1. Introduction
In the downlink of multi-user multiple-input and multiple-output (MU-MIMO) systems, the amount of channel-state information (CSI) available at the transmitter (CSIT) holds significant importance. Because individual users lack the capability to access the received signals of other users, joint signal processing among the received signals of different users is not feasible. As a result, the transmitter generally engages in appropriate signal processing to multiplex independent signals prepared for different users. Therefore, the accuracy of CSIT remains crucial and typically determines the achievable rate within the system [
1]. Securing precise CSIT presents difficulties in real-world systems, especially in frequency division duplex (FDD) systems. In FDD systems, tracking the downlink channel directly at the transmitter becomes challenging due to the separation of uplink and downlink transmission in the frequency domain. Consequently, it is a common practice in FDD-based systems to collect CSI feedback from users, establishing partial or limited CSIT. This approach, referred to as limited feedback, requires each user to estimate its own CSI through pilots, quantize the obtained information, and then provide feedback to the transmitter [
2].
To evaluate the effectiveness of MU-MIMO systems based on limited feedback in realistic situations, recent research has employed probabilistic models to simulate cellular networks. These models utilize stochastic geometry to depict the locations of both base stations (BSs) and users. This method allows for the calculation of average performance metrics in communication systems across the random point process. Consequently, it streamlines the mathematical analysis regardless of the number of BSs in the network. For example, research such as [
3,
4,
5] employed a stochastic geometry to model the positions of BSs. It was utilized to theoretically analyze the essential performance metrics in cellular networks, including the capacity and outage probability of MIMO systems, assuming a perfect CSI for both transmitters and receivers. In a different context, the authors in [
6] utilized a homogeneous
-Ginibre point process to introduce a cyber insurance framework that focuses on the physical layer security of wireless communication. In the works of [
7,
8], stochastic geometry played a fundamental role in modeling mmWave communication for heterogeneous networks. Furthermore, numerous investigations have delved into the downlink rate in cellular networks achieved by MU-MIMO systems with limited feedback. These studies leveraged stochastic geometry to simulate realistic cellular environments, as evidenced by [
9,
10,
11,
12]. The study in [
9] focused on estimating the optimal feedback rate that maximizes the net spectral efficiency. This analysis assumed the use of zero-forcing beamforming with limited feedback at each BS. The authors in [
10] extended the investigation to provide both approximate lower and upper bounds for the optimal feedback rate. They also delved into the exploration of associated estimation errors. In [
11], the focus is on identifying the optimal number of bits for quantization and feedback to achieve the best ergodic secrecy rate.
In [
12], the authors extended the analysis previously presented in [
9,
10] to encompass the scenarios where each user can employ multiple receive antennas for spatially multiplexing independent data streams tailored to that user. For instance, the investigation considered block diagonalization (BD) based on limited feedback. The primary emphasis was on analyzing the asymptotic behavior of the optimal feedback rate, particularly when the length of the coherent channel block, denoted by
, reached sufficiently large values. This was achieved by establishing asymptotic lower and upper bounds for the optimal feedback rate as
approaches infinity. In the derivation process, for the sake of analytical tractability, certain functions with slower growth rates in the number of feedback bits were neglected by assuming that
was sufficiently large. As a result, the obtained outcomes may be inaccurate when
assumes relatively smaller values.
In summary, although the authors in [
12] successfully depicted the asymptotic growth rate of the optimal number of feedback bits against the increase in
, the obtained estimates for the optimal feedback rate (utilized to illustrate the growth rate) exhibit discrepancies as
decreases. Considering the significant variability of
in practical scenarios, providing a consistently accurate estimate for the optimal feedback rate, irrespective of system parameters, remains highly crucial. In this context, this paper presents a more precise estimation of the optimal feedback rate in cellular networks, particularly when each base station implements limited-feedback-based BD. The main contributions of this study can be summarized as follows:
To attain a more precise estimation, we conduct an analysis of the derivative of the net spectral efficiency. This derivative comprises two functions demonstrating distinct growth rates with an increase in feedback bits. Unlike in previous studies, both functions are rigorously approximated through mathematical analysis.
