## 4 Experimental Results
### Comparison Results of Calculation Amount  
In order to quantify the computational complexity of the proposed approach, we perform a theoretical calculation of the real multiplications for the traditional approach used in references [35] (i.e., (12)) and the optimized approach presented in this paper. For simplicity in the calculation, only cases where \(L\), \(M\), \(L^{\prime}\), and \(M^{\prime}\) are powers of 2 are considered.  
When the calculation of CAF employs the equivalent pulse compression method [7], according to the principles of the fast fourier transform (FFT), the number of real multiplications for CAF results of size \(l\times m\) is given by  
\[C_{CAF}(l,m)=6m(2l-1)\log_{2}(2l-1)+2 ml\log_{2}(m)+4m(2l-1) \tag{38}\]  
The number of real multiplications for the traditional sparse representation approach is  
\[C_{1}=(LM+1)C_{CAF}(L,M) \tag{39}\]  
Figure 8: The sparse recovery results of the field signals. (**a**) Sparse recovery result with \(R=1,C=2\). (**b**) Doppler frequency direction of the sparse recovery result with \(R=1,C=2\). (**c**) Sparse recovery result with \(R=2,C=2\). (**d**) Doppler frequency direction of the sparse recovery result with \(R=2,C=2\). (**e**) Sparse recovery result with \(R=4,C=4\). (**f**) Doppler frequency direction of the sparse recovery result with \(R=4,C=4\).  
The number of real multiplications for the optimized sparse representation approach is  
\[{C^{\prime}}_{1}={C_{CAF}}(L,M)+{C_{CAF}}\big{(}2L^{\prime}-1,2M^{\prime}-1\big{)} \tag{40}\]  
Based on the method of computing the Moore-Penrose pseudoinverse using singular value decomposition in Matlab, the number of real multiplications for sparse recovery of \(l\times m\)-sized CAF results after \(k\) iterations is approximately  
\[{C_{R}}(l,m)\approx k\Big{(}40{{}^{3}}m^{3}+16l^{2}m^{2}+12 lm\Big{)} \tag{41}\]  
The number of real multiplications for sparse recovery using the traditional sparse representation approach is approximately  
\[{C_{2}}={C_{R}}(L,M) \tag{42}\]  
The number of real multiplications for sparse recovery using the optimized sparse representation approach is approximately  
\[{C^{\prime}}_{2}=RCC_{R}\big{(}L^{\prime},M^{\prime}\big{)} \tag{43}\]  
Therefore, the number of real multiplications for the traditional sparse representation and recovery approach is approximately  
\[{C_{all}}={C_{1}}+{C_{2}} \tag{44}\]  
The number of real multiplications for the optimized sparse representation and recovery approach is approximately  
\[{C^{\prime}}_{all}={C^{\prime}}_{1}+{C^{\prime}}_{2} \tag{45}\]  
Under different values of \(L\), \(M\), \(R\), and \(C\), we calculate the number of real multiplications for both the traditional and the optimized sparse representation approaches (according to (39) and (40)). The results are shown in Figure 9.  
From Figure 9, it can be observed that compared with the traditional sparse representation approach, the optimized sparse representation approach requires an order of  
Figure 9: The number of real multiplications for the sparse representation approach. (**a**) \(L=M\). (**b**) \(L=4M\). (**c**) \(4L=M\).  
magnitude fewer real multiplications. Moreover, as \(L\) and \(M\) increase, the optimized sparse representation approach saves even more on the number of real multiplications.  
Additionally, based on (44) and (45), we calculate the number of real multiplications for both the traditional and the optimized sparse representation and recovery approaches. The results are shown in Figure 10.  
From Figure 10, it can be observed that when \(R\) and \(C\) are each doubled, the number of real multiplications decreases by approximately one order of magnitude. Therefore, for any sparse recovery approaches with a time complexity of \(O\left(l^{3}m^{3}\right)\), the optimization approach can play a similar role in reducing the computational complexity.