Research Letter | Open Access
Improved Parameter Estimation for First-Order Markov Process
This correspondence presents a linear transformation, which is used to estimate correlation coefficient of first-order Markov process. It outperforms zero-forcing (ZF), minimum mean-squared error (MMSE), and whitened least-squares (WTLSs) estimators by controlling output noise variance at the cost of increased computational complexity.
Let us consider a linear multi-input multioutput model with an unknown parameter vector
where, is output vector,
is complex matrix, and is white Gaussian process noise vector with zero-mean and covariance matrix , where is identity matrix. The estimated unknown parameter vector may be defined as , where “” is linear transformation involving pseudoinverse. The application of ZF linear transformation to results in nonwhite noise with covariance matrix . On the other hand, MMSE linear transformation alleviates output noise variance by finding the optimum balance between data detection and noise reduction . However, the modification of least squares estimation is based on the concept of MMSE whitening; that is, WTLS performs well at low to moderate signal-to-noise ratios by using linear transformation , where with . It follows that
Substitution of the unique QR-decomposition in (3) leads to
where, is an matrix with orthonormal columns, is an real diagonal matrix whose diagonal elements are positive, and is an upper triangular matrix with ones on the diagonal (on contrary to ). Incorporation of in (4) yields , where is referred to as noise whitening-matched filter .
2. AR Parameter Estimation
In the presented exposition, the posited linear transformation is . Consequently,
with noise covariance matrix . This transformation also performs noise whitening. It is apparent that output noise variance is controlled and reduced, since for (i.e., row and column element of matrix ). For AR(1) correlation coefficient () estimation, the unknown parameter vector in (1) and (5) is replaced by with leakage coefficients . Thus, the estimated parametric value is
with for . For parameter values (true correlation coefficients) and (assumed), the simulation results depicted in Figure 1 demonstrate that the proposed technique outperforms other aforementioned linear transformations. However under similar conditions, the value of is found to be high in case of WTLS transform, which in turn increases the output noise variance.
The presented linear transformation based on a typical QR-decomposition () reduces output noise, which is utilized for the efficient estimation of correlation coefficient in first-order Markov process.
- S. Verdú, Multiuser Detection, Cambridge University Press, Cambridge, UK, 1998.
- Y. C. Eldar and A. V. Oppenheim, “MMSE whitening and subspace whitening,” IEEE Transactions on Information Theory, vol. 49, no. 7, pp. 1846–1851, 2003.
- A. K. Kohli and D. K. Mehra, “A two-stage MMSE multiuser decision feedback detector using successive/parallel interference cancellation,” Digital Signal Processing, vol. 17, no. 6, pp. 1007–1018, 2007.
- D. W. Waters and J. R. Barry, “Noise-predictive decision-feedback detection for multiple-input multiple-output channels,” IEEE Transactions on Signal Processing, vol. 53, no. 5, pp. 1852–1859, 2005.
Copyright © 2009 Deepak Batra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.