Exploiting multiple shift-invariances in harmonic retrieval: The incomplete data case

In the present paper, we propose a novel harmonic retrieval (HR) method for estimating the parameters of a damped harmonic mixture. Unlike existing approaches which rely on shift-invariance property of measurement data in the perfect uniform sampling case, here we also consider the incomplete data case where specific samples in the uniform sampling grid are unavailable. We develop a rank-reduction criterion which combines all possible shift-invariances contained in the measurements. For the complete data case, a simple and highly accurate polynomial rooting technique can be directly obtained from this rank-reduction criterion. In the missing sample case, however, we show that the rank-reduction estimator may suffer from ambiguities. To overcome this difficulties, we propose an unambiguous rank-reduction technique based on polynomial intersection. Our algorithm is search-free, exhibits low computational complexity and yet yield high resolution.