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Adaptive weight processing The most demanding processing stage in STAP is to calculate and apply the adaptive weights vectors. The weight vectors are then used to form beams in beam forming. This process is described in Figure 3. Beam Mb • • • Beam 2 Range Doppler Channel CPI Form Beams (range/doppler) Matrix Factorization Back Substitution W * R = s Steering Vectors, s Beam 1 W = R -1 s Weights Range Doppler Doppler Filter Figure 3. Adaptive processing Calculating such adaptive weights may be performed in several ways. The two most commonly used methods to solve this equation in STAP applications are Cholesky and QR decomposition. Cholesky method With the Cholesky matrix method we first solve the following equation, M cov * V weight = V steer We define M cov = Z * Z H and compute M cov Solving system V weight = M cov -1 * V steer determines the upper tri- angular R. We then use forward and backward substitution to compute weight vector. In the Cholesky approach, the data samples (i.e. voltages) are multiplied together to form the elements of the covariance matrix. The units of these elements are 'power', and thus the numerical dynamic range of the elements (in dB) are double that of the elements in the original data. As a result, especially in the presence of interference signals having a high interference-to-noise ratio, the resulting covariance matrix can rapidly become ill-conditioned requiring double-precision arithmetic to obtain an accurate decomposition. The QR decomposition approach de- scribed in the following section avoids this problem by working directly with the data in the 'voltage' domain, rather than with the 'power' ele- ments of the covariance matrix. QR decomposition method With the QR decomposition method you would apply the QR decompo- sition directly to the complex data matrix to obtain the desired upper triangular factor matrix R. In this method we decompose space-time matrix MxN for some range cells. We then use forward and backward substitution to compute weight vector. Once we have the weight vector we apply it using vector-matrix multiplication. A drawback with the QR decomposition method is that it requires nearly twice as much computation as Cholesky but given the finite-precision arithmetic inherent in all digital processors, the QR decomposition meth- od is numerically more stable than the Cholesky method. 3 Using the QR decomposition method the number of operations required for weights computation and application is described in Table 3. The parameters used are K Doppler FFT size (power of 2), M independent non-overlapping range blocks, L channels, Q processing order, and N R contiguous range cells per weight computation. Table 3. Adaptive processing Functional Block Functional Block Nr Operations Adaptive processing Weights computation K * M * 8 * [L*Q]²*(N R +1) Weights application K * M * (8 * L* Q * N R ) A first-order Doppler-factored STAP (Q = 1) represents a first-order post- Doppler adaptive DPCA algorithm. This is a basic post-Doppler STAP for clutter and interference suppression. It can be effective but since it only operates in a single temporal degree of freedom (DOF) it is not a true STAP. The processing demand for this algorithm is modest. A third-order Doppler-factored STAP provides performance approaching a fully adaptive system. This approach efficiently supresses clutter and interference. The processing demand for this algorithm can be high, es- pecially for a high channel count. This is where we will focus this study. STAP parameters and processing requirements In order to better understand the computational load related to these algorithms this section analyses the effect of some of these. For this exercise we have assumed the parameters listed in Table 4. Table 4. STAP Parameters Parameter Name Value IF sampling rate [MHz] F s 5 Time per pulse, T P = T CPI / P CPI [us] T P 504 Time per CPI T CPI = 32.25 ms [ms] T CPI 32.25 Range = C / (2*PRF) [km] Range 75 Pulses per second, PRF = 1/T P [kHz] PRF 1984 Samples per CPI, N CPI = P CPI * N N CPI 122,880 Nr channels L 22 Nr pulses per CPI P CPI 64 Nr pulses per Doppler processing block P D 64 Samples per pulse before decimation N 1920 Decimation factor D 4 Samples per pulse after decimation N D 480 FIR filter length used in video-to-I/Q K a 36 FIR filter length used in array calibration K c 3 FIR filter length used in pulse compression K p 63 Convolution length in calibration and pulse compression R cp 192 FFT size (power of 2) used by overlap-save fast convolution R 256 Number of blocks in the overlap-save fast convolution method B = N D / R cp B 3 Doppler FFT Size (power of 2) K 64 Number of independent non-overlapping range blocks M 2 Nr contiguous range cells per weight computation N R 240 Processing order Q 3

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