research letter improvedparameterestimationforfirst...

Hindawi Publishing CorporationResearch Letters in Signal ProcessingVolume 2009, Article ID 186250, 2 pagesdoi:10.1155/2009/186250

Research Letter

Improved Parameter Estimation for First-Order Markov Process

Deepak Batra,1 Sanjay Sharma,2 and Amit Kumar Kohli2

1 Department of Electronics and Communication Engineering, CITM, Faridabad 121002, India2 Department of Electronics and Communication Engineering, Thapar University, Patiala 147004, Punjab, India

Correspondence should be addressed to Amit Kumar Kohli, [email protected]

Received 18 June 2009; Accepted 20 July 2009

Recommended by A. G. Constantinides

This correspondence presents a linear transformation, which is used to estimate correlation coefficient of first-order Markovprocess. 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.

Copyright © 2009 Deepak Batra et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let us consider a linear multi-input multioutput modelwith an unknown N × 1 parameter vector −→x (n) =[a1a2 · · · aN ]T , such that

−→y = H−→x +−→v , (1)

where, −→y (n) = [h(n)h(n− 1) · · ·h(n−M + 1)]T is M × 1output 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)

⎤⎥⎥⎥⎦

(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 vectorwith zero-mean and covariance matrix σ2

v IM , where IM isM ×M identity matrix. The estimated unknown parametervector may be defined as x = T−→y , where “T” is lineartransformation involving pseudoinverse. The application ofZF 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 +σ2

v IN )−1HH alleviates output noise variance by find-ing the optimum balance between data detection and noisereduction [1]. However, the modification of least squaresestimation 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

xWTLS = B(HHH

)1/2−→x + B(HHH

)−1/2HH−→v . (3)

Substitution of the unique QR-decomposition H = QDM in(3) leads to

xWTLS = BDM−→x + BQH−→v (4)

where, Q = [q1,q2, . . . , qN ] is an M × N matrix withorthonormal columns, D is an N × N real diagonal matrixwhose diagonal elements are positive, and M is an N ×N upper triangular matrix with ones on the diagonal (oncontrary to [3]). Incorporation of BD = IN in (4) yieldsxWTLS =M−→x +D−1QH−→v , whereD−1QH is referred to as noisewhitening-matched filter [4].

2. AR(1) Parameter Estimation

In the presented exposition, the posited linear transforma-tion is TMZF =M(HHH)

−1HH . Consequently,

xMZF = TMZF−→y =M−→x + M

(HHH

)−1HH−→v (5)

with noise covariance matrix CMZF = σ2vM(HHH)

−1M

H.

This transformation also performs noise whitening. It is

2 Research Letters in Signal Processing

−40

−35

−30

−25

−20

−15

Mea

nsq

uar

eid

enti

fica

tion

erro

r(d

B)

−50 −40 −30 −20 −10 0 10 20 30 40 50

Process noise variance (dB)

MZFZF

MMSEWTLS

Figure 1: MSIE versus process noise variance (σ2v dB).

apparent that output noise variance is controlled andreduced, since 0 ≤ mi, j < 1 for i /= j (i.e., ith row andjth column element of matrix M). For AR(1) correlationcoefficient (a1) estimation, the unknown parameter vector−→xin (1) and (5) is replaced by x = [a1Δa2Δa3 · · ·ΔaN ]T withleakage coefficientsΔaj → 0. Thus, the estimated parametricvalue is

xMZF,1 = a1 = a1 + limΔaj → 0

N∑

j=2

m1, jΔaj + vMZF,1 (6)

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 1demonstrate that the proposed technique outperforms otheraforementioned linear transformations. However under sim-ilar conditions, the value of β is found to be high in case ofWTLS transform, which in turn increases the output noisevariance.

3. Conclusions

The presented linear transformation based on a typical QR-decomposition (i.e.,QDM) reduces output noise, which isutilized for the efficient estimation of correlation coefficientin first-order Markov process.

References

[1] S. Verdu, Multiuser Detection, Cambridge University Press,Cambridge, UK, 1998.

[2] Y. C. Eldar and A. V. Oppenheim, “MMSE whitening andsubspace whitening,” IEEE Transactions on Information Theory,vol. 49, no. 7, pp. 1846–1851, 2003.

[3] A. K. Kohli and D. K. Mehra, “A two-stage MMSE multiuserdecision feedback detector using successive/parallel interferencecancellation,” Digital Signal Processing, vol. 17, no. 6, pp. 1007–1018, 2007.

[4] D. W. Waters and J. R. Barry, “Noise-predictive decision-feedback detection for multiple-input multiple-output chan-nels,” IEEE Transactions on Signal Processing, vol. 53, no. 5, pp.1852–1859, 2005.

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