Mixed fourier transforms and image encryption

In this paper, we discuss a concept of the mixed Fourier transformation, when signals are transformed to the time-frequency domain, where the difference between the time and frequency is disappeared. Both cases of continuous-time and discrete-time signals are considered, and properties of the mixed transformations are described. The concept of mixed Fourier transform includes the fractional power of the discrete Fourier transform (DFT) which is referred to as a discrete version of the fractional Fourier transform. Mixed Fourier transformations can be used for calculation of roots of the Fourier and identity transformations. Examples of such root transforms for signal processing and image encryption are given.