Principles of Modern Radar - Volume 2 1891121537

470 CHAPTER 10 Clutter Suppression Using Space-Time Adaptive ProcessingFIGURE 10-7MVDR spectrum ofeigenvectorscorresponding to thedominant subspace,plus one noisesubspaceeigenvector.Doppler Frequency (Hz)250200150100500−50−100MVDR Spectrum0−5−10−15dB−150−20−200−250−20 −10 0 10 20−25Angle (Degrees)sequence:x k =NM∑m=1e k (m)q k (m); e k (m) = q H k (m)x k (10.43)The basis expansion in (10.43) follows from the Karhunen-Loève transform (KLT) [2,9];e k (m) is the m-th Karhunen-Loève (KL) coefficient. Contrasting the KLT to the Fouriertransform underlies the notion that a KL basis is data dependent, while the Fourier basisis fixed.Figure 10-7 shows the minimum variance distortionless response (MVDR) spectrumof the first four eigenvectors of the covariance matrix leading to the PSD shown in Figure10-5 with the addition of receiver noise with an output variance of 1 watt. The MVDRspectrum is a super-resolution view of the signal space-time characteristics; it representsthe output of the space-time beamformer when using the weight vectorw k =R −1k s s−t( f sp , ˜f d )s H s−t( f sp , ˜f d )R −1k s s−t( f sp , ˜f d )(10.44)The beamformer output power for the space-time filter with weight vector given by(10.44) isP o ( f sp , ˜f d ) =1s H s−t( f sp , ˜f d )R −1k s s−t( f sp , ˜f d )(10.45)The first three eigenvectors, corresponding to the three largest eigenvalues, point towardthe three different signal sources in angle and Doppler. In general, for all eigenvectorslying in the dominant subspace (i.e., the subspace not occupied by thermal noise),q k (m) ∈ span (s 1 ,s 2 ,...,s P ) (10.46)