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Top1. Introduction
Recent years multiple-input multiple-output (MIMO) technology has gained great attention due to its ability to increase the capacity linearly with the increase of antenna number (Goldsmith et al., 2003). The conventional channel models for MIMO systems have been proposed by organizations like Wireless World Initiative for New Radio (WINNERII) (WINNER+, 2003). However, these conventional channel models are two-dimension (2D) channel models for the reason that only degrees of freedom in the azimuth is considered. It is important for cross-polarized model to consider the azimuth and elevation angle together (Shafi, et al, 2006), and (Yong & Thompson, 2005) indicates that 2D channel model is not accurate to describe the realistic environment. To further enhance the performance of MIMO systems, researchers ((Pan, et al., 2013; Jassal, et al., 2014; Mondal, et al., 2015) are now attempting to extend 2D channel models to 3D channels model by exploiting degrees of freedom in the elevation except in the azimuth.
Generally speaking, MIMO channel models can be classified into two categories, including physical models which focus on propagation mechanisms between transmitter and receiver, and analytical models which can capture the antenna configuration as well as wave propagation by describing the transfer function between antenna arrays. Analytical models (Almers et al., 2007) therefore, have been successfully used for theoretical analysis and simulation of MIMO systems. This paper mainly studies 3D analytical models.
The notion of analytical models in 3D environments have been processed in some works. (Dao, et al., 2011) proposed a 3D correlation model which allowed to use Von Mises Fisher (VMF) distribution as angle distribution. The simulation results show that VMF distribution is suitable for 3D MIMO channel due to its character of spherical. But the 3D channel model in (Dao, et al., 2011) is based on correlation matrix which has to compute 
parameters (Almers et al., 2007). The Kronecker model (Almers et al., 2007) which is another analytical model just involves 
parameters instead of 
, where 
and 
represent transmit(Tx) and receive(Rx) antenna number respectively. Additionally, the model proposed by (Dao, et al., 2011) cannot derive the Tx(Rx) correlation conveniently. (Jiang, et al., 2014) get the expression of spatial fading correlation(SCF) in 3D environment. However, the closed-form expression of 3D Kronecker model is not derived.
Motivated by all of above, this paper mainly applies the 3D Kronecker modeling which is based on the Tx and Rx correlation matrix. Firstly, the 3D channel model is derived by extending the 2D channel model of WINNER. The closed form of Tx and Rx correlation matrix are derived to get 3D Kronecker channel model which facilitates the calculation for the correlation of Tx and Rx. Additionally, this paper also studies the capacity of 3D MIMO systems and 2D MIMO systems in multi-user cases, and the conclusion is proposed by this paper firstly.The remainder of the paper is structured as follows. Section 2 analyzes the 3D MIMO system and presents 3D channel model. The closed-form expression of 3D Kronecker model is derived in section 3. Section 3 also analyzes the capacity of multi-user systems. In section 4, the proposed 3D Kronecker model is validated by simulation. The conclusion is presented in section 5.