Pseudo-noise (PN) codes are widely used to characterize communication channels in acoustic communications and positioning systems, owing to their sharp autocorrelation properties. However, when either the transmitter or receiver is in motion, the Doppler effect alters the signal characteristics, and various methods have been proposed to compensate for this effect. Furthermore, the Doppler effect induced in the transmitted signal has also been utilized to estimate the velocity and direction of arrival of the moving source. When the fractional bandwidth of the transmitted signal is narrow, it is typically assumed that the frequency shift due to the Doppler effect is approximately uniform across the entire signal bandwidth. Under this assumption, the Doppler shift can be estimated by measuring the frequency shift of the carrier component. However, when the fractional bandwidth is large, the frequency shift varies across the spectrum, making Doppler correction via signal stretching necessary. We theoretically analyzed the cross-correlation characteristics of PN codes under Doppler compensation using signal stretching and clarified the relationship between signal parameters and correlation strength. Additionally, we investigated the cross-correlation behavior of the transmitted signal under the Doppler effect through simulations and compared the theoretical predictions with numerical results.
Tsuchiya et al. (2025) studied this question.