  For integrated sensing and communication (ISAC) systems, the channel
information essential for communication and sensing tasks fluctuates across
different timescales. Specifically, wireless sensing primarily focuses on
acquiring path state information (PSI) (e.g., delay, angle, and Doppler) of
individual multi-path components to sense the environment, which usually
evolves much more slowly than the composite channel state information (CSI)
required for communications. Typically, the CSI is approximately unchanged
during the channel coherence time, which characterizes the statistical
properties of wireless communication channels. However, this concept is less
appropriate for describing that for wireless sensing. To this end, in this
paper, we introduce a new timescale to study the variation of the PSI from a
channel geometric perspective, termed path invariant time, during which the PSI
largely remains constant. Our analysis indicates that the path invariant time
considerably exceeds the channel coherence time. Thus, capitalizing on these
dual timescales of the wireless channel, in this paper, we propose a novel ISAC
framework exploiting the recently proposed delay-Doppler alignment modulation
(DDAM) technique. Different from most existing studies on DDAM that assume the
availability of perfect PSI, in this work, we propose a novel algorithm, termed
as adaptive simultaneously orthogonal matching pursuit with support refinement
(ASOMP-SR), for joint environment sensing and PSI estimation. We also analyze
the performance of DDAM with imperfectly sensed PSI.Simulation results unveil
that the proposed DDAM-based ISAC can achieve superior spectral efficiency and
a reduced peak-to-average power ratio (PAPR) compared to standard orthogonal
frequency division multiplexing (OFDM).
