Space/time/frequency methods in adaptive radar - New Jersey ...

804.2.1 Analysis of Fixed TransformsFor fixed T, Equation 4.19 can be rewritten:where pr is the reduced-rank CSNR,and Pb is the bias in the optimal SINR introduced by the transformation T,Equations 4.20-4.22 clearly demonstrate the effect of reduced rank transformationon the SMI-MVB method. The linear transformation T preserves the Gaussiandistribution of the data, hence the reduced-rank CSNR, Pr, has a beta distributionwith parameters K and r (i.e., the density in Equation 4.18 with N replaced byr). Improved statistical stability is evident in the higher CSNR values associatedwith Pr. For example, for the full-rank SMI, E [p] = 0.5 for a number of snapshotsK = 2N — 3 [15], while for the reduced-rank SMI, E [Pr] = 0.5 for K = 2r — 3, i.e.,fewer samples are required for the same performance level. The higher CSNR valuesdue to improved statistical stability, are somewhat offset by the bias term ph , which isthe loss in the optimal SINR due to the rank reduction for known covariance matrix.This loss is the quantity µ/µmax analyzed in the previous section. The density of pis thin 171v11 in terms of the density of Pr by the expression:It is interesting to note that the linearly constrained (Frost-type) SNRmaximization problem also has a rank reduction property and the correspondingCSNR has the same distribution as Equation 4.18 with N replaced by N n 1,where n is the number of linear constraints on the weight vector [54].