This paper investigates jointly optimal array geometry and waveform designs for active sensing. Specifically, we focus on minimizing the Cramér-Rao lower bound (CRB) of the angle of a single target in white Gaussian noise. We first find that several array-waveform pairs can yield the same CRB by virtue of sequences with equal sums of squares, i.e., solutions to certain Diophantine equations. Furthermore, we show that under physical aperture and sensor number constraints, the CRB-minimizing receive array geometry is unique, whereas the transmit array can be chosen flexibly. We leverage this freedom to design a novel sparse array geometry that not only minimizes the single-target CRB given an optimal waveform, but also has a nonredundant and contiguous sum co-array—a desirable property when launching independent waveforms, with relevance also to the multi-target case.

The main focus of this paper is an active sensing application that involves selecting transmit and receive sensors to optimize the Cramér-Rao bound (CRB) on target parameters. Although the CRB is non-convex in the transmit and receive selection, we demonstrate that it is convex in the virtual array weight vector, which describes the multiplicity of the virtual array elements. Based on this finding, we propose a novel algorithm that optimizes the virtual array weight vector first and then finds a matching transceiver array. This greatly enhances the efficiency of the transmit and receive sensor selection problem.

In this paper, we show the mathematical equivalence of two popular modulation schemes: OSDM and OTFS. The former is mainly used in underwater acoustic communications, while the latter scheme is a promising modulation technique in radio-frequency communications. Although literature suggests a link between the two modulation schemes by connecting them to related modulation schemes like V-OFDM and A-OFDM, to the best of the authors’ knowledge, a direct mathematical comparison between the schemes has not been presented yet. The main purpose of this paper is therefore to show the mathematical equivalence of the two schemes. In addition, by combining the knowledge of acoustic and radio-frequency communications, we give insight in the performance of OSDM/OTFS in terms of intersymbol interference (ISI) and intercarrier interference (ICI) by analyzing its signal structure.

The sound field in a room can be represented by a weighted sum of room modes. To estimate the room modes, current literature uses on-the-grid, sparse reconstruction methods. However, these on-the-grid methods are known to suffer from basis mismatch. In this work, we investigate the use of a gridless framework for estimating room modes using atomic norm minimization, a gridless method. The advantage of this approach would be that it does not suffer from this basis mismatch problem. We derive a bound for the sound field reconstruction problem in a one-dimensional room with rigid walls and relate this to the frequency separation that is required by the atomic norm. We conclude that for perfect reconstruction based on the investigated gridless approach, additional prior knowledge about the signal model is required. We show how recovery is possible in a one-dimensional setting by exploiting both the structure of the sound field and the acquisition method.