## 1 Introduction
Finally, for efficient hardware implementation of the GS, the authors in [24] proposed the low-complexity inversion of the triangular matrix in the uplink system, and the authors in [25] proposed the parallel decoding scheme in the uplink system. According to several surveys, the common problems in [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25] are summarized as follows:  
First, the number of optimal iterations--which is one of the important parameters in the GS--is not estimated prior to signal transmission in the downlink system and signal detection in the uplink system, respectively. Second, all studies except for [15] evaluated the performances for the GS in only high \(\alpha\) systems such as \(\alpha=4\), \(\alpha=8\), or \(\alpha=16\). Third, all studies except for [14] considered the GS in only uplink systems. The GS updates the present symbol using the past symbol at each iteration and the error performance is improved as the number of iterations increases. Therefore, the number of iterations is a very important parameter in the GS and should be decided accurately prior to signal transmission in the downlink system and signal detection in the uplink system, respectively, for obtaining target error performance with minimum complexity. Let the number of required iterations for obtaining target error performance be \(\hat{i}\). The complexity for the GS is high when the number of iterations is larger than \(\hat{i}\), such as \(\hat{i}+1\), \(\hat{i}+2\), \(\cdots\). On the other hand, the error performance for the GS is degraded when the number of iterations is lower than \(\hat{i}\), such as \(\hat{i}-1\), \(\hat{i}-2\), \(\cdots\), \(1\), and it causes an error floor where the error performance is not improved despite infinite transmit power. However, all studies for the GS in [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25] did not consider a prior estimation of the number of iterations and obtained the number of optimal iterations by measuring error performances with Monte-Carlo simulations. In addition to problems of the prior estimation, the authors in [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25] considered the massive MIMO systems which have high \(\alpha\) values. The useful properties for the massive MIMO systems such as channel hardening effect and favorable propagation are more remarkable as \(\alpha\) is increased. However, the massive MIMO systems which have low \(\alpha\) should be researched for increasing cell capacity since the number of BS antennas cannot increase infinitely. Additionally, the GS solution is converged to the ZF solution regardless of an initial solution [26] when the number of iterations is sufficient since the gram matrix has the property of Hermitian positive definite [14]. Finally, in [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25],the GS was used in the uplink systems since the GS was originally developed for the purpose of solving linear equations, where the uplink signal detection is modeled as a linear equation problem at the BS. However, it should be also noted that the GS is performed well in the downlink system for the mobile communication system. Thus, this paper proposes an estimation scheme of the number of iterations for the GS pre-coding in the downlink massive MIMO-OFDM system for optimal error performance. The optimal error performance denotes the error performance for the ZF pre-coding and this paper uses the ZF pre-coding as a benchmark. In addition, various values of \(\alpha\) are considered from two to five since almost existing studies set \(\alpha\) as 4, 8, or 16, and the simulation results show that the proposed estimation scheme has a high accuracy even in low \(\alpha\) systems. This paper is organized as follows: Section 2 expresses the system model for downlink MU massive MIMO-OFDM. Sections 3 and 4 deal with conventional GS pre-coding and proposed estimation of the number of iterations. Section 5 shows the graphical performance evaluations for the estimation scheme proposed in Section 4.