# Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise
## 2 System Model  
In this paper, a non-reciprocal DL FDD MaMi communication system with single-cell scenario is considered, where the transmit BS employs antenna array of size \(N\) using either ULA or UPA. The BS serves \(K\) non-cooperative single antenna users with \(N\gg K\). The spatial correlation between the antenna elements would be increased in MAMi systems and each user could have distinct spatial channel characteristics. This paper considers a static block fading transmissions model (i.e., flat-fading wireless channels) based on Rayleigh-correlated fading model and colored noise environments. The overall coherence length for each transmission block in symbols is given by \(\tau\in\mathbb{Z}^{+}\). Figure 1 shows the BS with MaMi system and demonstrates the DL and UL transmissions within each coherence block length. In particular, the available coherence block length consists of resource transmission in time \(T_{\text{c}}\) and frequency \(B_{\text{c}}\) plane, which represents the time and the frequency intervals over which the channelresponses are approximately constant, as considered in the current long term evolution (LTE) wireless networks [35; 36]. According to the Nyquist-Shannon sampling theorem, the coherence block length is given as \(\tau=T_{\text{c}}\times B_{\text{c}}\). The available coherence length \(\tau\) for transmission in the UL and DL is freely divided between the training phase \(T_{\text{s}}\) and the data transmission phase \(\tau_{\text{d}}\), both enumerated in symbols. The available energy is divided between the DTS transmission and the useful data transmission.  
The received data transmission signal at the \(k\)-th user \(y_{k}\in\mathbb{C}^{K}\) can be expressed as [9; 10]  
\[y_{k}=\sqrt{\rho_{\text{d}}\,\xi\mathbf{h}_{k}^{\text{H}}\mathbf{Ws}+n_{k}}, \tag{1}\]  
where \(\mathbf{s}=[s_{1},\dots,s_{K}]^{\text{T}}\in\mathbb{C}^{K}\) is the per-user data symbol, which is modeled as a zero mean circularly symmetric complex Gaussian (CSCG) and satisfied \(\mathbb{E}[\mathbf{ss}^{\text{H}}]=\mathbf{I}_{K}\). The parameter \(n_{k}\) is the receiver colored noise, which satisfies \(\mathbf{n}_{k}=[n_{1},\dots,n_{K}]^{\text{T}}\in\mathbb{C}^{K}\sim\mathcal{CN }(\mathbf{0},\mathbf{R}_{n})\), where \(\mathbf{R}_{n}\) denotes the correlated colored noise covariance matrix. The complex precoder matrix at the BS is denoted by \(\mathbf{W}=[\mathbf{w}_{1},\dots,\mathbf{w}_{K}]\in\mathbb{C}^{N\times K}\), which is provided later in (7). The complex channels \(\mathbf{h}_{k}\), \(k=1,\dots,K\) are modeled as  
\[\mathbf{h}_{k}=\mathbf{R}_{\text{h}}^{1/2}\mathbf{v}_{k}, \tag{2}\]  
where \(\mathbf{R}_{\text{h}}\succeq\mathbf{0}_{N}\) denotes a deterministic Hermitian-symmetric positive definite correlation matrix at the BS-side and \(\mathbf{v}_{k}\), \(k=1,\dots,K\) depict the fast-fading instantaneous DL channels, which are modeled as uncorrelated wide-sense stationary independent and identically distributed CSCG random vectors with \(\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{\text{N}})\). As such, the DL channel \(\mathbf{h}_{k}\in\mathbb{C}^{N}\) satisfies \(\mathbf{R}_{\text{h}}=\mathbb{E}\big{[}\mathbf{h}_{k}\mathbf{h}_{k}^{\text{H }}\big{]}\in\mathbb{C}^{N\times N}\). The channel covariance matrix is denoted by the spatial correlation effects and the antenna array patterns. Further details about the channel covariance matrix designs can be found in Section 6.  
Parameter \(\xi=K/\mathbb{E}\left[\text{tr}\big{(}\mathbf{WW}^{\text{H}}\big{)}\right]\) in (1) is the precoding normalization factor, which is considered to ensure that \(\mathbb{E}\big{[}\|\sqrt{\xi}\mathbf{Ws}\|^{2}\big{]}=K\), as defined in [10]. Parameter \(\rho_{\text{d}}\) in (1) denotes the transmit DL signal-to-noise-ratio (SNR).  
Figure 1: Illustration of a base station (BS) with massive number of antenna elements that serves K users. It also shows the downlink (DL)/uplink (UL) pilot sequence and data transmissions in frequency division duplex (FDD) operation mode.