Subband decomposition ICA (SDICA), an extension of ICA, assumes that each source is represented as the sum of some independent subcomponents and dependent subcomponents, which have different frequency bands. In this article, we first investigate the feasibility of separating the SDICA mixture in an adaptive manner. Second, we develop an adaptive method for SDICA, namely band-selective ICA (BS-ICA), which finds the mixing matrix and the estimate of the source independent subcomponents. This method is based on the minimization of the mutual information between outputs. Some practical issues are discussed. For better applicability, a scheme to avoid the high-dimensional score function difference is given. Third, we investigate one form of the overcomplete ICA problems with sources having specific frequency characteristics, which BS-ICA can also be used to solve. Experimental results illustrate the success of the proposed method for solving both SDICA and the over-complete ICA problems.