As an important ultra-precision measurement method, white light interferometry is widely used in 3D measurements with nanometer resolution. In this paper, a white light interferometer is designed with a random phase noise insensitive algorithm. A discrete interferogram is established by analyzing the phase noise, which is modelized by the combination of random noise and systematic deviation. After Fourier analysis, the mathematical expression of the discrete interferogram in frequency domain is derived, where the random noise can be estimated by least square method and then be corrected. As a result, a more accurate relationship between phase distribution and surface height is established. To set up an stable system, the scanner of white light interferometer is driven by a precision step motor with scanning range 100 mm, and the travel range of the object stage in x and y directions is 60 mm. In the experiment, a step height standard (VLSI, 182.7±2.0 nm) and the end face of a multi-mode optical fiber are tested, where the repeatability error for the step height is less than 0.28%, which proves the measurement accuracy and robustness of the system.