We report on first-principles band-structure calculations of the semiconducting CuAlO₂ delafossite compound in the pure form and also with Cd impurity occupying either a Cu or Al position. The computational tool was a full-potential linear augmented plane-wave method, with the generalized gradient approximation accounting for the exchange and correlation effects. The changes caused by the presence of Cd are studied by the analysis of the electronic structure and the electric field gradient (EFG) in both Cd-doped and pure CuAlO₂ systems. Good agreement between the calculated and measured EFGs at Cd substituting for Cu or Al atoms in CuAlO₂ indicates that the calculations were able to correctly describe the ground state of the system containing the impurity. It is shown that a specific hybridization scheme, involving Cu (and Cd) s and d_z² orbitals and neighbouring O p_z orbitals, takes place at the Cu sites in CuAlO₂ as proposed earlier. The results of the calculations indicate that the Cd-doped system changes its electrical properties when Cd replaces Cu atoms (producing an n-type semiconductor), but not when it substitutes for Al atoms.