Journal of Electrical and Computer Engineering

Journal of Electrical and Computer Engineering / 2009 / Article

Research Letter | Open Access

Volume 2009 |Article ID 186250 | 2 pages | https://doi.org/10.1155/2009/186250

Improved Parameter Estimation for First-Order Markov Process

Academic Editor: A. G. Constantinides
Received18 Jun 2009
Accepted20 Jul 2009
Published09 Aug 2009

Abstract

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.

1. Introduction

Let us consider a linear multi-input multioutput model with an unknown 𝑁×1 parameter vector βƒ—π‘₯(𝑛)=[π‘Ž1π‘Ž2β‹―π‘Žπ‘]𝑇,suchthat

⃗⃗𝑦=𝐻⃗π‘₯+𝑣,(1) where, ⃗𝑦(𝑛)=[β„Ž(𝑛)β„Ž(π‘›βˆ’1)β‹―β„Ž(π‘›βˆ’π‘€+1)]𝑇 is 𝑀×1 output vector,

⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦𝐻(𝑛)=β„Ž(π‘›βˆ’1)β„Ž(π‘›βˆ’2)β‹―β„Ž(π‘›βˆ’π‘)β„Ž(π‘›βˆ’2)β„Ž(π‘›βˆ’3)β‹―β„Ž(π‘›βˆ’π‘βˆ’1)β„Ž(π‘›βˆ’π‘€)β„Ž(π‘›βˆ’π‘€βˆ’1)β‹―β„Ž(π‘›βˆ’π‘βˆ’π‘€+1)(2) is 𝑀×𝑁 complex matrix, and ⃗𝑣(𝑛)=[𝑣(𝑛)𝑣(π‘›βˆ’1)⋯𝑣(π‘›βˆ’π‘€+1)]𝑇 is white Gaussian process noise 𝑀×1 vector with zero-mean and covariance matrix 𝜎2𝑣𝐼𝑀, where 𝐼𝑀 is 𝑀×𝑀 identity matrix. The estimated unknown parameter vector may be defined as Μ‚π‘₯=𝑇⃗𝑦, where β€œπ‘‡β€ is linear transformation involving pseudoinverse. The application of ZF linear transformation 𝑇ZF=(𝐻𝐻𝐻)βˆ’1𝐻𝐻 to ⃗𝑦 results in nonwhite noise with covariance matrix 𝐢ZF=𝜎2𝑣(𝐻𝐻𝐻)βˆ’1. On the other hand, MMSE linear transformation 𝑇MMSE=(𝐻𝐻𝐻+𝜎2𝑣𝐼𝑁)βˆ’1𝐻𝐻 alleviates output noise variance by finding the optimum balance between data detection and noise reduction [1]. 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 𝑇WTLS=𝐡(𝐻𝐻𝐻)βˆ’1/2𝐻𝐻 [2], where 𝐡=diag[𝛽1,𝛽2,…,𝛽𝑁] with 𝛽=𝛽1=𝛽2=β‹―=𝛽𝑁=Tr{(𝐻𝐻𝐻)1/2}/Tr{𝐻𝐻𝐻}. It follows that

Μ‚π‘₯WTLS𝐻=𝐡𝐻𝐻1/2𝐻⃗π‘₯+π΅π»π»ξ€Έβˆ’1/2𝐻𝐻⃗𝑣.(3) Substitution of the unique QR-decomposition 𝐻=𝑄𝐷𝑀 in (3) leads to

Μ‚π‘₯WTLS=𝐡𝐷𝑀⃗π‘₯+𝐡𝑄𝐻⃗𝑣(4) where, 𝑄=[π‘ž1,π‘ž2,…,π‘žπ‘] is an 𝑀×𝑁 matrix with orthonormal columns, 𝐷 is an 𝑁×𝑁 real diagonal matrix whose diagonal elements are positive, and 𝑀 is an 𝑁×𝑁 upper triangular matrix with ones on the diagonal (on contrary to [3]). Incorporation of 𝐡𝐷=𝐼𝑁 in (4) yields Μ‚π‘₯WTLS=𝑀⃗π‘₯+π·βˆ’1𝑄𝐻⃗𝑣, where π·βˆ’1𝑄𝐻 is referred to as noise whitening-matched filter [4].

2. AR(𝟏) Parameter Estimation

In the presented exposition, the posited linear transformation is 𝑇MZF=𝑀(𝐻𝐻𝐻)βˆ’1𝐻𝐻. Consequently,

Μ‚π‘₯MZF=𝑇MZF⃗𝑦=𝑀⃗π‘₯+π‘€ξ€·π»π»π»ξ€Έβˆ’1𝐻𝐻⃗𝑣(5) with noise covariance matrix 𝐢MZF=𝜎2𝑣𝑀(𝐻𝐻𝐻)βˆ’1𝑀𝐻. This transformation also performs noise whitening. It is apparent that output noise variance is controlled and reduced, since 0β‰€π‘šπ‘–,𝑗<1 for 𝑖≠𝑗 (i.e., 𝑖th row and 𝑗th column element of matrix 𝑀). For AR(1) correlation coefficient (π‘Ž1) estimation, the unknown parameter vector βƒ—π‘₯ in (1) and (5) is replaced by π‘₯=[π‘Ž1Ξ”π‘Ž2Ξ”π‘Ž3β‹―Ξ”π‘Žπ‘]𝑇 with leakage coefficients Ξ”π‘Žπ‘—β†’0. Thus, the estimated parametric value is

Μ‚π‘₯MZF,1=Μ‚π‘Ž1=π‘Ž1+limΞ”π‘Žπ‘—π‘β†’0𝑗=2π‘š1,π‘—Ξ”π‘Žπ‘—+𝑣MZF,1(6) with 0β‰€π‘š1,𝑗<1 for 𝑗≠1. For parameter values π‘Ž1=0.95 (true correlation coefficients) and Ξ”π‘Ž2=Ξ”π‘Ž3=Ξ”π‘Ž4β‰ˆ0.0001 (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 𝛽 is found to be high in case of WTLS transform, which in turn increases the output noise variance.

3. Conclusions

The presented linear transformation based on a typical QR-decomposition (i.e.,𝑄𝐷𝑀) reduces output noise, which is utilized for the efficient estimation of correlation coefficient in first-order Markov process.

References

  1. S. VerdΓΊ, Multiuser Detection, Cambridge University Press, Cambridge, UK, 1998.
  2. Y. C. Eldar and A. V. Oppenheim, β€œMMSE whitening and subspace whitening,” IEEE Transactions on Information Theory, vol. 49, no. 7, pp. 1846–1851, 2003. View at: Publisher Site | Google Scholar
  3. A. K. Kohli and D. K. Mehra, β€œA two-stage MMSE multiuser decision feedback detector using successive/parallel interference cancellation,” Digital Signal Processing, vol. 17, no. 6, pp. 1007–1018, 2007. View at: Publisher Site | Google Scholar
  4. D. W. Waters and J. R. Barry, β€œNoise-predictive decision-feedback detection for multiple-input multiple-output channels,” IEEE Transactions on Signal Processing, vol. 53, no. 5, pp. 1852–1859, 2005. View at: Publisher Site | Google Scholar

Copyright Β© 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.


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