## 3 Analysis
### Adaptive Null Steering Performance  
In general, the radiation patterns of the array antenna elements are constructively and destructively added with adequate weights to construct a beam-forming pattern at the wanted signal direction and a beam-nulling pattern at the interference signal direction. From this common principle of a null pattern formation, we build a simulation model for an antijamming system using a conventional PI algorithm in order to estimate a practical pattern nulling performance. In the algorithm, the signal received by the proposed two-element array can modeled as  
\[\overline{X}\ =\ \overline{A}_{S}\overline{S}+\overline{A}_{J}\overline{J}+ \overline{N}, \tag{10}\]  
where \(\overline{X}\) is a column vector of the received signal, while \(\overline{S}\) and \(\overline{J}\) are column vectors of the wanted and jamming signals. \(\overline{A_{s}}\) and \(\overline{A_{J}}\) are the steering vector matrices of the wanted and jamming signals. \(\overline{N}\) is the vector of the additive white Gaussian noise to model a typical communication environment. We then apply the conventional nulling technique of the PI algorithm to adjust the weight vector \(\overline{W}\) with the received signals in order to suppress the interference. The output signal is \(y=\overline{W}^{H}\overline{X}\), which is weighted  
Figure 14: Comparisons of the ECCs for three types of array antennas. Figure 13: Conceptual figures for the interelement spacing \(d\): (**a**) \(d=0.08\lambda\); (**b**) \(d=0.25\lambda\); (**c**) \(d=0.5\lambda\). and summed of all the received signals, and we use the linearly constrained minimum variance (LCMV) method as a regular criterion to obtain the optimum weight vector of \(\overline{W}_{opt}\) as follows [45; 46]:  
\[\overline{W}_{opt}\ =\ \frac{\overline{R}^{-1}\overline{h}}{\overline{h}^{ \overline{T}}\overline{R}^{-1}\overline{h}}, \tag{11}\]  
where \(\overline{R}=\) E[\(\overline{X}X^{H}\)] is the covariance matrix of the received signal, and \(\overline{h}=\) [1, 0,..., 0]\({}^{T}\) is the common constraint matrix. For the interference mitigation scenario, we first suppose a scenario that the average powers of the wanted, interference, and noise signals are set to \(-\)83 dBm, \(-\)3 dBm, and \(-\)43 dBm, respectively. It is also assumed that the interference of the chirp signal arrives at an elevation angle \(\theta_{jam}=\) 45\({}^{\circ}\) at the frequencies ranging from 1.45 GHz to 1.65 GHz. This interference signal is then mitigated by the nulling process using the optimum weight \(\overline{W}_{opt}\) to examine the null depth, width, and DOA error. \[D_{null}\ =\ \big{|}G_{opt}-G_{ini}\big{|}, \tag{12}\]  
\[\Omega_{null}\ =\ \big{|}\Omega_{high}-\Omega_{low}\big{|}_{gain}\ =\ -10\ \text{ dBi}^{\prime} \tag{13}\]  
\[\Delta\theta_{null}\ =\ \big{|}\theta_{opt}-\theta_{jam}\big{|}. \tag{14}\]  
The null depth \(D_{null}\) is obtained by the difference between the initial array gain \(G_{ini}\) before the nulling process and the final array gain \(G_{opt}\) after the PI algorithm when the system is exposed to the same chirp interference signals. The null width \(\Omega_{null}\) is achieved by the angular difference between two points (\(\Omega_{high}\), \(\Omega_{low}\)) in the optimized nulling pattern where the gain is \(-\)10 dBi. We calculate the null DOA error \(\Delta\theta_{null}\) from the difference between the optimal null angle \(\theta_{opt}\) and the actual jamming angle \(\theta_{jam}\). Figure 15a shows the optimized null pattern of the proposed array in the UV domain when the interference signal is at 45\({}^{\circ}\). Figure 15b illustrates the initial and final power spectrums, confirming that the chirp interference signal can be suppressed significantly by applying the optimum null weights for the proposed antenna. The optimum null pattern of the proposed array has the \(D_{null}\) of \(-\)47.7 dB with a null width \(\Omega_{null}\) of 33.2\({}^{\circ}\), as presented in Figure 15c.