# Robust Space Time Adaptive Processing Methods for Synthetic Aperture Radar
## 4 The Piecewise Constrained Generalized Sidelobe Canceller  
Element space processing is a method of implementing STAP in the spatial frequency domain, and STAP can also be implemented as a generalized sidelobe canceller (GSC), which is equivalent to what element space approaches under ideal conditions [28]. This section presents a modified method of applying piecewise constraints to the GSC (PC-GSC) based on the same idea as Section 3.  
The processing flow of the GSC is shown in Figure 3, two processing paths are formed with the main beam \(d(t)\) in the desired direction and the reference beam \(x_{a}(t)\) formed by a transformation of the main beam with a blocking matrix \(B\) to remove the desired signal from the original echo data. The fixed weight vector of the upper processing path is given by \(w_{s}=a(\theta_{0})\), where \(\theta_{0}\) is the DOA of the desired target direction.  
The signal \(x_{a}(t)\) in the lower processing path then goes through an adaptive filter \(w_{a}\) to minimize the total output power, and is then subtracted from the main beam \(d(t)\). To mitigate the spatially nonstationary interference in SAR, piecewise adaptive processing is employed. If the aperture time is partitioned into \(M\) data sub-apertures, each of which consists of \(Q\) pulses (\(M=P/Q\)), the adaptive  
Figure 3: Processing diagram of the generalized sidelobe canceller.  
weight vectors \(w_{a}\) (\(m\)), \(m\) = 1,2,\(\cdots\),\(\mathcal{M}\) for the lower processing path in the GSC can be updated in each sub-aperture. The adaptive weight vectors \(w_{a}\)(\(m\)) are designed to minimize the scalar mean square error (MSE) of the difference between the upper and the lower processing path, which is given by the following formulation [24].  
\[\min E\big{(}\big{|}d(t)-w_{a}^{H}(m)x_{a}(t)\big{|}^{2}\big{)} \tag{20}\]  
\(t=(m-1)Q+1\),...,\(mQ\) and \(m\) = 1,2,\(\cdots\),\(\mathcal{M}\). \(d(t)\) is the output of the upper processing path of the GSC and \(x_{a}(t)\) is the \(N_{a}\)-dimensional data vector for the lower processing path of the GSC.  
The weight vector of the upper processing path of the GSC structure is fixed. The adaptive weight vector \(w_{a}\) (\(m\)) in the lower processing path needs to be changed in each sub-aperture. To alleviate the temporal modulation introduced by the varying \(w_{a}\) (\(m\)), constraints on the adaptive weight vectors \(w_{a}\) (\(m\)), \(m\) = 1,2,\(\cdots\),\(\mathcal{M}\) are required. Similar to (18), we define the vector as:  
\[C_{m}=\big{[}x(Q(m-1)-L/2+1),\cdots x(Q(m-1)+L/2)\big{]}^{H} \tag{21}\]  
Then the following constraints on the adaptive weight vector are used:  
\[w_{a}^{H}(m)C_{m}=w_{a}^{H}(m-1)C_{m}\ m=2,3,\cdots,M \tag{22}\]  
The adaptive weight vector \(w_{a}\) (\(I\)) in the first sub-aperture is calculated using (11) without any constraint. The solution to (20) is  
\[w_{a}(m)=R_{xx}^{-1}(m)r_{xd}(m)-(C_{m}^{H}R_{xx}^{-1}(m)C_{m})^{-1}\big{[}R_ {xx}^{-1}(m)r_{xd}(m)-w_{a}(m-1)\big{]} \tag{23}\]  
\(R_{xx}(m)\) is the spatial covariance matrix of the lower processing path of the GSC, and \(r_{xd}\) (\(m\)) is the cross-correlation vector between the lower and the upper processing path of the GSC.