# Improved Parameter Estimation for First-Order Markov Process

Authors: Deepak Batra; Sanjay Sharma; Amit Kumar Kohli
Journal: Research Letters in Signal Processing
 (2009)

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## 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.


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## Body

## 1. Introduction

Let us consider a linear multi-input multioutput model with an unknownN×1 parameter vector x⃗(n)=[a1a2⋯aN]T,suchthat(1)y⃗=Hx⃗+v⃗,
where, y⃗(n)=[h(n)h(n-1)⋯h(n-M+1)]T is M×1 output vector,(2)H(n)=[h(n-1)h(n-2)⋯h(n-N)h(n-2)h(n-3)⋯h(n-N-1)h(n-M)h(n-M-1)⋯h(n-N-M+1)]
is M×N complex matrix, and v⃗(n)=[v(n)v(n-1)⋯v(n-M+1)]T is white Gaussian process noise M×1 vector with zero-mean and covariance matrix σv2IM, where IM is M×M identity matrix. The estimated unknown parameter vector may be defined as x̂=Ty⃗, where “T” is linear transformation involving pseudoinverse. The application of ZF linear transformation TZF=(HHH)-1HH to y⃗ results in nonwhite noise with covariance matrix CZF=σv2(HHH)-1. On the other hand, MMSE linear transformation TMMSE=(HHH+σv2IN)-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(3)x̂WTLS=B(HHH)1/2x⃗+B(HHH)-1/2HHv⃗.
Substitution of the unique QR-decomposition H=QDM̅ in (3) leads to(4)x̂WTLS=BDM̅x⃗+BQHv⃗
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 M̅ is an N×N upper triangular matrix with ones on the diagonal (on contrary to [3]). Incorporation of BD=IN in (4) yields x̂WTLS=M̅x⃗+D-1QHv⃗, where D-1QH is referred to as noise whitening-matched filter [4].


## 2. AR(1) Parameter Estimation

In the presented exposition, the posited linear transformation isTMZF=M̅(HHH)-1HH. Consequently,(5)x̂MZF=TMZFy⃗=M̅x⃗+M̅(HHH)-1HHv⃗
with noise covariance matrix CMZF=σv2M̅(HHH)-1M̅H. This transformation also performs noise whitening. It is apparent that output noise variance is controlled and reduced, since 0≤mi,j<1 for i≠j (i.e., ith  row and jth column element of matrix M̅). For AR(1) correlation coefficient (a1) estimation, the unknown parameter vector x⃗ in (1) and (5) is replaced by x̅=[a1Δa2Δa3⋯ΔaN]T with leakage coefficients Δaj→0. Thus, the estimated parametric value is(6)x̂MZF,1=â1=a1+limΔaj→0∑j=2Nm1,jΔaj+vMZF,1
with 0≤m1,j<1 for j≠1. For parameter values a1=0.95 (true correlation coefficients) and Δa2=Δa3=Δa4≈0.0001 (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.Figure 1
MSIE versus process noise variance (σv2dB).


## 3. Conclusions

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



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*Source: 186250-2009-08-09.xml*