  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. To the
best of our knowledge, this is the first paper to rigorously derive OTFS
modulation from first principles. 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). The degree of localization of the DD
domain basis signals is inversely related to the time duration of the transmit
signal, which explains the trade-off between robustness to Doppler shift and
latency.
