Hybrid Beamforming and Adaptive RF Chain Activation for Uplink Cell-Free Millimeter-Wave Massive MIMO Systems

The performance CF massive MIMO systems in conventional sub-6-GHz frequency bands have been analyzed intensively [3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In particular, the closed-form expression of the achievable rate of CF massive MIMO systems is derived in [3, 4, 6, 7]. Furthermore, comparisons of CF massive MIMO and conventional small-cell MIMO systems in [3, 4, 8, 9, 7] show that the former is more robust to shadow fading correlation and significantly outperforms the latter in terms of throughput and coverage probability. In [10], a low-complexity power control technique with zero-forcing (ZF) precoding design is introduced for CF massive MIMO systems. In [11], an optimal power-allocation algorithm is proposed to maximize the total EE, which can double the total EE compared to the equal power control scheme. Furthermore, an AP selection (APS) scheme is proposed in [11], in which each user chooses and connects to only a subset of APs to reduce the power consumption caused by the backhaul links. Meanwhile, in [12], the EE maximization problem is considered under the effect of quantization distortion of the weighted received signals at the APs. In [13, 14], the authors propose AP-clustering approaches to overcome limitations upon the scalability of CF massive MIMO systems. Specifically, the APs are grouped in clusters and multiple CPUs are used to manage these clusters, leading to a reduction in the data distribution and computational complexity involved in channel estimation, power control, and beamforming.

Another line of work has attempted to investigate the performance of CF massive MIMO systems in mmWave channels [21, 22, 23, 5]. In particular, Femenias et al. introduced a hybrid beamforming (HBF) framework for CF mmWave massive MIMO systems with limited fronthaul capacity in [5], and the eigen-beamforming scheme is applied to generate precoders/combiners. Specifically, the phases of analog beamformers are obtained by quantizing those of dominant eigenvectors of the channel covariance matrix known at the APs. In [21], a hybrid precoding algorithm leveraging antenna-array response vectors is applied to distributed MIMO systems with partially-connected HBF architecture. Although the partially-connected HBF structure has lower power consumption than the fully-connected one, it cannot fully exploit the beamforming gains [24]. In [22] and [23], Alonzo et al. introduce an uplink multi-user estimation scheme along with low-complexity HBF architectures. Specifically, the baseband and analog precoders at each AP are generated by decomposing the fully digital ZF precoding matrix using the block-coordinate descent algorithm. Moreover, the problems of pilot assignment and channel estimation are considered in [25] and [26], respectively.

In mmWave communications, a large number of antennas at the APs not only provide beamforming gains to compensate for the propagation loss in mmWave channels, but also enhance favorable propagation [27]. In particular, it is shown in [27] that given a fixed total number of antennas at the APs, employing more antennas at fewer APs is more beneficial than deploying more APs with fewer antennas in terms of favorable propagation and channel hardening. Furthermore, large antenna arrays at the APs can achieve high beamforming gains to overcome the severe path loss in mmWave communication. However, an excessively high power consumption is required in this deployment because signal processing in the conventional digital domain requires a dedicated RF chain and analog-to-digital converter (ADC) for each antenna. Therefore, energy-efficient HBF schemes dedicated to CF mmWave massive MIMO systems are required, but only limited works exist in the literature focusing on the optimization of HBF for CF mmWave massive MIMO systems. Specifically, in [21, 22, 5, 23], the analog beamformers are all separately generated at the APs based on the local CSI, which are similar to HBF schemes for small-cell mmWave massive MIMO systems when each AP in the CF mmWave massive MIMO system is considered as a base station in the small-cell system. The antenna-selection (AS) schemes, which are proposed for the transceiver with a limited number of RF chains in [28, 29], can be applied to CF mmWave massive MIMO systems to reduce power consumption. However, they can cause performance degradation, especially for HBF in the highly correlated channels of mmWave communication [30].

