Principles of Modern Radar - Volume 2 1891121537

9.3 Adaptive Jammer Cancellation 423and plugging this into the optimum weight equation givesw opt = |S| 2 R −1x v (θ s) (9.25)So far, this version of the Wiener filter does not appear to have improved matters becausethe need to know the desired signal has been replaced only with a need to know the targetsignal power and angle of arrival. However, while a radar may not know a priori wherethe target signal is located, it does know where it is supposed to point the beam. Theremay or may not be a target at that location, but if θ s is replaced by the commanded beamlocation, θ 0 , then the adaptive weights will try to maintain gain in the commanded beamlocation while cancelling signals originating from any other direction. This results in thefollowing form of the Wiener filter:w opt = kR −1x v (θ 0) (9.26)where the signal power, |s| 2 , has been replaced with an arbitrary, nonzero scaling constantk because it has no impact on the output signal-to-interference-plus-noise ratio (SINR).9.3.2 Maximum SINRA more familiar choice of optimization criterion for the radar engineer is to maximizethe SNR or SINR. The signal power, P s , and the interference plus noise power, P i+n , aregiven byP s = ∣∣ w H sv (θ) ∣ ∣ 2 = |s| 2 w H v (θ) v H (θ)w = w H R s w (9.27)P i+n = w H R i+n w (9.28)and therefore by taking the ratio the SINR is equal toP s=∣ w H sv (θ) ∣ 2P i+n w H R i+n w(9.29)The optimum weight to maximize SINR can be found using a trick involving the Schwarzinequality. The first step is to insert R 1 2i+n R − 1 2i+n into the numerator∣ w H sv (θ) ∣ ∣ 2w H R i+n w∣ ∣ ∣∣∣∣ 1 1 ∣∣∣∣2sw H 2Ri+n R− 2i+n v (θ)= w H R i+n wThen employ the Schwarz inequality: |ab| 2 ≤ |a| 2 |b| 2(9.30)By setting a = w H R 1 2i+n and b = R − 1 2i+nv (θ), the SINR expression can be convertedinto the inequalityP sP i+n≤∣ ∣∣∣∣ 1|s| 2 w H 2Ri+n2 ∣ ∣ ∣∣∣ R − 1 ∣∣∣∣22i+n∣v (θ)w H R i+n wThe weight vector that achieves the bound is= |s| 2 v H (θ)Ri+n −1 v (θ) (9.31)w SNR = kRi+n −1 v (θ) (9.32)