We developed a theoretical model which describes the improvement of signal-to-noise ratio (SNR) by a grid in digital radiography. The model takes into account the effects of spatial variations in the scatter-to-primary ratio and in the large-area contrast over an image with structured background on quantum noise, and the effects of noise in the imaging system such as electronic noise and digitization noise. Based on the theoretical model, we analyzed the effects of these factors on the SNR when a grid is employed. We performed experimental measurements to evaluate the improvement in the SNR by a grid when quantum noise is the dominant noise source. It was found that the measured SNR improvement factor due to quantum noise agreed closely with that determined from the measured transmission values of a grid, as predicted from our theoretical model. In order to evaluate the relative performance of grids with various geometric design parameters for digital radiographic systems, we employed Monte Carlo calculations and determined the transmission values of a number of grids under various scatter conditions. The calculated SNR improvement factor, due to quantum noise, correlated well with the measured improvement of the SNR by the grids. Our model predicts that the SNR improvement factor depends strongly on the local contrast ratio and also on the scatter-to-primary ratio. The SNR improvement factor is higher in the underpenetrated regions than in the well-penetrated regions of an image.