Statistical Resolution Limits and the Complexified
Steven T. Smith · 2005
Array resolution limits and accuracy bounds on the multitude of signal parameters (e.g., azimuth, elevation, Doppler, range, cross-range, depth, frequency, chirp, polarization, ampli- tude, phase, etc.) estimated by array processing algorithms are es- sential 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 formula- tion of these bounds (from the perspective of derivations and anal- ysis) is presented for the case of deterministic and unknown signal amplitudes. A new derivation is given using the unknown com- plex signal parameters and their complex conjugates. The new for- mula is readily applicable to obtaining either symbolic or numer- ical solutions to estimation bounds for a very wide class of prob- lems encountered in adaptive sensor array processing. This for- mula is shown to yield several of the standard Cramer-Rao results for array processing, along with new results of fundamental in- terest. Specifically, a new closed-form expression for the statistical resolution limit of an aperture for any asymptotically unbiased su- perresolution algorithm (e.g., MUSIC, ESPRIT) is provided. The statistical resolution limit is defined as the source separation that equals its own Cramer-Rao bound, providing an algorithm-inde- pendent bound on the resolution of any high-resolution method. It is shown that the statistical resolution limit of an array or co- herent integration window is about SNR relative to the Fourier resolution limit of radians (large number of array elements). That is, the highest achievable resolution is pro- portional to the reciprocal of the fourth root of the signal-to-noise ratio (SNR), in contrast to the square-root SNR dependence of standard accuracy bounds. These theoretical results are con- sistent with previously published bounds for specific superresolu- tion 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 aper- ture wavelengths by wavelengths (assumed large), is approx- imately , in contrast to the resolution improvement of for a full aperture. Exact closed-form results for these prob- lems with their asymptotic approximations are presented.