Microscopy techniques using visible photons, x-rays, neutrons, and electrons have made remarkable impact in many scientific disciplines. The microscopic data can often be expressed as the convolution of the spatial distribution of certain properties of the specimens and the inherent response function of the imaging system. The x-ray grating interferometer (XGI), which is sensitive to the deviation angle of the incoming x-rays, has attracted significant attention in the past years due to its capability in achieving x-ray phase contrast imaging with low brilliance source. However, the comprehensive and analytical theoretical framework is yet to be presented. Herein, we propose a theoretical framework termed angular signal radiography (ASR) to describe the imaging process of the XGI system in a classical, comprehensive and analytical manner. We demonstrated, by means of theoretical deduction and synchrotron based experiments, that the spatial distribution of specimens' physical properties, including absorption, refraction and scattering, can be extracted by ASR in XGI. Implementation of ASR in XGI offers advantages such as simplified phase retrieval algorithm, reduced overall radiation dose, and improved image acquisition speed. These advantages, as well as the limitations of the proposed method, are systematically investigated in this paper. (C) 2016 Optical Society of America