We considered the problem of DOA estimation in both uniform linear array and nonuniform linear array scenarios. Numerical simulations are used to compare the proposed method with other well-known techniques. Simulation results for ULA are presented in Section 5.1 and NLA investigation in Section 5.2.

### Uniform linear array

In this part, several scenarios have been devised to compare the performance of DOA estimation method proposed with others using uniform linear array with *M* = 30 sensors. We consider three independent narrowband and far-field sources (*K* = 3) located at three distinct random angles. The angles of sources are uniformly selected from the range−90° to 90°. In each realization, new random angles are selected. The possible range is discretizes with a step size of 1°; subsequently, *N* = 181. The signals of the sources are determined from Gaussian distribution with zero mean and unit variance. Criterion used to assess the performance of algorithms is based on the empirical recovery rate, which is the probability of successful recovery of sources. It is assumed that *N*
_{
r
} realizations are achieved and in each realization *K* targets are found; therefore, estimation is determined for *K* × *N*
_{
r
} DOAs. The total number of successful recovered targets is represented by *N*
_{success} during the whole *N*
_{r} realizations, and ERR = *N*
_{success}/(*K* × *N*
_{
r
}).

Figure 1 shows the effect of the number of snapshots of the proposed algorithm, S-OMP, and BPDN. It is evident that more snapshots significantly improve the probability of source recovery using the proposed method. The reason is that SVD, which is used in the proposed algorithm, works better with more snapshots because the singular values are estimated more accurately. Increase of the number of snapshots marginally improves the performance of S-OMP and BPDN.

Figure 2 depicts ERR versus the number of sources using *L* = 70 and SNR = 30 dB for the proposed algorithm, S-OMP, and BPDN. It demonstrates that for *K* between 2 and 17, the proposed method achieves better recovery rate than S-OMP and BPDN. Failure rate of the proposed algorithm is comparable with the other two techniques for *K* ≥ 17*K* ≥ 17. In other words, by corrupting the sparsity (i.e., having more targets to estimate), the proposed algorithm fails to reconstruct sparse solution vector like the other compressed sensing methods.

In Fig. 3, a different scenario is assumed. Here, *M* = 30, *L* = 70, and three sources impinge on the array at 10°, 70°, and 71°. To verify the superiority of the proposed algorithm over S-OMP, *l*
_{2,1} minimization, and Root-MUSIC, RMSE indicator is utilized which is defined by:

$$ \mathrm{RMSE}=\frac{1}{K}{\displaystyle \sum_{k=1}^K}\sqrt{\frac{1}{N_r}{\displaystyle \sum_{j=1}^{N_r}}{\left({\widehat{\theta}}_{k j}-{\theta}_k\right)}^2} $$

(10)

where *N*
_{r} is the number of independent Monte Carlo realizations, *K* is the number of sources, *θ*
_{
k
} is the true DOA, and \( {\widehat{\theta}}_{kj} \) is the *j*-th estimation value for *θ*
_{
k
}.

Figure 3 demonstrates the efficiency of the proposed method in resolving adjacent angles in low SNRs. The proposed method outperforms the other algorithms and converges to the actual DOA values over a wider range of SNR values. The main advantage of proposed method is that it combines both SVD denoising and MMV approach to achieve better sparse signal recovery. In contrast, S-OMP is not able to discern adjacent targets even at an SNR = 30 dB where its root mean square error remains fixed at 3.54°. The high coherency of matrix Φ causes this method to be inaccurate in scenarios with closely spaced targets. Root-MUSIC and BPDN both are efficient for SNRs higher than 15 dB.

Running time investigation of the proposed algorithm, S-OMP, and BPDN are shown in Figs. 4 and 5 for *M* = 30, *L* = 70, and SNR = 30 dB. One can observe that, in contrast to S-OMP, the running time of the proposed algorithm increases negligibly with increase in *K*. Moreover, despite being insensitive for *K* > 5, *l*
_{2,1} minimization (BPDN) is a significantly slower algorithm, requiring 3 orders of magnitude longer times to resolve the targets. Figure 5 shows the running time against the number of snapshots. The results show once again that BPDN performed the worst requiring thousand-fold more time in comparison to the other two methods. The results also show the proposed method consumes less time to calculate SVD values than S-OMP because it deals with a reduced matrix size of *Y*
_{
red
} ∈ ℂ^{M × K} while S-OMP has to work with a larger matrix of *Y* ∈ ℂ^{M × L} .

