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We develop a rational-dilation wavelet transform for which the dilation factor, the Q-factor and the redundancy can be easily specified. The introduced transform contains Hilbert transform pairs of atoms, therefore it is also suitable for oscillatory signal processing. The transform may be modified to obtain a tight chirplet frame for discrete-time signals. A fast implementation, that makes use of an equivalent filter bank, makes the transform suitable for long signals. Examples on natural signals are provided to demonstrate the utility of the transform.