In this work, we investigate the HBF for uplink CF mmWave massive MIMO systems in two scenarios: the global CSI of all APs is available or unavailable at the CPU. Then, we propose an adaptive RF chain-activation (ARFA) scheme, which provides considerable power reduction while nearly maintaining the system’s total achievable rate; thus, the EE is remarkably improved. Our specific contributions can be summarized as follows:

• •

  We first propose the decentralized HBF (D-HBF) and semi-centralized HBF (SC-HBF) schemes. Both have digital beamformers generated at the AP, but their difference lies in the analog beamformer. Specifically, in SC-HBF, the analog beamforming matrices for all APs are generated at the CPU based on the global CSI. In contrast, that of the D-HBF is obtained at each AP based only on the local CSI. By exploiting the global CSI to jointly optimize the analog combiners at the CPU, SC-HBF is expected to outperform D-HBF. However, our analytical and numerical results show that D-HBF can perform approximately the same as SC-HBF while requiring substantially lower computational complexity and no global CSI.

• •

  In CF mmWave massive MIMO systems with $L$ APs and $N$ RF chains at each AP, the power consumption is approximately proportional to $LN$. Because $L$ is large in CF massive MIMO systems, the total power consumption can be excessively high. To overcome this challenge, we propose an ARFA scheme. In this scheme, the RF chains are selectively activated at the APs based on partial CSI, and the number of active RF chains at the APs is optimized so that the proposed scheme can significantly reduce the total power consumption while causing only marginal performance loss. Our numerical analysis reveals that in CF mmWave massive MIMO systems, the proposed ARFA scheme with a relatively small number of active RF chains can exhibit performance comparable to that of the conventional fixed-activation HBF scheme, which activates all the available RF chains. As a result, a considerable improvement in the EE is achieved.

• •

  In the proposed ARFA scheme, high computational complexity is required to find the optimal numbers of active RF chains at numerous APs with an exhaustive search. To reduce complexity, we propose a low-complexity near-optimal algorithms for the ARFA with SC-HBF. Furthermore, ARFA is incorporated with D-HBF in the proposed D-ARFA schemes, creating a singular value-based and path loss-based D-ARFA, wherein the CPU requires a very limited amount of information from the APs. Our simulation results show that the proposed algorithms perform very close to the conventional HBF scheme, in which all the available RF chains are turned on.

We note that an RF chain-selection (RFS) scheme is introduced in [31] for the conventional small-cell mmWave massive MIMO system. The RFS scheme and our proposed ARFA scheme are similar in exploiting a reduced number of RF chains for power reduction. However, the contributions in our work are novel in the following aspects. First, the number of active RF chains at the base station in the conventional small-cell system, which is a single integer, is optimized in the RFS scheme by solving the EE-optimization problem [31]. In contrast, we consider the CF mmWave massive MIMO system, and the numbers of active RF chains at numerous APs are jointly optimized by maximizing the system’s total achievable throughput. This results in unequal numbers of active RF chains at the APs, and an AP can even deactivate all RF chains, leading to significant power reduction. Second, [31] focuses on optimizing the number of RF chains at the transmitter rather than at the receiver to avoid the nontrivial integer programming problem. Our work fills in this hole by optimizing the numbers of active RF chains at the receivers, which are the APs in the uplink of CF mmWave massive MIMO systems. Because of these systematic differences, the algorithm presented in [31] cannot be leveraged for our proposed ARFA scheme, which thus requires novel algorithms as presented in Section IV.

The remainder of this paper is organized as follows: Section II introduces the system and channel models, whereas Section III describes the D-HBF and SC-HBF schemes. In Section IV, low-complexity ARFA algorithms are presented, and the power consumption of the proposed ARFA scheme is analyzed in Section V. Section VI presents simulation results, and the conclusion follows in Section VII.

For channel estimation, all $K$ UEs simultaneously transmit their pilot sequences to the APs. Let $\sqrt{\tau_{p}}\boldsymbol{\varphi}_{k}\in\mathbb{C}^{\tau_{p}\times 1}$ be the pilot sequence of the $k$th UE, where $\left\lVert\boldsymbol{\varphi}_{k}\right\rVert^{2}=1$, $k=1,\ldots,K$. Here, $\tau_{p}$ $(\tau_{p}<\tau_{c})$ is the length of $\boldsymbol{\varphi}_{k}$, where $\tau_{c}$ denotes the length of each coherence interval (in samples). When all $K$ UEs send their pilots, the received signal at the $l$th AP is given as:

where $\rho_{p}>0$ represents the average transmit power of each UE; $\textbf{{Z}}_{l}\in\mathbb{C}^{N_{r}\times\tau_{p}}$ is the receiver noise, whose entries are independent and identically distributed (i.i.d.) $\mathcal{CN}(0,\sigma^{2})$ random variables (RVs); and $\sigma^{2}$ is the noise power. To estimate $\textbf{{h}}_{kl}$, $\textbf{{Y}}_{l}$ is projected onto $\boldsymbol{\varphi}_{k}$, which yields

