Resource Allocation and Scheduling in Non-coherent User-centric Cell-free MIMO

As shown in Fig. 1, we consider the downlink of a time-division duplex (TDD) system comprising several RRHs, represented by the set $\mathcal{B}$, each equipped with $M$ antennas and jointly serving the active users, represented by the set $\mathcal{U}$. Both RRHs and users are randomly located in 2D space. The RRHs are controlled by a single control unit (CU), and as in [3], we assume a relaxed front-haul constraint, which can be realized through technologies like the radio stripes system [14].

For each user $u\in\mathcal{U}$, we define a cluster $\mathcal{C}_{u}$ that includes the RRHs that potentially can be selected to serve the user according to user-centric clustering. Specifically, $\mathcal{C}_{u}$ comprises the RRHs with strong average channels, i.e., $\mathcal{C}_{u}=\{r\mid\left(\psi_{ru}L(d_{ru})\right)\geq\rho\}$, where $\psi_{ru}$ denotes the shadowing, $L(d_{ru})$ accounts for the path loss; here, $d_{ru}$ is the distance between RRH $r$ and user $u$. If no RRH meets this criterion, the $\mathcal{C}_{u}$ for the user comprises the RRH with largest $\left(\psi_{ru}L(d_{ru})\right)$. Finally, we represent the users that may be served by RRH $r$ as $\mathcal{E}_{r}$.

Channel estimation is performed through an uplink pilot-training phase of length $\tau_{p}$. During this phase, we can write the signal ${\bf Y}_{r}\in\mathbb{C}^{M\times\tau_{p}}$ received at RRH $r$ as

where ${\bm{\Phi}}_{u}\in\mathbb{C}^{1\times\tau_{p}}$ is the unit norm (${\bm{\Phi}}_{u}{\bm{\Phi}}_{u}^{H}=1$) pilot sequence used by user $u$, $p_{u}$ is the transmit power of the user, and ${\bf Z}_{r}$ is the additive white Gaussian noise (AWGN) with entries $\sim\mathcal{CN}\left(0,\sigma_{Z}^{2}\right)$; ${\bf h}_{ru}\in\mathbb{C}^{M\times 1}$, the channel between RRH $r$ and user $u$ is modeled as ${\bf h}_{ru}\triangleq\sqrt{\psi_{ru}L(d_{ru})}{\bf g}_{ru}$, where ${\bf g}_{ru}\sim\mathcal{CN}({\bf 0},{\bf I}_{M})$ accounts for small-scale fading.

As in [2], we assume knowledge of the users’ transmit powers and large-scale fading. Hence, using ${\bf\breve{y}}_{r}=\mathrm{vec}\{{\bf Y}_{r}\}\in\mathbb{C}^{M\tau_{p}\times 1}$ and linear MMSE, the channel estimate ${\bf\hat{h}}_{ru},\forall u\in\mathcal{E}_{r}$ can be obtained as ${\bf\hat{h}}_{ru}={\bf R}_{ru}{\bf R}_{r}^{-1}{\bf\breve{y}}_{r},$ with ${\bf R}_{ru}=\sqrt{p_{u}}\psi_{ru}L(d_{ru})\left({\bm{\Phi}}_{u}^{*}\otimes{\bf I}_{M}\right)$ and

When the number of users $|\mathcal{U}|\geq\tau_{p}$, the available pilot sequences need to be reused by the users, adding pilot contamination. This results in the estimated channel ${\bf\hat{h}}_{ru}\sim\mathcal{CN}\left({\bf 0},{\bm{\Psi}}_{ru}\right)$, with the error covariance matrix given by [15]

where ${\bf D}_{ru}\in\mathbb{C}^{M\times M}$ is a diagonal matrix with diagonal entries $\left[{\bf D}_{ru}\right]_{mm}\triangleq\psi_{ru}L(d_{ru})$, and $\mathcal{U}_{u}$ is the set of users employing the same pilot sequence as user $u$ (including user $u$). It is known from MMSE that the channel estimation error $\mathrm{{\bf e}}_{ru}={\bf h}_{ru}-{\bf\hat{h}}_{ru}$ is uncorrelated with ${\bf\hat{h}}_{ru}$ and can be modeled as $\mathrm{{\bf e}}_{ru}\sim\mathcal{CN}\left({\bf 0},{\bm{\Theta}_{ru}}\right)$, where ${\bm{\Theta}_{ru}}\triangleq{\bf D}_{ru}-{\bm{\Psi}}_{ru}$.

