On the Poisson sum formula for the analysis of wave radiation and scattering from large finite arrays

Abstract

Poisson sum formulas have been previously presented and utilized in the literature [1]-[8] for converting a finite element-by-element array field summation into an alternative representation that exhibits improved convergence properties with a view toward more efficiently analyzing wave radiation/scattering from electrically large finite periodic arrays. However, different authors [1]-[6] appear to use two different versions of the Poisson sum formula; one of these explicitly shows the end-point discontinuity effects due to array truncation, whereas the other contains such effects only implicitly. It is shown here, via the sifting property of the Dirac delta function, that first of all, these two versions of the Poisson sum formula are equivalent. Second, the version containing implicit end point contributions has often been applied in an incomplete fashion in the literature to solve finite-array problems;it is also demonstrated here that the latter can lead to some errors in finite-array field computations.Index Terms-Array antennas, Poisson sum formula.

Field patterns of resonant noncircular closed-loop arrays: further analysis

“…The trapezoidal rule is, thus (12) where is the error of the trapezoidal rule and (13) Throughout this section, will mean . If the Fourier series of is (14) then (15) Furthermore, for all real , the periodic Fourier series in (14) converges to (is equal to) the periodic extension of at points of continuity, and to the mean value at points of finite discontinuity. 1 Therefore, for in we have (16) Now express and in terms of the Fourier coefficients .…”

“…But we have written Section IV as a stand-alone section in view of the usefulness of the PSF in other antenna problems (see [13], [14], and the references therein). In particular, Section IV supplements the recent discussions in [14] regarding the proper use of the PSF in the theory of finite arrays. The presentation in Section IV, which is original, is based on ideas that can be found in [15].…”

General Floquet mode characterization of phased array antennas using asymptotic techniques

“…Figure 1 shows the array configuration and its coordinates. We consider the radiation potential of the NFA, and employ the Poisson Sum formula [7] to first decompose the superposition of fields radiated from individual array elements into a sum of Floquet modes, where each Floquet mode becomes a radiation integral. This Floquet mode decomposition was previously employed for the characteristic investigation of FFA, and thus makes the current investigation more general with a good transition and reduction to the previous 978-1-4673-0335-4/12/$31.00 ©2012 IEEE investigation on FFA.…”

“…The investigation employs the Poisson sum formula [7] to decompose the antenna radiation into Floquet modes, which is a popular technique to quantize the propagation phenomena and has been previously used in the investigation of far field radiation. The asymptotic technique [8] is used to characterize the radiation phenomena in terms of ray propagation mechanisms.…”

General analysis of Floquet modes for an one-dimensional, infinite phased array antennas

“…The net radiation can be efficiently computed in terms of Floquet mode waves using the Poisson sum formula [3]:…”