Linear Boundary Extensions for Finite Length Signals and Paraunitary Two-Channel Filterbanks

In this paper, we introduce a novel and general matrix formulation of artificial linear boundary extension methods for removing border effects inherent to any paraunitary two-channel size-limited filterbank. This new characterization of the transformation operator allows us to prove that perfect reconstruction (PR) of finite signals can be ensured under some conditions without using extra subband coefficients; in other words, we characterize the signal extension methods that lead to nonexpansive transforms. The necessary and sufficient condition we find allows us to show that some traditional extension techniques that are being used in an expansive way, such as the polynomial extension, lead in fact to nonexpansive invertible transforms; moreover, we can also prove that in contradiction to previous literature, not every transformation matrix associated with a linear extension is invertible even if using prototype filters of the same length. Apart from these invertibility criteria, we propose the first algorithm for the design of all linear extensions and their associated biorthogonal boundary filters that lead to nonexpansive and invertible transforms. Analogously, we provide the first method for the design of all linear extensions that yield orthogonal transforms: We construct an infinite number of orthogonal extensions, apart from the commonly used periodic extension, and their associated orthogonal boundary filters. The final contribution of the paper is a new algorithm for the design of smooth orthogonal extensions, which keep the orthogonality property and overcome the main drawback of periodization, that is, the introduction of subband coefficients of great amplitude near the boundaries in the transform domain.