# Low-Complexity Channel Estimation in 5G Massive MIMO-OFDM Systems
## 3 System Model  
A typical FDD mode massive MIMO-OFDM system was established. The number of base station antennas in the cell was \(M\), and the number of single antenna users was \(K\), and \(M\gg K\), as shown in Figure 3. The total number of subcarriers of the OFDM symbol was \(N\), and the signal (including data and pilot) transmitted by the \(m\)th antenna of the base station was \(S_{m}\in\mathbb{C}^{N\times 1}\left(m=1,2,\ldots,\,M\right)\).  
Then, the signal received by the user side can be expressed as:  
\[y=\sum\limits_{m=\,1}^{M}S_{m}Fh_{m}+n \tag{5}\]  
After removing the guard interval and performing the DFT transform, the user side extracts the pilot information \(y_{r}\) in the \(r\)th OFDM symbol from the received information \(y\) according to the pilot allocation scheme, and \(y_{r}\) can be expressed as:  
\[\begin{array}{c}y_{r}=\sum\limits_{m=\,1}^{M}diag\{c_{m}\}F_{k}\bigg{[} \begin{array}{c}h_{m,r}\\ O_{\left(N\,-\,L\right)\,\times\,1}\end{array}\bigg{]}+n_{r}\\ =\sum\limits_{m=\,1}^{M}C_{m}F_{L}|b_{m,r}+n_{r}\\ =\sum\limits_{m=\,1}^{M}\Phi_{m}F_{L}h_{m,r}+n_{r}\end{array} \tag{6}\]  
where \(\delta\) denotes the position information of the subcarriers occupied by the pilot in one OFDM symbol.  
In this paper, the pilot scheme is a non-orthogonal pilot allocation scheme (see Section 4 for details) \(\Phi_{m}=C_{m}F_{L}|_{\delta}=C_{m}diag\{c_{m}\}\), where \(c_{m}\in\mathbb{C}^{N_{P}\,\times\,1}\) represents the set of pilot amplitudes of the \(m\)th antenna. \(F\) denotes a DFT matrix of size \(N\times N\), \(F_{L}\) denotes the first \(L\) rows of \(F\), \(F_{L}|_{\delta}\) denotes a submatrix according to \(F_{L}\), and \(n_{r}\) denotes an additive white Gaussian noise (AWGN) noise vector of the \(r\)th OFDM symbol in the channel. Simplifying Equation (6) gives:  
\[y_{r}=\Phi\widetilde{h}_{r}+n_{r} \tag{7}\]  
where \(\Phi=[\Phi_{1},\,\,\Phi_{2},\ldots,\Phi_{M}]^{T}\in\mathbb{C}^{N_{P}\,\times \,ML}\) and \(\widetilde{h}_{r}=\left[h_{1,r}^{T},\,\,h_{2,r}^{T},\ldots,h_{M,r}^{T}\right]^ {T}\in\mathbb{C}^{ML\,\times\,1}\).  
In massive MIMO-OFDM systems, \(N_{P}\ll ML\). Therefore, Equation (7) is an undetermined equation, which cannot be reliably estimated using a traditional channel estimation method such  
Figure 3: Massive MIMO-OFDM framework.  
1. Sparseness adaptation: In the case of unknown sparsity, the number of iterations can still be controlled by Equation (12), and finally the iteration is stopped, and the sparsity is obtained, thereby improving the ability of the algorithm in practical engineering applications.
2. Stop the dynamic setting of the iteration parameter \(\beta\): The influence of noise on the observation value will be different under different channel states, so the stop iteration parameter cannot be set to a fixed value but should be in different channel states. It is different, and the channel state is characterized by SNR, that is, the stop iteration parameter \(\beta\) is different under different SNRs.  
In [27], the method of using the threshold is also mentioned to make the algorithm adaptively obtain the sparsity, but the set stop iteration parameter is a fixed value, and the iterative parameters cannot be stopped according to the channel state change, and the proposed algorithm can dynamically get sparsity.