## 4 Experimental Results
### DOA Estimation  
As described by [PERSON] _et al._, for one snapshot case, the DOA problem of Equation (5) in [27] is equivalent to Equation (2) in this paper. With the assumption that the sources can be viewed as point sources and their number is small, the underling spatial spectrum is sparse and it can be solved via sparse recovery methods mentioned above. The first simulation considers the scenario that there are five uncorrelated signals impinging from [-47.7 \({}^{\circ}\), -28.5 \({}^{\circ}\), -9.2 \({}^{\circ}\), 11.6 \({}^{\circ}\), 30.1 \({}^{\circ}\)]. The grid is set to be within the range of -90 \({}^{\circ}\)to 90 \({}^{\circ}\)with 1 \({}^{\circ}\)spacing. A 30-element uniform linear array (ULA) spaced in half-wavelength units is used, and the signal-to-noise ratio (SNR) is set to be 20 dB. Here we only consider one snapshot case, and set \(H=1,L=S=8,Q=10\). Figure 4 depicts the solutions solved by different methods, and the vertical dashed lines mark the true directions. Obviously, our methods find the positions and the amplitudes exactly and achieve much sparser solutions than MP and BP. Moreover, they have a better amplitude estimation than H/1-DCD [40].  
Figure 4: Solutions of MP (**a**), BP (**b**), H\({}_{1}\)-DCD (**c**), HR-DCD (**d**), H\({\rm{H}}\)\({\rm{log}}\)-DCD (**e**), H\({\rm{H}}\)\({\rm{p}}\)-DCD (**f**) for off-grid DOAs. True DOAs are denoted by the vertical dashed lines.  
We give a time cost for the aforementioned methods in Table 3. From the results, we can see that our methods have the same time cost as the H\(l_{1}\)-DCD. Although they are a little longer than MP, but they are much less costly than BP. However, the significant advantages of our methods are that they are very easy to implement in a hardware platform because only bit-shifts are needed instead of multiplications similar as [37; 38; 39; 40], besides, they can achieve the similar reconstruction performance as BP without off-grid problem.  
Next we compare the mean squared error (MSE) of position and amplitude estimation performances by applying the proposed methods and several popular methods (MP, BP, FOCUSS, SBL, H\(l\)1-DCD, OSR), which are defined as follows:  
\[Position\_MSE=\frac{\left\|position(\mathbf{\hat{x}})-position\left(\mathbf{x} \right)\right\|_{2}^{2}}{\left\|position\left(\mathbf{x}\right)\right\|_{2}^{2}} \tag{20}\]  
\[Amplitude\_MSE=\frac{\left\|\mathbf{\hat{x}}-\mathbf{x}\right\|_{2}^{2}}{ \left\|\mathbf{x}\right\|_{2}^{2}} \tag{21}\]  
where \(\mathbf{\hat{x}}\) is an estimate of \(\mathbf{x}\).  
Figure 5a compares the MSEs of DOA estimation results of different methods under varying SNR, which are averaged by 100 Monte Carlo trials. From the curves of MSE of position _versus_ SNR shown in Figure 5a, it can be seen that H\(l_{1}\)-DCD and the proposed methods have similar results in position estimation, and outperform their counterparts. However, our methods have better amplitude estimation accuracy than H\(l_{1}\)-DCD from Figure 5b, and are even very close to the OSR performance.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **MP** & **BP** & **H\(l_{1}\)-DCD** & **HR-DCD** & **Hlog-DCD** & **H\(l_{p}\)-DCD** \\ \hline Time(s) & 0.013 & 0.721 & 0.130 & 0.122 & 0.129 & 0.126 \\ \hline \end{tabular}
\end{table}
Table 3: Time consumption of different methods.  
The MSEs of DOA estimation _versus_ number of signals shown in Figure 6 are obtained over 100 Monte Carlo trials with SNR equals to 20 dB. We can clearly see that our methods perform best compared to other methods with the same parameters, especially on the amplitude estimation results.