The fractional Fourier transform and its Wigner distribution function of a Gaussian-windowed complex chirp

A complex chirp signal is a sustained oscillation with infinite support that models linear frequency modulation, where the instantaneous frequency increases linearly over time. A Gaussian-windowed complex chirp signal shares this feature but offers the advantage of having an effectively finite support. The fractional Fourier transform (FRFT) provides a time-frequency representation of the signal along any axis at an arbitrary angle to the time axis in the combined time-frequency plane. The FRFT is related to a rotation of the Wigner distribution function (WDF), which is another representation of the signal in the time-frequency domain. The first main contribution of this paper is the derivation of closed-form expressions for both the FRFT of a Gaussian-windowed chirp and the WDF of this FRFT. When specialized to the case of the Fourier transform (FT), these results yield explicit formulas for the FT of a Gaussian-windowed chirp and its WDF. Since a Gaussian signal can be regarded as a limiting case of a Gaussian-windowed complex chirp, the general results are also specialized to this important case, providing explicit formulas for the FRFT of a Gaussian signal and its WDF. The case of a pure complex chirp signal is also examined, and closed-form expressions are derived for its FRFT and the WDF of this transform. The second major contribution is the evaluation of the fractional order of the transform and the variance of the Gaussian function used in the signal definition to minimize the spread function, which is defined as the variance of the Gaussian magnitude of the FRFT of the signal.