The last simulation of this part is dedicated to the case of two closely spaced sources with varying angular separation. Here, the ULA employs 20 sensors, and the number of snapshots *L* = 100 and SNR = 5 dB. The angle between two sources varies from 1° to 30°. The result for proposed algorithm, S-OMP, and root-MUSIC is shown in Fig. 6.

### Nonuniform linear array

In this section, DOA estimation is investigated in NLA scenarios where low-angle tracking is required to discern targets that are close to each other. By using NLA configurations, the accuracy of angle measurements can be improved in comparison to uniform linear array setup with the same number of sensors. In other words, NLA configurations have a narrower beamwidth by using a wider aperture and therefore achieve a better DOA resolution.

If one or more sensors in a ULA malfunction, then the array can be considered an NLA. As popular and efficient methods such as root-MUSIC cannot be used in NLAs, we have applied the CS approach to various configurations of NLA problem. An NLA with aperture *M′* and *M* sensors has been denoted by \( {\mathrm{NLA}}_{M^{\hbox{'}}, M} \). For example, NLA_{30,20} denotes 20 sensor linear arrays with an aperture length 30 × λ/2. In Fig. 7, the sensors of a NLA_{5,3} are located at positions corresponding to the vector *p* = [0,2,5].

The steering vector associated with the array in Fig. 7 can be written as \( a\left({\mu}_i\right)=\left[{e}^{j2{\mu}_i}\ {e}^{j5{\mu}_i}\right] \) where \( {\mu}_i=-\frac{2\pi}{\lambda}\ \Delta \sin \left({\uptheta}_{\mathrm{i}}\right) \) and Δ represents the distance between sensors. Regarding this nonuniform structure, the distance between the elements of vector increases; consequently, the coherency of vector decreases, and the process of recovery become more efficient. To evaluate the performance of recovery algorithms, some simulation tests have been carried out considering the scenario in Fig. 3 with NLA_{60,30}. The results in Fig. 8 show that for this case, the aperture has doubled in length while the number of sensors remains the same. The figure shows that the overall performance of all CS-based methods has improved in comparison to results in Fig. 3 where the ULA with 30 sensors using S-OMP method attains an error of 3.54° at SNR = 30 dB, while with NLA_{60,30}, an error limit is 1.44° at the same SNR. The improvement is because when the linear array has a nonuniform arrangement, the lower mutual coherency between array manifold columns helps reconstruction method pick up correct DOAs from sampling grid. NLA could be considered as a sampling machine taking random spatial samples from arriving signals. Increasing randomness in [10] is reported to tighten RIP property. Furthermore, owing to the doubled aperture length in Fig. 8, the ability of the array to discern closely located DOAs has been enhanced.

The second experiment concerns the effect of aperture length on DOA estimation in NLA scenario. We compare ULA_{15} to four NLAs with the same number of active sensors. The experiment has been carried as a Monte Carlo simulation with three randomly chosen angles in each of the iterations. Figure 9 shows that by increasing the aperture, while the number of sensors remains the same, the array’s ability to resolve DOAs improves significantly. This is because the longer aperture size provides narrow beamwidth. From the CS perspective, a longer aperture in the NLA arrangement decreases the matrix mutual coherency since the sensors can be located far from each other.

The last experiment investigates the performance of the array with fixed aperture and variable number of sensors. The results are summarized in Fig. 10. While NLA_{60,5} with only 5 sensors fail to achieve a satisfactory result for SNR < 20 dB, the other two alternatives with 10 and 15 sensors exhibit desirable performance. Indeed, the more sensors we exploit, the more measurements we get; consequently, the sparse signal will be recovered more accurately even at low SNRs.