In this section, we present the system model for a general RIS-aided downlink CF communication network. Subsequently, we formulate the joint transmit power and BS selection problem.
2.1. System Model
We consider a downlink multi-RIS-aided multi-user system consisting of
N single-antenna users, each of which would benefit from the collaboration of
M single-antenna BSs and
N RISs, as illustrated in
Figure 2. Each RIS
r equipped with elements
passively reflects the signals from BSs to assist a dedicated user
n, such that the system performance can be enhanced. For simplicity, the sets of users/RISs and BSs are, respectively, denoted by
and
.
In such a system, we assume that a CPU exists for network central processing, and carries out transmission scheduling according to the network information. To ensure accurate channel state information (CSI) at the CPU, various channel acquisition techniques are employed [
35,
36,
37].
The RIS comprises numerous cost-effective passive reflecting elements, each of which can be individually adjusted to manipulate the phase of incident electromagnetic signals using a programmable PIN diode. Thus, RISs can enable signals to be transmitted simultaneously over two links, i.e., direct and reflected links, resulting in more diverse transmissions. As shown in
Figure 2, the RISs receive the signals sent by the BSs, and then reflect these signals to the users. To simplify the problem, we assume that the reflection amplitude of each RIS element is maximized at 1 [
38]. Additionally, we assume that the signal is reflected ideally without any hardware defects. The phase shift of the
l-th RIS element of the RIS
r is denoted as
and belongs to the interval
, while
represents the frequency response of RIS
r and is defined as a diagonal matrix with diagonal elements
.
The channel between BS m and user n in the direct link is denoted as . The fading channels and are, respectively, the channel between the single-antenna BS b and RIS r, and between RIS r and the single-antenna user n. Under the hypothesis of independent Rayleigh fading channels, we have , and . The average power gains and depend on the distances between the BSs, RISs and users.
Moreover, we assume that, if RIS
associates with user
n, it adjusts its phase shift so that the reflected link, via itself, is aligned with the corresponding direct link of user
n. To streamline the practical design of BS selections, we introduce binary variables
to represent whether BS
m is selected by user
n (
) or not (
). We assume that each BS
m serves at most one user, while each user can be served by multiple BSs. Additionally, we ensure that there is at least one BS in
selected by each user
n to provide service coverage. Thus, we have
For coherent joint transmission, we assume, in this paper, the synchronization of all BSs associated with user n, such that user n could receive the exact same symbols from those BSs. Specifically, we denote the received symbol of each user n served by BS m as , where and represent the transmission by each user n and the precoding matrix, respectively. The power of is normalized, i.e., .
It is assumed, in this paper, that intended signals with predetermined phase shifts are reflected by each RIS
r purposefully for its associated user
n, while the unintended signals are scattered at random. Thus, each user
n receives the desired signal from its associated RIS
n as well as that scattered by other non-associated RISs [
31]. To account for the significant double path loss in the RIS-aided cascaded channel, we exclude any signals that are reflected or scattered more than once by the RIS. The channels from BSs to user
n consist of direct links from BSs as well as reflected and scattered links via RISs. Since user
n is served by multiple BSs simultaneously, each user
n could receive the effective signal composed of the aggregated signals originating from its selected BSs. Thus, the desired signal at user
n’s receiver can be expressed as
where
.
If BS
m sends information signals to user
n in the downlink communication,
is set to 1. In other words, we can consider the BSs selected by user
n as a single multi-antenna BS that employs precoding weights
’s. Consequently, each reflecting element on RIS
n adjusts its reflection phase to align the cascaded BS-RIS-user channel with the direct BS-user channel [
6]. To achieve this, the
l-th phase shift of RIS
n needs to be configured as
. Thus,
in (
3) can be rewritten as
Meanwhile, regardless of the desired signal from its own selected BSs, each user
n would also receive both the direct interference transmitted by other BSs and the interference reflected by RISs. Thus, each user
n would receive the following total interference:
In light of (
3) and (
5), we could further obtain the average SINR of each user
n as
where
is the Gaussian noise power. To make the analysis tractable and obtain more insights, we can rewrite (
6) to obtain a lower bound on it as follows:
where the inequality is valid as the function
is convex in
x for
, which is derived from Jensen’s inequality.
2.2. Problem Formulation
In this paper, we assume that any group of BSs use the maximum ratio transmission (MRT) [
39] to transmit signals to their served user. Then, we can write the precoded weight as
where
denotes the transmit power of the BS
m to user
n. Substituting (
3), (
5) and (
8) to (
6), we can obtain the average SINR received by user
n as
where
,
+
and
.
is the average path gain between BS
m and user
n via RIS
r. The derivation process can be referred to in Appendix A of [
32], which will not be repeated here for brevity.
According to the above, our objective is to optimize the transmit powers of the BSs and the BS selections to achieve the maximum weighted sum SINR for all users. Denote
as the weighting factor to control the scheduling priority of user
n and a larger value of
signifies a higher priority for information transmission to user
n. Based on (
9), the weighted sum SINR received by users can be expressed as
Accordingly, we can formulate the optimization problem as
where
represents the available transmit power budget of each BS. The BS’s transmit power and BS selections are denoted by
and
, respectively.
The weighted sum-SINR maximization problem (P0) is a complex task that involves optimizing a nonlinear sum-of-ratios function [
40], which involves maximizing a sum of several fractional functions. This non-convex optimization problem is notoriously difficult to solve optimally. Furthermore, the problem requires the joint optimization of the BS transmit power and BS selection, where the latter aspect involves binary/integer variables, i.e.,
. So, we decompose it into following two subproblems to solve this problem with low complexity.
BS Transmit Power Subproblem: Given the fixed BS selections , we focus on optimizing the transmit power of each BS, while ensuring it falls within the allocated budget . Then, the BS power problem can be formulated by