Using of the cross mutual spectral density of the partially coherent theory, the self-imaging of grating under illumination with arbitrary degree of partial coherence is developed. Both the temporal and the averaged distributions of diffracted intensities at the self-imaging distances are formulated, which can be applied to either continuous or pulsed illuminations with any kind of spectra. The theoretical result shows that the ultra-short laser pulse and the continuous polychromatic illuminations, whose spectrum distributions are the same, have the similar averaged self-image distribution. So, the Talbot effect may help to study the temporal and spectral characteristics of ultra-short laser pulses. An experiment using a light-emitting diode (LED) as the source of illumination is performed, and the experimental results confirm with the theoretical ones.