A discrete-time wavelet transform based on a continuous dilation framework

We present a new form of wavelet transform. Unlike the continuous wavelet transform (CWT) or discrete wavelet transform (DWT), the mother wavelet is chosen to be a discrete-time signal and wavelet coefficients are computed by correlating a given discrete-time signal with continuous dilations of the mother wavelet. The results developed are based on the definition of a discrete-time scaling (dilation) operator through a mapping between the discrete and continuous frequencies. The forward and inverse wavelet transforms are formulated. The admissibility condition is derived, and examples of discrete-time wavelet construction are provided. The new form of wavelet transform is naturally suited for discrete-time signals and provides analysis and synthesis of such signals over a continuous range of scaling factors.