# Robust Space Time Adaptive Processing Methods for Synthetic Aperture Radar
## 3 The Piecewise Constrained STAP Based on Sub-Apertures Framework  
The synthetic aperture time may be partitioned into \(M\) sub-apertures containing \(Q\) pulses (\(M=P/Q\)) and the adaptive weight vectors \(w_{m}\) for m = 1,2,\(\cdots\),\(M\) can be updated in each sub-aperture. The essence of the algorithm is to update the adaptive weight vector by subdividing the aperture to adapt to the changes of interference spatial non-stationarity, considering the DOA is stationary in the non-overlapping sub-apertures. In conventional sub-aperture processing, the adaptive weight vector of each sub-aperture is calculated according to (8):  
\[w_{m}=\left[v^{H}(\theta)R_{m}^{-1}v(\theta)\right]^{-1}\!R_{m}^{-1}v(\theta) \tag{10}\]  
where \(R_{m}\) is the covariance matrix obtained from the interference samples in sub-apertures, \(w_{m}\) is the adaptive weight vector of each sub-aperture and is used to beamform the multi-channel echo data of each sub-aperture respectively. It should be noted that the training data used to estimate the covariance matrix must be target-free in order to avoid the self-cancellation of the target itself [26; 27]. A low repetition frequency waveform is usually used in SAR, which means that the range is not ambiguous, so the training data can be sampled before the effective echo reaches the radar receiver in the fast time domain to obtain the target-free samples. Moreover, in order to estimate the covariance matrix accurately, it is necessary to have an adequate number of training data. In SAR, the sampling rate is relatively high. Therefore, it is easy to obtain enough interference samples to estimate the covariance matrix. The disadvantage of this method is discussed in Section 2. With different adaptive weight vectors in each sub-aperture, although the beam center of the mainlobe can be maintained by the constraint in the look direction, there are still variations in the other parts of the patterns in the mainlobe. The deformation of the antenna patterns would lead to the undermining of phase consistency in the slow time domain between the sub-apertures, which affect the azimuth resolution of the SAR images. Therefore, a sub-aperture STAP algorithm with data constraint is proposed. In this method, the adaptive weight vectors are calculated in each sub-aperture respectively, and the phase consistency in the slow time domain can be maintained by data constraints, so the azimuth resolution of SAR can be ensured while effectively suppressing the time-varying interference. In the first sub-aperture, the adaptive weight vector is obtained by the conventional method according to (10), and then the weight vector \(w_{1}\) can be used to beamform the array echo in the first sub-aperture. \[w_{1}=\left[v^{H}(\theta)R_{1}^{-1}v(\theta)\right]^{-1}\!R_{1}^{-1}v(\theta) \tag{11}\]  
In the second sub-aperture, the data constraint condition is added, which permits the vector to change for interference cancellation while guaranteeing that it would not affect the beamform output of the target echo in the second sub-aperture. In order to reduce the amount of computational complexity, in practice, only \(L\) pulses at the junction of two sub-apertures need to be selected to constrain the consistency of the echo. Then the constrain can be described as follows [23]:\[w_{1}^{H}x(t)=w_{2}^{H}x(t)\,t=Q-L/2+1\cdots Q+L/2 \tag{12}\]  
It can be approximately considered that \(w_{1}^{H}x(t)\approx w_{1}^{H}s(t),w_{2}^{H}x(t)\approx w_{2}^{H}s(t)\). Because the range bin numbers of each pulse are more than thousands in SAR, only range bins containing strong feature points need to be selected as constraints to reduce the computational complexity. The detection of range bins containing strong feature points can be performed in the fast time domain after the pulse compression process.