Unlike low-rank matrix decomposition, which is generically nonunique for rank greater than one, low-rank threeand higher dimensional array decomposition is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in the decomposition. Parallel factor (PARAFAC) analysis is a common name for low-rank decomposition of higher dimensional arrays. This paper develops Cramr--Rao Bound (CRB) results for low-rank decomposition of three- and four-dimensional (3-D and 4-D) arrays, illustrates the behavior of the resulting bounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simple-to-check necessary conditions for a unique low-rank decomposition are also provided. Index Terms---Cramr--Rao bound, least squares method, matrix decomposition, multidimensional signal processing. I.

Two-dimensional (2-D) and, more generally, multidimensional harmonic retrieval is of interest in a variety of applications, including transmitter localization and joint time and frequency offset estimation in wireless communications. The associated identifiability problem is key in understanding the fundamental limitations of parametric methods in terms of the number of harmonics that can be resolved for a given sample size. Consider a mixture of 2-D exponentials, each parameterized by amplitude, phase, and decay rate plus frequency in each dimension. Suppose that equispaced samples are taken along one dimension and, likewise, along the other dimension. We prove that if the number of exponentials is less than or equal to roughly , then, assuming sampling at the Nyquist rate or above, the parameterization is almost surely identifiable. This is significant because the best previously known achievable bound was roughly . For example, consider ; our result yields 256 versus 32 identifiable exponentials. We also generalize the result to dimensions, proving that the number of exponentials that can be resolved is proportional to total sample size. Index Terms---Array signal processing, frequency estimation, harmonic analysis, multidimensional signal processing, spectral analysis. I.

Consider a sum of exponentials in dimensions, and let be the number of equispaced samples taken along the th dimension. It is shown that if the frequencies or decays along every dimension are distinct and O PRQ SUT , then the parameterization in terms of frequencies, decays, amplitudes, and phases is unique. The result can be viewed as generalizing a classic result of Carathodory to dimensions. The proof relies on a recent result regarding the uniqueness of low-rank decomposition of-way arrays. Index Terms---Multidimensional harmonic retrieval, multiway analysis, PARAllel FACtor (PARAFAC) analysis, spectral analysis, uniqueness. I.