Double reflections Assisted RIS Deployment and Energy-efficient Group Selection in mmWaves D2D Communication

Nowadays RIS-assisted D2D communication forms a new research direction [14, 15, 16], and the majority of recent research focuses on RIS usage by assuming that RISs are already set up and accessible. There are few works of RIS deployment strategy and most of them deployed the RISs considering a single reflection, i.e., for RIS deployment, they considered that a device can communicate with another one via a single RIS. In practice, the secondary reflections are not insignificant, especially in metropolitan settings where the RISs are not widely dispersed [17, 18]. Therefore, in a few cases, double reflections can be impactful when a single reflection are insufficient for RIS-assisted communication. Additionally, by allowing double reflections, we can serve more obstructed device pairs without deploying more RIS. As a result, if we allow two or more consecutive RIS-assisted communications, using less number of RISs, we can get more coverage as well as sum throughput. In this work, we ignored the role of triple and higher order reflections due to significant effective path loss [17]. Here, we employ RISs to create an indirect LoS link between a transmitter and receiver pair in order to establish communication when there is no direct LoS link between them. After deploying the RISs, if a device pair wants to communicate with each other, multiple RIS may be available to assist their communication. Note that a subgroup may provide more energy-efficiency than the entire RIS [19]. Therefore, without using a full RIS, we select a specific energy-efficient subgroup for information transfer among all the available RISs. Our contribution is as follows.

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  By dividing the service area into regions, we model the environment by determining which places or regions are obscured from one another’s view by the known obstructions. If a pair of devices lie in two zones such that their LoS link is obstructed then we term them a blind pair. In this typical scenario, our proposed RIS deployment strategy helps to connect these blind pairs by forming an indirect LoS link via reflections from RISs. It identifies the blind pairs, determines candidate locations for RIS deployment and finally selects the exact locations based on a set cover formulation.

• •

  We observe that, it may not be possible to provide an indirect LoS via a single RIS reflections for a few device pairs due to the obstacles’s shape and density. In this case, it is possible to provide an indirect LoS via secondary reflections. Our RIS deployment strategy uses both single and double reflections to cover more blind pairs using less number of RISs.

• •

  Each RIS is divided into a number of non-overlapping subgroups to avoid the channel estimation overhead. After the RISs deployment, we propose an energy-efficient algorithm to find out the appropriate subgroups of an RIS to achieve higher sum throughput.

• •

  We have mathematically proved that in some specific scenarios, double reflections are more energy-efficient than single reflection, which is quite counter-intuitive.

Simulation results demonstrate that our proposed strategy can increase sum throughput significantly and reduce the number of overall RIS deployments in comparison to a random [14] as well as a recent deployment strategy [9].

This paper is organized as follows. In Section II, we have introduced the related and existing works. We have discussed the system model and preliminaries in Section III. The problem formulation is discussed in Section IV. RIS deployment strategy, the group selection strategy, and their analysis are discussed in Section V. Thereafter, in section VI, we have discussed the simulation results and compared this with the existing placement strategies. Finally, in Section VII, we give concluding remarks and the future direction of research.

Consider a wireless communication system that operates in a rectangular area that is partitioned into small squares or grids of unit size which is shown in Fig 1. The rectangular area consists of $M$ rows and $N$ columns. We assume that each unit square grid is identical with respect to their center and we also assume that the center of the leftmost corner square is the origin. It is noted that the device that will operate in this setting will have highly directed antennas. The position of a user is approximated to the center of the grid within which it lies. We assume that a device can communicate with another device directly if there is a LoS link between them and they are within a threshold distance $r$.

Now, we are presenting a few definitions below which will be used throughout our discussion.

We consider all the devices in this communication scenario to be pseudo-stationary [28], i.e., during communication time, a device does not move outside the grid. We also assume that a device follows any mobility model within a grid at any time instance. However, we presume that a device’s location in a unit grid is roughly determined by the grid’s center. It is noted that if we consider a device is within a grid that means it lies at the center of the grid.

Here, we position the obstacle inside the grids following the acquisition of the satellite images. We assume that the satellite images give the proper position of the obstacles. In Fig. 1, we consider that the black cells represent the location of the obstacles. We assume that, if a grid/cell is blocked, it means the whole cell is blocked and there will not be any partially block cells. Multiples of these blocked zones combine to build a polygon that closely resembles the shape and size of the obstruction. Within a blocked cell, a device can not lie, i.e., a device can lie only within the free cells. It is noted that an RIS can not be placed within a blocked cell, it will be strategically placed on poles in the free cells.

