Short-time Fractional Fourier Transform Research Articles

The logarithmic, Heisenberg's and short-time uncertainty principles associated with fractional Fourier transform

Firstly, based on the relation between the original function and definition of the fractional Fourier transform (FRFT), one novel logarithmic uncertainty principle and one Heisenberg's uncertainty principle in the FRFT domains are derived, which are associated with the FRFT parameter. Secondly, the uncertainty principle for the short-time fractional Fourier transform (STFRFT) is obtained. The novel logarithmic, Heisenberg's and the short-time uncertainty principles connect their bounds with the FRFT parameter, which can help us understand the physical interpretations of the uncertainty principle in nature. This discloses that for these new transforms involving the FRFT parameter the uncertainty principles continue to hold.

Short-time fractional Fourier methods for the time-frequency representation of chirp signals.

The fractional Fourier transform (FrFT) provides a valuable tool for the analysis of linear chirp signals. This paper develops two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals. Outputs have similar properties to the short-time Fourier transform (STFT) but show improved time-frequency resolution. The FrFT is a parameterized transform with parameter, a, related to chirp rate. The two short-time implementations differ in how the value of a is chosen. In the first, a global optimization procedure selects one value of a with reference to the entire signal. In the second, a values are selected independently for each windowed section. Comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains. For appropriately chosen FrFT orders, the derived fractional domain uncertainty relationship is minimized for Gaussian windowed linear chirp signals. The two short-time FrFT algorithms have complementary strengths demonstrated by time-frequency representations for a multicomponent bat chirp, a highly nonlinear quadratic chirp, and an output pulse from a finite-difference sonar model with dispersive change. These representations illustrate the improvements obtained in using FrFT based algorithms compared to the STFT.