### Abstract

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 I equispaced samples are taken along one dimension and, likewise, J along the other dimension. We prove that if the number of exponentials is less than or equal to roughly I J/4, 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 (I + J)/2. For example, consider I = J = 32; our result yields 256 versus 32 ide ntifiable exponentials. We also generalize the result to N dimensions, proving that the number of exponentials that can be resolved is proportional to total sample size.

Original language | English (US) |
---|---|

Pages (from-to) | 1849-1859 |

Number of pages | 11 |

Journal | IEEE Transactions on Signal Processing |

Volume | 49 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2001 |

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### Keywords

- Array signal processing
- Frequency estimation
- Harmonic analysis
- Multidimensional signal processing
- Spectral analysis

### Cite this

*IEEE Transactions on Signal Processing*,

*49*(9), 1849-1859. https://doi.org/10.1109/78.942615

**Almost-sure identifiability of multidimensional harmonic retrieval.** / Jiang, Tao; Sidiropoulos, Nikolaos; Ten Berge, Jos M.F.

Research output: Contribution to journal › Article

*IEEE Transactions on Signal Processing*, vol. 49, no. 9, pp. 1849-1859. https://doi.org/10.1109/78.942615

}

TY - JOUR

T1 - Almost-sure identifiability of multidimensional harmonic retrieval

AU - Jiang, Tao

AU - Sidiropoulos, Nikolaos

AU - Ten Berge, Jos M.F.

PY - 2001/9/1

Y1 - 2001/9/1

N2 - 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 I equispaced samples are taken along one dimension and, likewise, J along the other dimension. We prove that if the number of exponentials is less than or equal to roughly I J/4, 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 (I + J)/2. For example, consider I = J = 32; our result yields 256 versus 32 ide ntifiable exponentials. We also generalize the result to N dimensions, proving that the number of exponentials that can be resolved is proportional to total sample size.

AB - 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 I equispaced samples are taken along one dimension and, likewise, J along the other dimension. We prove that if the number of exponentials is less than or equal to roughly I J/4, 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 (I + J)/2. For example, consider I = J = 32; our result yields 256 versus 32 ide ntifiable exponentials. We also generalize the result to N dimensions, proving that the number of exponentials that can be resolved is proportional to total sample size.

KW - Array signal processing

KW - Frequency estimation

KW - Harmonic analysis

KW - Multidimensional signal processing

KW - Spectral analysis

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UR - http://www.scopus.com/inward/citedby.url?scp=0035446680&partnerID=8YFLogxK

U2 - 10.1109/78.942615

DO - 10.1109/78.942615

M3 - Article

VL - 49

SP - 1849

EP - 1859

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 9

ER -