Consequently, our proposed estimate surpasses the accuracy of estimates found in prior research, delivering precise approximations that are independent of and ultimately aim to maximize the net spectral efficiency.
Simulation results affirm that the proposed estimate consistently provides an accurate approximation of the optimal feedback rate. This is particularly notable in scenarios where assumes relatively smaller values, showcasing significantly improved precision compared to previous findings.
The remaining content of this paper is organized as follows:
Section 2 introduces the system model and preliminaries.
Section 3 details the performance analysis for the optimal feedback rate.
Section 4 offers simulation results validating the analysis presented in
Section 3, and
Section 5 provides the paper’s conclusion.
Appendix A summarizes the notations used in this paper.
3. Accurate Estimate for the Optimal Feedback Rate
As the optimal number of feedback bits is defined as the value of
B that maximizes
, we may solve the following problem to find it:
However, solving this problem directly poses a challenge since it is NP-hard. In prior studies, researchers addressed this complexity by extending the optimization domain to real numbers, facilitating mathematical analysis [
9,
10,
11]. The introduced analytical quantization model, known as QUB (detailed in
Section 2.6), enables such an extension, closely approximating the quantization performance while allowing
B to be an arbitrarily real number. By adopting this relaxation, the optimal value
can be determined within the real number domain. Moreover, since
is a continuously changing function of
B and the optimal value
generally corresponds to the rightmost critical point (or zero), as discussed in
Section 2.4, rounding
to the nearest integer provides a close approximation to the solution of (
27), as demonstrated in previous studies [
9,
10,
11]. Building upon these established findings, we identified the optimal solution for
B within the set of real numbers. That is, the optimal number
is obtained by solving the following problem:
Assuming
, finding the optimal point involves examining the critical points of
. Leveraging Remark 1 from [
12], we determine the optimum number of bits for feedback, maximizing
, through the analysis of critical points in
rather than utilizing
directly. The derivative of
can be derived as follows, incorporating (
18), (
21), and (
22):
where (a) is obtained using the differential of the determinant [
19]. In (
29), evaluating the first term on the right-hand side gives
where
follows from (
20). By applying approximations for
and
(in (
18) and (
21)) to (
30), we have
In, (
31), the expectations of off-diagonal elements in
,
, and
are all zero. To simplify the analysis, we approximate these off-diagonal elements with their respective expectations, leading to:
where
denotes the diagonal matrix whose diagonal entries are equivalent to those of
. Let
. Then,
with
denoting the
-th element of
,
representing the
-th element of
, and
denoting the
-th element of
. Gamma distribution with a shape
m and scale 1 characterizes the diagonal components of
. Consequently, both
and
are Gamma-distributed random variables with a scale parameter of 1. The shape parameters of
and
are
and
, respectively. Also, we have
, for a gamma-distributed random variable
with a shape parameter
and a scale parameter 1.
In (
33), the scalar
represents the only term related to the feedback rate. Furthermore, the term
stands out as the most dominant term in the denominator, given the interference-limited regime discussed in this paper. Consequently, we approximate all terms in the denominator, except
, with their respective expected values, resulting in
For further calculations, we consider the following lemma.
Lemma 1. We can obtain the following lower bound for the term on the rightmost side of (34). Combining (
29), (
32), (
34), and (
35), we obtain
where
is obtained by applying Lemma 2 in [
12].
Determination of the Optimal Feedback Rate
The critical points of can be numerically approximated by examining the zero-crossing points of . It is essential to highlight that the net rate, denoted by , is defined as , where R increases with B. However, as B increases, the rate of increase in R gradually diminishes, while the term shows a constant decreasing rate with increasing B. Consequently, if a critical point exists, it is likely that the rightmost critical point represents the local maximum of (or approximately of ). Therefore, the optimal B value is found by identifying the rightmost critical point if one exists, and is considered to be zero if no critical points are present.
4. Simulation Results and Discussions
In this section, we present the simulation results to verify the accuracy of the estimate introduced in the preceding section. To facilitate the comparison, the following notations are employed:
As previously denoted, represents the optimal number of feedback bits in terms of maximizing .
signifies the proposed estimate derived by numerically approximating the rightmost zero crossing point of , as outlined in Section Determination of the Optimal Feedback Rate.
stands for the lower bound of
obtained in [
12].
