Target parameter estimation and array features
1.3 Design factors for arrays
1.3.3.1 Sub-array design procedure for low sidelobe sum and difference beams
The design procedure of [18] is based on the observation that the difference beam is more sensitive to grating lobes than the sum pattern. The difference beam is therefore taken as the design criterion. Suppose we apply a weightingwfor sum beamforming at the elements. This may be a partial weighting only which needs a corresponding sub-array weightingm for a sum beam with the correct weighting and hence with the desired sidelobes.
We can now apply a difference weighting only at the sub-array outputs given this partial element weighting. This is not a serious restriction, because the typical low sidelobe difference weightings like Bayliss consist of a bell shaped amplitude taper similar to the sum beam taper times the optimal difference weighting according to the element positions, see e.g. [13 p. 54] or [14]. We now want to find a sub-array configuration that allows to realize two desired element level difference weightingsdx;dy2CN as good as possible at sub-array level. These desired ele- ment level taperingsdare functions of the element positionsdi¼fðxi;yiÞ. We may quantize the range of this function intoqsteps. This produces in the (x,y)-planeq sets, see Figure 1.6 for the case of 6 steps.
Note that, as we have already a fixed taperwapplied at the elements, we have to apply this set generating procedure for the taperd0¼ðdi=wiÞi¼1;N. We do this for both, the azimuth and elevation difference beam weightings. The inter-sections of the resulting sets represent sub-arrays that represent the best sets of elements for reconstructing the original weighting at sub-array level. However, to avoid some undesirable side effects the following refinement steps are necessary:
● The inter-sections of the two contour sets may create sub-arrays with only a single or very few elements. These sub-arrays can be ignored by merging them with a bigger neighbour sub-array.
● In the centre of the array fairly quadratic sub-arrays of nearly equal size will appear. These give rise to quasi-grating lobes and also, more important, quasi- grating notches with ABF. In a manual fine tuning step these regularities should be broken up.
Figure 1.7 shows an example of a design of a planar array with 902 elements on a triangular grid with 32 sub-arrays. The shape of the sub-arrays was optimized by the strategy of [20] such that the difference beams have low sidelobes approx- imating a desired35 dB Bayliss weighting applied at the elements. At the ele- ments a fixed partial 35 dB Taylor sum beam weighting is applied such that THT¼I. One can see that the regularity in the centre is broken up here by turning one ring of the inner sub-arrays by a small angle.
The positions of the super-array according to (1.21) are indicated by the larger symbols of the corresponding sub-array element symbols. The resulting super-array has sufficiently good symmetryPL
l¼1rx;lPL
l¼1ry;l0 and acceptable balance PL
l¼1rx;lry;l0, which is required for good monopulse performance, [14]. The resulting super-array pattern and the 32 sub-array patterns are seen in Figure 1.8 (azimuth cuts only). One can see that due to the optimized design the super-array
–0.15 –0.1 –0.05 0 0.05 0.1 0.15
Figure 1.6 Plot of azimuth difference beam taper for planar circular antenna with low sidelobes. Contour lines for 6 levels induce 6 corresponding sets in the (x, y)-plane
pattern has no grating lobes. The sub-array pattern cuts have of course quite different beamwidths in the azimuth direction due to the different extent of the sub-arrays. We will use this configuration as a generic array in the sequel for presenting various examples.
With the sub-array configuration given the remaining question is: what is the best difference weighting to apply at the sub-arrays for approximating the desired weightingd0? One can make a least squares approximation and seek a sub-array weightingd~with:
~
d¼arg min
r nkTrd0k2o
(1.23) The solution of this problem is known to be~d¼ ðTHTÞ1THd0. One can refine this solution by additional constraints, e.g. by requiringaH0Tr¼0, which makes the difference beam exactly zero in directionu0, or by using, instead of the Euclidean
Nu = 32
Figure 1.7 2D generic array with 902 elements grouped into 32 sub-arrays.
Super-array centres from35 dB Taylor weighting are shown by a large symbol of the corresponding sub-array
–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 –30
–25 –20 –15 –10 –5 0 5 10 15
u
dB
–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1
–30 –25 –20 –15 –10 –5 0 5
u
dB
(a)
(b)
Figure 1.8 Azimuth cuts of super-array and sub-array patterns of generic array of Figure 1.7. Phase shifter setting is for direction (u, v)¼(0.2, 0), the 3 dB sum beamwidth is shown by the shaded area. (a) Super-array pattern and (b) 32 sub-array patterns
square norm, the norm of the weighted patterns, i.e.kdk2¼R
WjaðuÞHdj2pðuÞdu, wherepis a directional weighting function. The solution of this refined approach is:
~d¼THCT1
TH Ia0aH0T T HCT1
TH aH0T T HCT1
THa0
!
Cd0 (1.24)
with the matrixC¼R
WaðuÞaðuÞHpðuÞdu and Wthe directions of view. For the full visible region and withp:1 one obtainsCI.
Figure 1.9 shows the resulting azimuth patterns for this array. Clearly, the sum beam is unaffected by the weighting and has exactly the35 dB Taylor shape. The difference pattern does not fully achieve the 35 dB Bayliss sidelobe level.
However, a maximum sidelobe level of30 dB is achieved. This is the price to be paid for the coarse discretization of only sub-array difference weighting.
An important feature of digital beamforming with sub-arrays is that the weighting for beamforming can be distributed between the element level (the weighting incorporated in the matrix T) and the digital sub-array level (the weightingm). This yields some freedom in designing the dynamic range of~ the amplifiers at the elements and of the level of the AD-converter input. This freedom also allows to normalize the power of the sub-array outputs such that
–1 –0.5 0 0.5 1
–60 –50 –40 –30 –20 –10 0
u
dB
Figure 1.9 Sum and difference pattern formed with sub-arrays resulting from configuration of Figure 1.7
THT¼I. As will be shown in Section 1.6, this is also a requirement for adaptive interference suppression to avoid pattern distortions.
1.3.3.2 Beam scanning at sub-array level and sub-array