The recently proposed orthogonal delay-Doppler division multiplexing (ODDM) modulation, which is a delay-Doppler (DD) domain multi-carrier (DDMC) modulation scheme based on the DD domain orthogonal pulse (DDOP), is studied. We first revisit the linear time-varying (LTV) channel model for the wireless channel, and review the conventional multi-carrier (MC) modulation schemes and their design guidelines for both linear time-invariant (LTI) and LTV channels. We then focus on the representation of the LTV channel in an equivalent sampled DD (ESDD) domain, and propose an impulse-function-based transmission strategy for the ESDD channel. Next, we take an in-depth look into the DDOP and show that it achieves orthogonality with respect to the fine time and frequency resolutions in the ESDD domain thus behaves like an impulse function. This allows us to unveil the unique input-output relation of the resultant ODDM modulation over the ESDD channel. We point out that the conventional MC modulation design guidelines based on the Weyl-Heisenberg (WH) frame theory can be relaxed without compromising its orthogonality or violating the WH frame theory. More specifically, for a practical communication system with bandwidth and duration constraints, MC modulation signals can be designed considering so-called local or sufficient (bi)orthogonality, which refers to the (bi)orthogonality among a WH subset for the MC signal within a specific 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.

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