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In the present paper, we propose a novel method for estimating one-dimensional damped and undamped harmonics. Our method utilizes the multiple shift-invariance property comprised in the signal model. We develop a new rank-reduction estimator which is formed as the weighted sum of the individual matrix polynomials obtained from individual shift-invariance equations. The uniqueness conditions for the proposed rank-reduction criteria are derived under the assumption that all samples are available. Moreover, a novel technique for the incomplete data case, where some samples are missing, is presented. In this case, the rank-reduction estimator may suffer from ambiguities. To overcome this problem, we propose an extension of the rank-reduction estimator that is based on polynomial intersection and the properties of the Sylvester matrix. The latter algorithm yields unique estimates of the damped harmonics. The proposed high-resolution techniques are search-free and therefore, they enjoy moderate computational complexity.