In this paper, we consider a matrix function generalization of the Laplace transform of a random variable, termed the matrix Laplace transform. We characterize the conditions under which the matrix Laplace transform exists, establish its relation to the higher order moments and CCDF of a random variable, and derive the matrix Laplace transform of general Poisson shot noise. Techniques leveraging matrix Laplace transforms can provide improved tractability in the analysis of wireless networks using stochastic geometry. In particular, when one considers the underlying point process of transmitters in the network to follow a Poisson Point Process (PPP), techniques exploiting matrix Laplace transforms provide tractable expressions for the coverage probability of the network when the fading power on the desired signal follows a general phase-type distribution, the metadistribution of the SINR when the fading power on the desired signal follows an exponential distribution, and the distribution of the interference power observed by the typical user in the network.

We develop tractable characterizations of the interference resulting from terrestrial cellular networks radiating towards passive satellite sensing receivers. Such a setting has important implications for the future allocation and terrestrial use of spectrum in the 100 to 300 GHz band. Building on a recently developed stochastic geometry approach, we focus on the outage probability experienced by to a constellation of satellite sensors, which depends upon the distribution of the interference experienced by a typical satellite sensor. The distribution is a function of spatial and temporal randomness. We obtain upper bounds on the outage probability using a large deviation technique for Poisson shot noise, which is a novel adaptation of the Chernoff technique. This analytical method allows for the distribution of the interference to be tightly and tractably bounded. Our analysis theoretically confirms that the satellite sensor's outage probability decreases exponentially as the interference constraint is relaxed, and allows bounding of very low outage probability values, which would be very difficult to simulate.

We consider a generalization of the Laplace transform of Poisson shot noise defined as an integral transform with respect to a matrix exponential. We denote this as the matrix Laplace transform and establish that it is in general a matrix function extension of the scalar Laplace transform. We show that the matrix Laplace transform of Poisson shot noise admits an expression analogous to that implied by Campbell’s theorem. We demonstrate the utility of this generalization of Campbell’s theorem in two important applications: the characterization of a Poisson shot noise process and the derivation of the complementary CDF (CCDF) and meta-distribution of signal-to-interference-and-noise (SINR) models in Poisson networks. In the former application, we demonstrate how the higher order moments of Poisson shot noise may be obtained directly from the elements of its matrix Laplace transform. We further show how the CCDF of this object may be bounded using a summation of the first row of its matrix Laplace transform. For the latter application, we show how the CCDF of SINR models with phase-type distributed desired signal power may be obtained via an expectation of the matrix Laplace transform of the interference and noise, analogous to the canonical case of SINR models with Rayleigh fading. Additionally, when the power of the desired signal is exponentially distributed, we establish that the meta-distribution may be obtained in terms of the limit of a sequence expressed in terms of the matrix Laplace transform of a related Poisson shot noise process.

From an information theoretic perspective, joint communication and sensing (JCAS) represents a natural generalization of communication network functionality. However, it requires the re-evaluation of network performance from a multi-objective perspective. We develop a novel mathematical framework for characterizing the sensing and communication coverage probability and ergodic rate in JCAS networks. We employ a formulation of sensing parameter estimation based on mutual information to extend the notions of coverage probability and ergodic rate to the radar setting. We define sensing coverage probability as the probability that the rate of information extracted about the parameters of interest associated with a typical radar target exceeds some threshold, and sensing ergodic rate as the spatial average of the aforementioned rate of information. Using this framework, we analyze the downlink sensing and communication coverage and rate of a mmWave JCAS network employing a shared waveform, directional beamforming, and monostatic sensing. Leveraging tools from stochastic geometry, we derive upper and lower bounds for these quantities. We also develop several general technical results including: i) a generic method for obtaining closed form upper and lower bounds on the Laplace Transform of a shot noise process, ii) a new analog of Hölder’s Inequality to the setting of harmonic means, and iii) a relation between the Laplace and Mellin Transforms of a non-negative random variable. We use the derived bounds to numerically investigate the performance of JCAS networks under varying base station and blockage density. Among several insights, our numerical analysis indicates that network densification improves sensing SINR performance – in contrast to communications.

Beamforming with high dimensional antenna arrays provides the gain needed to enable high bandwidth communication in the sub-terahertz (THz) band. The resulting narrow beams, however, come at the cost of increased sensitivity to beam alignment errors. A potential remedy to this problem is to introduce a form a macrodiversity through non-coherent joint transmission (NC-JT). We employ a stochastic geometry framework to analyze the performance of a THz network employing user-centric base station clustering and NC-JT. We derive semi-closed form lower bounds for the coverage probability and ergodic capacity experienced by a typical user in the network. Our model includes THz networks without joint transmission as a special case. Overall, our analysis indicates that joint transmission from a few base stations improves coverage and capacity by mitigating the impact of beam misalignment. Moreover, in certain settings, NC-JT is a more efficient usage a network access points compared to the non-cooperative case, and reduces the sensitivity of coverage and capacity to the choice of user beamwidth.

We develop a novel stochastic geometry framework to quantify the SINR coverage probability of a Terahertz (THz) cellular network. THz frequencies will require highly directional beams, leading to inevitably imperfect beam alignment. We derive a tractable and accurate semi-closed form lower bound for the coverage probability of a typical user in the network, introducing a novel approach of characterizing non line-of-sight (NLOS) links as equivalent LoS links using a non-homogeneous Poisson Point Process. We use the coverage bound to investigate the SINR scaling trends with base station density and array directivity at the base station and user. Dense base station deployments are required to achieve sufficient coverage and our analysis exposes a tradeoff between directivity (array gain) and loss in SINR due to misalignment.