61(e) = u3 I #(e, dB) I -3.01~ n 12 + L n u , ] 4 9 [P(e>lP(0)12npde (11) where the weighting function w3 dB can be adjusted to yield suitable results. The usefulness of this method is somewhat limited, however, in that the value for
Onull[which must still be specified in (ll)] severely restricts the range over which 6 3 d B may be influenced.Finally, the beamwidth may be controlled by keeping the current amplitudes ni fixed a t some set of values determined in the optimization program, and then merely varying the spacing between rings. This causes expansion or contraction of the radiation pattern along the horizontal scale, thus altering the beamwidth.I t should be noted here that increasing the spacing k E may introduce large sidelobes in the B= 90° region (since optimization only took pjace over a certain "visible region" in real space). Therefore, it is recommended that this approach be used only for the case where wider beamwidths are desired (i.e., with decreased spacings).As a last point, it would be instructive to consider the Abstract-A method for finding the coefficients of an nth-order linear recursive digital filter, which gives the best least squares approximation to a desired pulse response over a finite interval, is presented. A relationship is derived between the approximating error corresponding to an optimal set of numerator coefficients and the error produced by an overdetermined set of linear equations, which is a function of the denominator coefficients only. This relation provides a computational algorithm for calcularing the optimal coefficients by iteratively solving weighted sets of linear equations in terms of the denominator coefficients only. Both theoretical and numerical results are presented. Also, bounds are found on the interval in which the norm of the optimum error must lie.