useful to filter noise, but with the condition that it does not overlap with the desired signal in the time–frequency domain. Consider the following example. We cannot apply a filter directly to eliminate the noise, but with the help of the fractional Fourier transform, we can rotate the signal (including the desired signal and noise) first. We then apply a specific filter, which will allow only the desired signal to pass. Thus the noise will be removed completely. Then we use the fractional Fourier transform again to rotate the signal back and we can get the desired signal. Thus, using just truncation in the time domain, or equivalently low-pass filters in the frequency domain, one can cut out any convex set in time–frequency space. In contrast, using time domain or frequency domain tools without a fractional Fourier transform would only allow cutting out rectangles parallel to the axes. Fractional Fourier transforms also have applications in quantum physics. For example, they are used to formulate entropic uncertainty relations. They are also useful in the design of optical systems and for optimizing holographic storage efficiency. See also Least-squares spectral analysis Fractional calculus Mehler kernel Other time–frequency transforms: Linear canonical transformation Short-time Fourier transform Wavelet transform Chirplet transform Cone-shape distribution function Quadratic Fourier transform References Bibliography External links DiscreteTFDs -- software for computing the fractional Fourier transform and time–frequency distributions "Fractional Fourier Transform" by Enrique Zeleny, The Wolfram Demonstrations Project. Dr YangQuan Chen's FRFT (Fractional Fourier Transform) Webpages LTFAT - A free (GPL) Matlab / Octave toolbox Contains several version of the fractional Fourier transform. Fourier analysis Time–frequency analysis Integral transforms Articles containing video clips