Performance Relative to Cramer-Rao Lower Bound

We now plot the results:

• Since the Cramér-Rao lower bound on the variance of an estimator is given by $(1/J(\alpha))$, we plot this value along with the variance $\mathrm{var}(\hat{\alpha} - \alpha)$ of each set of estimates $\hat{\alpha}$ produced by each estimator.
• Figures 4 and 5 show plots for SNRs of 30dB and 15db, respectively. One thousand runs were used to determine the shown variance.
• Though the approximations invoked in creating the models used here lead to observable error variances, we note that the relative errors appear on the order of one to two percent.
• In applications where multiple sinusoidal signals and their parameters are detected [2], estimators such as those presented here are used as the first step in an iterative process, in general rendering such small errors inconsequential.

Figure 4: Each estimator is accurate in its intended range.

Figure 5: Relative estimator performance is similar at lower SNRs.