A 1.25-GHz high speed short-wavelength infrared single-photon detector (SPD) based on InGaAs/InP avalanche photodiode (APD) 1.25-GHz sine wave gating and Bessel LC low-pass filters (LPFs) was described. The avalanche signal amplitude distributions of the SPD at different reverse DC biases are studied experimentally by adjusting the discrimination level of the comparator circuit. As the discrimination level is raised, both detection efficiency and dark count rate reduce exponentially, while the after-pulse probability firstly rises to a peak value and then reduces. It indicates that the discrimination level should be as low as possible to obtain better SPD performance.