  The recently proposed orthogonal delay-Doppler division multiplexing (ODDM)
modulation, which is based on the new delay-Doppler (DD) domain orthogonal
pulse (DDOP), is studied. A substantial benefit of the DDOP-based ODDM or
general delay-Doppler domain multi-carrier (DDMC) modulation is that it
achieves orthogonality with respect to the fine time and frequency resolutions
of the DD domain. We first revisit the family of wireless channel models
conceived for linear time-varying (LTV) channels, and then review the
conventional multi-carrier (MC) modulation schemes and their design guidelines
for both linear time-invariant (LTI) and LTV channels. Then we discuss the
time-varying property of the LTV channels' DD domain impulse response and
propose an impulse function based transmission strategy for equivalent sampled
DD domain (ESDD) channels. Next, we take an in-depth look into the DDOP and the
corresponding ODDM modulation to unveil its unique input-output relation for
transmission over ESDD channels. Then, we point out that the conventional MC
modulation design guidelines based on the Wely-Heisenberg (WH) frame theory can
be relaxed without compromising its orthogonality or without violating the WH
frame theory. More specifically, for a communication system having given
bandwidth and duration, MC modulation signals can be designed based on a WH
subset associated with sufficient (bi)orthogonality, which governs the
(bi)orthogonality of the MC signal within the bandwidth and duration. This
novel design guideline could potentially open up opportunities for developing
future waveforms required by new applications such as communication systems
associated with high delay and/or Doppler shifts, as well as integrated sensing
and communications, etc.
