Sufficiently Informative Excitation for Estimation of Linear Responses Due to Sparse Scattering

In this paper, we are concerned with the identification of linear systems' impulse responses modeled as deterministic multipath channels. In the class of channels we study, we consider the effect of delays, Doppler shifts, and different angles of arrival and departure for each signal path. We explore the efficacy of techniques based on sparse signal recovery, which typically define a basis constructed using a finite quantization grid over the parameter space, and approximate the impulse response as a sparse linear combination over such basis. Our goal is to provide guidelines on the design of pilot sequences that are sufficiently informative (SI), i.e., those inputs that guarantee identifiability of system impulse responses that fit in the sparse model. Inputs that are SI provide minimal requirements for uniquely identifying the system response. However, a smaller class of inputs leads to good mean squared estimation error in the presence of noise and modeling errors, due to the finite precision of the parameter space quantization. To single out the class of robust designs, we provide a new metric, called localized coherence, in lieu of the so called mutual coherence, as a measure for ranking SI designs in terms of robustness to noise and to modeling errors.