# Compressive Sensing-Based Sparsity Adaptive Channel Estimation for 5G Massive MIMO Systems
## 3 Sparse Adaptive Matching Pursuit Algorithm  
### Sparseness Estimation  
Using compressed sensing to solve channel estimation can be equivalent to solving the following \(l_{0}\) norm minimum problem.  
\[\mathbf{\hat{h}}=\text{arg}\min\lVert\mathbf{h}\rVert_{0},\text{subject}\,\text{to} \lVert\mathbf{y}-\mathbf{A}\mathbf{h}\rVert_{2}\ \leq\ \varepsilon \tag{4}\]  
among them, \(\lVert\mathbf{h}\rVert_{0}\) is the vector \(l_{0}\) norm of the vector \(\mathbf{h}\) for the number of non-zero elements. Literature [28] has proved that when satisfied:  
\[\lVert\mathbf{h}\rVert_{0}\ <\ \frac{1}{2}\text{spark}(\mathbf{A}) \tag{5}\]  
Only the channel impulse response (\(\mathbf{h}\)) can be restored. Among them, \(\text{spark}(\mathbf{A})\) is the least linearly related column number in matrix \(\mathbf{A}\); it is easy to see \(2\leq\text{spark}(\mathbf{A})\leq\text{rank}(\mathbf{A})+1\). Since matrix \(\mathbf{A}\) is \(P\times L\) partial Fourier matrix and \(P<L\), then \(\lVert\mathbf{h}\rVert_{0}<\frac{1}{2}\ (P\ +\ 1)\).  
Because of the sparseness of the impulse response of the wireless communication channel, most of the energy is concentrated on a few taps while a small part of the energy distribution is below the noise threshold. The number of non-zero taps is much smaller than the channel length \(L\). Making full use of the sparse characteristics of the channel, we can use fewer pilot symbols to get the ideal channel estimation effect, so as to improve the spectrum utilization. An appropriate amount of pilot overhead satisfies Equation (5), so \[K=\left\{\begin{array}{cc}\frac{P}{2},&P\text{ is even}\\ \frac{P+1}{2},&P\text{ is odd}\end{array}\right. \tag{6}\]  
Based on the above analysis, the number of non-zero taps in the channel vector does not exceed \(K\), and at least \(L-K\) elements can be regarded as noise. Therefore, we first estimate the sparseness and then select the elements within this range. At higher signal-to-noise ratios, since the gain coefficients of the channel taps are higher than the noise amplitudes, the restored vector elements are arranged in descending order. The difference between adjacent elements is then used to determine the number of elements selected for this iteration and to further estimate the sparsity. The elements that precede the largest backward difference are selected for the support set because they may carry channel information.  
When the observation matrix satisfies certain conditions, the sparse signal restoration problem can be equivalent to the following convex optimization problem. Define the observation matrix \(\mathbf{A}\), the RIP parameter \(\delta_{k}\) is the minimum value \(\delta\) that satisfies Equation (7):  
\[\left(1-\delta\right)\left\|\mathbf{h}\right\|_{2}\leq \left\|\mathbf{A}\mathbf{h}\right\|_{2}\leq\left(1+\delta\right)\left\| \mathbf{h}\right\|_{2} \tag{7}\]  
among them, \(\mathbf{h}\) is the sparse signal for \(k\), if \(\delta_{k}<1\), the observation matrix \(\mathbf{A}\) satisfies the \(k\)th order RIP [29]. When matrix RIP parameters \(\delta_{k}<\sqrt{2}-1\), the reconstruction problem can be transformed into the following \(l_{1}\) norm minimum problem.  
\[\hat{\mathbf{h}}=\arg\min\left\|\mathbf{h}\right\|_{1},\text{ subject to}\left\|\mathbf{y}-\mathbf{A}\mathbf{h} \right\|_{2}\leq\varepsilon \tag{8}\]  
Due to partial Fourier matrix RIP parameters \(\delta_{k}<\frac{1}{2}\)[30], the regularization process based on convex optimization is introduced to improve the stability of the algorithm [23].