Nonseparable Nth-band filters as overlapping-subarray tapers

Abstract

Abstract-In a receive array "overlapped subarrays" refers to an array output formed as a weighted sum of subarray outputs themselves formed as identically weighted combinations of the outputs of overlapping element subsets. The system array factor becomes a product of two array factors with different periods, one associated with each sum, just as the frequency response of a two-stage IFIR filter in a DSP system is the product of two frequency responses with different periods. It is well known that an Nth-band filter can be the most efficient choice for an IFIR filter's first stage, but the corresponding array idea appears unknown.Design examples apply modern optimization to nonseparable tapers, with three of four examples featuring Nth-band subarray tapers. Whether an Nth-band approach is best is not settled here and is properly context dependent: designs should be carried out both ways and compared. Further, a very small subarray taper can easily be Nth-band entirely by accident, so designers should be aware of the associated array-factor features that necessarily result and cannot be optimized away. I. BACKGROUND AND INTRODUCTION A. Review of overlapped subarraysIn the overlapped-subarray receive-array architecture of Fig. 1, identically weighted sums of (typically few) signals from overlapping element subsets become subarray outputs. The array output is a final weighted sum of those (typically many) subarray outputs. While design of the detailed Fig. 2 example used an optimal approach discussed below, a simple two-step design approach is more typical.The first step designs the final array factor of Fig. 2(a), the Fourier transform of the final taper (summing weights) of Fig. 2(e), to fix the mainlobe shape. (More below about its apparently unreasonable width.) The small period of that final array factor replicates the mainlobe to create the two grating lobes of Fig. 2(a). The second step designs the subarray array factor, the Fourier transform of the subarray taper of Fig. 2(d), to suppress those grating lobes in the product of the final and subarray array factors, the system array factor of Fig. 2(b).Instead of one final sum there are typically several in parallel, creating several array outputs or "beams" . Their final array factors likely have narrower main beams than in Fig. 2(a) and are often identical except for horizontal translation. Five narrow mainlobes of five output beams might, for example, appear where the five peaks of the wide example mainlobe are in Fig. 2(a). Here that wide mainlobe stands in informally for such a cluster of five narrow mainlobes, with the subarray array factor suppressing the grating lobes of each. (More precise illustration of a beam cluster is deferred to the 2D examples of Figs. 4 through 7.) Element-output phase shifts not shown in Fig. 1 (but see vector kc in the Appendix's detailed formulation) can steer the cluster as a unit by translating all array factors in lockstep. B. IFIR filtersIt is well known that when subarraying is not used the mathematics of a uniform li...

On the weighting optimisation of the antenna with overlapped subarrays architecture

“…Thus we could remove Ω 2 from Ω 1 . Correspondingly, (6) and 7are transformed to be (16) and (17), respectively…”

“…The difference between (6) and (16) is that Ω 1 ∩ Ω 2 = Ω 2 (which means there is an overlap between regions Ω 1 and Ω 2 ) in (6), while Ω 1 ∩ Ω 2 = Ø (i.e. Ω 1 and Ω 2 are non-overlapped) in (16).…”

“…However fortunately, when we fixed h 1 (or h 2 ), there is a linear relationship between H(u) and h 2 (or h 1 ) (actually, it is a Fourier transform relationship, and can be calculated using FFT [14,15]). Therefore inspired by the literature [16], an alternating optimisation strategy could be used. Specifically, the procedure of the alternating optimisation can be simply summarised here…”

Uniform circular array pattern synthesis using second‐order cone programming

“…Antenna-relevant error measures for SOCP design of planar-array patterns have since been well developed, however. The basic case of constrained minimisation of SNR loss in a narrowband receive array of identical elements [20] has been adapted to overlapped subarrays [15] and to an array of non-identical and non-uniformly placed elements characterised by planar near-field measurements [12]. For wideband digital arrays FIR filters in the elements' individual processing channels can be jointly optimised to create a desired pattern as a function of the frequency [13,21] even with strict control of pattern polarisation when dual-pol elements are used [14].…”

“…The weight vector is optimised to minimise array taper loss subject to desired constraints on SLL. That work represented an extension of work done in [13][14][15] that allowed non-identical embedded element patterns while applying SLL constraints to arbitrarily polarised signals.…”