An Uncertainty Principle for Real Signals in the Fractional Fourier Transform Domain

Abstract—The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle in the time–fre-quency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signals is obtained, and it is shown that a Gaussian signal achieves the lower bound. The effect of shifting and scaling the signal on the uncertainty relation is discussed. An example is given in which the uncertainty relation for a real signal is obtained, and it is shown that this relation matches with that given by the uncertainty relation derived. Index Terms—Fractional Fourier transform, time–frequency analysis, uncertainty principle. I