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 technology of 6G wireless networks, in which a single waveform, aperture, and hardware platform is used to simultaneously transmit information and sense the physical environment. While the field has matured rapidly, its literature 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 is written to fill that gap. Starting from complex-baseband signal models, we build the mathematical foundations that a researcher entering ISAC needs. We develop the radar sensing toolkit (the matched filter, the ambiguity function, maximum-likelihood parameter estimation, the Fisher information matrix, the Cramer-Rao bound, and Neyman-Pearson detection), the communication toolkit (Shannon and MIMO capacity, water-filling, and multiuser precoding), and then unify them to derive the fundamental limits of ISAC: the Cramer-Rao bound-rate region, the capacity-distortion function, and the deterministic-random tradeoff. We build on these foundations to address waveform design (OFDM radar with 2D-FFT range-Doppler processing, OTFS, and joint waveform optimization), transmit beamforming optimization, and receiver-side estimation and detection. We then survey the architectures that scale ISAC to network level (mmWave and THz operation, reconfigurable intelligent surfaces, and cell-free networks) and the principal applications, standardization efforts, and security considerations. A fully worked MIMO-OFDM design example ties the mathematics together. Throughout, notation is unified and tabulated, key results are derived, 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.