# High velocity Doppler analysis of wideband radar signals

#### Abstract

A thorough literature review of wideband radar systems indicates that the Doppler Effect is mathematically modeled as a frequency shift proportional to normalized range rate [special characters omitted]. As the target reaches relativistic velocities, the relationship between the frequency shift and β becomes highly nonlinear. A more complete model for the Doppler effect sited in the literature involves the use of the compression factor [special characters omitted]. Time compression or dilation is explained through a change of variables from t to αt. In the case of perfect reflection, the working assumption is that the energy of the incident wave is equal to the energy of the reflected wave. Consequently, the amplitude of the reflected signal is scaled by a factor [special characters omitted]. In this work, we argue that for extremely high velocities, only a relativistic analysis based on the Lorentz transformation of the wave 4-vector and the electromagnetic field tensor provides correct and full description of the Doppler effect. Since the echo location problem involves transmission of the signal from a stationary frame and signal reflection from a moving frame, the wave 4-vector and the electromagnetic field tensor must be transformed twice to simulate a round trip. We show that for relative motion limited to the radial direction, the round-trip Lorentz transformation suggests a complete description of the Doppler effect that comprises phase scaling, amplitude scaling, and compression or dilation of the envelope, in proportion to α. To illustrate this, we analyze the effects of the Lorentz transformation on three different signals: continuous time harmonic signal; linear FM (chirp) signal; and linear period modulated signal. The analysis of these three signals includes a description of their instantaneous frequency and spectrum. Here we show that if we use a first-order approximation to model the Doppler of the reflected signal there exist mismatches at the amplitude, the frequency and the signal duration compared to using the full Doppler compensation. Furthermore, we analyze the ambiguity surface of the linear FM and the linear period modulated pulse to determine the impact on range and Doppler resolution. In addition, we analyze the effects of the Lorentz transformation on the output of the matched filter and the heterodyne radar. For the heterodyne radar we showed that the output is affected by the change in amplitude of the reflected signal. If we use a first-order approximation to model the reference signal of the heterodyne radar the output will be broadened due to the target’s velocity. For the matched filter we showed that the output is affected if we select the first-order approximation to model the reference signal; even more, we showed that by increasing the time-bandwidth product of the signal we contribute to augment the mismatches between the reference signal and the received signal. Finally, we show that the characteristics of a reflected signal with a large time-bandwidth product are highly sensitive to even relatively low relativistic velocities.

#### Subject Area

Mathematics|Electrical engineering

#### Recommended Citation

Ochoa, Hector A, "High velocity Doppler analysis of wideband radar signals" (2007). *ETD Collection for University of Texas, El Paso*. AAI3273990.

https://scholarworks.utep.edu/dissertations/AAI3273990