Statistical resolution limits and the complexified Crame´r-Rao bound

Array resolution limits and accuracy bounds on the multitude of signal parameters (e.g., azimuth, elevation, Doppler, range, cross-range, depth, frequency, chirp, polarization, amplitude, phase, etc.) estimated by array processing algorithms are essential tools in the evaluation of system performance. The case in which the complex amplitudes of the signals are unknown is of particular practical interest. A computationally efficient formulation of these bounds (from the perspective of derivations and analysis) is presented for the case of deterministic and unknown signal amplitudes. A new derivation is given using the unknown complex signal parameters and their complex conjugates. The new formula is readily applicable to obtaining either symbolic or numerical solutions to estimation bounds for a very wide class of problems encountered in adaptive sensor array processing. This formula is shown to yield several of the standard Crame´r-Rao results for array processing, along with new results of fundamental interest. Specifically, a new closed-form expression for the statistical resolution limit of an aperture for any asymptotically unbiased superresolution algorithm (e.g., MUSIC, ESPRIT) is provided. The statistical resolution limit is defined as the source separation that equals its own Crame´r-Rao bound, providing an algorithm-independent bound on the resolution of any high-resolution method. It is shown that the statistical resolution limit of an array or coherent integration window is about 1.2·SNR^-14/ relative to the Fourier resolution limit of 2π/N radians (large number N of array elements). That is, the highest achievable resolution is proportional to the reciprocal of the fourth root of the signal-to-noise ratio (SNR), in contrast to the square-root (SNR^-12/) dependence of standard accuracy bounds. These theoretical results are consistent with previously published bounds for specific superresolution algorithms derived by other methods. It is also shown that the potential resolution improvement obtained by separating two collinear arrays (synthetic ultra-wideband), each with a fixed aperture B wavelengths by M wavelengths (assumed large), is approximately (M/B)¹2/, in contrast to the resolution improv- ement of M/B for a full aperture. Exact closed-form results for these problems with their asymptotic approximations are presented.