  Delay-Doppler multiplexing has recently stirred a great deal of attention in
research community. While multiple studies have investigated pulse-shaping
aspects of this technology, it is challenging to identify the relationships
between different pulse-shaping techniques and their properties. Hence, in this
paper, we classify these techniques into two types, namely, circular and linear
pulse-shaping. This paves the way towards the development of a unified
framework that brings deep insights into the properties, similarities, and
distinctions of different pulse-shaping techniques. This framework reveals that
the recently emerged waveform orthogonal delay-Doppler multiplexing (ODDM) is a
linear pulse-shaping technique with an interesting staircase spectral
behaviour. Using this framework, we derive a generalized input-output
relationship that captures the influence of pulse-shaping on the effective
channel. We also introduce a unified modem for delay-Doppler plane
pulse-shaping that leads to the proposal of fast convolution based
low-complexity structures. Based on our complexity analysis, the proposed modem
structures are substantially simpler than the existing ones in the literature.
Furthermore, we propose effective techniques that not only reduce the
out-of-band (OOB) emissions of circularly pulse-shaped signals but also improve
the bit-error-rate (BER) performance of both circular and linear pulse-shaping
techniques. Finally, we extensively compare different pulse-shaping techniques
using various performance metrics.
