Principles of Modern Radar - Volume 2 1891121537

432 CHAPTER 9 Adaptive Digital Beamformingenough to cover all the noise eigenvalues without eclipsing any eigenvalues due to jammers.This is because the higher the diagonal loading level, the less jammer cancellationis achieved.9.3.4.3 Cholesky FactorizationThe adaptive weight estimation problem can be formulated as solving a system of linearequations.R x w = vCholesky factorization is a computationally efficient way of solving this problem inO(M 3 /3) operations [32]. The first step is to decompose the covariance matrix asR x = LDL Hwhere L is a lower triangular matrix, and D is a diagonal matrix. Software libraries forperforming this decomposition are widely available [33]. Using this decomposition of thecovariance matrix, the system of equations can be rewritten asLy = vwhere y = DL H w. Since L is a triangular matrix, and v is the known steering vector,it is easy to solve for y by applying back substitution. Once y is known, the weights arecomputed by solving the second triangular system using back substitution.L H w = D −1 yThe covariance based methods for estimating the adaptive weights have the disadvantagethat they involve squaring the data. This squaring causes an increase in dynamic range thatcan lead to numerical problems. If computing precision is limited, then there are methodsusing singular value decomposition or QR decomposition that solve for the adaptiveweights directly from the data without squaring it [34].9.3.5 Performance MetricsPerformance metrics are used to assess the effectiveness of the adaptive filter in cancellinga given jamming scenario. This section will define the most common metrics used forevaluating adaptive cancellation performance. Unfortunately, some of these metrics canbe precisely computed only in simulation and can be estimated or inferred only from hardwaremeasurements. Most of the performance metrics involve taking the ratio of variouscombinations of the signal, jammer, and noise powers with and without adaptive cancellation.Let S u ,J u , and N u be the unadapted signal, jammer, and noise powers, respectively,and S a ,J a , and N a , be the adapted signal, jammer, and noise powers, respectively:• N u = unadapted noise only• S u + N u = unadapted signal plus noise• J u + N u = unadapted jammer plus noise• J a + N a = adapted jammer plus noise• S a + J a + N a = adapted signal plus jammer plus noise