Orthogonal Delay-Doppler Division Multiplexing (ODDM) Modulation Over Highly-Spread Channels

This paper examines orthogonal delay-Doppler (DD) division multiplexing (ODDM) systems over highly-spread channels characterized by delay and Doppler shifts which can exceed, respectively, the sub-pulse separation and sub-tone separation of the DD orthogonal pulse employed by ODDM. We first derive an equivalent sampled DD (ESDD) channel model with on-grid delay and Doppler shifts specified by the sampling interval and frame duration of the transmission scheme. Our model accounts for off-grid delay and Doppler present in practical channels, samplewise pulse shaping at the transmitter, and matched filtering and windowing at the receiver. By examining their interaction, we show that the time-domain sequences of ODDM implemented approximately (digitally) using discrete Fourier transform (DFT) and inverse DFT (IDFT) and the discrete-time on-grid DD channel are finely coupled, and this leads to compact input-output (IO) relation for ODDM over on-grid channels. We then leverage the ESDD model to describe the IO relation of ODDM adopting more general pulses and windows over highly-spread off-grid channels, which reveals the potential folding (aliasing) of the ESDD channel and its implication on the signal model. We finally present numerical results. It is observed that pulses with shorter duration and windows with lower sidelobes in their spectrum lead to sparser ESDD channels.