Improved Parameter Estimation for First-Order Markov Process

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)=[𝑎₁𝑎₂⋯𝑎_𝑁]^𝑇,suchthat

𝑦⃗=𝐻𝑥⃗+𝑣,⃗(1) where, 𝑦(⃗𝑛)=[ℎ(𝑛)ℎ(𝑛−1)⋯ℎ(𝑛−𝑀+1)]^𝑇 is 𝑀×1 output vector,

𝐻(𝑛)=⎡⎢⎢⎣ℎℎℎ(((𝑛𝑛𝑛−−−𝑀12)))ℎ(ℎℎ𝑛−((𝑛𝑛𝑀−−23−))1)⋯⋯⋯ℎ(ℎ𝑛−(ℎ𝑛(𝑁−𝑛−𝑁−𝑁𝑀−)1+)1)⎤⎥⎥⎦(2) is 𝑀×𝑁 complex matrix, and 𝑣(⃗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 𝜎^2𝑣𝐼_𝑀, where 𝐼_𝑀 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 𝑇_ZF=(𝐻^𝐻𝐻)⁻¹𝐻^𝐻 to 𝑦⃗ results in nonwhite noise with covariance matrix 𝐶_ZF=𝜎^2𝑣(𝐻^𝐻𝐻)⁻¹. On the other hand, MMSE linear transformation 𝑇_MMSE=(𝐻^𝐻𝐻+𝜎^2𝑣𝐼_𝑁)⁻¹𝐻^𝐻 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 𝑇_WTLS=𝐵(𝐻^𝐻𝐻)^−1/2𝐻^𝐻 [2], where 𝐵=diag[𝛽₁,𝛽₂,…,𝛽_𝑁] with 𝛽=𝛽₁=𝛽₂=⋯=𝛽_𝑁=Tr{(𝐻^𝐻𝐻)^1/2}/Tr{𝐻^𝐻𝐻}. It follows that

𝑥̂_WTLS=𝐵𝐻^𝐻𝐻^1/2𝑥⃗+𝐵𝐻^𝐻𝐻^−1/2𝐻^𝐻𝑣.⃗(3) Substitution of the unique QR-decomposition 𝐻=𝑄𝐷𝑀 in (3) leads to

𝑥̂_WTLS=𝐵𝐷𝑀𝑥⃗+𝐵𝑄^𝐻𝑣⃗(4) where, 𝑄=[𝑞₁,𝑞₂,…,𝑞_𝑁] 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=𝐼_𝑁 in (4) yields 𝑥̂_WTLS=𝑀𝑥⃗+𝐷⁻¹𝑄^𝐻𝑣⃗, where 𝐷⁻¹𝑄^𝐻 is referred to as noise whitening-matched filter [4].

2. AR(𝟏) Parameter Estimation

In the presented exposition, the posited linear transformation is 𝑇_MZF=𝑀(𝐻^𝐻𝐻)⁻¹𝐻^𝐻. Consequently,

𝑥̂_MZF=𝑇_MZF𝑦⃗=𝑀𝑥⃗+𝑀𝐻^𝐻𝐻⁻¹𝐻^𝐻𝑣⃗(5) with noise covariance matrix 𝐶_MZF=𝜎^2𝑣𝑀(𝐻^𝐻𝐻)⁻¹𝑀^𝐻. This transformation also performs noise whitening. It is apparent that output noise variance is controlled and reduced, since 0≤𝑚_𝑖,𝑗<1 for 𝑖≠𝑗 (i.e., 𝑖th row and 𝑗th column element of matrix 𝑀). For AR(1) correlation coefficient (𝑎₁) estimation, the unknown parameter vector 𝑥⃗ in (1) and (5) is replaced by 𝑥=[𝑎₁Δ𝑎₂Δ𝑎₃⋯Δ𝑎_𝑁]^𝑇 with leakage coefficients Δ𝑎_𝑗→0. Thus, the estimated parametric value is

𝑥̂_MZF,1=𝑎₁̂=𝑎₁+lim_{Δ𝑎𝑗𝑁→0}_𝑗=2𝑚_1,𝑗Δ𝑎_𝑗+𝑣_MZF,1(6) with 0≤𝑚_1,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 (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.