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We consider the design of sufficiently informative inputs for estimating the parameters of a time-varying linear system. Specifically, we assume the output of the system may be modeled as linear combinations of known time-varying elements whose parameters are points in an appropriately chosen grid for the parameter space, such that these parameters are sparse in comparison with the grid size. The sufficiently informative inputs we study are signals that guarantee a unique sparsest solution thereby satisfying minimum requirements for numerically solving the estimation problem. Since in practice the parameters of interest may not lie on the grid constructed for the parameter space, we discuss how our input designs can be modified to provide robustness in the presence of modeling error and noise.