## 3 Blind Adaptive Beamforming Based on Direction Lock Loop
### DOA Tracking Based on DiLL  
Since the navigation signal is very weak and buried under the noise, the correlation accumulation with the locally copied pseudo-code signal is required to improve the signal-to-noise ratio. The correlation results can be modeled as  
\[\mathbf{r}=E[\hat{\mathbf{x}}(t)\cdot c_{L}^{*}(t-\hat{\tau})]\ =\ \mathbf{R}^{-1} \mathbf{a}(\theta_{0})r_{0}(\Delta\tau) \tag{10}\]  
where \(r_{0}(\Delta\tau)\) represents the correlation summation value between the received navigation signal and the local replica pseudo-code. \(c_{L}(t-\hat{\tau})\) is the local replica PRN code, and \(\Delta\tau=\tau_{0}-\hat{\tau}\) represents the estimation error of the time delay. The carrier is considered to be completely wiped off. From Equation (10), it can be seen that the correlation accumulation vector is closely related to the incident angle of the received signal. The array direction diagram is constructed by using this vector as the weight vector. Then, the gain is maximum in the direction of the signal incident. The normalized array pattern of the ULA is shown in Figure 2. The normalized spatial correlation function can be defined as  
\[R(\theta,\theta+\Delta\theta)=\frac{1}{N}\mathbf{a}^{H}(\theta+\Delta\theta) \mathbf{a}(\theta)=\frac{1}{N}\sum\limits_{n=1}^{N}e^{-j\pi(n-1)(\sin\theta- \sin(\theta+\Delta\theta))} \tag{11}\]  
where \(\left(\cdot\right)^{H}\) denotes the Hermitian transform and \(\Delta\theta\) is the direction bias. The array pattern is symmetrical around the central direction, so  
\[R(\theta,\theta+\Delta\theta)\approx R(\theta,\theta-\Delta\theta) \tag{12}\]  
The direction can be tracked as the same principle of the code DLL loop. Assuming that the estimated value of DOA at time \(k\) is \(\hat{\theta}_{k}\), the array guide vectors \(\mathbf{a}(\hat{\theta}_{k}-\Delta\theta)\) and \(\mathbf{a}(\hat{\theta}_{k}+\Delta\theta)\) of the left and right offset \(\Delta\theta\) of the current observed DOA are constructed. After correlating with the estimated signal vector, the spatial correlation functions can be obtained as  
\[z_{R}(k)=\frac{1}{N}\mathbf{a}^{H}(\hat{\theta}_{k}+\Delta\theta)\mathbf{r}= \frac{1}{N}r_{0}(\Delta\tau)\mathbf{a}^{H}(\hat{\theta}_{k}+\Delta\theta) \mathbf{R}^{-1}\mathbf{a}(\theta_{0})=r_{0}(\Delta\tau)R^{\prime}(\theta_{0},\hat{\theta}_{k}+\Delta\theta) \tag{13}\]  
\[z_{L}(k)=\frac{1}{N}\mathbf{a}^{H}(\hat{\theta}_{k}-\Delta\theta)\mathbf{r}= \frac{1}{N}r_{0}(\Delta\tau)\mathbf{a}^{H}(\hat{\theta}_{k}-\Delta\theta) \mathbf{R}^{-1}\mathbf{a}(\theta_{0})=r_{0}(\Delta\tau)R^{\prime}(\theta_{0},\hat{\theta}_{k}-\Delta\theta) \tag{14}\]  
where \(R^{\prime}(\theta_{1},\theta_{2})=\mathbf{a}^{H}(\theta_{2})\mathbf{R}^{-1} \mathbf{a}(\theta_{1})/N\) represents the normalized spatial correlation function after interference suppression. The discrimination error can be expressed as  
\[\varepsilon(k)=f[z_{R}(k),z_{L}(k)] \tag{15}\]  
where \(f(\cdot)\) denotes the discriminator function. The coherent DiLL and non-coherent DiLL are the two mainly used direction discriminating functions, which can be expressed as  
\[f_{coh}[R_{1},R_{1}]=\text{Re}[R_{1}-R_{2}] \tag{16}\]  
\[f_{noncoh}[R_{1},R_{2}]=\frac{|R_{1}|^{2}-|R_{2}|^{2}}{|R_{1}|^{2}+|R_{2}|^{2}} \tag{17}\]  
The S-curves of the coherent DiLL and non-coherent DiLL are shown in Figure 3, in which \(N=4,\ \Delta\theta=5^{\circ}\) and \(\theta=0^{\circ}\). As can be seen from Figure 3, the differential curve shows an approximate linear change near 0 degrees, so the DOA estimation error can be corrected by the discriminator. It also shown that the convergence of the non-coherent DiLL is larger than that of the coherent DiLL. After loop filtering, the direction discrimination error is sent to the beamforming device to update the DOA parameter value at the next time. Then, the estimated DOA value at time \(k+1\) is  
\[\hat{\theta}_{k+1}=\hat{\theta}_{k}+K_{0}\varepsilon(k)\otimes h(k) \tag{18}\]  
where \(h(k)\) stands for loop filter, \(\otimes\) represents a convolution operation.