On the Zak Transform-based Interpretation of OTFS Modulation
In this paper, we present a Zak transform-based development of the recently proposed orthogonal time frequency space (OTFS) modulation scheme. Unlike previous works, we focus on the interpretation of the spreading function as the Zak transform of the 'impulse train response' for general underspread linear time-varying (LTV) channels. For an underspread channel, the Zak transform of the output signal is given by the twisted convolution of the spreading function with the Zak transform of the input signal. This twisted convolution relationship provides a Zak domain input-output relationship for general underspread LTV channels. We extend these results to the discrete case, by presenting a development of the discrete Zak transform (DZT) similar to the one provided by Mohammed for the continuous Zak transform. We argue that the discrete Zak domain twisted convolution relationship for LTV channels provides a simple and concise input-output relationship for OTFS modulation, analogous to the frequency domain multiplication relationship for orthogonal frequency division multiplexing (OFDM) over linear time invariant (LTI) channels. Lastly, we discuss the impact of adding a cyclic prefix and zero-padding in delay and Doppler in the discrete Zak domain.
