Asymptotic Bias and Variance of Conventional Bispectrum Estimates for 2-D Signals

  • Jean-Marc Le Caillec
  • Rene Garello


In this paper, we derive the asymptotic bias and variance of conventional bispectrum estimates of 2-D signals. Two methods have been selected for the estimation: the first one – the indirect method – is the Fourier Transform of the weighted third order moment, while the second one – the direct method – is the expectation of the Fourier component product. Most of the developments are known for 1-D signals and the first contribution of this paper is the rigorous extension of the results to 2-D signals. The calculation of the bias of the direct method is a totally original contribution. Nevertheless, we did all calculations (bias and variance) for both method in order to be able to compare the results. The second contribution of this paper consists of the comparison of the theoretical bispectrum estimate bias and variance with the measured bias and variance for two 2-D signals. The first studied signal is the output of a non-minimal phase linear system driven by a non-symmetric noise. The second signal is the output of a non-linear system with Gaussian input data. In order to assess the results, we performed the comparison for both methods with different sets of parameters. We show that the maximum bias coefficient is the one of the 1-D case multiplied by the dimensionality of the signal for both methods. We also show that the estimate variance coefficient is the 1-D case coefficient with a power equal to the signal dimensionality.


bispectrum estimates variance estimates 2D symbols 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Dept Image et Traitement de l’InformationGET – ENST Bretagne, CNRS FRE 2658 TAMCIC – Equipe TIME, Technopôle Brest-Iroise – CS 83818Brest CedexFrance

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