Breaking the Kurtosis-Capacity Tradeoff in Random ISAC Signals via Short-Block Spherical and Clutter-Trench Constellation Shaping

Abstract

In integrated sensing and communications (ISAC), independent and identically distributed (i.i.d.) symbol signaling exhibits a fundamental trade-off between communication rate and random ranging sidelobes. Under i.i.d. modulation, the average squared autocorrelation and delay-Doppler ambiguity functions decompose into a deterministic mainlobe component ("the iceberg") and a random sidelobe variance floor ("the sea level") governed by the fourth-order amplitude kurtosis of the constellation. Constant-modulus phase-shift keying eliminates zero-Doppler modulation sidelobes at the expense of a 1.53 dB high-SNR shaping loss relative to the additive white Gaussian noise capacity, whereas Gaussian and probabilistically shaped constellations achieve capacity at the cost of a severe random sidelobe floor that masks weak radar targets. This paper breaks the kurtosis-capacity trade-off by introducing short-block spherical and subspace-simplex constellation shaping across the orthogonal frequency-division multiplexing (OFDM) symbols of a single 3GPP New Radio slot. By extending fourth-order moment-matrix theory to non-i.i.d. circularly symmetric ensembles, the delay-Doppler ambiguity variance is shown to reduce to a quadratic form of the symbol-intensity covariance matrix. While one-dimensional fast-time spherical shaping across subcarriers fails to suppress range sidelobes due to simplex exchangeability, two-dimensional slow-time subspace-simplex shaping projects stationary and low-Doppler clutter into the null space of the slow-time intensity covariance matrix while preserving near-Gaussian marginal kurtosis and communication rate. Finite-blocklength achievable rate and converse bounds are derived, together with a practical constant-composition distribution matching transceiver over concentric amplitude-phase-shift keying rings. Numerical evaluations over standardized 3GPP slot formats confirm that slow-time subspace-simplex shaping suppresses the zero- and low-Doppler clutter sidelobe floor by more than 32 dB and recovers weak targets masked under i.i.d. signaling while operating within 93% to 98% of the Gaussian channel capacity.