Given a sequence of moments {c n } n∈Z associated with an Hermitian linear functional L defined in the space of Laurent polynomials, we study a new functional L Ω which is a perturbation of L in such a way that a finite number of moments are perturbed.Necessary and sufficient conditions are given for the regularity of L Ω , and a connection formula between the corresponding families of orthogonal polynomials is obtained.On the other hand, assuming L Ω is positive definite, the perturbation is analyzed through the inverse Szeg´ó transformation.