Performance analysis of frequency estimators

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Thesis Type: Doctorate

Institution Of The Thesis: Uludağ Üniversitesi, Turkey

Approval Date: 2007

Thesis Language: Turkish

Student: ERSEN YILMAZ

Supervisor: ERDOĞAN DİLAVEROĞLU

Abstract:

This thesis deals with performance analysis of frequency estimators and analytical Cramér-Rao (C-R) bounds. Two time-series data models are considered, a real damped sinusoid in real white Gaussian noise (real damped model) and two damped cisoids in complex white Gaussian noise (complex damped model). Analytical C-R bounds are derived for estimating the amplitude, phase, damping factor and frequency parameters of both two models. The expressions give the bounds in terms of signal-to-noise ratio, total number of data samples, and a function dependent on the frequency difference between the sinusoids and the damping factors of the sinusoids (it is also dependent on phase difference between the sinusoids for real damped model). The analytical C-R bounds for real damped model are examined as the phase of the sinusoid varies, and simple expressions are obtained for the worst case and the best case C-R bounds and for the corresponding critical phase values. Expressions are then presented as simple closed-form expressions for the case of frequency difference is smaller than Fourier limit (low frequency case) and under the assumptions of low damping and sufficiently large number of data samples. The simple closed-form expressions are valid for low frequency case. The analytical C-R bounds for complex damped model are examined for the case of frequency difference is smaller than Fourier limit (close frequency case) and under the assumptions of low damping and sufficiently large number of data samples, and then presented as simple closed-form expressions, which are valid for whole range of frequency difference between the cisoids. Estimation of frequency parameters of both two models is performed by using Maximum Likelihood Estimator (MLE) and Damped MUSIC (DMUSIC) estimator. The performances of both two estimators are comparatively investigated, especially for the low/close frequency case. It is demonstrated numerically that performance of DMUSIC is close to performance of MLE in the case of low damping and sufficiently large number of data samples. First order analysis are provided for DMUSIC estimator and theoretical bias and variance expressions are derived for estimating the frequency parameters. The thoretical results are supported by numerical examples.