  This paper further investigates the role of the array geometry and redundancy
in active sensing. We are interested in the fundamental question of how many
point scatterers can be identified (in the angular domain) by a given array
geometry using a certain number of linearly independent transmit waveforms. We
consider redundant array configurations (with repeated virtual transmit-receive
sensors), which we have recently shown to be able to achieve their maximal
identifiability while transmitting fewer independent waveforms than
transmitters. Reducing waveform rank in this manner can be beneficial in
various ways. For example, it may free up spatial resources for transmit
beamforming. In this paper, we show that two array geometries with identical
sum co-arrays, and the same number of physical and virtual sensors, need not
achieve equal identifiability, regardless of the choice of waveform of a fixed
reduced rank. This surprising result establishes the important role the pattern
(not just the number) of repeated virtual sensors has in governing
identifiability, and reveals the limits of compensating for unfavorable array
geometries via waveform design.
