Optimizing RRH Placement Under a
Noise-Limited Point-to-Point Wireless Backhaul

We consider a cooperative distributed network comprising $Q$ cells. Assuming joint transmission with disjoint clustering, the RRHs found in each cell $q$ jointly serve the users inside their cell boundary $\mathcal{B}_{q}$ [14]. Each cell employs $N$ RRHs to serve $K$ users on the same time-frequency resource block using zero-forcing beamforming (ZF-BF). Each RRH uses $M$ antennas to serve the users, where $NM>K$. In each cell, a single CU, located at the cell center, controls the transmissions of the RRHs through point-to-point backhaul links. The users in the cell may be concentrated in (but not restricted) hotspots with a priori known traffic pattern. The backhaul and access links are spectrally orthogonal. To focus on network design, we assume perfect channel state information (CSI) is available. Our aim is to deploy the RRHs in the network so that we obtain the highest spectral efficiency over the access channel while controlling backhaul outage.

The signal received at a typical user $k$ located in cell $q$ is

where $s_{qk}\in\mathbb{C}$ is the power-limited signal sent to user $k$ by the serving RRHs in cell $q$, that is $\mathbb{E}\{{\bf s}_{q}{\bf s}_{q}^{H}\}=p{\bf I}_{K}$ for ${\bf s}_{q}=[s_{q1},\dots,s_{qK}]$, with $p$ being the total power budget for the RRHs in the cell, which is referred as vector normalization [15].

The vector ${\bf h}_{qn,qk}\in\mathbb{C}^{M}$ denotes the channel between RRH $n$ in cell $q$ and user $k$ in cell $q$ and accounts for the small-scale and large-scale fading. We have ${\bf h}_{qn,qk}=\sqrt{\ell_{k}(x_{qn},y_{qn})}{\bf g}_{qn,qk}$, where ${\bf g}_{qn,qk}\sim\mathcal{CN}(0,{\bf I}_{M})$ is a white complex Gaussian random vector representing the Rayleigh fading, while $\ell_{k}(x_{qn},y_{qn})=\left(1+d_{qn,qk}/d_{0}\right)^{-\alpha}$ is the path loss of the signal, with $d_{qn,qk}=\sqrt{(x_{qn}-\widetilde{x}_{k})^{2}+(y_{qn}-\widetilde{y}_{k})^{2}}$ being the distance between RRH $n$ and user $k$, $d_{0}$ is the reference distance, and $\alpha$ is the path loss exponent. The terms $x_{qn}$ and $y_{qn}$ are the $x,y$-coordinates of RRH $n$ and will be our optimization variables. Furthermore, ${\bf w}_{qnk}\in\mathbb{C}^{M}$ is the linear precoding vector used by RRH $n$ to serve user $k$ in its cell $q$, and $z_{qk}\sim\mathcal{CN}(0,\sigma_{z}^{2})$ is the independent additive white Gaussian noise (AWGN), with $\sigma^{2}_{z}$ being the noise power.

The RRHs use ZF-BF to serve the $K$ users in each cell. To construct the beamformer, we denote the concatenation of the channel matrix for all the users in cell $q$ as ${\bf H}_{q}=[{\bf h}_{q1}\dots\bf h_{qK}]\in\mathbb{C}^{NM\times K}$ with presumably linearly independent rows, with ${\bf h}_{qk}=[{\bf h}_{qk,qk}^{T}\dots\bf h_{qN,qk}^{T}]^{T}\in\mathbb{C}^{NM}$ being the concatenation of the access channels for user $k$. We can write ${\bf h}_{qk}={\bf D}_{qk}^{1/2}{\bf g}_{qk}$, where ${\bf D}_{qk}\in\mathbb{C}^{NM\times NM}$ is a diagonal matrix representing the large-scale fading between all the RRHs in the network and user $k$. Hence, $[{\bf D}_{qk}]_{mm}=\ell_{k}(x_{qn},y_{qn})$ where $n=\lceil{m/M}\rceil$ for $m=\{1,\dots,NM\}$. Furthermore, ${\bf g}_{qk}\in\mathbb{C}^{NM}$ represents the vector of the small-scale fading coefficients between the antennas of the RRHs and user $k$.

The precoding matrix ${\bf W}_{q}$ is designed for each cell $q$ assuming CSI at the RRHs is available. We have ${\bf W}_{q}=[{\bf v}_{q1}\dots\bf v_{qK}]\in\mathbb{C}^{NM\times K}$ with ${\bf v}_{qk}=[{\bf w}_{q1k}^{T}\dots\bf w_{qNk}^{T}]^{T}\in\mathbb{C}^{NM}$, where ${\bf w}_{qnk}\in\mathbb{C}^{N}$ is the precoding vector used at RRH $n$, and is given by

Here, $\bm{\mu}_{q}\in\mathbb{C}^{K\times K}$ is a diagonal matrix which satisfies the power budget $\mathbb{E}\left\{\text{tr}\left\{{\bf W}_{q}{\bf W}_{q}^{H}\right\}\right\}=p$, i.e., using average power normalization [16], and its $k^{th}$ entry is

with ${\bf\widetilde{v}}_{qk}=[{\bf H}_{q}\left({\bf H}_{q}^{H}{\bf H}_{q}\right)^{-1}]_{.k}$ is $k^{th}$ column of the matrix.

