Time-frequency plane behavioural studies of harmonic and chirp functions with fractional Fourier transform (FRFT)

The behaviour of harmonic and chirp functions was studied on the time-frequency plane with the help of fractional Fourier transform (FRFT). Studies were also carried out through simulation with different numbers of samples of the functions. Variations were observed in the maximum side-lobe level (MSLL), half main-lobe width (HMLW) and side-lobe fall-off rate (SLFOR) of the functions. The parameters of these functions were compared with a similar set of parameters of some of the popular window functions. It can thus be concluded that in the time-frequency plane, the chirp function provides better spectral parameters than those of Boxcar window function with some particular values of rotational angle. A similar type of inference can also be drawn for the harmonic function in the time-frequency plane. Of course the rotational angle might vary in this case and a comparative analysis was carried out with Fejer window and the cosine-tip window functions. This study may prove to be helpful in replacing these existing window functions in a variety of applications where a particular parameter or group of parameters of the harmonic and chirp functions are found superior.