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This paper proposes a new iterative framework for the correction of frequency response mismatch in time-interleaved analog-to-digital converters. Based on a general time-varying linear system model for the mismatch, we treat the reconstruction problem as a linear inverse problem and establish a flexible iterative framework for practical implementation. It encumbrances a number of efficient iterative correction algorithms and simplifies their design, implementation, and performance analysis. In particular, an efficient Gauss-Seidel iteration is studied in detail to illustrate how the correction problem can be solved iteratively and how the proposed structure can be efficiently implemented using Farrow-based variable digital filters with few general-purpose multipliers. We also study important issues, such as the sufficient convergence condition and reconstructed signal spectrum, derive new lower bound of signal-to-distortion-and-noise ratio in order to ensure stable operation, and predict the performance of the proposed structure. Furthermore, we propose an extended iterative structure, which is able to cope with systems involving more than one type of mismatches. Finally, the theoretical results and the effectiveness of the proposed approach are validated by means of computer simulations.