Asymptotic Bounds for Frequency Estimation in the Presence of Multiplicative Noise
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Research Article Asymptotic Bounds for Frequency Estimation in the Presence of Multiplicative Noise Zhi Wang and Saman S. Abeysekera School of Electrical and Electronic Engineering, Nanyang Technological University, Block S1, Nanyang Avenue, Singapore 639798 Received 29 January 2006; Revised 27 May 2006; Accepted 13 August 2006 Recommended by Vikram Krishnamurthy We discuss the asymptotic Cramer-Rao bound (CRB) for frequency estimation in the presence of multiplicative noise. To improve numerical stability, covariance matrix tapering is employed when the covariance matrix of the signal is singular at high SNR. It is shown that the periodogram-based CRB is a special case of frequency domain evaluation of the CRB, employing the covariance matrix tapering technique. Using the proposed technique, large sample frequency domain CRB is evaluated for Jake’s model. The dependency of the large sample CRB on the Doppler frequency, signal-to-noise ratio, and data length is investigated in the paper. Finally, an asymptotic closed form CRB for frequency estimation in the presence of multiplicative and additive colored noise is derived. Numerical results show that the asymptotic CRB obtained in frequency domain is accurate, although its evaluation is computationally simple. Copyright © 2007 Hindawi Publishing Corporation. All rights reserved.
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INTRODUCTION
The problem of frequency estimation from noisy signals is of fundamental importance in a variety of applications. Although the performance of frequency estimation in the presence of additive noise is rather well understood, the same can not be stated for frequency estimation in the presence of multiplicative noise. Recently, frequency estimation in the presence of multiplicative noise has received much attention, especially in fading multipath channels, backscatter radar signal processing, and array processing of spatial distributed signals [1–3]. A preliminary step in the development of estimation algorithms in these environments is to identify the fundamental limits of their performance. The Cramer-Rao lower bound (CRB) is a such fundamental lower bound on the variance of any unbiased estimate [4], and is also known to be asymptotically achievable when the number of observations is large. Computation of the exact CRB in the presence of multiplicative noise has been discussed in [3, 5]. However, the exact results are usually presented in matrix form that does not offer much insight into the estimation problem that one is dealing with. Furthermore, it is noticed that under high signal-to-noise ratio (SNR) conditions, the covariance matrix involved in CRB evaluation tends to be singular which makes the derivation of the exact CRB numerically unstable.
This effect is more prominent when the number of observations is large, and in certain multiplicative noise models (e.g., Jake’s model), the effect is quite apparent even at a low number of data samples. Similar singularity problems were also encountered in fading channel simulation, minimum mean square error (MMSE) multiuser de
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