All results in this section are acquired through Monte Carlo simulations to obtain , and a simulation guideline is provided as follows:
Initialization: Specify the values of the system parameters , , K, B, , and . Fix the network’s radius to a sufficiently large value. For our results, we used a radius of 5 km for the entire network, and the values of and are fixed as and .
BS Locations: At each frame, determine the locations of BSs based on a Poisson point process. The number of BSs in the current frame follows a Poisson distribution with the corresponding density. Then, the BSs are uniformly distributed within a 2D circle.
User and channel setup: Assuming the target user is located at the origin, identify the BS closest to the origin as . Calculate the distances between the BSs and the target user. Generate small-scale fading channel matrices for , where components are i.i.d. circularly symmetric Gaussian with a variance of one. denotes the channel matrix between and user , with user 1 as the target user. Note that, for simplicity, is denoted by throughout this paper.
Quantization and precoding: Each user obtains
by quantizing
in a distributed manner based on the QUB criterion described in
Section 2.6. Construct precoding matrices
using quantized channels,
collected from users, based on the BD criterion.
Inter-cell interference: Calculate matrices and following a similar approach used for obtaining and .
Net spectral efficiency: Compute the net spectral efficiency
at the current frame using Equations (
6)–(
8).
Monte Carlo simulation: Repeat steps (1)–(6) to obtain the ergodic average of the net spectral efficiency .
All results were obtained using MATLAB on an inter-core i9 processor without the assistance of a GPU. The total number of frames used to derive
exceeds 50,000 for all simulations. In (
36), the value of
is set to
for all simulations following the results in [
12].
Figure 1 and
Figure 2 depict the optimal feedback bits varying with the coherent channel block length
, considering two different setups of
. The black dashed line representing
obtained through exhaustive search, while the blue dotted line indicates its analytical estimate,
, proposed in this paper. In accordance with our analysis in
Section 3, the proposed estimate (dotted line) closely approximates the optimal number of feedback bits found through an exhaustive simulation. In contrast, the red solid lines depict two approximations proposed in [
12], showing less accuracy when
is relatively small (the discrepancy decreases as
increases, consistent with the assumption of a large
in the derivation of [
12]). In summary, the proposed estimate in this study generally provides a much more accurate estimation compared to previous results.
Figure 3 directly compares the estimation errors of
and
in estimating
. The corresponding errors are depicted against the number of transmission antennas. As the number of transmission antennas increases,
generally rises due to the increased requirement for feedback bits to attain a comparable CSIT accuracy for precoding. The bounds derived in previous studies, including
, necessitate ‘sufficiently large’ feedback bits to closely approximate
; however, the range of ‘sufficiently large’ increases with
since
also increases with
. Consequently,
becomes less accurate with the increasing
, and this trend is more pronounced when
is smaller, as depicted in
Figure 3. In contrast, the proposed estimate
generally maintains accuracy across varying
.
In
Figure 3, a noticeable amount of estimation error for the proposed estimate might be observed, especially for larger values of
(e.g., when
and
). However, around the optimal feedback bits, the rate of increase in
decreases as
increases. Consequently, the estimation error of
depicted in
Figure 3 remains tolerable regardless of
, as illustrated in
Figure 4. Remarkably,
at
and
exhibit nearly identical values across varying
and
K, while the previously proposed lower bound
demonstrates inaccurate approximation, particularly when
is small. When considering the maximization of the net sum rate, our proposed estimate provides a much closer approximation to
.
Simulation results show that the optimal number of feedback bits is quite large, especially when and have relatively large values. Thus, in practice, realizing such a huge codebook may be infeasible because the memory of a mobile device can be limited. Nevertheless, establishing a maximum achievable rate offers valuable insights into network operation during the design of corresponding wireless communication systems. It is important to note the substantial difference between designing a system with the knowledge of the performance upper bound and designing without such knowledge. In this context, we contend that our analysis results yield meaningful insights irrespective of the magnitude of the optimal B, even if the practical implementation of the entire optimal B value may pose challenges. While acknowledging that the optimal value of B may be too large for implementation by some network operators, they can still consider the increasing B as much as possible, given the awareness of the upper bound.