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In this paper, a unified formulation is made for the optimization of directivity and signal-to-noise ratio of an arbitrary array, with or without a constraint on the array Q-factor. When there is a constraint, the solution is reduced to that of a polynomial; when there is no constraint, the solution is given in a very simple form. First, it is shown that for a given array geometry there exists a finite permissible range of the Q-factor and this range reduces to zero for large spacings. Second, a detailed comparison between four well-known excitations (uniform, Hansen-Woodyard, optimum cophasal, and optimum) is made and the main results are as follows, 1) The Hansen-Woodyard excitation yields a directivity higher than that of the uniform only when the element spacing is somewhat smaller than a half wavelength (λ/2), but at the price of much higher Q. On the other hand, it is much lower than that of the optimum excitation. 2) For spacing less than λ/2, the optimum excitation is strongly tapered toward the ends of the array and approximately antiphasa (i.e., 0, 180° 0, 180°, ...); whereas for spacing greater than or equal to λ/2, it is nearly uniform and cophasal. 3) For large spacings, the directivity of uniform excitation is nearly optimum. For small spacings, the optimum directivity becomes much higher than all others, but is always associated with an enormously large Q-factor. Therefore in this case a constraint of the Q-factor is important. 4) Hansen-Woodyard and uniform arrays have the interesting property that their sensitivity factors are independent of spacing. The optimization of signal-to-noise ratio is also demonstrated. In particular, the result shows that although an improvement in gain over the uniform excitation is very difficult to realize in practice, a substantial improvement in signal-to-noise ratio is entirely practical. Other numerical results and some extensions of the theory to aperture antennas are also included.