2-D and 1-D multipaired transforms: Frequency-time type wavelets

In the present paper, a concept of multipaired unitary transforms is introduced. These kinds of transforms reveal the mathematical structure of Fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images. Considering paired transforms, we analyze simultaneously the splitting of the multidimensional Fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one-dimensional (1-D) "signals," that determine such splitting. The main properties of the orthogonal system of paired functions are described, and the matrix decompositions of the Fourier and Hadamard transforms via the paired transforms are given. The multiplicative complexity of the two-dimensional (2-D) 2(r) x 2(r) -point discrete Fourier transform by the paired transforms is 4(r) / 2(r - 7/3) + 8/3 - 12 (r > 3), which shows the maximum splitting of the 2-D Fourier transform into the number of the short 1-D Fourier transforms. The 2-D paired transforms are not separable and represent themselves as frequency-time-type wavelets for which two parameters are united: frequency and time. The decomposition of the signal is performed in a way that is different from the traditional Haar system of functions.