Exact Criteria for Capacity-Lossless Sensing in Noncoherent ISAC

In this paper, we consider a single-antenna transmitter that sends data over a block-Rayleigh-fading channel with coherence length T, whose gains are known to neither end, while it estimates a fluctuating target from its echo, with the normalized minimum mean-square error as the sensing cost. We derive exact results on the rate-sensing region at finite signal-to-noise ratio (SNR). We prove that, for every T>=3 and every target strength, the best sensing and the capacity are achieved together, so that sensing incurs no rate loss, exactly when the SNR is at least the threshold from which constant-energy signaling achieves capacity. This threshold does not depend on the target, it is the same for a target constant over several blocks, and it lies below -1.22102~dB for every T>=17. Below the threshold, sensing incurs a rate loss, and, when the echo is as strong as the received signal, we show that every input on the boundary strictly between its two end points uses exactly two energy levels, on-off or two positive energies, at every T>=3, and interval-arithmetic certificates locate a change between the two types. Finally, for the coherent trade-off between rate and the Cramer-Rao bound, we show that, for every block length L>=2, the boundary-achieving inputs are not independent across the block wherever the sensing constraint binds, which corrects a recent claim.