A modified algebraic method of mathematical signal processing in radar problems

Abstract

A new method for approximate solution of ill-posed inverse problems of reconstructing images of objects with angular resolution exceeding the Rayleigh criterion is proposed and justified, i.e. with super resolution. Angular super-resolution allows you to obtain images of objects with increased clarity and distinguish previously invisible details of images of complex objects. In addition, on this basis, the probability of correct solutions to problems of object recognition and identification increases. Mathematically, the problem is reduced to solving the linear Fredholm integral equation of the first kind of convolution type. Solutions are sought with additional conditions in the form of restrictions on the location and size of the desired radiation source, which makes it possible to regularize the problem. The method is a development of one of the parameterization methods the algebraic method. The solution is sought in the form of a representation of the desired function in the area where the source is located in the form of a series expansion over the input sequence of orthogonal functions with unknown coefficients. Thus, the inverse problem is parameterized and reduced to searching for expansion coefficients. The presented method is based on the use of a priori information about the localization area of the radiation source, or on an estimate of the location and size of this area obtained by scanning the viewing sector with a goniometer system. Using zero values of the function describing the source outside this region, for systems based on antenna arrays it is possible to find tens and even hundreds of expansion coefficients of the desired function in a Fourier series. The solution is constructed in the form of an iterative process with a consistent increase in the number of functions used in the expansion until the solution remains stable. The adequacy and stability of the solutions was verified during numerical experiments using a mathematical model. The results of numerical studies show that the presented methods of digital processing of received signals make it possible to achieve an effective angular resolution 3–10 times higher than the Rayleigh criterion. The proposed method makes it possible to miniaturize the antenna system without degrading its characteristics. Compared to known ones, it is relatively simple, which allows it to be used by systems in real time.