We consider point-to-point single-input single-output (SISO) noise-limited wireless transmission between the CUs and their RRHs (representing an upper-bound for the performance of a SISO solution). We assume that the CU implements directional antennas, providing parallel Rician channels between the CU and the $N$ RRHs.

The downlink average achievable rate between CU $q$ and its served RRHs is defined as

where $\rho_{c}=\frac{p_{c}}{\sigma^{2}}$, $\bar{\ell}_{q}(x_{qn},y_{qn})=\left(1+\bar{d}_{qn}/d_{0}\right)^{-\alpha}$ is the path loss on the backhaul link, and it depends on the distance between CU $q$ and RRH $n$ denoted as $\bar{d}_{qn}=\sqrt{(x_{qn}-\bar{x}_{q})^{2}+(y_{qn}-\bar{y}_{q})^{2}}$, where $\left(\bar{x}_{q},\bar{y}_{q}\right)$ are the coordinates of the CU. The term $\bar{g}_{qn}$ is the small-scale fading of the channel, which follows a Rician fading model. This Rician channel is characterized by the Rician parameter (the Rician K-factor) $\mathcal{K}=\eta_{1}^{2}/\eta_{2}^{2}$, where $\eta_{1}^{2}$ and $\eta_{2}^{2}$ represent the power of the line-of-sight (LoS) and NLoS components respectively.

We use a traffic probability density function (PDF) that is a combination of both a uniform distribution, chosen with probability $P_{0}$, and a number $N_{h}$ of bivariate normal distributions, representing hotspots (a choice of small $P_{0}$ means more dense hotspots). The number of these hotspots is uniformly distributed, i.e., $N_{h}\sim U\left(N_{h}^{\text{min}},N_{h}^{\text{max}}\right)$. This number can be generated for the whole network or for each cell separately. These hotspots are centered at locations $\left[{\bf\widetilde{x}}_{h},{\bf\widetilde{y}}_{h}\right]=\left[\left[{\widetilde{x}}_{h1},\dots,{\widetilde{x}}_{hN_{h}}\right]^{T},\left[{\widetilde{y}}_{h1},\dots,{\widetilde{y}}_{hN_{h}}\right]^{T}\right]\in\mathbb{R}^{N_{h}\times 2}$, and they have equal variances $\sigma_{h}^{2}$ on both the $x$ and $y$ axis, without any correlation between the two axis. Therefore, the cell traffic distribution PDF bounded by the cell boundary $\mathcal{B}_{q}$ is defined as

where $B_{q}$ (different from cell boundary $\mathcal{B}_{q}$) is the area of cell $q$, $f_{0}$ is a normalizing factor that can be calculated numerically to normalize the PDF. This traffic model is flexible in the sense that it can be constructed from a traffic survey that identifies the locations of hotspots in the network.

On the access channel, the ZF beamformer is formed per cell $q$. Hence, the intra-cluster interference found in (II-B) will be completely canceled. As a result, the mean achievable spectral efficiency over the access channel for user $k$ in cell $q$ is

The concatenated form of the RRHs’ locations in the network is written as $\left[{\bf x},{\bf y}\right]$, where the the x-coordinates are ${\bf x}=[{\bf x}_{1}^{T},\dots,{\bf x}_{q}^{T},\dots,\dots,{\bf x}_{Q}^{T}]^{T}\in\mathbb{C}^{NQ}$, for $q\in\{1,\dots,Q\}$ (similarly for ${\bf y}$). These will form our optimization variables. We use $[{\bf x}_{q},{\bf y}_{q}]$ to refer to the locations of the RRHs in cell $q$ and $[{\bf x}_{-q},{\bf y}_{-q}]$ to denote the locations of all the RRHs in the network except those in cell $q$.

The spectral efficiency over the access channels depends on the distances between all the RRHs found in the network and the user $k$, which in its turn depends on the locations of all the RRHs. Given the distances to the serving and interfering RRHs, the lower-bound of spectral efficiency can be written as [10]

with

where $\rho=\frac{p}{\sigma_{z}^{2}}$, and $\text{ICI}_{q^{\prime}k}({\bf x}_{q^{\prime}},{\bf y}_{q^{\prime}})$ denotes the inter-cell interference (ICI) contributed by cell $q^{\prime}$ to the user under test and $\xi(q^{\prime},j)=\text{tr}\{{\bf D}_{q^{\prime}j}\}=M\sum_{m=1}^{N}\ell_{j}(x_{q^{\prime}m},y_{q^{\prime}m})$. As can be seen, this interference depends on the locations of the users in the interfering cells because the beamformers in each cell is designed based on the channels of the users in these cells.

Using the traffic distribution PDF, $f_{q}(\widetilde{x}_{k},\widetilde{y}_{k})$, in (III-A), we further average this achievable rate over $f_{q^{\prime}}(\widetilde{x}_{k},\widetilde{y}_{k})$ in the interfering cells by writing the ICI term as

where, as indicated earlier, $\mathcal{B}_{q^{\prime}}$ is the boundary of cell $q^{\prime}$.