Thus, the minimum mean-square error (MMSE) estimate of $\textbf{{h}}_{kl}$, denoted by $\hat{\textbf{{h}}}_{kl}$, is given by

Assume that knowledge of the correlation matrix $\boldsymbol{\Psi}_{kl}$, $\forall k$, is available at the $l$th AP [9], from which $\hat{\textbf{{h}}}_{kl}$ can be determined. Note that in (4), we assume that the signals received at all the antennas of the AP are available for MMSE estimation. As a result, the estimate of the full CSI associated with all the antennas can be obtained via the low-complexity MMSE estimator. In a very slow-fading and sparse channel, the full CSI can be obtained by a compressed sensing-based approach [32], but with the high complexity required for a large number of compressed-sensing measurements [38].

Let $\hat{\textbf{{H}}}_{l}\triangleq[\hat{\textbf{{h}}}_{1l},\ldots,\hat{\textbf{{h}}}_{Kl}]\in\mathbb{C}^{N_{r}\times K}$ denote the estimated channel matrix between the $K$ UEs and $l$th AP. Furthermore, we define $\hat{\textbf{{H}}}\triangleq\left[\hat{\textbf{{H}}}_{1}^{T},\ldots,\hat{\textbf{{H}}}_{L}^{T}\right]^{T}\in\mathbb{C}^{LN_{r}\times K}$ as the composite estimated channels between all the APs and UEs. In the next section, $\hat{\textbf{{H}}}_{l}$ and $\hat{\textbf{{H}}}$ are employed for the hybrid beamforming design of the uplink data transmission.

Denote by $x_{k}$ the symbol sent from the $k$th UE to all the APs, such that $\mathbb{E}\{\left|x_{k}\right|^{2}\}=1$, $\forall k$. The signal input-output relationship at the $l$th AP can be expressed as

where $\rho$ represents the average transmit power, and $\textbf{{z}}_{l}$ is the noise vector, whose elements are i.i.d. $\mathcal{CN}(0,\sigma^{2})$ RVs. Furthermore, $\textbf{{F}}_{l}\in\mathbb{C}^{N_{r}\times N}$ is the analog combining matrix at the $l$th AP. Its $n$th column, i.e., $\hskip 2.84544pt\textbf{{f}}_{ln}=[f_{ln}^{(1)},\ldots,f_{ln}^{(N_{r})}]^{T}$, is the analog weight vector corresponding to the $n$th RF chain at the $l$th AP, and $f_{ln}^{(i)}=\frac{1}{\sqrt{N_{r}}}e^{j\theta_{ln}^{(i)}}$ is the $i$th element of $\hskip 2.84544pt\textbf{{f}}_{ln}$. $\textbf{{W}}_{l}\in\mathbb{C}^{N\times K}$ denote the digital combining matrix at the $l$th AP. Then, the APs send $\textbf{{r}}_{l},\forall l$ to the CPU via a fronthaul network to perform the final signal detection. In this work, we assume a simple centralized decoding scheme at the CPU, which requires minimal information exchange between the APs and CPU. In this scheme, the final decoded signal at the CPU is given as the average of the local estimates, that is, $\frac{1}{L}\sum_{l=1}^{L}\textbf{{r}}_{l}$ [9].

