The linear propagation characteristics of the hyperbolic secant optical pulse with initial linear and nonlinear frequency chirps are numerically studied in the anomalous-dispersion regime of a single mode fiber by use of the split-step Fourier method. It is found that the linear chirped hyperbolic secant pulse gradually evolves into near Gaussian pulse for |C|>0.1, and evolves into near hyperbolic secant pulse for 0≤|C|≤0.1. The smaller |C| is, the more the waveform approaches to hyperbolic secant curve. The effect of the negative linear chirp on the pulse broadening is greater than that of the positive chirp. The effect of the initial linear chirp on broadening of hyperbolic secant pulse is greater than that of Gaussian pulse for |C|≥0.5. The temporal waveform splitting of the hyperbolic secant pulse with nonlinear chirp is more obvious than that of Gaussian pulse during linear propagation. Furthermore, the expression of the time-bandwidth product of the pulse with the linear chirp is given.