The downlink signal received at user $u$ can be modeled as

where $\{x_{ru}:r\in\mathcal{C}_{u}\}$ are the symbols transmitted by the serving RRHs for user $u$ with $\mathbb{E}\{|x_{ru}|^{2}\}=1$, ${\bf w}_{ru}\in\mathbb{C}^{M\times 1}$ is the precoding vector used by RRH $r$ to serve user $u$, and $z_{u}\sim\mathcal{CN}(0,\sigma_{z}^{2})$ is the AWGN.

Properly assigning the pilots to the users is clearly pivotal to decrease pilot contamination.

The HAC algorithm is more consistent than the K-means and Gaussian mixture models, and it is not sensitive to the choice of the used distance-metric [16]. Also, as this algorithm does not require selecting the number of clusters needed, it allows us to easily define the cluster based on an upper limit of the number of users belonging to it, i.e., relate it to $\tau_{p}$.

To decode the data streams from the RRHs, the users employ successive interference cancellation (SIC). Under the assumption of perfect SIC, the effective achievable rate¹¹1This expression is based on using the famous Jensen’s Inequality to write down a lower-bound for the data rate with an expectation over the unknown instantaneous channel state information (CSI) error $\{\mathrm{{\bf e}}_{ru}:r\in\mathcal{B},u\in\mathcal{E}_{r}\}$, i.e., $\mathbb{E}_{\mathrm{{\bf e}}}\left\{\log\left(1+\widetilde{\gamma}_{u}\right)\right\}\geq\log\left(1+1/\mathbb{E}_{\mathrm{{\bf e}}}\left\{\widetilde{\gamma}_{u}^{-1}\right\}\right)$, then using SIC to decode the received data streams at the user. Note that this expression is only used to perform the resource allocation, however, when we plot the performance, we use the actual achievable rate using the actual channels. for user $u$ can be modeled by the CU as [13]

where $\tau_{d}$ is the channel coherence time, $\mathcal{S}=\{{\bf s}_{1},\dots,{\bf s}_{|\mathcal{B}|}\}$ is the set of binary scheduling variables at the RRHs with ${\bf s}_{r}=[s_{ru_{1}}\ \dots\ s_{ru_{|\mathcal{E}_{r}|}}]^{T}\in\mathbb{B}^{|\mathcal{E}_{r}|\times 1}$, i.e, if $s_{ru}=1$, user $u$ is scheduled by RRH $r$, else it is not. Similarly, the set of the beamformers is $\mathcal{W}=\{{\bf W}_{1},\ \dots,\ {\bf W}_{|\mathcal{B}|}\}$ with ${\bf W}_{r}=[{\bf w}_{ru_{1}},\ \dots,\ {\bf w}_{ru_{|\mathcal{E}_{r}|}}]\in\mathbb{C}^{M\times|\mathcal{E}_{r}|}$. The term ${\bm{\Theta}_{ru^{\prime}}}=\mathbb{E}\{\mathrm{{\bf e}}_{ru^{\prime}}\mathrm{{\bf e}}_{ru^{\prime}}^{H}\}$ is the covariance of the estimation error of the channel between RRH $r$ and user $u^{\prime}$, and including it in the model allows to construct a robust beamforming.

We formulate the following WSR problem on the CU

where $\delta_{u}$ denotes the proportional fair weights for user $u$. The term $A_{u}$ is defined in (III-A). Problem (6) optimizes the decision variables $\mathcal{S}$ and $\mathcal{W}$ which determine the user-scheduling and beamforming weight vectors, respectively, such that the total utility in (6a) is maximized. We ignore the pre-log pilot training overhead factor because it is a constant. Constraints (6b) prevent the RRHs from simultaneously serving more than $M$ users on the same channel. Constraints (6c) satisfy the power budget of the RRHs, and (6e) show that a user $u$ can be scheduled or not. Constraints (6d) define the effective signal to interference and noise ratio (SINR) as an auxiliary variable.

Problem (6) is a mixed-integer non-convex problem and obtaining a global optimum is mathematically prohibitive.

