Most motion control systems consist of a desired trajectory generator, a motion controller such as a conventional PID controller, and a plant to be controlled. The desired trajectory generator as well as the motion controller is very important to achieve a good tracking performance. Especially, if the desired trajectory is generated actively utilizing the maximum velocity, acceleration, jerk and snap as given system specifications, the tracking performance would be better. For this, we make use of the properties of convolution operator in order to generate a smooth (S-curve) trajectory satisfying the system specifications. Also, the proposed trajectory generation method is extended to more general cases with arbitrary initial and terminal conditions. In addition, the suggested trajectory generator can be easily realized for real-time implementation. Finally, the effectiveness of the suggested method is shown through numerical simulations.