In this article, based on the example of antenna array consisting of radiating strips in a two dimensional periodical antenna array located at the distance of 
above an impedance structure (fig. 1), boundary effects caused by finite size of the aperture of the antenna array are analyzed. As an exterior, the wave propagating in the feeders uniform-linearly with linear phase incursion.
Fig.1
To pass on from the model of infinite antenna array [1] to the model of finite antenna array we will use the following method, which allows determining characteristics of the finite antenna array in an infinite screen with impedance 
(fig.1) based on the solution of the boundary value problem of electrodynamics for an infinite antenna array [2].
Let’s assume that the radiators do not reveal themselves physically in the meaning of changing the characteristics of the antenna array if the currents on them are equal to zero [3]. In this case, the voltages set on their inputs form the boundary wave [4]. Let’s consider that each radiator of the antenna array consists of 
elements (for example, bi-polarized antenna array, multiple-frequency antenna array etc.). Then, the following method is used to analyze the resulting antenna array:
— to radiators with numbers 
we connect infinite resistors (idling) through virtual feeders with wave impedance W;
— radiators with numbers 
can be connected either to matched load or generators through feeders with the same wave impedance W.
Here N — is the finite set of radiator numbers which limits the fragment of the infinite periodical antenna array inside which the radiators of the finite array which are of interest are located. We will use the following notation:
— 
– coefficient of interdependence between the l−th element of the m−th radiators and the s−th element of the n−th radiator;
— 
– amplitude of the wave of fundamental type propagating in the direction of input of the s−th element of the n−th radiator;
— 
– amplitude of the wave of fundamental type propagating in the direction from input of the l−th element of the m−th radiator.
Then the following equation is true, written for simplicity as applicable for a one-dimensional antenna array (a line of radiators):
 |
(1) |
We write amplitudes of the incident field as follows:
 |
(2) |
where 
— are reflection coefficients of the load enabled in the s−th excited element of the n−th radiator, 
— the same for non−excited elements, 
— is the set of numbers of radiators where there is at least one excited element.
Let 
(i.e. there is no dependency on the radiator number). Then, substituting (2) into (1), applying Fourier transform and convolution theorem and passing on to the matrix form, it is easy to show that amplitudes 
of the wave of fundamental type propagating in the direction from radiator inputs satisfy the following equation of second kind:
 |
(3) |
where the following notation is used for matrixes
S — is the square matrix of dispersion between radiator elements, 
, 
, 
— column vector 
, Е — unitary matrix, 
. Dimensions of all matrixes are determined by the M value.