The composite received signal available at the CPU can be expressed as

Let $\textbf{{F}}\triangleq\text{diag}\left\{\textbf{{F}}_{1},\ldots,\textbf{{F}}_{L}\right\}\in\mathbb{C}^{LN_{r}\times LN}$ and $\textbf{{W}}\triangleq\text{diag}\left\{\textbf{{W}}_{1},\ldots,\textbf{{W}}_{L}\right\}\in\mathbb{C}^{LN\times LK}$ be block-diagonal matrices containing the analog and digital combiners for all $L$ APs. In this work, we refer to F and W as global combiners, whereas $\left\{\textbf{{F}}_{1},\ldots,\textbf{{F}}_{L}\right\}$ and $\left\{\textbf{{W}}_{1},\ldots,\textbf{{W}}_{L}\right\}$ for the signal combined at APs $\{1,\ldots,L\}$ are referred to as the local combiners. Furthermore, let $\textbf{{H}}_{l}\triangleq\left[\textbf{{h}}_{1l},\ldots,\textbf{{h}}_{Kl}\right]\in\mathbb{C}^{N_{r}\times K}$ denote the channel matrix between the $K$ UEs and the $l$th AP. We define $\textbf{{r}}\triangleq[\textbf{{r}}_{1}^{T},\ldots,\textbf{{r}}_{L}^{T}]^{T}\in\mathbb{C}^{LK\times 1}$, $\textbf{{x}}\triangleq[x_{1},\ldots,x_{K}]^{T}\in\mathbb{C}^{K\times 1}$, $\textbf{{z}}\triangleq[\textbf{{z}}_{1}^{T},\ldots,\textbf{{z}}_{L}^{T}]^{T}\in\mathbb{C}^{LN_{r}\times 1}$, and $\textbf{{H}}\triangleq[\textbf{{H}}_{1}^{T},\ldots,\textbf{{H}}_{L}^{T}]^{T}\in\mathbb{C}^{LN_{r}\times K}$. Then, (6) can be rewritten in a more compact form as

where $\tilde{\textbf{{z}}}\triangleq\textbf{{W}}^{H}\textbf{{F}}^{H}\textbf{{z}}\sim\mathcal{CN}(0,\textbf{{R}}_{\tilde{z}})$, and $\textbf{{R}}_{\tilde{z}}\triangleq\sigma^{2}\textbf{{W}}^{H}\textbf{{F}}^{H}\textbf{{F}}\textbf{{W}}$.

We note that the analog processing is separately performed at the APs because the ADCs, RF chains, and PSs are installed at the APs. However, $\left\{\textbf{{F}}_{1},\ldots,\textbf{{F}}_{L}\right\}$ can be generated either at the APs based on their local CSI or at the CPU based on the global CSI. Once the analog combiner is obtained, digital processing can be carried out at the corresponding AP. We follow the common assumption in [4, 19] that the digital combining is performed at the APs individually. Therefore, in this work, the D-HBF scheme refers to the HBF with analog combiners generated at each AP separately, whereas SC-HBF implies that the analog combiners are computed at the CPU based on the global CSI. In this regard, we note that the perfect local/global CSI is not available at neither the APs nor the CPU, respectively. Therefore, to perform D-HBF, the AP, say the $l$th AP, treats its own estimated channels $\hat{\textbf{{H}}}_{l}$ as the true channel and employs it to generate the hybrid combiners $\textbf{{F}}_{l}$ and $\textbf{{W}}_{l}$. Similarly, in SC-HBF, the CPU exploits the global estimated CSI $\hat{\textbf{{H}}}$ to obtain $\left\{\textbf{{F}}_{1},\ldots,\textbf{{F}}_{L}\right\}$.

It is evident from (11) and (12) that $\mathcal{R}^{\mathrm{a}}_{l}$ depends not only on $\hat{\textbf{{H}}}_{l}$, but also on $\hat{\textbf{{H}}}_{l-1},\hat{\textbf{{H}}}_{l-2},\ldots,\hat{\textbf{{H}}}_{1}$. Therefore, finding $\textbf{{F}}_{l}^{\star}$ requires not only $\hat{\textbf{{H}}}_{l}$ but also $\hat{\textbf{{H}}}_{l-1},\hat{\textbf{{H}}}_{l-2},\ldots,\hat{\textbf{{H}}}_{1}$. This is similar to the requirements for determining analog beamformers for sub-arrays in the partially-connected HBF architecture [40, 24]. As a result, solving $\left\{\textbf{{F}}_{1}^{\star},\textbf{{F}}_{2}^{\star},\ldots,\textbf{{F}}_{L}^{\star}\right\}$ requires the CSI of the channels between all $L$ APs and $K$ UEs, i.e., $\left\{\hat{\textbf{{H}}}_{1},\hat{\textbf{{H}}}_{2},\ldots,\hat{\textbf{{H}}}_{L}\right\}$, which can be available at the CPU; hence, finding $\textbf{{F}}^{\star}$ based on (13) requires a SC-HBF scheme.

