The matrix operator in the equation (3) exists only if for all differential phase shifts in interval . Therefore, when (idling mode in virtual feeders) in case of antenna arrays with dielectric coating the operator exists only if there are losses in layers.

For the norm of the operator in discrete space using the Cauchy−Bunyakovsky inequality [5] the following estimate can be found:

where — are elements of the square matrix .

In case when *q* < 1 ((the operator *K* is a compression operator) to solve the equation (3) an iterative procedure can be used. In this case the following estimate is true for convergence speed [6]:

where — is vector of amplitudes on the *k*-th iteration step. As a zero approximation, the solution for the infinite antenna array can be used.

For antenna arrays with small period *Т* (, — wavelength) the convergence speed becomes unsatisfactory. In this case, the problem must be reformulated and the equation written not in relation of wave amplitudes in feeders but amplitudes of the total boundary wave [4]. If the Polozhy method [7], we get the following equation instead of (3):

where — is the matrix of the sought amplitudes of boundary wave,

— is the second iterated matrix core (), — Kronecker symbol.

After finding the sought amplitudes are determined from the equation , and the directional diagram of the finite array is determined from the formula:

where — row vector {}, and — is the directional diagram of the *s*−th element of the zero radiator in the infinite array, found using the assumption that all other elements in all radiators are loaded with matched load.

It is not hard to generalize the received result for the two-dimensional case.

Fig. 2 Directional diagram of the radiating strip of the finite array in an infinite screen in *H*−plane depending on its position in the *OX* axis (1 — = = 0; 2 — = 4, = 0; 3 — = 12, = 0).

Fig. 3 Directional diagram of the radiating strip of the finite array in an infinite screen in *H*−plane depending on its position in the *OY* axis (1 — = 0, = 4; 2 — = 0, = 12; 3 — radiator in the infinite array).

Fig. 4 Directional diagram of the radiating strip of the finite array in an infinite screen in *E*−plane depending on its position in the *OX* axis (1 — = = 0, 2 — = 4, = 0, 3 — = 12, = 0).

Fig. 5 Directional diagram of the radiating strip of the finite array in an infinite screen in *E*−plane depending on its position in the *OY* axis (1 — = 0, = 4; 2 — = 0, = 12; 3 — radiator in the infinite array).