Sufficient conditions for the existence of optimum beam patterns for unequally spaced linear arrays with an example

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Dolph's method for determining the optimum element currents for half-wavelength equispaced discrete linear arrays is generalized to symmetric discrete linear arrays. The theorem proved gives sufficient conditions for the existence of optimum beam patterns for arrays with elements symmetrically positioned about the array center, but with fixed unequal spacings between the elements. The conditions are such that the Remes exchange algorithm for minimax approximation of functions can be employed to compute the optimum element currents corresponding to an optimum beam pattern directly from the given spacings of the elements. Half-wavelength spaced linear arrays satisfy the conditions of the theorem; therefore, it provides a new method of calculating the well-known Dolph-Chebyshev element currents. An example with unequal spacings is included to show the utility of the method even when the hypotheses of the theorem may not be met.