Abstract
A simple model of deconvolution can be described as observing {x(t)} which is a convolution of a signal {s(t)} with a filter {f(j)}, x = s * f. More specifically, we have
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Andrews, B., Davis, R.A., and Breidt, F. J. (2006). “Maximum Likelihood Estimation for All-Pass Time Series Models”. J. Multivariate Analysis. 97, 1638–1659
Andrews, B., Davis, R.A., and Breidt, F. J. (2007). “Rank Estimation for All-Pass Time Series Models”. Annals of Statistics, 35, 844–869.
Andrews, B., Calder, M. and Davis, R.A. (2009). “Maximum likelihood estimation for a-stable autoregressive processes”. Annals of Statistics, 37, 1946–1982.
Brillinger, D. R., Rosenblatt, M. (1967a) “Asymptotic theory of k-th order spectra.” In Spectral Analysis of Time Series, Ed. B. Harris, 153–188. New York, Wiley.
Brillinger, D. R., Rosenblatt, M. (1967b) “Computation and interpretation of k-th order spectra.” In Spectral Analysis of Time Series, Ed. B. Harris, 189–232. New York, Wiley.
Brillinger, D. (1977). “ The Identification of a Particular Nonlinear Time Series System,” Biometrika, 64, 509–515.
Breidt, F. J., Davis, R. A., Lh, K. S., and Rosenblatt, M. (1991). “Maximum Likelihood Estimation for Noncausal Autoregressive Processes”, J. Multivariate Analysis, 36, 175–198.
Breidt, F.J., Davis, R.A., and Trindade, A. (2001). “Least Absolute Deviation Estimation for All-Pass Time Series Models”. Annals of Statistics 29, 919–946
Brockwell, P. J., and Davis, R. A. (1987). Time Series: Theory and Methods. Springer-Verlag, New York.
Cheng, Q. (1990). “Maximum standardized cumulant deconvolution of non-Gaussian linear processes”, Annals of Statistics, 18, 1774–1783.
Donoho, D. (1981). “On minimum entropy deconvolution”, In Applied Time Series Analysis, II. Ed. by Findly, D. F. 565–608. Academic Press New York.
Gamboa, F., and Gassiat, E. (1996). “Blind deconvolution of discrete linear systems”, Annals of Statistics, 24, 1964–1981.
Jian, H., and Pawitan, Y. (1999). “Consistent estimation for non-Gaussian non-causal autoregressive processes”, J. of Time Series Analysis, 20, 417–423.
Kannnan, G., Milani, A. A., Panahi, I., and Briggs, R. (2008). “Equalizing secondary path effects using the periodicity of fMRI accoustic noise”, 30th Annual Intrernational IEEE EMBS Conference, Cancouver, British Columbia, Canada, August 20–24, 2008, 25–28.
Lanne, M., and Saikkonen, P. (2009). “Noncausal vector autoregression”, Bank of Finland Research, Discussion papapers 18, 2009.
Larue, A., Mars, J. I., and Jutten, C. (2006). “Frequency-domain blind deconvolution based on mutual information rate”, IEEE Transactions on Signal Processing, 54, 1771–1781.
Li, T. H. (1995). “Blind deconvolution of linear systems with multilevel stationary inputs”, Annals of Statistics, 23 690–704.
Li, T. H., and Lii, K. S. (2002). “ A joint estimation approach for tw-tone image deblurring by blind deconvolution”, IEEE Tansactions on Image Processing, 11, 847–858.
Lii, K. S., Rosenblatt, M., and Van Atta, C. (1976). “Bispectra measurements in turbulance”, Part 1, J. Fluid Mechanics, 77, 45–62.
Lii, K. S., and Rosenblatt, M. (1982).“Deconvolution and estimation of the transfer function phase and coefficients for non-Gaussian linear processes,” Ann. Statist., 10, 1195–1208.
Lii, K. S., and Rosenblatt, M. (1984). “Remarks on nonGaussian linear processes with additive Gaussian noise”. Lecture Notes in Statistics, V. 26, 185–197. In “Robust and Nonlinear Time Series Analysis” Ed. by Franke, J., Hardle, W., and Martin, D.
Lii, K. S., and Rosenblatt, M. (1985). “A fourth order deconvolution technique for nonGaussian linear processes”. In Krishnaiah, P. R. (eds), Multivariate Analysis VI, 395–410.
Lii, K. S., and Rosenblatt, M. (1988). “Nonminimum phase non-Gaussian deconvolution”, J. Multivariate Analysis, 27, 359–374.
Lii, K. S., and Rosenblatt, M. (1988).“Estimation and Deconvolution When the Transfer Function Has Zeros,” J. Theoretical Probability, 93–113.
Lii, K. S., and Rosenblatt, M. (1990).“Cumulant spectral estimates: Bias and covariance,” Proc. Third Hungarian Colloquium on Limit Theorems in Probability and Statistics (held at PECS), 365–405.
Lii, K. S., and Rosenblatt, M. (1990).“ Asymptotic normality of cumulant spectral estimates,” J. Theoretical Probability, 3, 367–385.
Lii, K. S., and Rosenblatt, M. (1992). “An Approximate Maximum Likelihood Estimation for Non-Gaussian Non-Minimum Phase Moving Average processes”, J. Multivariate Analysis, 43, 272–299.
Lii, K. S., and Rosenblatt, M. (1993). “Bispectra and phase of nonGaussian linear processes”, J. Theoretcal Probability, 6, 579–593.
Lii, K. S., and Rosenblatt, M. (1996). “Maximum Likelihood Estimation for NoGaussian NonMinimum Phase ARMA Sequences”, Statistica Sinica, 6, 1–22.
Lii, K. S., and Rosenblatt, M. (1996).“NonGaussian autoregressive sequences and ran-dom fields”. In Adler, R. J., Muller, P., and Rozovskii, B. (Eds). Stochastic Modeling in Physical Oceanography, 295–309.
Matsuoka, T., and Ulrych, T. J. (1984). “Phase Estimation Using Bispectrum”, Proc. IEEE, 72, 1403–1411.
Nikias, C. L., and Petropulu, A. P. (1993) Higher-order Spectra Analysis, a Nonlinear Signal Process Framework, Prentice Hall, Englewood Cliffs, NJ.
Oppenheim, A. V., and Lim, J. S. (1981). “ The importance of phase in Signals,” Proc. IEEE, 69, 529–541.
Rosenblatt, M., and Van Ness, J. W. (1965). “Estimation of bispectra”, Annals of Mathematical Statistics, 36, 1120–1136.
Rosenblatt, M. (1980). “Linear processes and Bispectra”, J. Appl. Probab., 17, 265–270.
Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhauser, Boston.
Rosenblatt, M. (2000). Gaussian and Non-Gausian Linear Time Series and Random Fields. Springer, New York.
Scargle, J. D. (1981). “Phase-sensitive deconvolution to model random processes with special reference to astronomical data”. In Applied Time Series Analysis, II. Ed. by Findly, D. F. 549–564. Academic Press New York.
Wiggins, R. A. (1978). “Minimum entropy deconvolution”, Geoexploration 16, 21–35.
Xia, B., and Zhang, L. (2007). ”Blind deconvolution in nonminimum phase systems using cascade structure”, EURASIP J. on Advances in Signal Processing, V. 2007, Article ID 48432, 10 pages.
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Davis, R.A., Lii, KS., Politis, D.N. (2011). Rosenblatt’s Contributions to Deconvolution. In: Davis, R., Lii, KS., Politis, D. (eds) Selected Works of Murray Rosenblatt. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8339-8_4
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