On the Conjugate Gradient Matched Filter

Abstract

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 ( +1)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 ( +1)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.Index Terms-Conjugate gradient method, Krylov subspace, matched filter, space-time adaptive processing (STAP).

Fast image inpainting using exponential-threshold POCS plus conjugate gradient

“…[24][25][26] The main advantage of the CG is that it can minimise the objective function with a limited number of iterations, which is not bigger than the matrix rank. However, the non-local regularisation solved by the CG, named as CG for simplicity, is not good at reconstructing big area of missing pixels.…”

“…Then, this result is taken as an initial input for other methods that are good at recovering the detailed information. The nonlocal restoration 20-23 solved with the conjugate gradient (CG) [24][25][26] is selected for the purpose of refining the high frequency components, since it is powerful in reconstructing the details. The main drawback of this method is that it requires many more iterations to reconstruct wide areas of missing pixels (i.e.…”

Conjugate directions based order recursive implementation of post-Doppler adaptive target detectors

“…Recently, some methods utilising the conjugate gradients for the reduced-rank filtering have been proposed [17,18]. The methods based on the conjugate gradients are known to produce a solution in the Krylov subspace generated by the clutter and noise covariance matrix (R cn matrix) and the steering vector.…”

“…Different from these techniques, we utilise the 'conjugate directions' to solve for the optimal coefficients in the reduced dimensional space. In contrast to the conjugate gradient-based methods, such as [17], the 'conjugate directions' based methods present a solution in a desired subspace which is not necessarily a Krylov space. Recently, the constant false alarm property of the conjugate gradient filters along with some other properties, such as their optimality in the residing Krylov space, have been shown in [17].…”

“…In contrast to the conjugate gradient-based methods, such as [17], the 'conjugate directions' based methods present a solution in a desired subspace which is not necessarily a Krylov space. Recently, the constant false alarm property of the conjugate gradient filters along with some other properties, such as their optimality in the residing Krylov space, have been shown in [17]. The equivalence of the conjugate gradient filters (with a proper initialisation) and the multi-stage Wiener filters has also been shown in [19].…”