We define our problem of optimizing the locations of the RRHs in the network as

where

The terms $\omega$, $\omega_{c}$ are the bandwidth allocated for the access channel and the backhaul, respectively, and $\epsilon$ is the allowed backhaul outage probability for each CU-RRH link.

The formulation in (11) maximizes the average spectral efficiency of typical user $k$ in the network, and this spectral efficiency is averaged over the traffic distribution in the user’s cell as shown in (11a), which means it maximizes the spectral efficiency for the system. Additionally, the $N$ constraints in (11b) place an upper bound on the sum of the rates over the access channel with respect to the backhaul achieved capacity. If this constraint is not respected the backhaul will experience an outage.

For notational simplicity we define the following term.

We can write the Lagrangian formulation of our problem as

where the $\bm{\lambda}\succcurlyeq 0$ denotes the vector of Lagrange multipliers. For a specific RRH $m\in\{1,\dots,N\}$ in cell $q$, we differentiate the Lagrangian formulation with respect to the x-coordinate $x_{qm}$ of RRH $m$, set it to zero, and obtain an iterative formula for $x_{qm}$ that can be written as

The same formulation applies by differentiating the Lagrangian with respect to the y-coordinate.

where

The terms $A_{1}(\widetilde{x}_{k},\widetilde{y}_{k}),A_{2},A_{3}$ and $A_{4}$ depend on the RRHs locations, but we do not write the RRHs ${\bf x}$ and ${\bf y}$ coordinates as parameters to minimize the notation. Here ${}_{0}F_{1}(;.;.)$ is the regularized confluent Hypergeometric function, and we can write it in an alternate form as a function of the modified Bessel function of first kind as ${}_{0}F_{1}\left(;1;\frac{\eta_{1}^{2}}{2\eta_{2}^{2}}\left(J_{1_{qm}}({\bf x},{\bf y})\right)^{2}\right)=I_{0}\left(\frac{\sqrt{2}\eta_{1}}{\eta_{2}}J_{1_{qm}}({\bf x},{\bf y})\right)$, i.e., ${}_{0}F_{1}\left(;1;z\right)=I_{0}\left(2\sqrt{z}\right)$.

The derivative of $\mathcal{L}({\bf x},{\bf y},\bm{\lambda})$ in (16) with respect to $\lambda_{m}$ is

Using the batch gradient descent, we can obtain an iterative formula for $\lambda^{(i+1)}_{m}$ as

where $\left[\cdot\right]^{+}=\max\left(\cdot,0\right)$, and $\nu\in\mathbb{R}_{+}$ is a step size chosen small enough to guarantee convergence. Based on this analysis, we can construct Algorithm 1 to obtain the optimal locations of the RRHs in the network as described below.

The algorithm starts by choosing random locations for the RRHs (Step 1). Then, we define a $d_{\text{max}}$ as a distance large enough to start the locations update. For the RRHs location update in each algorithm iteration, we perform a cell wrap-around (Step 1) to place the cell of these RRHs at the center, hence eliminating network border effect. After that, we update the locations of the RRHs in this cell using the indicated equations in Step 1, and we calculate the maximum RRHs location change. We do one iteration for each cell at a time until we iterate through all the network cells. Hence, the total number of iterations will be the same for all the cells, and the updated RRHs locations in the interfering cells will be used, which is very reliable. At last, the algorithm convergence is determined when the maximum change in the RRHs locations ($d_{\text{max}}$ in Step 1) is lower than a small distance $d_{\text{cvg}}$. We note that our scheme is sub-optimal mainly because of the complex traffic distribution shown in (III-A).

Choosing an appropriate step size $\nu$ for ${\bf\lambda}$ update in (24) can be tricky especially when the constraint (11b) is tight. To address this issue, we define an equivalent approach that is distance-based to solve problem (11) by formulating it as

where $\bar{d}_{\text{out}}$ is the maximum allowed backhaul distance to guarantee that the outage is lower than $\epsilon$. When the achievable spectral efficiency over the access channel is fixed, $\bar{d}_{\text{out}}$ can be easily obtained using a bisection search to obtain $P_{qn}\left({\bf x},{\bf y}\right)=\epsilon$ defined in (III-C). Once $\bar{d}_{\text{out}}$ is found, we can write an iterative formula for $x_{qm}$ and $y_{qm}$ as

Similarly for the y-coordinates:

To update $\lambda_{m}^{i}$, we perform another bisection search, such that $\bar{d}_{qm}=\bar{d}_{\text{out}}$. This method eliminates the need for a step size $\nu$ to update $\bm{\lambda}$, and at the same time (26) and (27) guarantee that the RRHs will be placed in the locations that maximize (25a) as we will see in the results section. Consequently, we can obtain the optimal locations of the RRHs in our network using Algorithm 1, but with replacing Step 1 with two steps; one that obtains $\bar{d}_{\text{out}}$ using bisection search and the other updates $x_{qm}$ and $y_{qm}$ using (26) and (27) respectively.