Principles of Modern Radar - Volume 2 1891121537

3.5 Constrained Optimum MIMO Radar 105Magnitude (dB)100−10−20−30−40−50−60FIGURE 3-13Example of a linearlyconstrainedoptimization inwhich two interferersare removed via theprojectionoptimizationapproach.−70−80*−90−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5Normalized Angle*The keep-out constraints have the formwhere0 =[Gs] s′= I1s ′ sI 2⎡ ⎤1e j2π ¯θ Iks Ik =⎢⎣ .⎥⎦e j2π(N−1)¯θ Ik(3.56)(3.57)Figure 3-13 shows the resulting constrained optimum transmit pattern for the case where¯θ T = 0, ¯θ I1 =−0.25, ¯θ I2 = 0.4. As expected a peak is placed in the desired target directionwith nulls simultaneously placed in the keep-out directions.Case 3.2: Nonlinear ConstraintsIn practice other generally nonlinear constraints may arise. One family of such constraintsrelates to the admissibility of transmit waveforms, such as the class of constant modulusand stepped frequency waveforms [12], to name but a few.For example, if it is desired to transmit a waveform that is nominally of the LFM type(or any other prescribed type) but that is allowed to modestly deviate to better match thechannel characteristics, then the nonlinear constrained optimization has the formmax{s}∣ s ′ H ′ Hs ∣ ∣ (3.58)subject to:‖s − s LFM ‖ ≤ δ (3.59)