Principles of Modern Radar - Volume 2 1891121537

422 CHAPTER 9 Adaptive Digital BeamformingThen the covariance matrix has the form⎡⎤|S| 2 + σn2 |S| 2 e j2π f 0 d sin θc ··· |S| 2 e j2π f 0(M−1) d sin θc|S| 2 e − j2π f 0 d sin θc |S| 2 + σn 2 ··· |S| 2 e j2π f 0(M−2) d sin θcR x =.⎢ .... ⎣. ⎥⎦|S| 2 e − j2π f 0(M−1) d sin θc |S| 2 e − j2π f 0(M−2) d sin θc ··· |S| 2 + σn2(9.19)where σn 2 is the noise power. Notice that the noise contributes only along the diagonalof the covariance matrix. This is because the diagonal elements of the covariance matrixcorrespond to the autocovariance of each spatial channel, which is equal to the sum ofthe signal power and the noise power. The off-diagonal elements of the covariance matrixare due to plane wave sources and contain information about the angle of arrival. It is theinformation in these off-diagonal terms that is exploited by adaptive algorithms to canceljammers and by high-resolution angle estimation algorithms to locate sources.9.3.1 Wiener FilterIf the optimization criterion for the adaptive filter is to minimize the mean square error,then the resulting solution for the adaptive weights is known as the Wiener filter. Recallthat the error at the output of the beamformer can be written asε (t) = d (t) − y (t) = d (t) − w H x (t) (9.20)where d(t) is a reference signal that is highly correlated with the target signal of interest.The MSE is given byMSE = E [ |ε (t)| 2] = |d(t)| 2 − w H r dx − r H dx w + wH R x w (9.21)where R x is the covariance matrix andr dx = E [ d ∗ (t) x (t) ] (9.22)is the cross-correlation vector between the desired signal and the array data vector. Minimizingthe MSE with respect to the weight vector results in an optimal weight vectorgiven byw opt = R −1x r dx (9.23)This form of the Wiener filter is not typically used for radar applications because it willattempt to form beams in the direction of any signals that are correlated with the desiredsignal. This can result in highly distorted beampatterns when there is specular multipath orcoherent jamming present. To eliminate the need for the desired signal, consider the idealcase where the desired signal is set exactly equal to the target signal, s(t). Then assumingthat the target signal is uncorrelated with the jamming and the noise, the cross-correlationvector reduces tor dx = E [ S ∗ (t) x (t) ] = |S| 2 v (θ s ) (9.24)