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Performance of Mean-Frequency Estimators for Doppler Radar and Lidar

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  • 1 Cooperative Institute for Research in the Environmental Sciences (CIRES), University of Colorado, Boulder, Colorado
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Abstract

The performance of mean-frequency estimators for Doppler radar and lidar measurements of winds is presented in terms of two basic parameters: Φ, the ratio of the average signal energy per estimate to the spectral noise level; and Ω, which is proportional to the number of independent samples per estimate. For fixed Φ and Ω, the Cramer-Rao bound (CRB) (theoretical best performance) for unbiased estimators of mean frequency (normalized by the spectral width of the signal), signal power, and spectral width are essentially independent of the number of data samples M. For Φ, the estimators of mean frequency are unbiased and the performance is independent of M. The spectral domain estimators and covariance based estimators are bounded by the approximate periodogram CRB. The standard deviation of the maximum-likelihood estimator approaches the exact CRB, which can be more than a factor of 2 better than the performance of the spectral domain estimators or covariance-based estimators for typical Ω. For small Φ, the estimators are biased due to the effect of the uncorrelated noise (white noise), which results in uniformly distributed “bad” estimates. The fraction of bad estimates is a function of Φ and M with weak dependence on the parameter Ω. Simple empirical models describe the standard deviation of the good estimates and the fraction of bad estimates. For Doppler lidar and for large Φ, better performance is obtained by using many low-energy pulses instead of one pulse with the same total energy. For small Φ, the converse is true.

Abstract

The performance of mean-frequency estimators for Doppler radar and lidar measurements of winds is presented in terms of two basic parameters: Φ, the ratio of the average signal energy per estimate to the spectral noise level; and Ω, which is proportional to the number of independent samples per estimate. For fixed Φ and Ω, the Cramer-Rao bound (CRB) (theoretical best performance) for unbiased estimators of mean frequency (normalized by the spectral width of the signal), signal power, and spectral width are essentially independent of the number of data samples M. For Φ, the estimators of mean frequency are unbiased and the performance is independent of M. The spectral domain estimators and covariance based estimators are bounded by the approximate periodogram CRB. The standard deviation of the maximum-likelihood estimator approaches the exact CRB, which can be more than a factor of 2 better than the performance of the spectral domain estimators or covariance-based estimators for typical Ω. For small Φ, the estimators are biased due to the effect of the uncorrelated noise (white noise), which results in uniformly distributed “bad” estimates. The fraction of bad estimates is a function of Φ and M with weak dependence on the parameter Ω. Simple empirical models describe the standard deviation of the good estimates and the fraction of bad estimates. For Doppler lidar and for large Φ, better performance is obtained by using many low-energy pulses instead of one pulse with the same total energy. For small Φ, the converse is true.

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