## 6 Simulations  
In this section, we consider simulations for arrays with \(64\) and \(320\) sensor elements; the arrays are ULA with regular \(\lambda_{c}/2\) spacing between the sensor elements. The covariance matrix \(\hat{\mathbf{R}}\) is obtained by time-averaging recursions with \(N=1,\ldots,120\) snapshots. The DoA mismatch is also considered in order to verify the robustness of various beamforming algorithms. For the robust designs, we use the spherical uncertainty set, and the upper bound is set to \(\epsilon=140\) for \(64\) sensor elements and \(\epsilon=800\) for \(320\) sensor elements, respectively. There are four incident signals; while the first is the SoI, the other three signals' relative power with respect to the SoI and their DoAs in degrees are detailed in Table 1. The algorithms are trained with \(120\) snapshots, and the signal-to-noise ratio (SNR) is set to \(10\) dB for all of the simulations. In Figure 2, we compare various beamforming techniques with a steering array of 64 elements. We introduce a maximum of two degrees of DoA mismatch, which is independently generated by a uniform random generator in each simulation run. The proposed IOVP-RLS and IOVP-SG algorithms are implemented in both non-robust MVDR [2] and robust RCB [4] schemes, respectively. The competitors including two conventional full-rank beamformers, such as MVDR-RLS and RCB-RLS, as well as two reduced-rank beamformers, such as MVDR-Krylov and RCB-Krylov [14]. In this simulation, we select \(D=2\) for all reduced rank schemes, including MVDR-Krylov, RCB-Krylov, MVDR-IOVP-RLS/SG and RCB-IOVP-RLS/SG. A non-orthogonal Krylov projection matrix \(\mathbf{S}_{\mathbf{D}}[i]\in\mathbb{C}^{64\times 2}\) and a non-orthogonal IOVP rank reduction matrix are also generated for rank reduction. It is also important to note that the projection matrix \(\mathbf{S}_{\mathbf{D}}[i]\) can be initialized as \(\mathbf{S}_{D}[0]=[\mathbf{I}_{D}^{T}\), \(\mathbf{0}_{D\times(M-D)}^{T}]\), and the inverse of the covariance matrix \(\hat{\mathbf{R}}^{-1}[i]\) for each snapshot can be obtained by using the proposed RCB-IOVP-RLS algorithm. In Figure 3, we choose a similar scenario, but without DoA mismatch. We can see from the plots that the IOVP and Krylov algorithms have a superior SINR performance to other existing methods, and this is particularly noticeable for a reduced number of snapshots. By comparing the curves in Figures 2 and 3, \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Snapshots** & **Signal 1 (SoI)** & **Signal 2** & **Signal 3** & **Signal 4** \\ \hline
1 to 120 & 10/90 & 20/35 & 20/135 & 20/165 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Interference and direction of arrival (DoA) scenario, P(dB)relative to the desired User 1/DoA (degree). SoI, signal of interest. we can see that by introducing the DoA mismatch, the conventional MVDR-RLS and RCB-Krylov-RLS schemes have about a 10-dB SINR loss; their performances are prone to steering vector mismatch. In contrast, all of the proposed IOVP reduced-rank schemes experience less than 2.5 dB of performance loss, which implies that these schemes are robust to steering vector mismatch. On the other hand, by comparing the performance of their robust rivals (such as RCB-RLS, MVDR-Krylov-RLS), the proposed schemes may provide higher SINR performance and much higher convergence speed. In Figure 4, we compare the output SINRs of the Krylov and the proposed IOVP rank reduction technique using a spherical constraint in the presence of steering vector errors with \(320\) sensor elements. We assume a DoA mismatch with two degrees and four interferences with the profile listed in Table 1. With Krylov and IOVP rank reduction, the MVDR-Krylov, MVDR-IOVP, RCB-Krylov and RCB-IOVP have superior SINR performance and a faster convergence compared to their full-rank rivals. Figure 2: Signal-to-interference plus noise ratio (SINR) performance _vs_. the number of snapshots, with steering vector mismatch due to 2\({}^{\circ}\)DoA mismatch.