On the Fundamental CRB-Rate Tradeoff for GNSS-Integrated Sensing and Communication

We derive a closed-form tradeoff region between the ranging Cramér-Rao bound (CRB) and the achievable communication rate for a single-antenna integrated sensing and communication (ISAC) system that transmits a GNSS-style spread-spectrum ranging waveform jointly with a Gaussian data signal over a shared additive white Gaussian noise (AWGN) channel under a total power budget. The region is parameterized in closed form by a scalar power-split α ∈ [0,1], and the Pareto boundary is convex with a one-line closed-form optimal power allocation. As a specialization, setting α = 1 recovers the classical multi-satellite GNSS position-domain accuracy formula via the geometric dilution of precision (GDOP), providing what is, to the author’s knowledge, the first formal demonstration that classical GNSS dilution of precision is a boundary case of an ISAC CRB-Rate region. Numerical evaluation at a GPS L5 / Galileo E5a ranging bandwidth shows that the closed-form joint design exceeds conventional time-division and frequency-division separation benchmarks by 9%-80% depending on the power split, with gains of approximately 24% and 21% at a representative low-ranging-fraction operating point (α ≈ 0.22, SNR = 40 dB). The result establishes a closed-form analytical baseline for the sensing–communication tradeoff in GNSS-integrated ISAC, and is intended as a building block for the broader research program on integrated positioning, sensing, and communication (IPSC) toward 7G.