The beamforming vectors are constructed for users that are actually scheduled on the channel and hence

where $\left\|\cdot\right\|_{0}$ is the $\ell_{0}$-norm. Using the literature of compressive sensing, the $\ell_{0}$-norm of a vector ${\bf x}$ can be approximated as a weighted convex $\ell_{1}$-norm $\|{\bf x}\|_{0}\simeq\sum_{m}\alpha_{m}|x_{m}|=\|{\bm{\alpha}}{\bf x}\|_{1}$ [17], where $\alpha_{m}$ are positive weights that penalize the nonzero coefficients $x_{m}$, and ${\bm{\alpha}}=\textbf{diag}\{\alpha_{1},\alpha_{2},\dots\}$ is a diagonal matrix. For our case, ${\bf x}=\left\|{\bf w}_{ru}\right\|_{2}$ which is scalar. We can construct an iterative process to find these weights at each iteration $i$ as suggested in [17]

where $\epsilon>0$ provides stability and ensures that a zero-valued component in $\left\|{\bf w}_{ru}\right\|_{2}^{2}$ does not strictly prohibit a nonzero estimate at the update in the next iteration.

As a result, our problem can be formulated as follows

where $B_{u}\left(\mathcal{W}\right)=A_{u}\left({\bf 1},\mathcal{W}\right)$. We use the Lagrangian for the equality constraints in (9d)

When $\mathcal{W}$ is fixed, we evaluate the first optimality condition of the SINR auxiliary variable $\gamma_{u}$ by setting the derivative of (III-B) with respect to $\gamma_{u}$ to zero, which results in a value for $\nu_{u}$ that satisfies this optimality. Substituting $\nu_{u}$ back into (III-B):

Setting the derivative of (III-B) to zero, we obtain the expected optimal formula for $\gamma_{u}$ in (9d), which means they are equivalent.

Hence, our new reformulated problem can be written as

Note that we are not writing the dual problem here, but rather we are introducing SINR auxiliary variables ${\bm{\gamma}}$ that act as a proxy to account for the changes of the other variables.

When the variables other than ${\bm{\beta}}_{r}$ are fixed, the optimal value of the auxiliary variable $\beta_{ru}$ can be obtained from its corresponding first-order optimality condition from (16) as

Similarly for the beamformers ${\bf w}_{ru}$, we can write the Lagrangian formulation using the new objective function (16) and the constraints in (III-B), then evaluating the corresponding first-order optimality condition to write ${\bf w}_{ru}$ as

where the Lagrangian multipliers $\lambda_{r}\geq 0$ and $\mu_{r}\geq 0$ correspond to the capacity (9b) and power (9c) constraints. Importantly, both these constraints relate to the power used at RRH $r$, i.e., both cannot be tight simultaneously. From complementary slackness, therefore, one of these Lagrange multipliers, both corresponding to RRH $r$, must be zero.

Unfortunately, we do not know a priori which constraint will remain tight. As we will see in our algorithm section, we propose a heuristic that, at each iteration of the algorithm, checks for whether the capacity constraint is satisfied (allowing $\lambda_{r}=0$); if it is not satisfied, we update set $\lambda_{r}$ to a small value and update $\mu_{r}$ using a bisection search to meet the power constraint. Our results show that after a few iterations, $\lambda_{r}$ always converges to zero; we will comment on this in the results section.

We construct Algorithm 1 to allocate the resources for the users. The algorithm initializes some variables (Step 1) (e.g., conjugate beamforming to initialize ${\bf w}_{ru}$). Then, it updates the variables ${\bm{\gamma}}$, ${\bm{\beta}}$, $\mathcal{W}$, and ${\bm{\alpha}}$ iteratively one at a time until convergence.

The complexity of updating ${\bm{\gamma}}$, ${\bm{\beta}}$, and ${\bm{\alpha}}$ is $\mathcal{O}\left(|\mathcal{U}|\right)$, $\mathcal{O}\left(|\mathcal{U}|C_{\textrm{avg}}\right)$, and $\mathcal{O}\left(|\mathcal{U}|C_{\textrm{avg}}\right)$, respectively, where $C_{\textrm{avg}}$ is the average cluster size per user, and it is affected by both the density of the active users and the large scale fading threshold $\rho$. The complexity of the beamforming using a weighted MMSE [18] is $\mathcal{O}\left(|\mathcal{U}_{s}|^{2}M^{2}+|\mathcal{U}_{s}|M^{3}\right)$, where $\mathcal{U}_{s}$ is the set of scheduled users. Hence, leading to a total algorithm complexity of at most $\mathcal{O}\left(M^{3}|\mathcal{B}|^{2}+M^{4}|\mathcal{B}|+|\mathcal{U}|C_{\textrm{avg}}\right)$, where the number of the scheduled users is at most $|\mathcal{U}_{s}|\leq M|\mathcal{B}|$.