Principles of Modern Radar - Volume 2 1891121537

482 CHAPTER 10 Clutter Suppression Using Space-Time Adaptive ProcessingSetting g = 1 is known as the distortionless response and leads to the weight vector in(10.44) [9]. An expression for the output power of this beamformer is given by (10.45).We find that (10.77) is of the form w k = βR −1k s s−t( f sp , ˜f d ), whereβ =g ∗s H s−t( f sp , ˜f d )R −1k s s−t( f sp , ˜f d )(10.78)Hence, (10.77) likewise yields maximal SINR.As in Section 10.5.1, in practice the adaptive processor substitutes v s−t as a surrogatefor s s−t ( f sp , ˜f d ) and ˆR k as an estimate of R k .10.5.3 Generalized Sidelobe CancellerThe GSC is a formulation that conveniently converts the minimum variance constrainedbeamformer described in the preceding section into an unconstrained form [9,19]. Someprefer the GSC interpretation of STAP over the whitening filter-warped matched filterinterpretation of Section 10.5.1; since the GSC implements the MV beamformer for thesingle linear constraint [19], this structure likewise can be shown to maximize outputSINR. However, this approach has computational drawbacks: as the reader will see, theinterference-plus-noise covariance matrix is recomputed every time the target steeringvector—and, hence, the blocking matrix—changes.Figure 10-15 provides a block diagram of the GSC. The top leg of the GSC providesa quiescent response by forming a beam at the angle and Doppler of interest. A blockingmatrix, B G = null(s s−t ( f sp , ˜f d )), prevents the target signal from entering the lower leg;essentially, the blocking matrix forms a notch filter tuned to ( f sp , ˜f d ). With the desiredsignal absent from the lower leg, the processor tunes the weight vector to provide a minimalmean square error (MMSE) estimate of the interference in the top leg. In the final step,the processor differences the desired signal, d o/k , with the estimate from the lower leg,ˆd o/k , to form the filter output, y k = d o/k − ˆd o/k . Ideally, any large residual at the outputcorresponds to an actual target.The desired signal is given asThe signal vector in the lower leg isd o/k = s H s−t ( f sp, ˜f d )x k (10.79)x 0/k = B G x k ; B G ∈ C (NM−1)xNM ; x 0/k ∈ C (NM−1)x1 (10.80)Forming the quiescent response of (10.79) uses a single DoF, resulting in the reduceddimensionality of x 0/k . By weighting the signal vector in the lower leg, the GSC forms aFIGURE 10-15GSC block diagram.S s-t (f sp , f˜d )d o/kX k+ y k−X 0/kB d ˆ o/kG W MMSE/k