In case of AA coupling value analyzing, usually the following construction is taken [1] (fig.1): L active transmitters of two–dimensional array are connected to first outcome of emitting subarray beam former (BF_{1}), М emitters are connected to certain secondary radiator (for example, it`s loaded to matched resistance) of decoupling subarray, N secondary radiators are connected to second emitting subarray beam former (BF_{}2). The S_{2} and S_{22} scattering matrix choice, then BF_{1} and BF_{2} characteristic of active and secondary subarrays [1…3] fixed, is worthy of notice. BF_{1} and BF_{2} characteristic influence and mutual coupling magnitude between emitters of emitting and secondary subarrays are of less importance.
Figure 1. Antenna system scheme.
In the article bypassing magnitude calculation results for special cases of gain–phase distribution (GPD) in BF_{1} and BF_{2} outcomes are given. It can be gotten if wideband microstrip antenna array (WMAA) [4] would be used as single transmitter–receiver curtain. Suppose that both BF have isolated and matched outputs and decoupling subarray is absent (M = 0). Then coupling coefficient expression looks like this [1]:
(1) 
where Y_{12} — dimension matrix 1xN of complex transfer constant coefficients from BF_{2} outcomes to it`s input; X_{21} — dimension matrix Lx1 of complex transfer constant coefficients from BF_{1} input to it`s outcomes; C_{31} — Toeplitz matrix of coupling coefficients between emitters of emitting and secondary subarrays.
Suppose, that all subarrays are belong to infinite two–dimensional AA and all feeder line have equal wave resistance. Failing loss in BF they can be circumscribed with unitary scattering matrix with the zero–order terms on principal diagonal. Let BF generates in outcomes GPD, which in general case can be described with complex–valued functions , i=1,2. Indices i=1,2 conform with BF_{1} and BF_{2}.
For emitting subarray under unit amplitude of principle wave in input, in BF_{1} outcomes we`ll get follows amplitudes of same wave:
(2) 
At that wave amplitudes in secondary subarrays radiator will equal to:
(3) 
where C_{pt} — mutual coupling coefficient between infinite AA radiators and wave amplitude in BF_{2} input is:
(4) 
Then formula (1) taking into account (2), (3), (4) can be written as:
where P_{1} is connected with infinite AA radiators interaction, P_{2} is determined from gain distribution on BF outcomes and doesn’t depend on phase distribution, radiator`s type and their position in array.
Taking into account linkage between coefficient C_{st} and reflectance , depending on equal–amplitude feed of infinite AA with linear phase progression, expression can be written as follows:
(5) 
where * — complex conjugation symbol
(6) 
, — differential phase shift lengthwise OX and OY. f_{1} and f_{2} functions represent subarray multiplier and possess only the finite data modulo for realizable gain–phase distribution.
On the basis of expression analysis (5) it’s possible to develop the follows relation approach of AA and BF for wideband bypassing increase, concerned with radiators interaction (in case then subarrays workspaces are in etch angle sector):
— in etch angle sector and given frequency band of infinite AA radiator reflectance Г should take on a lower–range value modulo (in the limit — zero value for all ):
— BF_{1} and BF_{2} have to generate in outcomes GPD, then "basis ray" of f_{1}, f_{2} functions should be narrow at most and "side lobes" should have lower–range value modulo (in the limit — zero value for all ) in the invisible angle sector (where Г = 1 in case of diffraction lobe of higher order lack):
then
To obtain first principle the usual variants of waveguilding, vibrator, resonant microstrip and other frequency–dependent radiators are not matched up, because required Г behavior in the sector of broad band frequency (octave and more) and wide angle sector (about ±60^{0}), is not provided.
Let us examine a WMAA, described in the work [4]. It represents grid composed of narrow strip band, length , lying on axe OX on the magneto dielectric layer, there thickness is and penetrations are . Here T_{x}, T_{y} — АA periods lengthwise ОХ и OY axes. In the etch angle sector and the closest to it invisible angle sector input resistance of such radiator can be written as follows [4]:
where W=120π, and reflectance looks like:
where , . Angles , determined direction, in which WMAA radiators are adjusted.
The second principle achievement leads to task solution of physically feasible GPD synthesis on conditions that directivity level of both subarrays is the highest [5].
As an example lets examine the follows gain distributions in BF outcomes [6]:

P_{2} takes on a minimal value in case of equalamplitude distribution:

In the table 2 estimated on the basis of IBM by formulas (5) and (6) decoupling levels between emitting and secondary subarrays, size of each is 5λx5&lambda (АA periods T_{x}=T_{y}=0.14λ) are an example in the case of emitting subarray phase angle =–60^{0} situated in the E–plane (φ=0^{0}) for line radiators based on the magneto dielectric layer. Radiators are matched in the direction , . Based on the computation, character of gain distribution in H–plane (φ=90^{0}) doesn’t have an influence on coupling magnitude in the E–plane. That’s why equal–amplitude distribution in both subarrays is admitted to H–plane everywhere.


There are subarrays (size 10λx10λ) computation results of decoupling levels in the table 3 and the same computation results for subarrays (size 20λx20λ) in the table 4.
As it’s necessary to numerically calculate sums for gain distributions (6), that concerned with plenty of computer time spending, the Euler–MacLaurin summation formula has been used in this paper [7]. At that term of N_{0} series quantity in the expression for probability integral of complex argument [8], which is computed with the EulerMacLaurin summation formula, doesn`t depend on radiators quantity in subarrays. Beside it’s substantially smaller than amount, computed with formula (6), especially in the case with great number of radiators.



There are GPD of type decoupling levels for close located emitting and secondary subarrays of different sizes in tables 5…7 (for subarrays with size 5λx5λ, 10λx10λ, 20λx20λ accordingly, for emitting subarray phase angles =–60^{0}). In this connection N_{0} series quantity in all cases is less than 15, and mean time of decoupling level computation using IBM РСАТ80286/287 for 93x93 points of numerical integration of , amounts 15 minutes.
Based on the results obtained in this paper we can conclude:
— the WMAA decoupling level with gain distributions (AE=0.66, SLL=–31.5dB) are greater than with gain distributions (AE=0.808, SLL=–32.1dB). It relates to directivity decrease in second case due to slower NL decline moving off main lobe;
— in both cases decoupling level decreases than , because of radiators interaction in invisible angle sector of differential phase shifts. Interaction increases as far as visible ray of diagram approximates verge of etch and invisible sectors;
— for gain distributions using WMAA decoupling level no less than 100dB in ±60^{0} angle sector lying in E–plane, is gain on conditions that and subarray sizes no less than 10λx10λ. At that each subarray AE is no more than 0.66.