For an \(n\)-dimensional Stein manifold \(V\) and a closed ideal \(I \subset \mathcal{O}(V)\), it is known that if an analytic functional \(T\) satisfies \(Th=0, \forall h\in I\), then there exists a compactly supported \((n,n)\) current \(\widetilde{T}\), s.t. \(\widetilde{T} \phi = 0, \forall \phi \in IE^{n,n}(V)\) and \(Tf=\widetilde{T}f \forall f \in \mathcal{O}(V)\). This paper gives an explicit construction of \(\widetilde{T}\) in terms of the residual currents in the case \(I\) is generated by a complete intersection \(f_{1},...,f_{p} \in \mathcal{O}(V)\), or is a locally Cohen-Macaulay ideal. Besides, for the case of noncomplete intersection, by using integral representation formula the analytic functionals orthogonal to \(I\) are also constructed by currents annihilated by a power of the local integral closure of \(IE(V)\).