Granularity of beam positions in digital phased arrays

Many phased arrays are designed to use digital or digitalized phase shifters in which the phase shift varies in discrete steps rather than continuously. In such cases, the beam can be steered only in discrete steps. Granularity, defined as the finest realizable increment between adjacent beam positions, is an important parameter in the description of the performance of a phased array antenna using digital phase shifters. Two approaches to the problem of determining granularity are developed in this paper. The first is based upon the expansion by its moments of complex aperture distribution functions; the second is based upon approximating the actual discrete phase distribution by a linear continuous one. Results of the two approaches are shown to be in excellent agreement, with a variation of less than 3.6 percent being obtained. Arrays employing binary digital phase shifters are explicitly considered, but the techniques can be generalized to other types of digital, or digitalized, phase shifters. For other than the first beam position, there are alternate phase distributions which might be used to obtain a given beam position. The selection of a specific phase distribution depends, at least in part, upon the computer logic and round-off procedures. It is shown that these alternate phase distributions do not produce large differences in the beam position. The spread in beam position is between 2 and 3 percent for the alternate distributions. The effect of an amplitude taper on granularity is also considered. It is found that when the edge elements are less intensely illuminated than those at the center, the first beam position off-broadside is nearer broadside than for an array having the same phase distribution but uniform amplitude. The average granularity of a phased array has been determined, and is plotted against the number of elements with the number of phase bits as a parameter.