PDF's, confidence regions, and relevant statistics for a class of sample covariance-based array processors

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Abstract

In this paper, we add to the many results on sample covariance matrix (SCM) dependent array processors by i) weakening the traditional assumption of Gaussian data and ii) providing for a class of array processors additional performance measures that are of value in practice. The data matrix is assumed drawn from a class of multivariate elliptically contoured (MEC) distributions. The performance measures include the exact probability density functions (pdf's), confidence regions, and moments of the weight vector (matrix), beam response, and beamformer output of certain SCM-based (SCB) array processors. The array processors considered include the SCB: i) maximum-likelihood (ML) signal vector estimator ii) linearly constrained minimum variance beamformer (LCMV) iii) minimum variance distortionless response beamformer (MVDR) iv) generalized sidelobe canceller (GSC) implementation of the LCMV beamformer. It is shown that the exact joint pdf's for the weight vectors/matrices of the aforementioned SCB array processors are a linear transformation from a complex multivariate extension of the standardized t-distribution. The SCB beam responses are generalized t-distributed, and the pdf's of the SCB beamformer outputs are given by Kummer's function. All but the beamformer outputs are shown to be completely invariant statistics over the class of MEC's considered.

Original languageEnglish (US)
Pages (from-to)1779-1993
Number of pages215
JournalIEEE Transactions on Signal Processing
Volume44
Issue number7
DOIs
StatePublished - Dec 1 1996
Externally publishedYes

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Parallel processing systems
Statistics
Probability density function
Covariance matrix
Linear transformations
Maximum likelihood

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering

Cite this

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abstract = "In this paper, we add to the many results on sample covariance matrix (SCM) dependent array processors by i) weakening the traditional assumption of Gaussian data and ii) providing for a class of array processors additional performance measures that are of value in practice. The data matrix is assumed drawn from a class of multivariate elliptically contoured (MEC) distributions. The performance measures include the exact probability density functions (pdf's), confidence regions, and moments of the weight vector (matrix), beam response, and beamformer output of certain SCM-based (SCB) array processors. The array processors considered include the SCB: i) maximum-likelihood (ML) signal vector estimator ii) linearly constrained minimum variance beamformer (LCMV) iii) minimum variance distortionless response beamformer (MVDR) iv) generalized sidelobe canceller (GSC) implementation of the LCMV beamformer. It is shown that the exact joint pdf's for the weight vectors/matrices of the aforementioned SCB array processors are a linear transformation from a complex multivariate extension of the standardized t-distribution. The SCB beam responses are generalized t-distributed, and the pdf's of the SCB beamformer outputs are given by Kummer's function. All but the beamformer outputs are shown to be completely invariant statistics over the class of MEC's considered.",
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