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In engineering practice, including signal compression sensing, image processing, and speech processing, the construction of wavelet and framelet systems becomes important.In particular, the construction problem of orthogonal wavelets and tight framelets is transferred to the Paraunitary Filter Bank(PUFB) design problem based on the unitary extension principle and the oblique extension principle.In the context of designing a symmetric paraunitary filter bank with a fast implementation framework, we are interested in the parameterization of a symmetric paraunitary filter bank based on the lattice factorization of a polyphase matrix.In this lattice factorization there are unitary matrices corresponding to the order of the polyphase matrix, which can be chosen arbitrarily.Hence, this book presents a new proposal for symmetric paraunitary matrix extension that provides a fast implementation framework with lattice factorization-based parameterization.Based on these methods, we also consider methods for constructing symmetric orthogonal wavelets and symmetric tight framelets that satisfy the shortest support condition.