We investigated radio acoustic sounding system (RASS) echoes backscattered from refractive index fluctuations produced by an acoustic pulse linearly modulated in frequency (a chirped acoustic pulse). We have numerically simulated errors of the Doppler frequency shift which are caused by the fact that the widths of the acoustic and radar pulse are finite. We analytically showed that a RASS with a chirped acoustic pulse can correctly measure a wide range of Doppler shifts. The error of the Doppler shift measurement is found to be a function of the radar range resolution, the rate at which acoustic frequency is swept, and the width of the acoustic pulse. The error is also shown to be almost independent of the radar frequency and the lapse rates of the atmospheric temperatures. We have also numerically showed that when the chirped acoustic wave frequency is 50 Hz/s, errors are within 0.018°C when the radar frequency is 1357.5 MHz and the range resolution is 75 m. At the chirped wave frequency, errors are within 0.055°C for a 46.5-MHz radar with a 150-m range resolution. We also propose a method for designing a chirped acoustic pulse to measure atmospheric temperature.