Algorithm 1 presents the proposed SC-HBF scheme to obtain $\left\{\textbf{{F}}_{1}^{\star},\textbf{{F}}_{2}^{\star},\ldots,\textbf{{F}}_{L}^{\star}\right\}$. In particular, in steps 3–6, the combining vector $\hskip 2.84544pt\textbf{{f}}_{ln}^{\star}$ is obtained by quantizing $\textbf{{u}}_{ln}^{\star}$ based on (14), which ensures that the resultant analog combiners belong to the feasible set $\mathcal{F}$. Then, $\textbf{{F}}_{l}^{\star}$ is found in step 7 and $\textbf{{G}}_{l}$ is computed in step 8, followed by $\hat{\textbf{{Q}}}_{l}$ being updated in step 9. In step 10, $\mathcal{R}^{\mathrm{a}}_{l}$ corresponding to the $l$th AP is computed. Furthermore, the digital combiner is computed at each AP, as in step 12. We note that in Algorithm 1, steps 1–11 are performed at the CPU, whereas step 12 is performed at the APs.

Let $\left\{\tilde{\textbf{{u}}}^{\star}_{l1},\ldots,\tilde{\textbf{{u}}}^{\star}_{lN}\right\}$ be the $N$ singular vectors corresponding to the $N$ largest singular values of $\hat{\textbf{{H}}}_{l}$, which are in decreasing order. Furthermore, define

Then, in the D-HBF scheme, the optimal local analog combiner generated at the $l$th AP based on $\hat{\textbf{{H}}}_{l}$ can be given as $\tilde{\textbf{{F}}}_{l}^{\star}=\left[\tilde{\hskip 2.84544pt\textbf{{f}}}_{l1}^{\star},\ldots,\tilde{\hskip 2.84544pt\textbf{{f}}}_{lN}^{\star}\right]$ [21]. Let $\tilde{\textbf{{F}}}^{\star}=\text{diag}\left\{\tilde{\textbf{{F}}}_{1}^{\star},\ldots,\tilde{\textbf{{F}}}_{L}^{\star}\right\}$. In the following theorem, we show that the total achievable rate achieved by analog combining in the D-HBF scheme is approximately equal to that in SC-HBF.

It is observed that D-HBF can be performed with considerably lower computational complexity than SC-HBF. Specifically, only $N$ singular vectors corresponding to the $N$ largest singular values of the channel matrix are required to form the analog combiner. In contrast, additional matrix inversions, multiplications, and additions are performed in steps 4, 8, and 9 of Algorithm 1 for the SC-HBF scheme. Notably, despite the simpler implementation and lower complexity of the D-HBF scheme, it can approximately achieve the performance of SC-HBF, as stated in Remark 1. Furthermore, the D-HBF scheme requires less information exchange between the APs and CPU. Specifically, only $K$ complex numbers in $\textbf{{r}}_{l}$ are sent to the CPU on the fronthaul link to perform the final soft detection [4, 19], whereas the exchange of the CSI and analog combining matrix is not required, in contrast to the SC-HBF scheme

Remark 1 indicates that an efficient analog beamforming matrix can be designed based on the local CSI available at each AP of CF mmWave massive MIMO systems. However, this does not mean that further information exchange via the fronthaul links is completely useless. Indeed, the information exchange between the APs and CPU in CF massive MIMO systems can also be exploited to improve the EE. In the next section, it is discussed that by adaptively activating RF chains based on global CSI in SC-HBF or on limited information in D-HBF, the power consumption can be reduced, which leads to improved EE.

The global analog combiner F can be expressed as

and the following facts are noted:

• •

  Based on Theorem 1, the total achievable rate obtained by analog combining can be expressed as a sum of $\{\bar{\mathcal{R}}^{\mathrm{a}}_{1},\ldots,\bar{\mathcal{R}}^{\mathrm{a}}_{L}\}$ corresponding to APs $\{1,\ldots,L\}$. In CF mmWave massive MIMO systems, the APs are distributed in a large area, and their communication channels experience different path losses and shadowing effects. Therefore, the contributions of the local analog combiners at different APs to the total achievable rate are of different significances.

