Exact analysis of radiation patterns using the expansion of the Fourier sum

The time required to compute radiation patterns of linear arrays given in a form of a Fourier sum depends on the number of array elements. This paper presents a fast algorithm for computing Fourier sums. The radiation pattern given by this sum can be replaced by an infinite series whose terms depend on the envelope of the excitation function, w(x), and its derivatives at the edges of the linear array. In cases when w(x) has a few nonzero derivatives, this infinite series can be replaced by a finite sum which can be evaluated more rapidly than the original Fourier sum, making the method especially suitable for real-time applications. The effect of critical point is also investigated. Some sample case studies are included.