The continuous generalized wavelet transform (GWT) which is regarded as a kind of time-linear canonical domain\n(LCD)-frequency representation has recently been proposed. Its constant-Q property can rectify the limitations of the\nwavelet transform (WT) and the linear canonical transform (LCT). However, the GWT is highly redundant in signal\nreconstruction. The discrete linear canonical wavelet transform (DLCWT) is proposed in this paper to solve this\nproblem. First, the continuous linear canonical wavelet transform (LCWT) is obtained with a modification of the GWT.\nThen, in order to eliminate the redundancy, two aspects of the DLCWT are considered: the multi-resolution\napproximation (MRA) associated with the LCT and the construction of orthogonal linear canonical wavelets. The\nnecessary and sufficient conditions pertaining to LCD are derived, under which the integer shifts of a chirp-modulated\nfunction form a Riesz basis or an orthonormal basis for a multi-resolution subspace. A fast algorithm that computes\nthe discrete orthogonal LCWT (DOLCWT) is proposed by exploiting two-channel conjugate orthogonal mirror filter\nbanks associated with the LCT. Finally, three potential applications are discussed, including shift sampling in\nmulti-resolution subspaces, denoising of non-stationary signals, and multi-focus image fusion. Simulations verify the\nvalidity of the proposed algorithms.
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