• •

  In a local analog combiner $\textbf{{F}}_{l}$, combining vectors $\{\hskip 2.84544pt\textbf{{f}}_{l1},\ldots,\hskip 2.84544pt\textbf{{f}}_{lN}\}$ have different contributions to the sub-rate $\bar{\mathcal{R}}^{\mathrm{a}}_{l}$ given in (11). Specifically, the contribution of $\hskip 2.84544pt\textbf{{f}}_{ln}$ is more significant than that of $\hskip 2.84544pt\textbf{{f}}_{lm}$ if $n<m$ because $n$ and $m$ are the indices of the ordered singular values of $\hat{\textbf{{H}}}_{l}\hat{\textbf{{Q}}}_{l-1}^{-1}\hat{\textbf{{H}}}_{l}^{H}$ in SC-HBF and of $\hat{\textbf{{H}}}_{l}$ in D-HBF.

As a result, it is likely that a subset of analog combining vectors in $\{\hskip 2.84544pt\textbf{{f}}_{11},\ldots,\hskip 2.84544pt\textbf{{f}}_{LN}\}$ are insignificant and can be removed from the global combiner F without causing considerable performance loss. We note that at an AP, an analog combining vector represents the effect of $N_{r}$ PSs connected to an RF chain, followed by an ADC. Therefore, an insignificant analog combining vector can be removed from signal combining by turning off its corresponding RF chain, ADC, and PSs, which results in a reduction in the total power consumption. Motivated by this, we propose an ARFA scheme that selectively activates RF chains at the APs. Let $\textbf{{n}}=\{n_{1},\ldots,n_{L}\}$, where $n_{l}$ is the number of turned-on RF chains out of $N$ RF chains installed at the $l$th AP, $0\leq n_{l}\leq N$. We note that for $n_{l}=0$, all the RF chains at the $l$th AP are turned off, and this AP does not consume any power for signal combining. It is also noted that in the uplink training phase of the next coherent time, the deactivated RF chains are reactivated for channel estimation. The optimal activation of RF chains at the APs can be performed based on the following remark.

The HBF architecture with the proposed ARFA scheme is illustrated in Fig. 1. As an example, at the $l$th AP, only $n_{l}$ out of $N$ RF chains are turned on. Furthermore, the ADCs and PSs connected to the inactive RF chains are also turned off. Consequently, the local combiner $\textbf{{F}}_{l}$ consists of only $n_{l}$ analog combining vectors, i.e., $\textbf{{F}}_{l}=\{\hskip 2.84544pt\textbf{{f}}_{1},\ldots,\hskip 2.84544pt\textbf{{f}}_{n_{l}}\}$.

Unlike the conventional fixed-activation HBF, in the proposed ARFA scheme, the global combiner F, $\mathcal{R}^{\mathrm{a}}_{l}$, and $\mathcal{R}\triangleq\sum_{l=1}^{L}\mathcal{R}^{\mathrm{a}}_{l}$ depend on n. Therefore, in this section, they are expressed as functions of n, i.e., $\textbf{{F}}(\textbf{{n}})$, $\mathcal{R}^{\mathrm{a}}_{l}(\textbf{{n}})$, and $\mathcal{R}(\textbf{{n}})$, respectively. We limit the total number of turned-on RF chains in the ARFA scheme to $L\bar{n}$, i.e., $\sum_{l=1}^{L}n_{l}=L\bar{n}$, where $\bar{n}(\leq N)$ is the average number of activated RF chains at each AP. Based on Remark 2, the optimal activation of RF chains at the APs in the ARFA scheme can be performed by solving

where $\mathcal{S}=\left\{\textbf{{n}}:0\leq n_{l}\leq N,\sum_{l=1}^{L}n_{l}=L\bar{n}\right\}$ is the feasible set of n. The optimal $\textbf{{n}}^{\star}$ in (18) can be found by exhaustive search over the entire feasible set $\mathcal{S}$. However, in CF mmWave massive MIMO systems, $L$ is large; thus, an excessively large number of candidates for n need to be examined in the exhaustive search, which is almost computationally prohibitive. In the following subsections, we propose three low-complexity algorithms to find $\textbf{{n}}^{\star}$. By abuse of notation, we use $\textbf{{F}}^{\star}$ for the global combiner found by the proposed ARFA algorithms. Furthermore, we note that Algorithm 1 can be easily modified for the ARFA scheme by replacing $N$ in steps 3, 7, and 10 with $n_{l}$, reflecting a dynamic number of analog combining vectors for the local combiner $\textbf{{F}}_{l}$ at the $l$th AP.

