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.

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.