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In this paper, we propose a decoupled root-MUSIC algorithm adapted to the multidimensional harmonic model, which is widely used in MIMO channel sounding. The optimization criterion of the proposed algorithm is based on multidimensional orthogonal condition testings between a tensor steering manifold parameterized by the parameters of interest and a set of orthogonal projectors associated with each dimension. This criterion can be viewed as a set of decoupled estimation subproblems and allows the use of fast polynomial rooting techniques. In consequence, the proposed algorithm is highly scalable, parallelizable and avoids costly enumerative-based search. However, decoupling property implies to correctly pair the estimated model parameters. So, we propose a fast automatic pairing procedure based on the exploitation of the Vandermonde-structure preserving property of the alternating least squares candecomp/parafac (ALS-CP) algorithm. In addition, we study in a first time the case of a single snapshot and we generalize our algorithm to the multiple snapshots scenario. Finally, by means of numerical simulations, we show that the proposed scheme is efficient for one order of magnitude less complex than other standard algorithms.