# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 1 Introduction  
A multimodal system can be used to provide significant performance improvements in tracking multiple objects by integrating information from asymmetric sensors. When tracking multiple but similar objects, receiver processing at a given time requires estimation of information that the objects have in common, such as object states, object label and time-varying cardinality. Approaches used for this problem include random finite set methods with probability hypothesis density filtering, multi-Bernoulli or labeled multi-Bernoulli filtering, and more recently, nonparametric Bayesian methods to model state priors [1; 2; 3; 4; 5; 6; 7]. When tracking objects with different types of unknown information, applicable sensing modalities can be appropriately designed to increase overall system performance. In some cases, however, performance can suffer from operational conditions that result in system interference. For example, both target position and user communication messages must be estimated in a system with both radar and wireless communication modalities [8; 9]. However, if the modalities operate in adjacent bands, the ever-increasing demands on the operational spectrum can cause growing levels of interference. Currently, spectrum congestion has affected weather radars [10], airport surveillance radars [11], remote sensing systems [12], and military radar and communications systems [13; 14; 15; 16]. A recent approach to deal with spectrum congestion is system spectrum sharing while designing system parameters to reduce interference. Various methods have been considered in designing the transmit waveforms of different systems that share spectra. Codesign methods construct the transmit waveform of one system to minimize strong interference caused by other systems [17; 18; 19; 20; 21]. Coexistence methods, on the other hand, design a joint transmit waveform for multiple spectrum sharing systems. Various such approaches were developed for coexisting radar and communications systems [18; 22; 23; 24; 25; 26; 27; 28; 29]. For example, both systems shared orthogonal frequency division multiplexing signaling in References [22; 23] and linear frequency-modulated chirp signaling in References [24; 25; 26]. In Reference [27], radar and communications coexistence was achieved using joint channel estimation and adaptive parameter optimization, whereas maximization of the combined mutual information was used in Reference [28]. Note that coexistence has also been considered in heterogeneous systems, such as narrowband and ultrawideband networks and wireless networks [30; 31; 32; 33; 34; 35; 36; 37]. In this manuscript, we propose a new signaling scheme for use in tracking a moving target and a time-varying number of user messages using a coexisting radar and multiuser wireless communications system. The scheme adapts a transmit waveform with the same nonlinear phase function and with parameters that are selected to optimize performance metrics under the constraint of given system criteria. Such criteria include fixed range resolution for estimating target position and fixed gross data rates for communications transmission. The performance metrics considered are interference between users, interference between systems, and error in estimating object states. The transmit waveforms of the communications users were first designed to reduce multiple access user interference. Then, the transmit waveform of the radar was selected so that the interference between the two systems is minimized. A second waveform design approach was considered that uses multiobjective (Pareto) optimization to jointly minimize overall system interference and mean-square error of object state estimation. Tradeoffs in system performance were demonstrated using a pulse-Doppler radar that shares the same spectrum with a multiple phase-shift keying communications system. The rest of the paper is organized as follows. In Section 2, we describe the common transmit waveform used by the two systems.