# The Role of Correlation in the Performance of Massive MIMO Systems
## 2 System Model  
In this study, we consider a scenario of a single-cell block-fading model over which the channel coefficients are timely invariant. In particular, in the typical cellular networks, the time and frequency resource blocks are allocated to different user equipments (UEs) simultaneously [24; 25; 26; 27; 28]. In a block-fading model, the channels are considered to be frequency flat, which remain unchanged during a constant period of time that corresponds to a channel coherence time of [29]. Two common transmission modes are used in the current generation of wireless communications systems, namely time-division duplex (TDD) and frequency-division duplex (FDD) [30; 31; 32; 33]. The canonical m-MIMO systems are typically considered to operate in TDD mode, where the uplink and downlink transmissions share the same frequency band. TDD operation relies on the assumption that the channels in the uplink and downlink are reciprocal. In this case, the uplink channel can be used for designing the downlink precoder. Specifically, the channel reciprocity allows the uplink channel estimation to be used for downlink precoding without the requirement for downlink CSI estimation. Hence, the uplink channel estimation depends only on the number of UEs, but it is independent of the number of base station antennas, where (), hence making the m-MIMO systems overhead free and fully scalable with respect to the number of base station antennas. For this reason, this study concentrates on investigating the performance of m-MIMO systems operating with TDD transmission mode only.  
This study considers a base station that is equipped with antenna elements, which serves UEs simultaneously over the same time and frequency resources. Each UE is equipped with a single antenna. The UEs are perfectly synchronized. This study follows the typical assumption of an m-MIMO system, in which the number of base station antennas is much larger than the number of UEs (i.e., ). In each coherence block, the energy is freely divided between the useful data transmission and the pilot sequences depending on the availability of total energy at the transmitter. During the channel estimation phase, the UEs transmit UL pilot sequences of length with training power denoted by per channel coherence block. The remaining duration is dedicated to data transmission, which is defined as. During the data transmission phase, the received signal at the -th UE can be expressed as  
(1)  
where is the complex true channel vector intended between the base station and the -th UE, while denotes the additive white Gaussian noise, which is modeled as a zero mean independent additive noise. The downlink transmit vector in Equation (1) is given as  
(2)where \(\mathbf{V}=[\mathbf{v}_{1},\ldots,\mathbf{v}_{U}]\in\mathbb{C}^{M\times U}\) is the precoding matrix that is utilized at the base station to beam-form the useful data in the next transmission phase. Parameter \(\lambda\) denotes the normalization constant that is used in order to ensure that the average transmit power of the base station is constant per UE during the data transmission, i.e, \(\mathbb{E}\big{[}\mathbf{d}^{\mathrm{H}}\mathbf{d}\big{]}=U\). As such, \(\lambda\) is defined as  
\[\lambda=\frac{1}{\mathbb{E}\Big{[}\frac{1}{U}\Big{(}\text{trace}\big{(} \mathbf{V}\mathbf{V}^{\mathrm{H}}\big{)}\Big{)}\Big{]}}. \tag{3}\]  
The following section discusses the received signal using linear precoding at the transmitter.