Principles of Modern Radar - Volume 2 1891121537

10.3 Space-Time Properties of Ground Clutter 473zyxk( ϕ,θ(FIGURE 10-9Geometry forspace-time clutterpatch calculation(after [8], c○ 2003IEEE)Iso-RangeClutter PatchR a (m,n; k) = σc/m,n;k 2 1 MN; 1 MN is the Hadamard identity, an NM × NM matrix filled withones, and σc/m,n;k 2 represents the clutter power for the m − n-th patch at the k-th range.Under such circumstances, (10.49) simplifies toN aR c/k = E [ c k ckH ] ∑N c∑= σc/m,n;k 2 s s−t( f sp/m,n , ˜f d/m,n ; k)ss−t H ( f sp/m,n, ˜f d/m,n ; k)m=1 n=1(10.50)Since ground clutter at a given angle has a specific Doppler frequency dependent onthe projection of the platform velocity vector onto the direction vector pointing at theclutter patch (i.e., v p · k(φ m,n ,θ m,n ), as seen from (10.48)), clutter resides on a ridgewhen viewing the angle-Doppler power spectrum [4-8]. Specifically, each clutter patchyields a sinc-like response in the angle-Doppler domain—similar to the responses shownin Figure 10-5—and thus the clutter contribution over all azimuth and at a particulariso-range traces out a ridge. In the case of a perfectly sidelooking array (SLA), andbarring any angle or Doppler aliasing, the angle-Doppler clutter ridge assumes an S-shaped structure. The clutter patch at broadside exhibits a response at zero angle andzero Doppler, while points at azimuths toward the direction of flight (positive spatialfrequencies) exhibit increasingly positive Doppler frequency, and points away from thedirection of flight have negative spatial and Doppler frequencies. The forward-lookingarray (FLA) represents the extreme situation from the SLA. When considering an FLA,the clutter ridge exhibits increased curvature and symmetry, since clutter at two pointson either side of the direction of flight have equal but opposite spatial frequencies yetidentical Doppler frequency.In [8], we validate the clutter model of (10.47) using measured data taken fromthe Multi-Channel Airborne Radar Measurements (MCARM) Program [12]. Specifically,using auxiliary variables characterizing the platform attitude and radar settings, we used(10.47) to simulate clutter data and then compared the results against the MCARM data.Overall, we show extraordinary correspondence between measured and simulated data in[8,13], thereby justifying the form given in (10.47).