The experiments of second harmonic generation (SHG) of 57.4 fs pulses at 1550 nm wavelength for type Ⅰ(o+o-e) and type 0 (e+e-e) are achieved using Mg-doped lithium niobate crystal with mole fraction of 5% of magnesium. For SHG of type Ⅰ, the second harmonic pulse with spectrum width of 28 nm and pulse width of 79 fs is obtained at the peak density of 4.3 GW/cm2, and the conversion efficiency is up to 54%. For SHG of type 0, the second harmonic pulse with spectrum width of 2.1 nm is obtained at the peak density of 3.7 GW/cm2, and the conversion efficiency is up to 40%. The key points of the evolution for fundamental pulse and second harmonic pulse in SHG process are analyzed, which are the phase-match of femtosecond pulse multi-wavelength components of fundamental wave in frequency domain and the group-velocity-match of fundamental pulse and second harmonic pulse in time domain. It is found that the spectral width can keep constant in the propagation when the phase-match of multi-wavelength components is satisfied. However, the spectral width will gradually become narrower with the increase of the propagation distance when the phase-match of the center wavelength is satisfied only.