Lapped Hadamard transforms and filter banks

In this paper, we generalize the Hadamard transform to the case of lapped transform. A matrix A(z) is a lapped Hadamard transform if it satisfies A^T(z^-1)A(z)=αI for some integer α and all the entries of its coefficient matrices are ±1. Many methods have been proposed to construct lapped Hadamard matrices. In this paper, we study these matrices using the theory of paraunitary filter bank. This approach not only greatly simplifies the analysis of lapped Hadamard transform but also gives rise to new construction methods that can generate a much wider class of lapped Hadamard matrices.