# Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems
## 2 System Model  
### System Model  
Without loss of generality, we consider a typical mmWave point-to-point hybrid massive MIMO system, as shown as Figure 1. Let \(N_{T}\) and \(N_{R}\) denote the number of antennas at the transmitter and receiver, respectively. For simplicity, we assume the number of RF chains deployed at both sides is equal, i.e., \(N_{RF}^{T}=N_{RF}^{R}=N_{RF}\). Since the RF chains are costly and due to power consumption, the number of RF chains is much smaller than that of antennas, i.e., \(\min\left\{N_{T},N_{R}\right\}\gg N_{RF}\). For such an mmWave massive MIMO system with hybrid structure, the transmitter and receiver communicate via \(N_{s}\) streams where \(N_{s}\leq N_{RF}\). Specifically, the transmitter employs a baseband precoder \(\mathbf{F}_{BB}\in\mathbb{C}^{N_{RF}\times N_{s}}\) followed by an analog phase shifter precoder \(\mathbf{F}_{RF}\in\mathbb{C}^{N_{T}\times N_{RF}}\). After the combined transmitter precoding matrix \(\mathbf{F}=\mathbf{F}_{RF}\mathbf{F}_{BB}\in\mathbb{C}^{N_{T}\times N_{s}}\), the transmitted signals can be written as \(\mathbf{x}=\mathbf{F}\mathbf{s}\in\mathbb{C}^{N_{T}\times 1}\), where the \(i\)th element of \(\mathbf{x}\) is the transmitted signal at the \(i\)th antenna, \(\mathbf{s}\) is the transmitted symbol vector which satisfies \(\mathrm{E}\left\{\mathbf{s}\cdot\left(\mathbf{s}\right)^{H}\right\}=\frac{P} {N_{s}}\mathbf{I}_{N_{s}}\), and \(P\) is the total power of transmitted symbols [12]. During the channel estimation phase, it is worth noting that the pilot power and pilot pattern also affect the performance of sparse channel estimation, which need to be carefully designed. However, to simplify expression, we assume the pilot symbols are evenly placed and transmitted with equal power. Figure 1: Illustration of a hybrid MIMO architecture with the fully connected phase shifting network. At the receiver, the received signals on all antennas are combined with a combiner \(\mathbf{W}\in\mathbb{C}^{N_{R}\times N_{R}}\) composed of the RF combiner \(\mathbf{W}_{RF}\in\mathbb{C}^{N_{R}\times N_{RF}}\) and baseband combiner \(\mathbf{W}_{BB}\in\mathbb{C}^{N_{RF}\times N_{R}}\), which can be expressed as  
\[\mathbf{y}=\mathbf{W}^{H}\mathbf{H}\mathbf{F}\mathbf{s}+\mathbf{W}^{H}\mathbf{n} \tag{1}\]  
where \(\mathbf{H}\in\mathbb{C}^{N_{T}\times N_{R}}\) denotes the channel matrix discussed in the next subsection and \(\mathbf{n}\sim CN\left(\mathbf{0},\sigma^{2}\mathbf{I}_{N_{t}}\right)\) denotes the symmetric complex Gaussian distributed additive noise vector at the receiver. In a practical communication system, the non-ideal characteristics of components, including nonlinear power amplifier [14], inphase/quadrature-phase imbalance [15], phase noise [16] and carrier frequency offset [17], will greatly affect the estimation performance. Especially in mmWave band, the use of cheap equipment is encouraged, which leads to unignorable hardware impairments. To analyze the effect of hardware impairments on the performance of channel estimation algorithms, we take the residual transmitter hardware impairments into consideration and model them as the additive distortion noise [18]. Specifically, the additive distortion noise denoted by \(\mathbf{e}_{I}\in\mathbb{C}^{N_{T}\times 1}\) can be modeled as a Gaussian distributed random variable and its power is proportional to the pilot symbols power. That is, \(\mathbf{e}_{I}\sim CN(\mathbf{0},\mathbf{\Lambda})\) and \(\mathbf{\Lambda}\overset{\Delta}{=}k_{e}\left(q_{11},\cdots,q_{N_{T}N_{T}} \right)^{T}\), where \(q_{ii}\) denotes the \(i\)th element of covariance matrix \(\mathbf{Q}\overset{\Delta}{=}E\left(\mathbf{x}\mathbf{x}^{H}\right)\) and \(k_{e}\) denotes the proportionality coefficient [19]. In addition, we assume that the hardware impairments at the transmitter are uncorrelated with the transmitted signals \(\mathbf{x}\) and ignore the hardware impairments at the receiver as they can be simply treated as additive noise during the channel estimation phase.