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Research Letters in Signal Processing
Volume 2009, Article ID 186250, 2 pages
http://dx.doi.org/10.1155/2009/186250
Research Letter

Improved Parameter Estimation for First-Order Markov Process

1Department of Electronics and Communication Engineering, CITM, Faridabad 121002, India
2Department of Electronics and Communication Engineering, Thapar University, Patiala 147004, Punjab, India

Received 18 June 2009; Accepted 20 July 2009

Academic Editor: A. G. Constantinides

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.

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 𝑥(𝑛)=[𝑎1𝑎2𝑎𝑁]𝑇,suchthat

𝑦=𝐻𝑥+𝑣,(1) where, 𝑦(𝑛)=[(𝑛)(𝑛1)(𝑛𝑀+1)]𝑇 is 𝑀×1 output vector,

𝐻(𝑛)=(𝑛1)(𝑛2)(𝑛𝑁)(𝑛2)(𝑛3)(𝑛𝑁1)(𝑛𝑀)(𝑛𝑀1)(𝑛𝑁𝑀+1)(2) is 𝑀×𝑁 complex matrix, and 𝑣(𝑛)=[𝑣(𝑛)𝑣(𝑛1)𝑣(𝑛𝑀+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 ̂𝑥=𝑇𝑦, where “𝑇” is linear transformation involving pseudoinverse. The application of ZF linear transformation 𝑇ZF=(𝐻𝐻𝐻)1𝐻𝐻 to 𝑦 results in nonwhite noise with covariance matrix 𝐶ZF=𝜎2𝑣(𝐻𝐻𝐻)1. On the other hand, MMSE linear transformation 𝑇MMSE=(𝐻𝐻𝐻+𝜎2𝑣𝐼𝑁)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=𝐵(𝐻𝐻𝐻)1/2𝐻𝐻 [2], where 𝐵=diag[𝛽1,𝛽2,,𝛽𝑁] with 𝛽=𝛽1=𝛽2==𝛽𝑁=Tr{(𝐻𝐻𝐻)1/2}/Tr{𝐻𝐻𝐻}. It follows that

̂𝑥WTLS𝐻=𝐵𝐻𝐻1/2𝐻𝑥+𝐵𝐻𝐻1/2𝐻𝐻𝑣.(3) Substitution of the unique QR-decomposition 𝐻=𝑄𝐷𝑀 in (3) leads to

̂𝑥WTLS=𝐵𝐷𝑀𝑥+𝐵𝑄𝐻𝑣(4) where, 𝑄=[𝑞1,𝑞2,,𝑞𝑁] 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 𝐵𝐷=𝐼𝑁 in (4) yields ̂𝑥WTLS=𝑀𝑥+𝐷1𝑄𝐻𝑣, where 𝐷1𝑄𝐻 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)1𝐻𝐻. Consequently,

̂𝑥MZF=𝑇MZF𝑦=𝑀𝑥+𝑀𝐻𝐻𝐻1𝐻𝐻𝑣(5) with noise covariance matrix 𝐶MZF=𝜎2𝑣𝑀(𝐻𝐻𝐻)1𝑀𝐻. This transformation also performs noise whitening. It is apparent that output noise variance is controlled and reduced, since 0𝑚𝑖,𝑗<1 for 𝑖𝑗 (i.e., 𝑖th row and 𝑗th column element of matrix 𝑀). For AR(1) correlation coefficient (𝑎1) estimation, the unknown parameter vector 𝑥 in (1) and (5) is replaced by 𝑥=[𝑎1Δ𝑎2Δ𝑎3Δ𝑎𝑁]𝑇 with leakage coefficients Δ𝑎𝑗0. Thus, the estimated parametric value is

̂𝑥MZF,1=̂𝑎1=𝑎1+limΔ𝑎𝑗𝑁0𝑗=2𝑚1,𝑗Δ𝑎𝑗+𝑣MZF,1(6) with 0𝑚1,𝑗<1 for 𝑗1. For parameter values 𝑎1=0.95 (true correlation coefficients) and Δ𝑎2=Δ𝑎3=Δ𝑎40.0001 (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.

186250.fig.001
Figure 1: MSIE versus process noise variance (𝜎2𝑣dB).

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

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