative, cell-free (CF) multiple-input multiple output (MIMO) systems have emerged as a potential solution to improve the performance and satisfy throughout requirements of future wireless networks [5, 6].

Compared to conventional BS-based networks, CF MU-MIMO systems employ multiple APs distributed geographically over the area of interest. A central processing unit (CPU), which may be located at the cloud server, coordinates the APs. The distributed deployment of CF networks yields higher coverage than the BSs with collocated antennas [7]. In addition, CF multiuser MIMO (MU-MIMO) has been shown to provide increased throughput per user [8, 9] as well as better performance in terms of energy efficiency [10, 11, 12].

Further, CF MU-MIMO employs the same time-frequency resources to provide service to multiple users as BS-based systems. To avoid the multiuser interference (MUI) in the downlink, a precoder is often implemented at the transmitter. Prior works on CF MU-MIMO have focused on linear precoding techniques such as matched filter (MF), zero-forcing (ZF) [13], and minimum mean-square error (MMSE) [14] techniques. However, it is well-known that nonlinear precoders [15, 16, 17] have the potential to outperform their linear counterparts [18, 19, 20, 21, 22, 23].

State-of-the-art in CF MU-MIMO systems has proposed network-wide (NW) precoders [24, 13, 14, 25] but these techniques entail a very high signaling load. Moreover, NW approaches demand high computational complexity because they require the inversion of a matrix whose size increases with the number of APs and users. To mitigate this problem, NW precoders that employ APs and user clusterization have been proposed [26] for lower computational complexity and signaling load. For instance, in [20] the number of APs is curtailed to reduce the signaling load. In [27], scalable MMSE combiners and precoders are developed. Very recently, a regularized ZF precoder based on subsets of user was proposed in [28] to judiciously use the available resources.

Unlike previous works [29, 20, 23], we propose nonlinear precoding techniques for CF MU-MIMO systems. The proposed techniques are based on the well-established Tomlinson-Harashima precoder (THP) [30], which may be interpreted as the transmit analog of the successive interference cancellation (SIC) employed at the receiver [31]. Essentially, THP employs a nonlinear modulo operation that reduces the power penalty associated with the linear precoders thereby enhancing the overall performance. Additionally, a cluster-based approach is devised based on a user selection matrix, resulting in a user-centric nonlinear precoder and addressing the gap in nonlinear structures for cluster-based precoders in CF networks. The resulting precoder is sparse and its complexity is reduced by employing clusters of users, thereby reducing the amount of signaling and the computational load. Our numerical experiments show that the TH precoding techniques outperform their linear counterparts.

The rest of this paper is organized as follows. In the next section, we describe the ssytem model of the CF MU-MIMO communications. We derive the proposed cluster-based nonlinear precoding techniques in Section III. We introduce the metric to evaluate the performance of the proposed precoders in Section IV. We validate our model and methods via numerical experiments in Section V. We conclude in Section VI.

Consider the downlink of a CF MIMO system, where $N$ geographically distributed APs serve $K$ users equipped with a single omnidirecitonal antenna. A central processing unit (CPU) located at the cloud server is connected to the APs. The data are transmitted over a flat-fading channel $\mathbf{G}\in\mathbf{C}^{N\times K}$. The $(n,k)$-th element of matrix $\mathbf{G}$ is the channel coefficient between the $n$-th AP and $k$-th user, i.e., $g_{n,k}=\sqrt{\zeta_{n,k}}h_{n,k}$, where $\zeta_{n,k}$ is the large-scale fading coefficient that models the path loss and shadowing effects, and $h_{n,k}$ represents the small-scale fading coefficient. The coefficients $h_{n,k}$ are modeled as independently and identically distributed (i.i.d.) random variables with complex Gaussian distribution $\mathcal{CN}\left(0,1\right)$.

Denote the transmit signal by $\mathbf{x}\in\mathbb{C}^{N}$, which obeys the transmit power constraint $\mathbb{E}\left[\lVert\mathbf{x}\rVert^{2}\right]\leq P_{t}$, where $\mathbb{E}[\cdot]$ denotes the statistical expectation. Then, the $K\times 1$ received signal vector is

where $(\cdot)^{T}$ is the conjugate transpose and $\mathbf{n}\in\mathbb{C}^{K}$ is the additive white Gaussian noise (AWGN) that follows the distribution $\mathcal{CN}\left(\mathbf{0},\sigma_{n}^{2}\mathbf{I}\right)$.

