Coherent radar imaging: Signal processing and statistical properties

The recently developed technique for imaging radar scattering irregularities has opened a great scientific potential for ionospheric and atmospheric coherent radars. These images are obtained by processing the diffraction pattern of the backscattered electromagnetic field at a finite number of sampling points on the ground. In this paper, we review the mathematical relationship between the statistical covariance of these samples, 〈$\widehat{f}\widehat{f}$^†〉, and that of the radiating object field to be imaged, 〈FF^†〉, in a self-contained and comprehensive way. It is shown that these matrices are related in a linear way by 〈$\widehat{f}\widehat{f}$^†〉 = aM〈FF^†〉M^† a^∗, where M is a discrete Fourier transform operator and a is a matrix operator representing the discrete and limited sampling of the field. The image, or brightness distribution, is the diagonal of 〉FF^†〉. The equation can be linearly inverted only in special cases. In most cases, inversion algorithms which make use of a priori information or maximum entropy constraints must be used. A naive (biased) “image” can be estimated in a manner analogous to an optical camera by simply applying an inverse DFT operator to the sampled field $\widehat{f}$ and evaluating the average power of the elements of the resulting vector $\widehat{F}$. Such a transformation can be obtained either digitally or in an analog way. For the latter we can use a Butler matrix consisting of properly interconnected transmission lines. The case of radar targets in the near field is included as a new contribution. This case involves an additional matrix operator b, which is an analog of an optical lens used to compensate for the curvature of the phase fronts of the backscattered field. This “focusing” can be done after the statistics have been obtained. The formalism is derived for brightness distributions representing total powers. However, the derived expressions have been extended to include “color” images for each of the frequency components of the sampled time series. The frequency filtering is achieved by estimating spectra and cross spectra of the sample time series, in lieu of the power and cross correlations used in the derivation.