**4. Simulation**

in [33].

 =

In this section, a half-wavelength spacing ULA with *M* = 10 was considered. We assumed that there existed an SOI impinging from the direction of *θ*0 = 3◦ and two interferences impinging from *θ*1 = −35◦ and *θ*2 = 42◦. The additive noise was presumed to be a complex circularly symmetric Gaussian zero-mean unit-variance spatially and temporally white process. All these sources were narrowband and assumed to be independent to the noise. To obtain each output SINR point, 200 Monte Carlo trials were used in each simulation. The proposed method was compared with the RAB method based on INCM reconstruction and steering vector estimation (INCM-SVE) in [6], subspace-decomposition and SV adjustment (SDA) in [30], MEPS-IPNC in [31], INCM reconstruction via orthogonality of subspace (INCM-OS) in [32], desired signal eigenvalue replacement (DSEB) in [33], SPCMR in [34] and INCM reconstruction via the intersection of subspaces (INCM-IS) in [35]. For all methods involved in the comparisons, the angular region was presumed to be **Θ** = **Θ***s* = (*<sup>θ</sup>*0 − 6◦, *θ*0 + 6◦) and **— Θ** = **<sup>Θ</sup>***p* = (−90◦, *θ*0 − 6◦) (*<sup>θ</sup>*0 + 6◦, <sup>90</sup>◦). and the interference angular region to be **Θ***i* = (*θi* − 6◦, *θi* + <sup>6</sup>◦). The number of non-dominant eigenvectors of the matrix *C* was set as *L* = 7, and the RCB boundary was = √0.1 in [6]. The*N*7dominanteigenvectorsofmatrix*B*employedfor*B*1in[32].ξ0.95

 were

 =  as

 points

*L* = 5M and S = 20 were as in [31]. The scale factor μ = 0.1 and the sampling points *N* = 2μ/0.01 were as in [30]. For the proposed method, the amplitude and phase mismatch boundary were set as = 0.3 and *Φ* = 6◦, respectively, the depth of the iteration was set as depth = 10, and the saturation value was *ϕ* = 0.05. Furthermore, the MATLAB CVX toolbox was used to solve the QCQP optimization problem in [6]. In our simulations, the optimal output SINR can be calculated by:

$$SINR\_{opt} = \sigma\_0^2 a\_0^H \mathcal{R}\_{i+n}^{-1} a\_0. \tag{22}$$

#### *4.1. Example 1: Mismatch Due to the Amplitude and Phase Error of the SV*

In the first example, the influence of the SVs with arbitrary amplitude and phase errors on the beamformer output SINR was considered. The relationship between the *m*th element of the nominal SV and the actual SV was modeled as *am* = *<sup>α</sup>mam<sup>e</sup>jβ<sup>m</sup>* , where the arbitrary amplitude error, *αm*, and phase error, *β<sup>m</sup>*, on each array sensor, respectively, followed the Gaussian distribution *N*1, 0.05<sup>2</sup> and *N* 0,(5◦) 2 [6]. Figure 3a depicts the output SINR of the tested methods versus the input SNR for the fixed number of snapshots *K* = 100. It was observed that the proposed method had a similar performance among the tested methods except in [32,33] at high SNRs. In addition, the performance of the proposed method was only lower than in [6] when the SNR was low. However, the computational complexity of our method was obviously lower than that in [6]. In Figure 3b, the output SINRs are shown versus the number of snapshots for the fixed *SNR* = 30 dB and *INR* = 20 dB. The proposed method had a similar performance to the tested methods in [6,30,34,35], and the number of snapshots did not affect the performance of our proposed method.

**Figure 3.** Output SINRs in the case of amplitude and phase errors versus (**a**) input SNR with *K* = 100; (**b**) the number of snapshots with *SNR* = 30 dB.

#### *4.2. Example 2: Mismatch Due to the Random Look Direction Error*

In the second example, the influence of the random look direction errors on the beamformer output SINR was considered. Assuming that the look direction mismatch of both the SOI and interferences were uniformly distributed in ( <sup>−</sup>5◦, <sup>5</sup>◦). That is means that the DOA of the SOI was uniformly distributed in ( <sup>−</sup>2◦, <sup>8</sup>◦), and the DOAs of the two interference were uniformly distributed in ( <sup>−</sup>40◦, 30◦) and (37◦, <sup>47</sup>◦). Note that the random DOAs of the SOI and interferences changed in each trial while remaining constant over snapshots. Figure 4a shows the output SINRs of the tested methods versus the input SNRs with the fixed snapshots *K* = 100. It was observed that our proposed method was only inferior to that in [6] in the performance at low SNR and inferior to that in [6,35] at high SNRs. Figure 4b depicts the output SINRs of the tested methods against the snapshot number at *SNR* = 30 dB and *INR* = 20 dB. It was observed that the performance of our proposed method was similar to that in [30] when *K* > 40. In addition, the methods in [33,34] were significantly affected by mismatches due to the look direction error.

