Abstract
This paper deals with the identification of an autoregressive (AR) process disturbed by an additive moving-average (MA) noise. Our approach operates as follows: Firstly, the AR parameters are estimated by using the overdetermined high-order Yule–Walker equations. The variance of the AR process driving process can be deduced by means of an orthogonal projection between two types of estimates of AR process correlation vectors. Then, the correlation sequence of the MA noise is estimated. Secondly, the MA parameters are obtained by using inner–outer factorization. To study the relevance of the resulting method, we compare it with existing algorithms, and we analyze the identifiability limits. The identification approach is then combined with Kalman filtering for channel estimation in mobile communication systems.
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Notes
In this case, the exterior of the unit disk in the z-plane plays the role of the unit disk when using the z-transform.
More particularly, the square of its 2-norm defined by: \(\parallel .\parallel _2^2=\tilde{\mathbf {a}}\tilde{\mathbf {a}}^T\) and its \(\infty \)-norm defined by \(\parallel .\parallel _{\infty }=max \lbrace (a_i-\hat{a}_i)\rbrace _{i=1 \ldots p}\).
It should be noted that the channel is approximated with high-order AR model. Here, we assume a second-order AR model to approximate the channel as an example.
References
Nehorai, A., Stoica, P.: Adaptive algorithms for constrained ARMA signals in the presence of noise. IEEE Trans. Acoust. Speech Sign. Process. 36(8), 1282–1291 (1988)
Petitjean, J., Diversi, R., Grivel, E., Guidorzi, R., Roussilhe, P.: Recursive errors-in-variables approach for AR parameter estimation from noisy observations. Application to radar sea clutter rejection. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 3401–3404 (2009)
Wu, W., Chen, P.: Adaptive AR modeling in white Gaussian noise. IEEE Trans. Sign. Process. 45(5), 1184–1191 (1997)
Chung, W., Un, C.: Iterative autoregressive parameter estimation in presence of additive white noise. IEEE Electron. Lett. 27(20), 1800–1802 (1991)
Davila, C.: A subspace approach to estimation of autoregressive parameters from noisy measurements. IEEE Trans. Sign. Process. 46(2), 530–534 (1998)
Petitjean, J., Grivel, E., Bobillet, W., Roussilhe, P.: Multichannel AR parameter estimation from noisy observations as an errors-in-variables issue. Signal, Image Video Process. 4(2), 209–220 (2010)
Bobillet, W., Diversi, R., Grivel, E., Guidorzi, R., Najim, M., Soverini, U.: Speech enhancement combining optimal smoothing and Errors-In-Variables identification of noisy AR processes. IEEE Trans. Sign. Process. 55(12), 5564–5578 (2007)
Zheng, W.: Estimation of the parameters of autoregressive signals from colored noise-corrupted measurements. IEEE Sign. Process. Lett. 7(7), 201–204 (2000)
Mahmoudi, A., Karimi, M.: Parameter estimation of autoregressive signals from observations corrupted with colored noise. Sign. Process. 90(1), 157–164 (2010)
Mahmoudi, A., Karimi, M., Amindavar, H.: Parameter estimation of autoregressive signals in presence of colored AR(1) noise as a quadratic eigenvalue problem. Sign. Process. 92(4), 1151–1156 (2012)
Diversi, R., Ijima, H., Grivel, E.: Prediction error method to estimate the AR parameters when the process is disturbed by a colored noise. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 6143–6147 (2013)
Gibson, J., Boneung, K., Gray, S.: Filtering of colored noise for speech enhancement and coding. IEEE Trans. Sign. Process. 39(8), 1732–1742 (1991)
Durbin, J.: Efficient estimation of parameters in moving average models. Biometrika 46(3/4), 306–316 (1959)
Stoica, P., McKelvey, T., Mari, J.: MA estimation in polynomial time. IEEE Trans. Sign. Process. 48(7), 1999–2012 (2000)
Dumitrescu, B., Tabus, I., Stoica, P.: On the parameterization of positive real sequences and MA parameter estimation. IEEE Trans. Sign. Process. 49(11), 2630–2639 (2001)
Moses, R., Liu, D.: Optimal nonnegative definite approximations of estimated moving average covariance sequences. IEEE Trans. Sign. Process. 39(9), 2007–2015 (1991)
Stoica, A., Moses, R., Stoica, P.: Enforcing positiveness on estimated spectral densities. Electron. Lett. 29(23), 2009–2011 (1993)
Bhansali, R.: A simulation study of autoregressive and window estimators of the inverse correlation function. J R Stat Soc Appl Stat 32(2), 141–149 (1983)
Rusek, F., Anderson, J.: Constrained capacities for faster-than-Nyquist signaling. IEEE Trans. Inf. Theory 55(2), 764–775 (2009)
Box, G., Jenkins, D.: Time Series Analysis, Forecasting and Control. Hoden-Day, San Francisco (1970)
Ljung, L.: System Identification—Theory for the User. Prentice Hall, Englewood Cliffs, NJ (1999)
Merchan, F., Turcu, F., Grivel, E., Najim, M.: Rayleigh fading channel simulator based on inner-outer factorization. Sign. Process. 90(1), 24–33 (2010)
Lukacs, E., King, E.: A property of the normal distribution. Ann. Math. Stat. 25(2), 389–394 (1954)
Najim, M.: Modeling Estimation and Optimal Filtering in Signal Processing. Chapter 3. ISTE Ltd and Wiley, New York (2008)
Hoffman, K.: Banach Spaces of Analytic Functions. Prentice Hall Inc., Englewood Cliffs NJ (1962)
Nikias, C., Mendel, J.: Signal processing with higher order spectra. IEEE Sign. Process. Mag. 10(3), 10–37 (1993)
Lii, K., Rosenblatt, M.: Deconvolution and estimation of transfer function phase and coefficients for nonGaussian linear processes. Ann. Stat. 10(4), 1195–1208 (1982)
Grivel, E., Gabrea, M., Najim, M.: Speech enhancement as a realization issue. Sign. Process., Elsevier 82(12), 1963–1978 (2002)
Grolleau, J., Grivel, E., Najim, M.: Subspace identification method for Rayleigh channel estimation. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), vol. 3, pp. 449–452 (2007)
Jamoos, A., Abdou, A., Abdel-Nour, H., Grivel, E.: Two cross-coupled H\(\infty \) filters for fading channel estimation in OFDM systems. In: Proceedings of the Computer, Information, Systems Sciences and Engineering (CISSE), pp. 349–353. Springer, Heidelberg (2010)
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Abdou, A., Turcu, F., Grivel, E. et al. Identifying an autoregressive process disturbed by a moving-average noise using inner–outer factorization. SIViP 9 (Suppl 1), 235–244 (2015). https://doi.org/10.1007/s11760-015-0803-3
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DOI: https://doi.org/10.1007/s11760-015-0803-3