Principles of Modern Radar - Volume 2 1891121537

428 CHAPTER 9 Adaptive Digital BeamformingFIGURE 9-28Generalized sidelobecanceller example.0−10GSC BeamformerSINR_a = 30.8 dBSINR_q = −7.7 dB−20Gain (dB)−30−40−50−60−80 −60 −40 −20 0 20 40 60 80AOA (Degrees)9.3.3.3 Beamformer Constraint DesignIn addition to point constraints, other types of constraints are possible to control the antennapattern response over extended angular regions. Two constraints of this type are derivativeconstraints and eigenvector constraints.Derivative Constraints As the name implies, a point constraint will fix the pattern responseat a specified point in the antenna pattern (e.g., the center of the mainbeam), butit in no way ensures that there will be a peak at that point. A derivative constraint is away to force an antenna pattern peak to be located in the constraint direction [26,27]. Thederivative constraint is implemented by adding the derivative of the steering vector withrespect to angle (evaluated at the constraint angle) as a column in the constraint matrixand adding a zero to the response vector to force a relative maximum in that direction.Note that a point constraint in the look direction is still necessary to maintain gain in thatdirection.[v(θ 0 ), ∂v (θ ] H [ ]0) 1w =∂θ 0} {{ } }{{}CgAnother application for using derivative constraints is in the difference pattern for monopulseto maintain the monopulse slope at the beam center. In this case, instead of setting thecorresponding value of the response vector to zero, it is set equal to the nominal value ofthe difference pattern slope at beam center. Also, for large arrays, derivative constraintsshould be used with caution because the derivative of the steering vector can become verylarge and cause the constraint matrix to lose rank and lead to numerical problems whencomputing the weights.Eigenvector Constraints One approach to constraining the antenna pattern over an angularregion (as shown in Figure 9-29) might be to place a number of point constraintsdistributed over the angles in that region. This tends to be a highly inefficient use of degreesof freedom, so a better approach is to employ eigenvector constraints [28]. The constraintsare computed by first forming a constraint covariance matrix, which is computed as the