**Figure 4.** Output SINRs in the case of the look direction error versus (**a**) input SNRs with *K* = 100; (**b**) the number of snapshots with *SNR* = 30 dB.

#### *4.3. Example 3: Mismatch Due to the Incoherent Local Scattering Error*

(a)

In the third example, the influence of the incoherent local scattering error on the beamformer output SINRs was considered. The SOI was assumed to have a time-varying signature, which was modeled as:

$$\mathfrak{X}\_{\mathfrak{s}}(k) = \mathfrak{s}\_{0}(k)\mathfrak{a}\_{0} + \sum\_{p=1}^{4} \mathfrak{s}\_{p}(k)\overline{\mathfrak{a}}(\theta\_{p}),\tag{23}$$

(b)

where *a*0 denotes the SOI SV. *<sup>a</sup><sup>θ</sup>p*(*<sup>p</sup>* = 1, 2, 3, 4) denotes the incoherent scattering signal SV, and the DOAs, *<sup>θ</sup>p*, are independently distributed in a Gaussian distribution drawn from a random generator *<sup>N</sup>*(*<sup>θ</sup>*0, 4◦) in each trial. *sp*(*k*) are independently and identically distributed zero-mean complex Gaussian random variables independently drawn from a random generator, *N*(0, <sup>1</sup>). In addition, *θp* changes from trial to trial, while it remains fixed over the samples. At the same time, *sp*(*k*) changes both from trial to trial and from sample to sample. In this case, the SOI covariance matrix is no longer a rank-one matrix and the output SINR should be expressed as [5]:

$$SINR\_{opt} = \frac{w^H \mathbf{R}\_{\mathcal{l}} w}{w^H \mathbf{R}\_{\mathcal{l}+\mathcal{U}} w}. \tag{24}$$

The optimal weight vector can be obtained by maximizing the SINR [5]:

$$\mathfrak{w}\_{opt} = P\left\{ \mathbf{R}\_{i+n}^{-1} \mathbf{R}\_{\mathfrak{s}} \right\},\tag{25}$$

where *P*{·} denotes the principal eigenvector of a matrix. Figure 5a shows the output SINRs of the tested methods versus the input SNRs with the fixed snapshots *K* = 100. It was observed that the performance of our proposed method was similar to that in [6,30] at high SNRs and only lower than in [33] at low SNRs. However, the method in [33] had severe performance degradation at high SNRs. Figure 5b depicts the output SINRs of the tested methods against the snapshot number at *SNR* = 30 dB and *INR* = 20 dB. It was observed that the proposed method had a similar performance with the tested methods in [6,30,35], and the number of snapshots did not affect the performance of our proposed method.

**Figure 5.** Output SINRs in the case of incoherent local scattering error versus (**a**) input SNRs with *K* = 100; (**b**) the number of snapshots with *SNR* = 30 dB.

#### *4.4. Example 4: Mismatch Due to the Coherent Local Scattering Error*

(a)

In the fourth example, the influence of the coherent local scattering mismatch on the beamformer output SINRs was considered. The coherent local scattering mismatch usually occurs in multipath propagation scenarios. Assume that the SOI is distorted by local scattering and consists of four coherent paths; the actual SV is formed as:

$$\mathfrak{a}\_0 = \mathfrak{a}\_0 + \sum\_{p=1}^4 \mathfrak{e}^{j\Phi\_p} \overline{\mathfrak{a}}(\theta\_p),\tag{26}$$

(b)

where *a*0 denotes the SOI SV. *<sup>a</sup><sup>θ</sup>p*(*<sup>p</sup>* = 1, 2, 3, 4) denotes the coherent signal path from *<sup>θ</sup>p*. *θp* are independently distributed in a Gaussian distribution drawn from a random generator, *<sup>N</sup>*(*<sup>θ</sup>*0, <sup>4</sup>◦), in each trial. *<sup>Φ</sup>p* denotes the path phase and uniformly distributed in (0, <sup>2</sup>*π*) from trial to trial. *θp* and *<sup>Φ</sup>p* change from trial to trial, while it remains fixed over the samples. Figure 6a shows the output SINRs of the tested methods versus the input SNRs with the fixed snapshots *K* = 100. It was observed that the performance of the optimal beamformer had an approximately 6 dB increment in output SINRs due to the extra paths. The performance of our proposed method was similar to that in [6,30,35] at high SNRs. The method in [6] achieved the best performance at the cost of the highest complexity compared with the others. Figure 6b depicts the output SINRs of the tested methods against the snapshot number at *SNR* = 30 dB and *INR* = 20 dB. It was observed that the proposed method had a small impact on the number of snapshots.

**Figure 6.** Output SINRs in the case of coherent local scattering error versus (**a**) input SNR with *K* = 100; (**b**) the number of snapshots with *SNR* = 30 dB.
