On the Spectral Response of ODDM Signals

The orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation has been proposed as a promising paradigm for ensuring reliable communications in doublyselective channels. Although ODDM functions as a multicarrier modulation on the DD plane, academia and industry are more accustomed to understanding and altering the time and frequency resources of signals and waveforms. Recognizing this, in this work, we investigate the time and spectral occupancies of the transmitted signals in both analog (direct) and digital (approximate) implementations of ODDM systems. We begin by explicitly determining the basis functions for both ODDM systems in time and frequency domains, and highlight their phase term difference in the time domain and their envelope difference in the frequency domain. We then derive the spectral responses of their transmit signals. We reveal that while the spectral response of signals in analog ODDM systems exhibits a step-wise behavior in its transition regions, the spectral response of signals in digital ODDM systems is confined to that of the ODDM sub-pulse. Finally, through numerical results, we verify our findings and show that out-of-band-emission of ODDM systems can be further improved by increasing the duration of the ODDM sub-pulse.