Hilbert transform associated with the fractional Fourier transform

The analytic part of a signal f(t) is obtained by suppressing the negative frequency content of f, or in other words, by suppressing the negative portion of the Fourier transform, f/spl circ/, of f. In the time domain, the construction of the analytic part is based on the Hilbert transform f/spl circ/ of f(t). We generalize the definition of the Hilbert transform in order to obtain the analytic part of a signal that is associated with its fractional Fourier transform, i.e., that part of the signal f(t) that is obtained by suppressing the negative frequency content of the fractional Fourier transform of f(t). We also show that the generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.