Integrated Sensing and Communications: A Tutorial on Mathematical Foundations, Signal Processing and System Design for 6G

Abstract

Integrated sensing and communications (ISAC) is a cornerstone of sixth-generation (6G) wireless networks: a single waveform, aperture, and hardware platform simultaneously conveys information and senses the physical environment. Its literature, however, is spread across radar signal processing, estimation and information theory, and wireless communications, which makes it difficult for a newcomer to assemble the full mathematical picture. This tutorial fills that gap. Starting from complex-baseband signal models, we build the required foundations step by step, with self-contained derivations. We develop the sensing toolkit (matched filter, ambiguity function, maximum-likelihood estimation, Fisher information, the Cram\'er-Rao bound (CRB), and Neyman-Pearson detection) and the communication toolkit (Shannon and MIMO capacity, water-filling, and multiuser precoding), and then unify them into the fundamental limits of ISAC: the CRB-rate region, the capacity-distortion function, and the deterministic-random tradeoff. On these foundations we treat waveform design (OFDM radar with two-dimensional FFT processing, OTFS, and jointly optimized waveforms), transmit beamforming as explicit optimization, including semidefinite relaxation, and receiver-side estimation and detection. We then survey the architectures that scale ISAC to the network level (millimeter-wave and terahertz operation, reconfigurable intelligent surfaces, and cell-free networks), together with applications, standardization, and security. A worked MIMO-OFDM design example with concrete numerology ties the mathematics together. Throughout, notation is unified and tabulated, key results are derived rather than merely cited, and seminal contributions are placed in context, so that the reader finishes with both the mathematical fluency and the literature map required to begin research in ISAC.