Generalizing Carathèodory's Uniqueness of Harmonic Parameterization to N dimensions

N. D. Sidiropoulos

doi:10.1109/18.923759

Generalizing Carathèodory's Uniqueness of Harmonic Parameterization to N dimensions

N. D. Sidiropoulos

Research output: Contribution to journal › Letter › peer-review

50 Scopus citations

Abstract

Consider a sum of F exponentials in N dimensions, and let I_n be the number of equispaced samples taken along the nth dimension. It is shown that if the frequencies or decays along every dimensions are distinct and ∑_n=1^N I_n ≤ 2F + (N - 1), then the parameterization in terms of frequencies, decays, amplitudes, and phases is unique. The result can be viewed as generalizing a classic result of Carathéodory to N dimensions. The proof relies on a recent result regarding the uniqueness of low-rank decomposition of N-way arrays.

Original language	English (US)
Pages (from-to)	1687-1690
Number of pages	4
Journal	IEEE Transactions on Information Theory
Volume	47
Issue number	4
DOIs	https://doi.org/10.1109/18.923759
State	Published - May 2001
Externally published	Yes

Keywords

Multidimensional harmonic retrieval
Multiway analysis
PARAllel FACtor (PARAFAC) analysis
Spectral analysis
Uniqueness

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10.1109/18.923759

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title = "Generalizing Carath{\`e}odory's Uniqueness of Harmonic Parameterization to N dimensions",

abstract = "Consider a sum of F exponentials in N dimensions, and let In be the number of equispaced samples taken along the nth dimension. It is shown that if the frequencies or decays along every dimensions are distinct and ∑n=1N In ≤ 2F + (N - 1), then the parameterization in terms of frequencies, decays, amplitudes, and phases is unique. The result can be viewed as generalizing a classic result of Carath{\'e}odory to N dimensions. The proof relies on a recent result regarding the uniqueness of low-rank decomposition of N-way arrays.",

keywords = "Multidimensional harmonic retrieval, Multiway analysis, PARAllel FACtor (PARAFAC) analysis, Spectral analysis, Uniqueness",

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T1 - Generalizing Carathèodory's Uniqueness of Harmonic Parameterization to N dimensions

AU - Sidiropoulos, N. D.

PY - 2001/5

Y1 - 2001/5

N2 - Consider a sum of F exponentials in N dimensions, and let In be the number of equispaced samples taken along the nth dimension. It is shown that if the frequencies or decays along every dimensions are distinct and ∑n=1N In ≤ 2F + (N - 1), then the parameterization in terms of frequencies, decays, amplitudes, and phases is unique. The result can be viewed as generalizing a classic result of Carathéodory to N dimensions. The proof relies on a recent result regarding the uniqueness of low-rank decomposition of N-way arrays.

AB - Consider a sum of F exponentials in N dimensions, and let In be the number of equispaced samples taken along the nth dimension. It is shown that if the frequencies or decays along every dimensions are distinct and ∑n=1N In ≤ 2F + (N - 1), then the parameterization in terms of frequencies, decays, amplitudes, and phases is unique. The result can be viewed as generalizing a classic result of Carathéodory to N dimensions. The proof relies on a recent result regarding the uniqueness of low-rank decomposition of N-way arrays.

KW - Multidimensional harmonic retrieval

KW - Multiway analysis

KW - PARAllel FACtor (PARAFAC) analysis

KW - Spectral analysis

KW - Uniqueness

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SN - 0018-9448

VL - 47

SP - 1687

EP - 1690

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 4

ER -

Generalizing Carathèodory's Uniqueness of Harmonic Parameterization to N dimensions

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Keywords

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