Obstacles prevent a device pair from directly communicating with one another. Therefore, to provide an indirect LoS between them, we use single and double reflections via one and two consecutive RISs. Let us assume that an RIS $R_{i}$ consists of $M_{r}$ rows and $N_{r}$ columns which are effectively controlled to adjust both the amplitude and phase of the incident waveform. However, only the phase is tuned or optimized, and the amplitude factor is fixed to unity for the purposes of simplicity and mathematical tractability [13]. We use a grouping approach in order to minimize the channel estimation overhead [12]. Furthermore, for a large RIS, the phase shift computation is hard, and a minimum energy is required for each phase shift. Therefore, we divide $R_{i}$ into $K_{g}$ number of non-overlapping subgroups $R^{t}_{i}$ to minimize the energy consumption where $1\leq t\leq K_{g}$, and each $R^{t}_{i}$ consists of $N_{g}\times N_{g}$ number of patches, i.e, $K_{g}\times N_{g}\times N_{g}=M_{r}\times N_{r}$. It is noted that the number of partitions and the ensuing sub-surfaces are decided upon beforehand [13], and each subgroup is capable of providing a desired throughput. In this scenario, $R^{t}_{i}$ can be in one of two states: ON or OFF. An incident signal’s phase can be altered to a desired direction when an element is in the ON state; when it is in the OFF state, it cannot reflect [13]. Here, we use one particular subgroup instead of using total RIS, and we also assume that a subgroup can serve a single request at a particular instant [29]. Our objective is to select the energy-efficient subgroups for information transfer.

In this communication scenario, let there be $n_{d}$ number of devices that lie within the free zones. We assume that any two devices $u$ and $v$ want to communicate with each other and they lie within the free zone $z_{1}$ and $z_{2}$, respectively. Now, depending on the position of the obstacles, $u$ and $v$ can communicate in three different ways as follows: (i) directly, $u$ and $v$ can communicate directly if there is no obstacle in between them, i.e., there is a LoS link between them. (ii) via single reflection, and (iii) via double reflections. We suppose that the wireless link experiences both small-scale block fading and large-scale path loss effects. The direct channel from $u$ to $v$ exhibits small-scale fading and their corresponding path loss factor is $\rho_{L}^{\frac{1}{2}}d^{-\frac{\alpha}{2}}_{uv}$, where $\rho_{L}$ is the path loss at one meter distance, $\alpha$ is the path loss exponent and $d_{uv}$ denote the distance between $u$ and $v$.

We also assume that each group $R^{t}_{i}$ of an RIS $R_{i}$ consists of $N_{g}\times N_{g}$ number of elements. Let $\mathbf{h}_{uR^{t}_{i}}\in\mathbb{C}^{N_{g}\times 1},\mathbf{h}_{R^{t}_{i}R^{s}_{j}}\in\mathbb{C}^{N_{g}\times N_{g}}$ and $\mathbf{h}_{R^{t}_{i}/R^{s}_{j}v}\in\mathbb{C}^{1\times N_{g}}$ be the channel matrix from $u$ to $R^{t}_{i}$, $R^{t}_{i}$ to $R^{s}_{j}$ and, $R^{t}_{i}/R^{s}_{j}$ to $v$, respectively. Note that, unlike the conventional RIS-based approach, the grouping-based approach does not involve a diagonal phase shift matrix of non-zero $N_{g}\times N_{g}$ elements [13]. A common phase shift is applied on the incoming signals, and the product of each point-to-point link determines the total path loss for each of these channel matrices [17]. As a result, the effective channel gain for single and double reflection is given by $\mathbf{h}_{R^{t}_{i}v}\mathbf{h}_{uR^{t}_{i}}\times e^{j\phi_{i,t}}$ and $\mathbf{h}_{R^{s}_{j}}\mathbf{h}_{R^{t}_{i}R^{s}_{j}}\mathbf{h}_{uR^{t}_{i}}\times e^{j(\phi_{i,t}+\phi_{j,s})}$, where $e^{j\phi_{i,t}}$ and $e^{j\phi_{j,s}}$ are the common phase shift of $R^{t}_{i}$ and $R^{s}_{j}$, respectively.

Here we define a metric, data rate that will be used to quantify the performance of our proposed strategy. Here we assume orthogonal frequency-division multiplexing (OFDM) [30] is used in our communication scenario. Let $P$ be the transmitted power, then the signal-to-noise-ratio (SNR) at the receiver for single RIS-assisted communication is

and SNR for double RISs reflected communication is

where $\sigma^{2}$ is the variance of the circularly symmetric zero mean additive white Gaussian noise. Note that, we adjust the common phase shift of (1) and (2) to attain the optimal SNR. Therefore, from Shannon’s capacity formula, we can get the throughput $R(\gamma)=\log(1+\gamma_{\rm{sr/dr}})$. Hence, by calculating the throughput we can compare the performances.

