  The direction of arrival (DOA) estimation algorithms are crucial in
localizing acoustic sources. Traditional localization methods rely on
block-level processing to extract the directional information from multiple
measurements processed together. However, these methods assume that DOA remains
constant throughout the block, which may not be true in practical scenarios.
Also, the performance of localization methods is limited when the true
parameters do not lie on the parameter search grid. In this paper we propose
two trajectory models, namely the polynomial and bandlimited trajectory models,
to capture the DOA dynamics. To estimate trajectory parameters, we adopt two
gridless algorithms: i) Sliding Frank-Wolfe (SFW), which solves the Beurling
LASSO problem and ii) Newtonized Orthogonal Matching Pursuit (NOMP), which
improves over OMP using cyclic refinement. Furthermore, we extend our analysis
to include wideband processing. The simulation results indicate that the
proposed trajectory localization algorithms exhibit improved performance
compared to grid-based methods in terms of resolution, robustness to noise, and
computational efficiency.
