Some properties of an adaptive filtering structure that employs an analysis filterbank to decompose the input signal and sparse adaptive filters in the subbands are investigated in this paper. The necessary conditions on the filterbank and on the structure parameters for exact modeling of an arbitrary linear system with finite impulse response (FIR) are derived. Then, based on the results obtained for the sparse subfilter structure, a new family of adaptive structures with critical sampling of the subband signals, which can also yield exact modeling, is obtained. Two adaptation algorithms based on the normalized LMS algorithm are derived for the new subband structures with critical sampling. A convergence analysis, as well as a computational complexity analysis, of the proposed adaptive structures are presented. The convergence behavior of the proposed adaptive structures is verified by computer simulations and compared with the behavior of previously proposed algorithms.