In SC-ARFA, the ARFA scheme is incorporated with SC-HBF, and the optimal numbers of active RF chains at the APs are found at the CPU based on the global CSI. The idea to find $\textbf{{n}}^{\star}$ is to turn on/off as many RF chains as possible at the APs corresponding to the largest/smallest $\mathcal{R}^{\mathrm{a}}_{l}$, as presented in Algorithm 2. In steps 1–3, all elements of n are set to $\bar{n}$, then $\textbf{{F}}(\textbf{{n}}),\mathcal{R}^{\mathrm{a}}_{l}(\textbf{{n}}),\forall l$, and $\mathcal{R}(\textbf{{n}})$ are computed. In step 4, the elements of n are ordered to obtain $\{n_{[1]},\ldots,n_{[L]}\}$ in the decreasing order of sub-rates $\{\mathcal{R}^{\mathrm{a}}_{1}(\textbf{{n}}),\ldots,\mathcal{R}^{\mathrm{a}}_{L}(\textbf{{n}})\}$. Therefore, $n_{[i]}$ is the number of turned-on RF chains at the AP with the $i$th largest sub-rates, i.e., $\mathcal{R}^{\mathrm{a}}_{[i]}(\textbf{{n}})$.

In step 5, we initialize $i=1$ and $k=L$. In steps 6–16, $n_{[i]}$ is increased by one, whereas $n_{[k]}$ is decreased by one, in each iteration. We note that in step 13, $n_{[i]}$ and $n_{[k]}$ are updated simultaneously to guarantee $\sum_{l=1}^{L}n_{[l]}=L\bar{n}$. The updates of $n_{[i]}$ and $n_{[k]}$ result in a new candidate n. Hence, $\textbf{{F}}(\textbf{{n}})$ and $\mathcal{R}(\textbf{{n}})$ are found in step 14, and $\textbf{{F}}^{\star}$ is updated if the performance is improved, as shown in step 15. Once $n_{[i]}$ reaches the maximum, i.e., $N$, the number of turned-on RF chains at the AP associated with the $(i+1)$th largest sub-rate, i.e., $n_{[i+1]}$, is considered, as shown in steps 7–9. In contrast, if $n_{[k]}$ reaches the minimum, i.e., zero, $n_{[k-1]}$ is considered next, as shown in steps 10–12. This iterative process is terminated if $i\geq k$, for which we have $\mathcal{R}^{\mathrm{a}}_{[i]}(\textbf{{n}})\leq\mathcal{R}^{\mathrm{a}}_{[k]}(\textbf{{n}})$ and the increase (decrease) in $n_{[i]}$ $(n_{[k]})$ is unlikely to provide performance improvement. Once all the analog combiners are determined and sent to the APs, the digital combiner at each AP is determined, as in step 17.

We note that the ARFA process needs to be performed at the CPU to jointly optimize the numbers of RF chains at all APs. In the SC-ARFA schemes, the global CSI is exploited to evaluate the candidates for n. However, the employment of SC-HBF in these schemes requires high computational complexity and a large amount of information exchanged between the CPU and APs, as discussed in Section III-B. This motivates us to propose an ARFA scheme incorporated with D-HBF in the next subsection.

Without global CSI, the ARFA scheme can be performed if the CPU knows the qualities of the available combining vectors or the path loss corresponding to each AP. The former idea relies on the fact that a combining vector is obtained by quantizing a singular vector of the channel matrix, as shown in (16). Therefore, the quality of a combining vector can be evaluated based on its corresponding singular value. In contrast, the latter idea for D-ARFA is motivated by the observation that the AP with more significant path loss should have fewer activated RF chains because it is more likely to have a low sub-rate.