Yu.I. Choni – Ph.D.(Eng.), Associate Professor, 

Kazan National Research Technical University named after A.N. Tupolev - KAI E-mail: tchoni@rambler.ru

Inverse problems, which include the synthesis of antennas, are often incorrect. According to Hadamard, for a correctly posed task, three conditions must be true: 1) a solution exists; 2) it is unique; 3) it changes continuously as the initial conditions change. If any of the conditions is not true, the task becomes invalid. Moreover, even a correctly stated task can be ill conditioned in the sense that small deviations of the right side of the equation can lead to a drastic change in the solution.

Regularization as a universal and robust mathematical method allows us to cope with ill-posed tasks. Regularization is adequate to the meaning of a huge number of applied problems arising, for example, in probing the underlying surface, in tomography, in retrieving the ARP from measurements in non ideal conditions, etc. The fact is that in such situations, the right side of the equation actually contains a noise component.

The situation with the synthesis of antennas is so different that we should turn the regularization upside down. The fact is that the required ARP F0(r°) is a precisely defined one: it must be optimal for a particular radio system. Therefore, the engineer does not care how the solution behaves with variations of F0(r°). However, what he is deeply interested in is that the solutions obtained are not too sensitive to the errors δI(ρ), which inevitably accompany the implementation of the calculated solution. That is, he is interested in the stability of the solution of the synthesis equation not from right to left according to Hadamard, but from left to right according to practical needs and common sense.

The value of the relative mean square deviation (MSD) σ2 is usually an estimate of the accuracy of the solution. Two ARP differing only by a factor, correspond to different values of σ2, although in practical terms these ARP are equivalent. Therefore, it would be correct to calculate the MSD σ2, taking into account only the shape of the ARP (the angular dependence). The paper discusses the corresponding formula. It is clear that for solutions minimizing the MSD, the values of σ2 and σ2 coincide. However, when assessing the accuracy of measured ARP or those obtained by iterative algorithms, the value σ2 provides a significantly more adequate estimation. The work convincingly demonstrates this on synthesizing a circular antenna array for a desired ARP in the form of a δ-function. It is important that the use of the MSD by form in the synthesis of the antenna according to a requied pattern, taking into account the finite accuracy of its implementation, shifts the optimal solution towards a less reactive solution.

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