Principles of Modern Radar - Volume 2 1891121537

3.4 Optimum MIMO Design for Target Identification 103Multitarget Case Given L targets in general, we wish to ensure that the L-target responsespheres are maximally separated (an inverse sphere packing problem [22]). Toaccomplish this, we would like to jointly maximize the norms of the set of separations{‖d ij ‖|i = 1:L; j = i + 1:L}:maxsL∑L∑i=1 j=i+1∣∣d ′ ijd ij∣ ∣ (3.46)Since, by definition, d ij is given byd ij =(HTi − H Tj) s = Hij s (3.47)(3.46) can be rewritten asmaxss ′ ⎛⎝L∑L∑H ij ′ H iji=1 j=i+1⎞⎠ s = s ′ K s (3.48)Since K ∈ C N×N is the sum of positive semidefinite matrices, it shares this same property,and thus the optimum transmit input satisfiesK s opt = λ max s opt (3.49)EXAMPLE 3.5Multitarget IdentificationFigure 3-12 depicts the impulse responses of three different targets, two of which are thesame as in Example 3.4. Solving (3.48) and (3.49) yields an optimally separating waveformNormalized Voltage10.80.60.40.20−0.2−0.4−0.6−0.8Target 1Target 2Target 3FIGURE 3-12Target impulseresponses used forthe three-targetidentificationproblem.−10 5 10 15 20 25 30 35 40 45 50Time