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Two new eigenvector methods for identifying the poles of a linear system from observations of its transient response are presented. The methods have the desirable properties of completely eliminating the spurious zeros associated with the degeneracy of the noise subspace and, at the same time, using this degeneracy to improve the quality of the pole estimates. This is accomplished either by forming appropriate linear combinations of the noise subspace eigenvectors or by working with the signal subspace. The relationship of these methods to MUSIC is discussed and some simulation results are presented.