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 x⃗(n)=[a1a2⋯aN]T,suchthat

y⃗=Hx⃗+v⃗,
where, y⃗(n)=[h(n)h(n-1)⋯h(n-M+1)]T is M×1 output vector,

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

x̂WTLS=B(HHH)1/2x⃗+B(HHH)-1/2HHv⃗.
Substitution of the unique QR-decomposition H=QDM̅ in (3) leads to

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].

In the presented exposition, the posited linear transformation is TMZF=M̅(HHH)-1HH. Consequently,

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

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.

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.

VerdúS.EldarY. C.OppenheimA. V.MMSE whitening and subspace whiteningKohliA. K.MehraD. K.A two-stage MMSE multiuser decision feedback detector using successive/parallel interference cancellationWatersD. W.BarryJ. R.Noise-predictive decision-feedback detection for multiple-input multiple-output channels