The procedure for calculating the spectrum of discrete periodic operators perturbed by periodic operators of smaller dimensions is obtained. Some properties of the corresponding algebra of operators with defects are considered. These results yield explicit expressions for propagative, guided and other localized spectra of different physical operators on periodic structures coupled with various defects. As shown in the example of the Schrodinger operator on 2D lattice with linear and point defects, the explicit formula for the localized spectrum can be helpful to solve inverse problems such as a recovery of point defects.