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.