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Analysis of the conjugate gradient matched filter

We consider the conjugate gradient (CG) algorithm for the calculation of the weight vector of the optimum matched filter (MF). As an iterative algorithm, it produces a series of approximations to the optimum MF weight vector, each of which can be used to filter the test signal and form a test statistic. This effectively leads to a family of detectors, referred to as the CG-MF detectors, which are indexed by k the number of iterations incurred. We first consider a general case involving an arbitrary covariance matrix of the disturbance (including interference, noise, etc.) and show that all CG-MF detectors attain constant false alarm rate (CFAR) and, furthermore, are optimum in the sense that the k-th CG-MF detector yields the highest output signal-to-interference-and-noise ratio (SINR) among all linear detectors within the k-th Krylov subspace. We then consider a structured case frequently encountered in practice, where the covariance matrix of the disturbance contains a low-rank component (rank-r) due to dominant interference sources, a scaled identity due to the presence of a white noise, and a perturbation component containing the residual interference and/or due to the estimation error. We show that the (r + 1)-st CG-MF detector achieves CFAR and an output SINR nearly identical to that of the optimum MF detector which requires complete iterations of the CG algorithm till reaching convergence. Hence, the (r + 1)-st CG-MF detector can be used in place of the MF detector for significant computational saving when r is small.