Derivation of OTFS Modulation From First Principles

Orthogonal Time Frequency Space (OTFS) modulation has been recently proposed to be robust to channel induced Doppler shift in high mobility wireless communication systems. However, to the best of our knowledge, none of the prior works on OTFS have derived it from first principles. In this paper, using the ZAK representation of time-domain (TD) signals, we rigorously derive an orthonormal basis of approximately time and bandwidth limited signals which are also localized in the delay-Doppler (DD) domain. We then consider DD domain modulation based on this orthonormal basis, and derive OTFS modulation. We show that irrespective of the amount of Doppler shift, the received DD domain basis signals are localized in a small interval of size roughly equal to the inverse time duration along the Doppler domain and of size roughly equal to the inverse bandwidth along the delay domain (time duration refers to the length of the time-interval where the TD transmit signal has been limited). With sufficiently large time duration and bandwidth, there is little interference between information symbols modulated on different basis signals, which allows for joint DD domain equalization of all information symbols. This explains the inherent robustness of DD domain modulation to channel induced Doppler shift when compared with Orthogonal Frequency Division Multiplexing (OFDM).