On a class of optimum aperture distributions for pattern shaping

A class of optimum aperture distributions that gives the best mean-square approximation with weight factor (
1 - \eta^{2})^\alpha
to any given pattern space factor for an arbitrarily prescribed value of a generalized superdirectivity ratio
\gamma_{\alpha}
is derived for all
\alpha > -1
. For
\alpha = 0
the parameter
\gamma_{\alpha}
becomes Taylor's superdirectivity ratio
\gamma
and the
\gamma_{\alpha}
-constrained solution reduces simply to a
\gamma
- constrained solution published earlier. But for
\alpha = -\frac{1}{2}, \frac{1}{2}
, or 1 the solution becomes the best, or near best,
Q
-constrained solution in the sense of least radiated power in the error pattern for the case of
E
-plane strip source antennas or of
H
-plane strip or line-source antennas, respectively. Thus, for strip or line source antennas the new solution possesses an important physical interpretation that the earlier solution did not.