Principles of Modern Radar - Volume 2 1891121537

10.5 STAP Fundamentals 479It is convenient to interpret the optimal space-time filtering operation in two steps:whitening and then warped matched filtering. If, for instance, we let R k = σn 2I NM, theoptimal weight vector simplifies to w k = s s−t ( f sp , ˜f d ), where we have chosen β = σn 2;we recognize the resulting filter as the matched filter. In the more general case, using theoptimal weight vector in (10.51) yieldsy k = w H k x k = s H s−t ( f sp, ˜f d )R −1k x k =(s H s−t ( f sp, ˜f d )R −1/2k)(R −1/2k x k)= s H x k (10.67)where s = R −1/2k s s−t ( f sp , ˜f d ) and x k = R −1/2k x k . The covariance matrix of x k isR k = E [ x k xkH ] −1/2 = R k E [ x k xkH ] −1/2 R k = R −1/2k R k R −1/2k = I NM (10.68)By assuming the form of an identity matrix, R k indicates the elements of x k are uncorrelated.White noise likewise has a covariance matrix equal to an identity matrix scaled bythe single-element power, σn 2I NM. Since R −1/2k whitens x k , R −1/2k is known as a whiteningfilter. The processor also applies the whitening transformation to the target signal. For thisreason, the matched filter must be modified accordingly; s is the corresponding warpedmatched filter. Substituting ˆR k for R k and v s−t for s s−t ( f sp , ˜f d ) in the adaptive case leadsto a similar interpretation.STAP is a data domain implementation of the optimal filter with weight vector givenby (10.66). In practice, both R k and s s − t ( f sp , ˜f d ) are unknown. The processor substitutesan estimate for each quantity to arrive at the adaptive weight vectorŵ k = ˆβ ˆR −1k v s−t (10.69)where ˆβ is a scalar, v s−t is a surrogate for s s−t ( f sp , ˜f d ), and ˆR k is an estimate of R k .This approach is known as sample matrix inversion (SMI) [15,16]. The surrogate steeringvector, v s−t , may differ from the exact steering vector due to slightly different steeringangle and Doppler, or as a result of system errors; the argument, ( f sp , ˜f d ), is dropped fromv s−t to acknowledge potential mismatch.Some selections for ˆβ are more useful than others. For example, it is shown in [17,18]that√ˆβ = 1/ vs−t H −1 ˆR k v s−t (10.70)yields a constant false alarm rate (CFAR) characteristic under certain operating characteristics.It is most common to calculate the space-time covariance matrix estimate as [15]ˆR k = 1 PP∑x m xm H (10.71)m=1{x m }m=1 P are known as secondary or training data. If the training data are independentand identically distributed (IID) and there is no steering vector mismatch (i.e., v s−t =s s−t ( f sp , ˜f d )), Reed, Mallett, and Brennan (RMB) showed L s,2 ( f sp , ˜f d ) is beta-distributedwith mean [15]E [ L s,2 ( f sp , ˜f d ) ] =(P + 2 − NM)(P + 1)(10.72)