On the Conjugate Gradient Matched Filter

The conjugate gradient (CG) algorithm is an efficient method for the calculation of the weight vector of the matched filter (MF). As an iterative algorithm, it produces a series of approximations to the 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 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 th CG-MF detector yields the highest output signal-to-interference-and-noise ratio (SINR) among all linear detectors within the 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- ) due to dominant interference sources, a scaled identity due to the presence of a white noise, and a perturbation component containing the residual interference. We show that the st CG-MF detector achieves CFAR and an output SINR nearly identical to that of the MF detector which requires complete iterations of the CG algorithm till reaching convergence. Hence, the st CG-MF detector can be used in place of the MF detector for significant computational saving when is small. Numerical results are presented to verify the accuracy of our analysis for the CG-MF detectors.