# Low-Complexity Channel Estimation in 5G Massive MIMO-OFDM Systems
## 3 System Model
### Proposed SSA-CoSaMP Algorithm  
By exploiting the characteristics of the sparse channel, the reconstruction performance is improved from the aspects of structuring and dynamic sparsity adaptation. The specific steps of the sparse adaptive CoSaMP (SSA-CoSaMP) is proposed in Algorithm 1. ```
Input: Signal observation value \(Y\), observation matrix \(\Psi\), maximum channel length \(L\), number of antennas configured on the base station \(M\), number of consecutive OFDM symbols \(R\), algorithm stop parameter \(\beta\). Output: Estimated CIR matrix \(\hat{H}\). 1: Initialization: Initial residual \(V_{0}=Y\); sparsity \(S=1\); number of cycles \(k=1\); support set \(\Omega_{0}=\phi\) (Equations (1) and (2))
2:\(Z=\Psi^{H}\)\(V_{k}\) (using the method of Reference [14])
3: Combine multiple vectors of the remainder into a vector using massive MIMO common space-time sparsity, i.e., \(c(\tau)=\sum\limits_{\tau=\ 1,\ i=0}^{R,\ M-1}\left|Z^{(\tau+\ \hat{L},\tau)}\right|^{2}\), \(0\leq\tau\leq L-1\) where \(c(\tau)=[c(0),\ \ldots,\ c(L-1)]^{T}\), \(Z^{(m,n)}\) is the \(m\)th row and \(n\)th column of \(Z\) (using Equation (6) and Reference [9])
4:\(\Omega=\Omega\cup\text{supp}\ \{c\}_{2K}]\) merged set (from Equation (2) and References [9; 14; 15])
5:\(\Gamma=\Omega\cup[\Omega+L|\cup,\ldots,\ \cup[\Omega+L(M-1)]]\) merged set
6:\(\hat{H}_{r}=\hat{S}_{r}\)
7:\(V_{k}=y-\Psi^{H}\Delta\hat{h}\) updates the margin (using the method of References [8; 9; 10; 11; 12; 28])
8: Stop iteration if \(\|V_{k}\|_{2}\leq\beta\times\|Y_{2}\|\) is satisfied (using Equation (12)); otherwise let, \(k=k+1\), goto Step 2. ```  
**Algorithm 1** SSA-CoSaMP  
From the improved SSA-CoSaMP algorithm flow, the SSA-CoSaMP algorithm does not stop the iteration according to the sparsity given in advance but establishes the relationship between the residual and the observation matrix. When the set condition is met, the iteration is automatically stopped. Therefore, the algorithm can adaptively acquire the sparsity, which provides the possibility of engineering application of the algorithm in actual channel estimation. From the SSA-CoSaMP process, it is mainly divided into two parts. The first part is the determination of the sparsity \(S\). The algorithm sets the sparsity \(S=1\) in the initialization phase, and then uses the loop iteration method to finally obtain the sparsity \(S\) that satisfies the stop iteration. See Steps 1 and 8 in Algorithm 1. The second part is to use the space-time common sparsity feature of the wireless massive MIMO channel in the case of the preset sparsity \(S\). In the process of reconstructing and estimating \(\Phi\), multiple sparse vectors can be simultaneously applied to each iteration. The update corresponds to Steps 2-7 in the SSA-CoSaMP algorithm. If the CoSaMP algorithm only uses space-time common sparsity without considering the sparsity adaptive environment, we call it the structured CoSaMP (i.e., S-CoSaMP) algorithm. The S-CoSaMP algorithm can only perform the number of iterations according to the given sparsity, instead of judging whether to end the algorithm by the actual channel state. There are two consequences of this:1. The algorithm has reconstructed the result, but the algorithm itself needs to continue the loop iteration in order to satisfy the given sparsity, resulting in a waste of time and energy. 2. The result of the algorithm reconstruction is not accurate enough, but the given sparsity has already met the number of loop iterations, and the iteration stops, resulting in an insufficient accuracy of the estimation. Section 5 compares the performance of S-CoSaMP and SSA-CoSaMP algorithms. In the SSA-CoSaMP algorithm, \(\beta\) is different under different SNRs. If the value of \(\beta\) is fixed, rather than dynamically changing according to the channel state, it will directly have a very serious impact on the reconstruction result.