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This paper presents a general framework of constructing a large family of lapped transforms with symmetric basis functions by adding simple time-domain pre- and post-processing modules onto existing block discrete cosine transform (DCT)-based infrastructures. A subset of the resulting solutions is closed-form, fast computable, modular, near optimal in the energy compaction sense and leads to an elegant boundary handling of finite-length data. Starting from these solutions, a general framework for block-based signal decomposition with a high degree of flexibility and adaptivity is developed. Several simplified models are also introduced to approximate the optimal solutions. These models are based on cascades of plane rotation operators and lifting steps, respectively. Despite tremendous savings in computational complexity, the optimized results of these simplified models are virtually identical to that of the complete solution. The multiplierless versions of these pre- and post-filters when combined with an appropriate multiplierless block transform, such as the binDCT, which is described in an earlier paper by Liang and Tran (see IEEE Trans. Signal Processing, vol.49, p.3032-44, Dec. 2001), generate a family of very large scale integration (VLSI)-friendly fast lapped transforms with reversible integer-to-integer mapping. Numerous design examples with arbitrary number of channels and arbitrary number of borrowed samples are presented.