Multidimensional Harmonic Retrieval via Coupled Canonical Polyadic Decomposition—Part I: Model and Identifiability

Abstract

Multidimensional Harmonic Retrieval (MHR) is a fundamental problem in signal processing. We make a connection with coupled Canonical Polyadic Decomposition (CPD), which allows us to better exploit the rich MHR structure than existing approaches in the derivation of uniqueness results. We discuss both deterministic and generic conditions. We obtain a deterministic condition that is both necessary and sufficient but which may be difficult to check in practice. We derive mild deterministic relaxations that are easy to verify. We also discuss the variant in which the generators have unit norm. We narrow the transition zone between generic uniqueness and generic non-uniqueness to two values of the number of harmonics. We explain differences with one-dimensional HR.Index Terms-coupled canonical polyadic decomposition, tensor, Vandermonde matrix, multidimensional harmonic retrieval. I. IntroductionDuring the past two decades Multidimensional Harmonic Retrieval (MHR) has become an important problem in signal processing. MHR is a fundamental problem that appears in a wide range of applications in traditional signal processing, such as radar, sonar, wireless communication and channel sounding, see [32], [16], [19], [24], [22], [23], [13], [37] and references therein. The MHR structure can be due to Doppler effects, structured receive and/or transmit antenna arrays, sinusoidal carriers, carrier frequency off-sets, and so on. Another classical signal processing application of MHR is multidimensional NMR spectroscopy (e.g. [21]). More recent MHR applications in signal processing include sampling of parametric nonbandlimited 2D signals [25], phase retrieval of parametric 2D signals [31], phase retrieval

Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects

“…This is analogous to the previously mentioned result (Section III-C2) for a single tensor, that increasing its order N relaxes the bound on R [57], [91]. Adding assumptions such as individual uniqueness of one of the involved CPDs, full column rank of the shared factor C, or a specific structure such as a Vandermonde matrix, also reinforces the uniqueness of the whole decomposition [90], [124]. Finally, all these results can be extended to more elaborate tensor decompositions that are not limited to rank-1 terms [90].…”

“…5.1.1]. Coupled tensor decompositions have already proven useful in telecommunications [125], multidimensional harmonic retrieval (MHR) [124], chemometrics and psychometrics [8], [99], and more. See Fig.…”

“…II]. This type of optimization is used in group NMF [186], coupled NMF [182], certain SCA-based methods [98], [176]- [178], and numerous coupled tensor decompositions; see, e.g., [16], [99], [124], and references therein. In some cases, it may be better to tailor loss functions individually to each data set, and use norms other than Frobenius [8], [153].…”

“…The link is established via the shared ''patients'' mode [155]. As a second example, certain properties of a communication system in a coupled CPD framework may be expressed using a factor with a Vandermonde structure [124]. This can be implemented in SDF with z f a vector of p scalars and x f ðz f Þ a p  q Vandermonde matrix constructed thereof.…”

Analysis of the photic driving effect via joint EEG and MEG data processing based on the coupled CP decomposition

“…Such applications include multirate sampling for array signal processing [5], [6], data fusion with heterogeneous data sets of multiple sources, i.e., social sites or review sites can be processed jointly [7] and data clustering [8]. Moreover, biomedical data analysis can benefit from coupled tensor decompositions because often EEG and MEG recordings are performed simultaneously.…”