# Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems
## 2 System Model
### Channel Model  
In a practical massive MIMO system, the transmitters, such as base stations, are typically located at high elevation with fewer surrounding scatterers, while the receivers, such as user equipment, are surrounded by rich scatterers at low elevation. Thus, each path departing from the transmitter may arrive at several reception directions [11]. Moreover, the channel is time-varying as the multipath delay and frequency offset are common and it is often useful to represent the channel response in frequency domain, which can be expressed as [20; 21]  
\[\mathbf{H}\left(\mathbf{t},f\right)\mathbf{=}\sum_{l=1}^{L}\sum_{q=1}^{L_{A0 \Lambda}}a_{l,q}e^{l2\pi\left(v_{l}a^{l}-\tau_{l,q}f\right)}\mathbf{a}_{R} \left(\theta_{l,q}^{R}\right)\mathbf{a}_{T}^{H}\left(\theta_{l}^{T}\right) \tag{3}\]  
where \(L\) denotes the number of paths that depart from the transmitter with different AoDs; \(L_{A0\Lambda}\) denotes that each of departure path will contribute \(L_{A0\Lambda}\) reception directions; \(a_{l,q}\), \(v_{l,q}\), \(\tau_{l,q}\) and \(\theta_{l,q}^{R}\), respectively, denote the complex gain, Doppler shift, delay and the AoA of the \(q\)th arrival path of the \(l\)th departure path; \(\theta_{l}^{T}\in[0,2\pi)\) denotes the AoD of the \(l\)th path; and \(\mathbf{a}_{T}\left(\theta_{l}^{T}\right)\) and \(\mathbf{a}_{R}\left(\theta_{l,q}^{R}\right)\) denote the antenna array response vector at the transmitter and receiver, respectively. For a uniform linear array (ULA) antenna, the array response vectors at transmitter and receiver can be presented as  
\[ \begin{split}\mathbf{a}(\theta_{l,q}^{T})&=\left[1,e ^{l2\pi d\sin\theta_{l,q}^{T}/\lambda},\cdots,e^{l2\pi(N-1)d\sin\theta_{l,q}^{T }/\lambda}\right]^{T}\\ \mathbf{a}(\theta_{l}^{R})&=\left[1,e^{l2\pi d\sin \theta_{l}^{R}/\lambda},\cdots,e^{l2\pi(N-1)d\sin\theta_{l}^{R}/\lambda} \right]^{T}\end{split}  \tag{4}\]  
where \(\lambda\) is the wavelength of signal and \(d\) is the antenna spacing, which is typically assumed as \(d=0.5\lambda\). It is difficult to estimate such a channel matrix as five parameters need to be estimated in each path: the complex gain \(a_{l}\), the angle of arrival \(\theta_{l,q}^{R}\), the angle of departure \(\theta_{l}^{T}\), the Doppler shift \(v_{l}\) and the delay \(\tau_{l}\). Besides, although there are only 2-8 paths on average in the mmWave band [22],the actual number of paths also needs to be estimated, which makes it more difficult to estimate the CSI. Therefore, it is necessary to simplify the channel model according to the practical scenarios. Suppose that the channel is slowly varying over the signal duration of interest \(T\), which means the channel is quasi-static and the Doppler frequency shift can be ignored [21; 23]. Based on such assumption, we can rewrite Equation (3) as  
\[\mathbf{H}=\sum\limits_{l=1}^{L}\sum\limits_{q=1}^{L_{QQ}}a_{l,q}e^{-j2\pi \gamma_{l,q}}\mathbf{a}_{R}\left(\theta_{l,q}^{R}\right)\mathbf{a}_{T}^{H} \left(\theta_{l}^{T}\right) \tag{5}\]  
In addition, if we further assume that the bandwidth B of channel is sufficiently small (i.e., \(B\tau_{l}\ll 1\ \forall l,l=1,\cdots,L\)), the channels of mmWave massive MIMO system can be simplified as a narrowband frequency-flat physical channel model [11]. Then, Equation (5) can be simplified as  
\[\mathbf{H}=\sum\limits_{l=1}^{L}\sum\limits_{q=1}^{L_{4Q}A}a_{l,q}\mathbf{a}_{ R}\left(\theta_{l,q}^{R}\right)\mathbf{a}_{T}^{H}\left(\theta_{l}^{T}\right) \tag{6}\]  
Although there are abundant scatterers surrounding the ends, the penetration and reflection loss are also significant at mmWave band. Thus, a geometric channel model is widely used in the prior work [9; 10; 24], which can be expressed as  
\[\mathbf{H}=\sum\limits_{l=1}^{L}a_{l}\mathbf{a}_{R}\left(\theta_{l}^{R}\right) \mathbf{a}_{T}^{H}\left(\theta_{l}^{T}\right) \tag{7}\]  
To simplify the expression, we adopt the multipath narrowband frequency-flat channel model in Equation (7) to compare three categories channel estimation algorithms.