Here, we define the energy efficiency metric $(\rm E_{eff})$ [19], which will be used to measure how well our suggested approach performs. Let $R(\gamma)$ be the throughput obtained at the receiver end. Additionally, it requires a total $\rm E^{i}_{c}$ amount of energy for information transfer via a single RIS $R_{i}$, where

For two consecutive RIS ($R_{i}$ and $R_{j}$) assisted communication the total energy consumption is denoted by $\rm E^{i,j}_{c}$ where

$\beta$ is the number of packets each with $\phi$ bits, $P$ is the transmitted power, and $P_{\rm phase}(R_{i})$ is phase shift power for RIS $R_{i}$.

Therefore, energy efficiency for single reflection and double reflections are defined by

respectively. Moreover, we use this metric to select an appropriate subgroup for the information transfer of each blind pair.

In the first part, we want to formulate an optimization problem to cover a maximum number of blind pairs using the least number of RISs. Note that we do not consider those device pairs that have direct LoS and are totally blind. Let $\mathcal{B}$ be the set of all coverable blind pairs, i.e., no element of $\mathcal{B}$ remains uncovered after deploying the RISs. Moreover, to formulate the optimization problem, we define a few notations below.

Let $i$ and $j$ be two locations and $a_{ij}$ be an indicating variable that describes the status of having LoS between them, where $1\leq i,j\leq n^{2}$. That is,

Additionally, let $S_{i}$ be the set of all blind pairs covered by an RIS at the $i$-th cell. That is,

Note that a blind pair of $\mathcal{B}$ may be visible via a single or double reflections. Therefore, we also consider $D_{ij}$ as a set of all blind pairs that are coverable via double reflections using two RISs located at $i$-th and $j$-th cell but not coverable by either $i$-th or $j$-th RIS via a single reflection. Hence, $D_{ij}$ can be represented as

However, we define $Z_{ij}$ as a set of all blind pairs that are coverable via single as well as double reflections using RIS placed at $i$-th and $j$-th locations. That is,

We now introduce the following binary variable:

Hence, we formulate the optimization problem as follows:

This integer program can be linearized by using an intermediate binary variable $y_{ij}$ as follows:

Note that the above integer linear program (ILP) is nothing but a classical set cover problem which is a well-known NP-hard problem. Therefore, in Section V-A, we present a greedy solution for the RIS deployment problem.

In the second part, for group selection, we assume that RISs have already been deployed. Moreover, we also know which blind pair will be covered by which RISs. Note that, a blind pair may be covered by single as well as double reflections. Hence, for a blind pair, more than one subgroup may be available to complete the information transfer. However, we allow a single subgroup for information transfer because of the scenario of energy constraints. Therefore, our primary objective is to identify a specific energy-efficient subgroup for each blind pair. Let us assume that there are $n_{b}$ number of blind pairs, and $R$ number of RISs are deployed in the surroundings and each of them is subdivided into $l$ number of subgroups. Let $\rm G^{s}_{i}$ be the set of all subgroups that can provide an indirect LoS link to the $i$-th blind pair via single reflection. Hence, we can represent $\rm G^{s}_{i}$ as

Similarly, let $\rm G^{d}_{i}$ be the set of all subgroups that can provide an indirect LoS link to the $i$-th blind pair via double reflections. That is,

Hence, if a blind pair is visible by single reflection, our problem is to find a more energy-efficient subgroup $R^{k^{*}}_{j^{*}}$ from $\rm G^{s}_{i}$ such that

If a blind pair is visible via double reflections, we select $R^{p^{*},q^{*}}_{l^{*},m^{*}}$ as energy-efficient subgroups from $\rm G^{d}_{i}$ such that

Note that if a blind pair is visible via single as well as double reflections, we will select the more energy-efficient case for them. In Section. V-B, we have discussed the group selection strategy in detail.

In our proposed RIS deployment strategy, we first identify which blind pairs are present in the surroundings. After identifying the blind pairs, we will find the candidate locations for RIS deployment, and finally select the candidate locations for final deployment. All these steps are now discussed in detail below.

In section IV above, we have strategically found $R$, the set of all deployed RISs, and the set of all coverable blind pairs $\mathcal{B}=B\setminus B_{un}$. A blind pair $(u,v)\in\mathcal{B}$ can be covered in three different ways: i) via only single reflection ii) via only double reflections, and iii) via both single and double reflections. Since energy consumption is a very important parameter in wireless communication scenarios, we discuss in the following the process to find the energy-efficient group considering the above-mentioned three cases.