Principles of Modern Radar - Volume 2 1891121537

3.4 Optimum MIMO Design for Target Identification 101whered = y 1 − y 2= H T1 s − H T2 s= ( H T1 − H T2) s= Hs(3.42)and whereSubstituting (3.42) into (3.41) yieldsH = H T1 − H T2 (3.43)max{s}∣ s ′ H ′ Hs ∣ ∣ (3.44)This is precisely of the form (3.10) and thus has a solution yielding maximum separationgiven by( H ′ H ) s opt = λ max s opt (3.45)It is noted that (3.45) has an interesting interpretation: s opt is that transmit input thatmaximally separates the target responses and is thus the maximum eigenfunction of thetransfer kernel H ′ H formed by the difference between the target transfer matrices (i.e.,(3.43)). Again if the composite target transfer matrix is stochastic, H ′ H is replaced withits expected value E { H ′ H } in (3.45).EXAMPLE 3.4Two-Target Identification ExampleLet h 1 [n] and h 2 [n] denote the impulse responses of targets #1 and #2, respectively (Figure 3-9).Figure 3-10 shows two different (normalized) transmit waveforms—LFM and optimum(per (3.46))—along with their corresponding normalized separation norms of 0.45 and 1,Normalized Votage10.80.60.40.20−0.2Target 1Target 2FIGURE 3-9Target impulseresponses utilizedfor the two-targetidentificationproblem.−0.4−0.6−0.80 5 10 15 20 25 30 35 40 45 50Time