## 1 Introduction
The seriousness of channel estimation error which is used in this paper will be shown in Section 5. We also show that the derived closed-form approximations are quite simple and effective, and in good agreement with the simulation results. Second, based on the derived closed-form approximation of SE, and using the exact inverse of the SE closed-form approximation, we propose a simplified determination scheme of the number of antennas needed in each group. Without this scheme, it is very difficult to apply antenna grouping-based RS overhead reduction scheme in a real system. The change of parameters must be reflected in real time to effectively determine the number of antennas in each group. The presented system block diagram and flow chart can be greatly helpful to real implementation of the proposed scheme. In a word, based on closed-form approximation of SE, to support distributed IoT devices, we propose a simplified determination scheme of the number of antennas needed in each group. More specifically, our main contributions are summarized as follows:  
* We present the closed-form approximations of achievable SE with channel estimation error for two representative linear precodings. We used the error factor which reflects the degree of channel estimation error and the error matrix which has the same statistical characteristic but independent of the channel matrix. The range of error factor is between 0 and 1, and it is a function of the number of antennas in each antenna group, and the adjustable factor which reflects the seriousness of the channel estimation error depending on the applied reference signal (RS) overhead reduction technique. We prove the derived SEs are quite simple and effective using extensive simulation results. * Based on the closed-form approximations of achievable SEs, we derive the determination criteria of the number of antennas needed in each antenna group. The determination criteria are derived from the inverse functions of achievable SEs. * We provide the system block diagram and algorithmic steps to apply the proposed determination scheme. The system block diagram is consisted of 5 main blocks, The central management unit, operation parameters unit, minimum data rate unit, decision unit, and UE. There are 3 thresholds we should consider for the systematic operation, i.e., power consumption, data rate, and the number of iterations. * We provide the extensive simulation results of the proposed scheme with various parameters, and show that the derived approximations are very effective and well-matched with the simulation results. The paper is organised as follows: the system description is presented in Section 2. Massive MIMO model, related precoding techniques, and RS overhead model are given in Section 2. In Section 3, the performance analysis and closed-forms approximation of achievable SE are presented. In Section 4, the determination scheme of the number of antennas in each group is shown. In Section 5, the proposed model is numerically analyzed. With monte-carlo (MC) simulations, we show that the analysis is well-matched with simulation results. Finally, concluding remarks are given in Section 6. _Notation:_ In the rest of the paper, boldface lower-case and upper-case characters denote vectors and matrices, respectively. The operators \((\cdot)^{H}\) and \(\mathbb{E}[\cdot]\) denote conjugate transpose and expectation, respectively. The \(N\times N\) identity matrix is denoted \(\mathbf{I}_{N}\), and the \(N\times N\) zero matrix is denoted \(\mathbf{0}_{N}\). \(\mathbf{X}\sim N(\mathbf{0}_{N},\ \mathbf{V}_{N})\) is the complex Gaussian distributed vector with mean zero and covariance \(\mathbf{V}_{N}\). \(\log_{2}(\cdot)\) denotes the common logarithm and \(\|\cdot\|_{F}\) represents the Frobenius norm operator. We use \(\circ\) to denote the componentwise product of the matrices. we use \(\text{diag}\left[d_{1},\cdots,d_{N}\right]\) for \(N\times N\) diagonal matrix with \(d_{i}\) as the \(i\)th diagonal element.