A general theoretical description of the effect of the second-order dipolar coupling between spin-1/2 and quadrupole nuclei is presented based on the density operator formalism rather than the Shrodinger equation as in previous publications. The main task of this formalism is to diagonalize the evolution operator which can be performed analytically when the quadrupole nucleus is spin-1 or spin-3/2; no approximations such as the adiabatic assumption, perturbation expansion and the average Hamiltonian treatment are assumed. Therefore, it is general and can be used for all nuclear spins and any value quadrupole coupling constant and in the case of magic-angle spinning (MAS) for any rotor spinning speed; it can also include the case when more than one species of spin-1/2 nuclei is coupled to the quadrupole nucleus. The effects of indirect dipolar coupling, chemical shift anisotropy and sample spinning speeds on Nuclear Magnetic Resonance (NMR) lineshapes can be uniformly incorporated in the formalism. Lineshape simulations based on this formalism can yield structural and electronic parameters of compounds and materials with high accuracy. Experimental results for several typical compounds of different complexities are demonstrated and are shown to be in good agreement with the theoretical spectral simulations. (C) 1997 American Institute of Physics.