The article discusses the method of mathematical modeling of characteristics of scattering of ultrashort-pulse signals of complex shape objects, taking into account the shape of the envelope of the ultrashort-pulse radio signal and the features of radiation of an ultrawideband antenna. To estimate the reflecting properties of objects, the determination of the ultra-short-pulse radar cross section of complex shape objects with their radar location by non-harmonic signals is given. The mathematical model of radar scattering characteristics of location objects under ultra-short-pulse signals takes into account the shape of the envelope of the signal generated by a transmitter of a single-chip microchip NVA 6100 with duration of 300 ps and obtained through direct measurements. The electric scattering field of a location object in the frequency domain is determined on the use of a polygonal model of the object and asymptotic methods of diffraction theory. The transfer function of the Vivaldi antenna to radiation obtained by numerical modeling using the method of moments was used in the electric field calculations. Verification of the developed mathematical model was carried out on the basis of determining the ultra-short pulse radar cross section of perfect electric conductive sphere. A diagram of the ultra-short-pulse radar cross section of the GPS satellite was obtained, taking into account the radiation of the ultra-short-pulse signal by the Vivaldi antenna. The diagram of ultra-short pulse radar cross section of GPS satellite does not contain interference maxima and minima, and the averaged values ultra-short radar cross section of GPS satellite from lateral angles are on average 3..8 times less than radar cross section at the center frequency of the spectrum. The mathematical model of radar scattering characteristics of location objects under ultra-short-pulse signals is intended for preliminary estimates of the ultra-short-pulse radar cross section at an early step of designing ultra-wideband radars. The obtained estimates of the ultra-short-pulse radar cross section of the GPS satellite make it possible to identify the main features in the scattering of ultra-short-pulse signals at the location of extended targets.