On protecting the planet against cosmic attack: Ultrafast real-time estimate of the asteroid׳s radial velocity

Abstract

A new method for the line-of-sight velocity estimation of a high-speed near-Earth object (asteroid, meteorite) is suggested. The method is based on use of fractional, one-half order derivative of a Doppler signal. The algorithm suggested is much simpler and more economical than the classical one, and it appears preferable for use in orbital weapon systems of threat response. Application of fractional differentiation to quick evaluation of mean frequency location of the reflected Doppler signal is justified. The method allows an assessment of the mean frequency in the time domain without spectral analysis. An algorithm structure for the real-time estimation is presented. The velocity resolution estimates are made for typical asteroids in the X-band. It is shown that the wait time can be shortened by orders of magnitude compared with similar value in the case of a standard spectral processing.

Best Approximation of the Fractional Semi-Derivative Operator by Exponential Series

“…It has been shown in [3,4] that the right-hand side of Equation (1) can be represented by the ratio of the square of the fractional derivative to the square of the signal's norm (energy), which allows for us to find the centroid of the spectrum without using the Fourier transform:…”

“…The pulse response (7) is identical to the kernel of the fractional differentiation operator. It follows from Equation (2) that for a limited observation interval of signal (T − T 0 < ∞), the processing time x(t) can be substantially reduced if we calculate the fractional derivative of the order of 1 ⁄2 in real time [3,4]. Estimating the location of the centroid of the spectrum (2) can be undertaken without spectral processing as the reflected signal reaches the detector and is obtained at the end of the observation interval T. For example, in the problem of the rapid estimation of the radial velocity of a potentially dangerous near-Earth asteroid, the real computational speed can be increased by a million times [4].…”

“…An explicit expression for the pulse response of the Liouville operator of a semi-derivative is presented in [3,4]:…”

“…In [3,4], the definition of a fractional Riemann-Liouville derivative was used. Here we confine ourselves to its particular limiting case, the Liouville derivative [16] The Liouville derivative (3) has more habitual and evident properties, such as the identity vanishing of the derivative of a constant.…”

“…A complex movement of an object can take place if the object 1 ⁄2 has been proposed, which allows for the estimation of the centroid of the spectrum of the input signal to be obtained in real time without spectral processing. It has also been shown that the use of such a fractal-processing algorithm makes it possible to increase the rate by which an estimate can be obtained by up to six orders of magnitude [4].…”