## 5 Numerical Examples
### Performances of \(\hat{\rho}\)  
In Figure 5 we show the estimated squared bias of the SNR estimator \(\hat{\rho}\) as a function of the number of observations \(K\) at different SNRs, obtained by changing \(S\) with \(N=1\). Results reveal that, in certain cases, the SNR estimator exhibits some bias that depends on both \(S\) and \(N\) to be estimated and the number of samples observed over which the estimate of the SNR is computed. In fact, for a sufficiently large number of observations, the squared bias can be made very small, even if that means that for high values of \(S\) the number of observations required for an unbiased estimator is very large; e.g., with an SNR of \(76.9897\) dB the squared bias is still greater than one with \(2^{22}\) observations. Figure 3: Estimated squared bias \(b^{2}(\hat{N})\) of the noise power estimator \(\hat{N}\) versus the number of observations \(K\) for different values of SNR obtained by changing the signal power \(S\) and with noise power \(N=1\). Each subfigure shows estimated squared biases for different SNRs as reported in the legend. Figure 4: Asymptotic and estimated variances of the noise power estimator \(\hat{N}\) versus the number of observations \(K\) for different values of SNR obtained by changing the signal power \(S\) and with noise power \(N=1\). Each subfigure shows asymptotic (thvar) and estimated (sim) variances for different SNRs along with CRLBs. Estimated variances, denoted with (sim), at different SNRs as functions of the number of observations are shown in Figure 6 where asymptotic variances, denoted with (theor) and corresponding CRLBs are also shown. Figure 5: Estimated squared bias \(b^{2}(\hat{\rho})\) of the SNR estimator \(\hat{\rho}\) versus the number of observations \(K\) for different values of SNR obtained by changing the signal power \(S\) and with noise power \(N=1\). Each subfigure shows estimated squared biases for different SNRs as reported in the legend. In Figure 6a,b we show results for SNRs in the range from \(-3.0103\) dB to \(3.0103\) dB obtained by varying \(S\) with \(N=1\). Within this range the SNR estimator becomes quickly unbiased and the asymptotic performances are reached for small sizes of observation windows. The SNR estimator  
Figure 6: Asymptotic and estimated variances of the SNR estimator \(\hat{\rho}\) versus the number of observations \(K\) for different values of SNR obtained by changing the signal power \(S\) and with noise power \(N=1\). Each subfigure shows asymptotic (thvar) and estimated (sim) variances for different SNRs along with corresponding CRLBs. is consistent but not efficient, as the estimator's variance and the corresponding CRLB are not superimposed. We note that such a gap becomes smaller as S increases. In Figure 6c,d we show results for SNRs within the range from 10.9691 dB to 36.9897 dB. The estimated variances reach the corresponding asymptotic variances, though a greater number of samples is needed. In this case we see that the difference between the estimator's variance and the corresponding CRLBs increases as \(S\) increases. Note that as the SNR increases the dominant term in (23) becomes  
\[\left(\frac{10}{\ln 10}\right)^{2}\frac{1}{K} \tag{39}\]  
that does not depend on the SNR to be estimated, while estimator's asymptotic variances given, in general, by (24), and plotted in Figure 6, still depend on \(S\) and \(N\). Performances degrades as \(S\) increases with \(N=1\). In Figure 6e,f we consider increasing SNRs from 44.9485 dB to 76.9897 dB, a range for which the estimator is biased even for relatively large number of samples. The estimated variances reach the asymptotic performances only when the number of samples is high enough and the SNR estimator becomes substantially unbiased. For example, when the SNR is equal to 76.9897 dB and \(N=1\) we need more than \(2^{22}\) samples to get the asymptotic variance. Most importantly, for higher values of \(S\) the asymptotic performances show that to get acceptable accuracies the number of samples must be increased even further.