# Low-Complexity Channel Estimation in 5G Massive MIMO-OFDM Systems
## 2 Space-Time Common Sparsity Modeling  
### The Spatial Common Sparsity  
In the actual environment of signal transmission in massive MIMO-OFDM systems, a large number of scatterers are distributed to form a Rayleigh multipath transmission channel, and most of the energy of the wireless channel is concentrated on the very small components of the CIR [14], so that the CSIs formed by the user side and the base station naturally exhibit sparsity [15]. At the same time, when the signal from each transmitting antenna on the base station reaches the receiving antenna on the user side, the scatters passing through are very similar and almost identical. That is, the single antenna on the user has the characteristics of the spatial sparsity of all antennas on the base station. In addition, since the path delay varies much more slowly than the gain associated with the time channel, this sparsity is almost constant during the coherence time [16], forming a temporal correlation.  
A typical single-cell downlink multi-user massive MIMO-OFDM wireless communication system has common spatial sparsity, which is caused by many factors, such as multipath effect, antenna physical characteristics, and wireless transmission environment. In a wireless transmission environment, the scatterer causes the signal to form multiple paths from the same transmit antenna to the receive antenna, creating a multipath effect. However, in these paths, the time from the transmitting antenna to the receiving antenna is also inconsistent [17]. Therefore, most of the energy with CIR is concentrated on a very small part of the multipath component; that is, the CIR energy is not evenly distributed over all the taps of the channel, which exhibits sparsity. In particular, the CIR formed between the \(m\)th transmit antenna of a massive MIMO and the user of a single antenna can be expressed as:  
\[h_{m,r}=\left[h_{m,r}(1),\;h_{m,r}(2),\;\ldots,\;h_{m,r}(L)\right]^{T} \tag{1}\]  
where \(r\) represents the \(r\)th OFDM symbol and \(L\) represent the equivalent channel length, \(1\leq m\leq M\).  
The support set of \(h_{m,r}\) can be expressed as:  
\[\Gamma_{m,r}=\text{supp }\left\{h_{m,r}\right\}=\left\{\ell:\left|h_{m,r} \middle[\ell\right]\right|>\eta_{\text{min}},1\leq\ell\leq L\;\right\} \tag{2}\]  
where \(\eta_{\text{min}}\) represents the minimum noise in the channel. The sparsity \(\lambda_{m,r}=\left|\Gamma_{m,r}\right|_{c}\) is affected by the actual propagation environment and the sparse nature of multipath propagation. In general, \(\lambda_{m,r}\ll L\).  
In a typical massive MIMO, the antenna array on the base station is dense, and the spacing between the antennas is relatively small compared to the distance traveled by the signals arriving at the user-side single antenna, so that is negligible, and there is a common scatterer in the channel between each transmitting and receiving antenna. Therefore, each antenna on the base station exhibits a highly similar path delay to the user's single antenna [18; 19], as shown in Figure 1.  
The sparse characteristics of the CIR between different pairs of transmitting antennas are mostly coincident. This is the common spatial sparsity of massive MIMO channels. That is, for the \(r\)th OFDM symbol, different transmit antennas maintain the same sparsity, which can be expressed as:  
\[\Gamma_{1,r}=\Gamma_{2,r}=\cdots=\Gamma_{M,r} \tag{3}\]