Velocity Information Geometry of Coherent Intra-CPI Waveform Agility

Abstract: Spectrum sharing forces radars to vary carrier frequency and bandwidth on a pulse-to-pulse basis within a coherent processing interval (CPI). While the resulting range-Doppler distortion is well-studied, the corresponding velocity estimation limit is not. We show that in the resolved-bin slow-time model of coherent agile-CPI processing, the effective Fisher information for radial velocity is the SNR-weighted energy of the carrier-time lever arm that survives projection out of the range and phase nuisance subspace. The carrier sequence thus sets the projection geometry, while the bandwidth sequence enters only through SNR weighting. Two consequences follow. First, the carrier sequence inflates the bound by a closed-form factor governed by the correlation between carrier offset and slow time: randomized or orthogonalized hops are nearly harmless, while ramp-correlated hops can severely degrade velocity information. Second, under matched filtering at equal pulse energy, the velocity Cramer-Rao bound (CRB) is invariant to the bandwidth sequence; a corollary recasts the output-SNR loss of agile-CPI mismatched filtering as a processing cost entering only through a per-pulse mismatch loss. The bound is verified against a brute-force Fisher matrix and Monte-Carlo maximum-likelihood estimation. The result yields a design principle: carrier hopping should be chosen not only for spectral coexistence but also to preserve the velocity-information residual.