Perfect-translation-invariant 3-dilation complex discrete wavelet transform based on 3-dilation orthogonal basis

In this paper, firstly, we prove a useful theorem to investigate behavior of discrete wavelet transforms in the frequency domain. Next, based on this theorem, we propose a 3-dilation orthogonal basis, constructed from two types of symmetric and antisymmetric wavelets arrayed alternately. Finally, based on this orthogonal basis, we propose a 3-dilation complex discrete wavelet transform having perfect translation invariance.