  In this paper, we analyze the recently discovered delay-Doppler plane
orthogonal pulse (DDOP), which is essential for delay-Doppler plane
multi-carrier modulation waveform. In particular, we introduce a local
orthogonality property of pulses corresponding to Weyl-Heisenberg (WH) subset
and justify the DDOP's existence, in contrast to global orthogonality
corresponding to WH set governed by the WH frame theory. Then, sufficient
conditions for locally-orthogonal pulses are presented and discussed. Based on
the analysis, we propose a general DDOP design. We also derive the frequency
domain representation of the DDOP, and compare the DDOP-based orthogonal
delay-Doppler division multiplexing (ODDM) modulation with other modulation
schemes, in terms of TF signal localization. Interestingly, we show perfect
local orthogonality property of the DDOP with respect to delay-Doppler
resolutions using its ambiguity function.
