Space-time signal processing in MIMO antennas system with the Earth surface reflections

. The generalized correlation integral for space-time signal processing at the output of a MIMO antenna system radio altimeter in the Earth’s surface reflections presence is obtained. The results of calculation of one-dimensional cross-section of the generalized correlation integral at an elevation angle with a known target range and two-dimensional cross- section of the generalized correlation integral «horizontal range - height» for specific surface reflection coefficient are presented.


Introduction
Since the beginning of the 2000s, MIMO technology based on usage of antenna arrays with transmitting elements, which emit orthogonal signals used in information transmission systems, has found application in small-size radars creating [1,2]. Probe signal components scattered from the target can be separated at the outputs of transmitting elements according to their orthogonality. Similar solutions took place in the long-range radar RIAS [3], produced in 80-90s of the last century, when there was no MIMO technology in information transmission systems. Usage of the MIMO antenna systems leads to a significant reduction of the total number of elements in antenna array, simplifying the element feeding and microwave wiring of signals, which is important in small-sized radars development. Resolution performance of MIMO radar in case of monitoring targets in free space were analysed and well explored in numerous papers, for example in [1][2][3].
In this paper we enounce new theoretical results in the context of monitoring pin-point target in presence of earth or water surface reflections. In this paper we derived and quantitatively analyzed the expression for generalized correlation integral for space-time signal processing of orthogonal signals scattered by the target and its "antipode" [4], simulating earth surface reflections.
Some example of calculating the cross sections of the generalized correlation integral by elevation and "distance-height" coordinates accompanied by specific surface reflection coefficient for a radar altimeter with MIMO antenna system are given in this paper.

Generalized correlation integral over spatial coordinates for MIMO radio altimeter
To simplify the presentation of the theoretical analysis method in case of monitoring a target above the Earth surface in MIMO radar, we considered the case in which the earth * Corresponding author: valch2008@yandex.ru surface is flat and the roughness coefficient  lies within the limit 0<  <1. As it is known, the definition of an «antipode», which allows to substitute the reflection of probe signal from the reflecting surface with scattering signals from both the target and the «antipode», is introduced for a target situated above a reflecting surface [4]. The main feature of the MIMO radar is that this fact is true for all of the pairs of receiving and transmitting elements, specifically for the orthogonal component of probe signals on propagation ways such as «the n-th transmitting element -target -the k-th receiving element and the n-th transmitting element -antipode -the k-th receiving element. Figure 1 illustrates the ray path geometry for the pair «n-th transmitting element» and «k-th receiving element» for the target (T) and its antipode (A).
In this paper we considered a MIMO antenna system consists of linear transmitting and receiving subarrays, when the radar operates in radar altimeter mode. This fact made it possible to approximately substitute each of the subarrays with its phase center, which has a wide radiation pattern in vertical plane, and to present the whole antenna system as a vertical linear MIMO antenna array which consists of t N transmitting and The shape of the complex amplitude spectrum modulus () u G  is close to rectangular [5] and the spectrums of orthogonal partial probe signals (1) practically do not overlap in case of chirp signals with a large time-frequency base and a frequency deviation equal to the frequency step   in (1). In the presence of surface reflections, pin-point target T and its antipode A are considered with coordinate vectors ,, x y z  R

and
,, x y z   R , where z is a height of the target, and x, y are the horizontal plane coordinates.
In case of using broadband probe signals of the form (1) and, in particular, the chirp probe signals, the analysis of interference surface reflections has a certain peculiarity in  (2) Taking into account the interference from the surface with a complex reflection coefficient  , the signal from the n-th transmitting subarray at the target with consideration of the shape of the probe signal (1) will be where 0 U is an amplitude conditioned by the propagation. The signal at the output of the kth receiving subarray, corresponding to the reflected target signal, due to the n-th transmitting subarray, is equal to where 1 U is an amplitude also conditioned by the propagation. Substituting (3) into (4) and adding up over all transmitting subarrays, we obtain a signal corresponding to the output signal of the k-th receiving subarray for a pair of reflectors «target at the point with coordinates R -antipode at the point with coordinates R'»: Here we introduced and denoted sums of delays nk   in a form of (2) for different propagation ways from phase centers of subarrays for the target and the antipode with vectors depending on the Cartesian coordinates x, y, z, respectively considering the fact that As a reference signal, let us choose a signal corresponding to the reference point in space with a vector of coordinates where symbol «*» means complex conjugate value.