## 4 Simulation Results and Discussions  
In this section we conducted some experimental simulations to investigate the performance of the proposed GARLM algorithm, comparing it to the typical greedy algorithm GESPAR in various situations, such as recovery probability, robustness to noise, and effectiveness. We considered a random vector \(\boldsymbol{x}\) of length \(n\) as the original unknown signal with randomly chosen elements, the sparsity level of this signal as \(k\), and the magnitude square of its \(N\)-point DFT as \(\boldsymbol{y}\), as the sampling measurement. Our goal was to recover the unknown signal--which is known to be sparse--from the magnitude square of its Fourier transform. To achieve this target, we listed some important system parameters. The number of maximum indices in the index set \(S\) is \(n=64\) and the number of the sampling measurement \(y_{i}\) is \(N=128\), and the two algorithms both use \(\tau=10^{-6}\) and \(T=6400\) to recover the signal \(\boldsymbol{x}\). All the experiments were performed in MATLAB on a desk station with a two 2.3 GHz Intel Core i5 CPU and 4 GB of ROM. We chose to use signal recovery probability to measure the effectiveness of different algorithms for signal recovery. The signal recovery probability was defined as the ratio of successful recovery signals in 100 simulations. Figure 1 shows the comparison of the effectiveness of the signal recovery probability of different algorithms, and the probability of successful recovery is shown by different sparsity levels. Here we compare the signal recovery probabilities of the GARLM and GESPAR algorithms at the same sparsity, and in each simulation both the original signal and its support are randomly reset. We observed that both the GARLM and GESPAR algorithms had a high successful recovery probability, the signal recovery probability was close to 100% before the signal sparsity level \(k\) was less than or equal to 13, and the performance of GARLM was better because its successful recovery probability began to decline up to \(k=16\). As the sparsity level \(k\) continued to increase, we observed that the probability of successful recovery of the two algorithms decreased, but the performance of the GARLM algorithm was still obviously better than the GESPAR algorithm. Although it was expected that our algorithm should outperform GESPAR in the measurement setup in a noise-free environment, in real life the measurements we get are always accompanied by noise, which degrades the performance of the algorithm. It is therefore necessary to consider the  
Figure 1: Recovery probability versus sparsity. GARLM is described in Section 3, and GESPAR is the method presented in [44]. effectiveness of the algorithm in the presence of noise. We wanted to observe its normalized mean square error (NMSE) performances with different signal-to-noise ratio (SNR) values. In our simulation, we still chose the random vector \(\boldsymbol{x}\) of length \(n\) with randomly chosen elements as the original unknown signal, its sparsity level as \(k\), and the magnitude square of its \(N\)-point DFT as \(\mathbf{y}\). We considered adding random white Gaussian noise \(\mathbf{n}\) to the measurements \(\mathbf{y},\ \mathbf{y}=\left|\mathbf{F}\mathbf{x}\right|^{2}+\mathbf{n}\). To recover the unknown signal, our algorithm GARLM chose \(n=64,\ N=128,\ \tau=10^{-6},\ T=10000\), and \(\text{SNR}=\left\{10,20,30,40\right\}\) dB, as well as GESPAR. Figure 2 shows the NMSE performances versus sparsity level \(k\) at different SNR values--the NMSE defined as \(\text{NMSE}=\frac{\left\|\mathbf{x}-\mathbf{\hat{x}}\right\|_{2}}{\left\| \mathbf{x}\right\|_{2}}.\) We observed that when \(\text{SNR}=10\), the overall performance of the GARLM algorithm was better than the GESPAR algorithm. With the increase of sparsity \(k\), the noise robustness of the GARLM algorithm was better than the GESPAR algorithm, except for \(k=9.\) This situation also occurred in other SNR values.