## 2 System Model Description  
This paper considers a single-cell wireless communications system, where the BS is equipped with an \(N\) transmit antenna that serves \(K\) single antenna users. We consider a non-line-of-sight (NLOS) Rayleigh fading channels over a single-frequency band with an overall coherence time denoted by \(T\in\mathbb{Z}^{+}\) and enumerated in symbols per transmission block. Figure 1 demonstrates a BS with m-MIMO system and illustrates the DL and UL transmissions within each coherence block length. The received signal during the data transmission phase at the \(k\)-th user is given as:  
\[y_{k}=\sqrt{\rho_{\text{d}}\,\lambda}\,\mathbf{h}_{k}^{\text{H}}\mathbf{V} \mathbf{s}+z_{k} \tag{1}\]  
where \(\rho_{\text{d}}\) denotes the per-user signal-to-noise-ratio (SNR) during the data transmission phase and \(\lambda\) refers to the normalization constant, which is defined as \(\lambda=K/\mathbb{E}[\text{tr}(\mathbf{V}^{\text{H}})]\), and \(\mathbf{V}=[\mathbf{v}_{1},\dots,\mathbf{v}_{K}]\in\mathbb{C}^{N\times K}\) is the precoding matrix. The term \(\mathbf{s}=[s_{1},\dots,s_{K}]^{\text{T}}\in\mathbb{C}^{K}\) is independently and identically distributed (IID) with a zero mean circularly symmetric complex Gaussian (CSCG) vector of data symbols, which satisfies \(\mathbb{E}[\mathbf{s}\mathbf{s}^{\text{H}}]=\mathbf{I}_{K}\), and \(z_{k}\)denotes the additive noise, which is modeled as a zero mean unit variance CSCG random variable. The DL instantaneous channel vector is given by \(\mathbf{h}_{k}=\mathbf{R}_{k}^{1/2}\tilde{\mathbf{h}}_{k}\), where the elements of \(\tilde{\mathbf{h}}_{k}\in\mathbb{C}^{N}\) are IID with zero mean and unit variance and \(k\)-th user's correlation matrix, \(\mathbf{R}_{k}\) satisfies \(\mathbf{R}_{k}=\mathbb{E}\big{[}\mathbf{h}_{k}\mathbf{h}_{k}^{\mathrm{H}} \big{]}\in\mathbb{C}^{N\times N}\). It is worth noting that \(\mathbf{R}_{k}\) depends on large-scale statistics, i.e., angles of arrival and departure or spatial/temporal correlation, which are considered to be frequency-invariant, and thus, can be efficiently obtained in the FDD or TDD systems [34]. To this end, the received signal-to-interference-plus-noise ratio (SINR), denoted \(\tau_{k}\), is given as:  
\[\gamma_{k}=\frac{\lambda\mid\mathbb{E}[\mathbf{h}_{k}^{\mathrm{H}}\mathbf{v}_{ k}]\mid^{2}}{\frac{1}{\rho_{\mathrm{d}}}+\lambda\mathbb{E}\big{[}|\mathbf{h}_{k}^{ \mathrm{H}}\mathbf{v}_{k}-\mathbb{E}[\mathbf{h}_{k}^{\mathrm{H}}\mathbf{v}_{ k}]\mid^{2}\big{]}+\lambda\sum_{j\
eq k}^{K}\mathbb{E}\big{[}|\,\mathbf{h}_{k}^{ \mathrm{H}}\mathbf{v}_{l}\mid^{2}\big{]}} \tag{2}\]  
Accordingly, the DL achievable sum rate, \(C\), for an FDD m-MIMO system under consideration can be expressed as:  
\[C=\left(1-\frac{T_{\mathrm{tr}}}{T}\right)\sum_{k=1}^{K}\log_{2}\!\left(1+ \gamma_{k}\right)\quad\left[\mathrm{bit/s/Hz}\right] \tag{3}\]  
where parameter \(T_{\mathrm{tr}}\) represents the training sequence length used during the training-phase. This paper uses \(T_{\mathrm{tr}}\) that maximizes the achievable sum rate. The received SINR term in (2) depends on the channel statistics, the channel estimates, and the linear precoding technique used at the BS. We consider two commonly prevailing types of linear precoders, named the BF and the RZF, as defined in [33]. The expectation (i.e., the averaging process) in (2) is taken with respect to different channel realizations, which are obtained separately by means of Monte Carlo simulation. As the purpose of the present paper is to concentrate on the DL training sequence design, the UL feedback time and associated error rate are assumed to be zero. Efficient feedback schemes can be considered in the future, see, e.g., [35, 36, 37, 38]. Besides, advanced signal processing techniques can also be considered for feedback, see e.g., [39, 40]. A large system limit based on random matrix theory methods is used to provide asymptotic  
Figure 1: A base station with \(N\) m-MIMO that serves K users in FDD mode.