Principles of Modern Radar - Volume 2 1891121537

98 CHAPTER 3 Optimal and Adaptive MIMO Waveform Designarbitrary, it does provide a mechanism for controlling null depth, that in practice is limited bythe amount of transmit channel mismatch [20]. Note the presence of transmit antenna patternnulls in the directions of the competing targets as desired.EXAMPLE 3.3Optimal Pulse Shape for Maximizing SCRIn this simple example, we rigorously verify an intuitively obvious result regarding pulse shapeand detecting a point target in uniform clutter: the best waveform for detecting a point target inindependent and identically distributed (i.i.d) clutter is itself an impulse (i.e., a waveform withmaximal resolution), a well-known result rigorously proven by Manasse [21] using a differentmethod.Consider a unity point target, arbitrarily chosen to be at the temporal origin. Its correspondingimpulse response and transfer matrix are respectively given byandh T [n] = δ[n] (3.30)H T = I N×N (3.31)where I N×N denotes the N × N identity matrix. For uniformly distributed clutter, the correspondingimpulse response is of the formh c [n] =∑N−1k=0˜γ k δ[n − k] (3.32)where ˜γ i denotes the complex reflectivity random variable of the clutter contained in the i-thrange cell (i.e., fast-time tap). The corresponding transfer matrix is given by⎡⎤˜γ 0 0 0 ··· 0˜γ 1 ˜γ 0˜H c =˜γ 2 ˜γ 1 ˜γ 0(3.33)⎢⎣....⎥⎦˜γ N−1 ˜γ N−2 ˜γ N−3 ··· ˜γ 0Assuming that the ˜γ i values are i.i.d., we haveE { ˜γ ∗i ˜γ j}= Pc δ[i − j] (3.34)and thus{ [E ˜H c ′ ˜H] } {0, i ≠ jc =i, j (N + 1 − i)P c , i = j(3.35)where [] i, j denotes the (i, j)-th element of the transfer matrix. Note that (3.35) is also diagonal(and thus invertible), but with nonequal diagonal elements.Finally, substituting (3.31) and (3.35) into (3.26) yieldsE { ˜H ′ c ˜H c} −1s= λs (3.36)