The system employs the time division duplexing (TDD) protocol and therefore the channels can be estimated employing the channel reciprocity property and pilot training [32]. After receiving the pilots, the CPU computes the channel estimate $\mathbf{\hat{G}}^{\text{T}}=\left[\mathbf{\hat{g}}_{1},\mathbf{\hat{g}}_{2},\cdots,\mathbf{\hat{g}}_{k}\right]^{\text{T}}\in\mathbb{C}^{K\times N}$, whose $(n,k)$-th element is

where $\hat{g}_{n,k}$ is the channel estimate between the $n$-th AP and the $k$-th user; $\tilde{h}_{n,k}$ are i.i.d complex Gaussian random variables that follow the distribution $\mathcal{CN}\left(0,1\right)$ (independent from $h_{n,k}$) and model the errors in the channel estimates; and $\sigma_{e}$ represents the quality of the channel state information (CSI). The error affecting the channel estimate $\hat{g}_{n,k}$ is $\tilde{g}_{n,k}=\sigma_{e}\sqrt{\zeta_{n,k}}\tilde{h}_{n,k}$.

To enhance the performance of the system while reducing the signaling load and computational complexity of NW precoders, we propose cluster-based nonlinear precoders. To this end, we form clusters of APs and users. These clusters are defined based on the large-scale channel coefficients given by $\zeta_{n,k}$. Since only small subsets of APs transmit the most relevant signals for reception, the contribution of the remaining APs is not significant and the transmission over such APs is avoidable. The upshot of this technique is that we discard the APs whose processing is cost-ineffective to reduce the signaling load.

To evaluate the proposed nonlinear schemes, we employ the ergodic sum-rate (ESR) defined as

where $\bar{R}_{k}=\mathbb{E}\left[R_{k}|\hat{\mathbf{G}}\right]$ is the average rate and $R_{k}$ is the instantaneous rate of the $k$-th user. The rate $\bar{R}_{k}$ averages out the effects of the imperfect CSIT because the instantaneous rates are not achievable. Considering Gaussian codebooks, the instantaneous rate is

where $\gamma_{k}$ is the signal-to-interference-plus-noise ratio (SINR) at user $k$.

Denote the SINR for the centralized and decentralized structures by $\gamma_{k}^{\left(\text{c}\right)}$ and $\gamma_{k}^{\left(\text{d}\right)}$, respectively. Then, depending on the specific THP structure used, we employ $\gamma_{k}^{\left(\text{c}\right)}$ or $\gamma_{k}^{\left(\text{d}\right)}$ in (25) to obtain the instantaneous rate. To compute the SINR, we obtain the mean powers of the received signal at user $k$ for centralized and decentralized structures as, respectively,

and

where $d_{g}^{\left(\text{c}\right)}=\lvert\tilde{\mathbf{g}}_{k}\mathbf{p}^{\left(\text{c}\right)}_{k}\rvert^{2}-2\text{Re}{\left(\tilde{\mathbf{g}}_{k}\mathbf{p}^{\left(\text{c}\right)}_{k}\right)}$ and $d_{g}^{\left(\text{d}\right)}=c_{k,k}^{2}\lvert\tilde{\mathbf{g}}_{k}\mathbf{p}^{\left(\text{d}\right)}_{k}\rvert^{2}-2\text{Re}{\left(c_{k,k}\tilde{\mathbf{g}}_{k}\mathbf{p}^{\left(\text{d}\right)}_{k}\right)}$. This yields

and

We assess the performance of the proposed TH precoders via numerical experiments. Throughout the experiments, the large scale fading coefficients are set to

where $P_{k,n}$ is the path loss and the scalar $10^{\frac{\sigma^{\left(\textrm{s}\right)}z_{k,n}}{10}}$ include the shadowing effect with standard deviation $\sigma^{\left(\textrm{s}\right)}=8$. The random variable $z_{k,n}$ follows Gaussian distribution with zero mean and unit variance. The path loss was calculated using a three-slope model as

For all experiments, we have $128$ APs randomly distributed over a square with side equal to $20$ km. The APs serve a total of $24$ users, which are geographically distributed. We considered a total of 10,000 channel realizations to compute the ESR. Specifically, we employed $100$ channel estimates and, for each channel estimate, we considered $100$ error matrices. It follows that the average rate was computed with $100$ error matrices.

We first compare the ESR of the proposed precoders with their linear counterparts. We consider that the error in the channel coefficient estimate has a variance of $0.01$. Fig. 1 shows the sum-rate performance of the proposed nonlinear precoders against their conventional linear counterparts. The dTHP with sparse channel estimate or “dTHP-SP” performs the best, even better than the linear ZF precoder that employs all the APs (ZF-NW).

In the second experiment, we assessed the sum-rate performance at SNR$=15$ dB with respect to CSIT quality (Fig. 2). The proposed dTHP with reduced dimensions or “dTHP-RD” outperforms the ZF-RD precoder. The RD precoding techniques have reduced computational complexity than the corresponding SP precoders. We observe that our proposed nonlinear cluster-based precoders generally yield better ESR than their linear counterparts.

We proposed clustered nonlinear precoders based on the noninear THP algorithm. Our proposed THP-SP reduces the signaling load and the THP-RD additionally lowers the computational complexity at the expense of performance. Note that the reduction of the computational complexity is critical for practical applications. Numerical experiments showed that the proposed cluster-based nonlinear precoders yield better performance and robustness against CSIT uncertainties than the conventional linear precoders.