Z-transform theory for linear array analysis

The author wishes to amplify Christiansen's recent communication [P.L. Christiansen, "On the closed form of the array factor for linear arrays," IEEE Trans. on Antennas and Prop. (Commun.), vol. AP-11, p. 198, March, 1963.] on the use of Z-transform theory for obtaining a closed form for the array polynomial of linear antenna arrays. He is convinced that his proposed approach provides a concise and convenient way for linear array analysis, and his new results will be useful. In his communication Christiansen stressed that the Z-transform approach does not require that the envelope function of the current distribution be periodic, nor is the approach limited to envelope functions with a linear phase relation. That a unit gate function could be used in cases where the amplitude envelope is not representable by a periodic function was pointed out previously by the present author [1960]. Christiansen gave the array factor of a very complicated amplitude distribution function without showing the intermediate steps. In an effort to clarify the situation, two examples are provided that may prove to be helpful to interested readers.