Fourier ptychographic microscopy (FPM) is a promising computational imaging technique with high resolution, wide field-of-view (FOV) and quantitative phase recovery. So far, a series of system errors that may corrupt the image quality of FPM has been reported. However, an imperceptible artifact caused by edge effect caught our attention and may also degrade the precision of phase imaging in FPM with a cross-shape artifact in the Fourier space. We found that the precision of reconstructed phase at the same subregion depends on the different sizes of block processing as a result of different edge conditions, which limits the quantitative phase measurements via FPM. And this artifact is caused by the aperiodic image extension of fast Fourier transform (FFT). Herein, to remove the edge effect and improve the accuracy, two classes of opposite algorithms termed discrete cosine transform (DCT) and periodic plus smooth image decomposition (PPSID) were reported respectively and discussed systematically. Although both approaches can remove the artifacts in FPM and may be extended to other Fourier analysis techniques, PPSID-FPM has a comparable efficiency to conventional FPM algorithm. The PPSID-FPM algorithm improves the standard deviation of phase accuracy as a factor of 4 from 0.08 radians to 0.02 radians. Finally, we summarized and discussed all the reported system errors of FPM within a generalized model.