## 5 Numerical Examples
### Performances of \(\hat{S}\)  
First, we verified that \(\hat{S}\) is asymptotically unbiased, as expected from first-order approximation analysis. Results of simulations run at different SNRs show that the estimated squared bias is very small even with a small number of observations. As an example, we report in Figure 1 the estimated squared bias for \(\hat{S}\) at different SNRs obtained by increasing \(S\) and with \(N=1\). Though the squared bias grows with increasing \(S\), and then with the SNR in this case, it can be made quickly very small by increasing the number of observed samples. Some bias is noticeable at high SNR; e.g., for 76.9897 dB with a low number of samples. In Figure 2 we show the results of estimated variance of the estimator \(\hat{S}\) at different SNRs, denoted with (sim), and we compare the results with the corresponding CRLBs that we have derived in Section 3 and asymptotic variances, denoted with (theor), derived in Section 4. Variances are reported on a logarithmic scale as \(10\log_{10}\text{var}\) versus the \(\log_{2}K\), where \(K\) is the number of observed samples. We note that estimated variances reach the corresponding asymptotic performances in the whole range of SNRs under consideration, confirming the validity of the asymptotic performances derivation. However, the number of samples required to get the asymptotic performances depends on the current value of \(S\): for low values of \(S\), the estimator reaches quickly the asymptotic variance, while for high values of \(S\) the number of required sample results to be quite large. Unfortunately, it is not possible to predict when the asymptotic result is valid as the assumption made in performance derivation is to have a large number of samples. Plots in Figure 2 show also how the CRLBs for \(\hat{S}\) depend on the current value of \(S\) and become very high as \(S\) increases. In general, the CRLB for \(\hat{S}\) (16) depends on the product \(SN\) and then for fixed \(N\) it becomes larger as \(S\) increases; i.e., the best accuracy that can be obtained worsens with increasing  
Figure 1: Estimated squared bias \(b^{2}(\hat{S})\) of signal power estimator \(\hat{S}\) versus the number of observations \(K\) for different values of signal-to-noise ratio (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. signal power. On the other hand, the asymptotic variance depends on both \(S\) and \(N\) through terms of the type \(A^{2k}\sigma^{2m}\), with \(k\) and \(m\) non negative integers, that are present in both the derivatives of the function that maps the moments to the estimate, i.e., \(g\left(\mu_{2},\mu_{4}\right)\), as defined in (29), and its derivatives (30) and (31), and the covariance matrix (26). Figure 2: Asymptotic and estimated variances of the signal power estimator \(\hat{S}\) 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. In Figure 2a,b we see that for SNRs equal to \(-3.0103\) dB and \(3.0103\) dB the estimator \(\hat{S}\) is not asymptotically efficient though it is consistent, as the asymptotic variance and the corresponding CRLB are not superimposed. Note that the difference between the asymptotic variance and the CRLB, that cannot be filled by simply increasing the number of observations, can be calculated by means of the results derived in Sections 3 and 4. On the other hand, for higher values of \(S\) with \(N=1\), the asymptotic variance is superimposed to the corresponding CRLB, showing that for a large range of SNRs under consideration the estimator is efficient and that such efficiency is reached in many cases with a relatively small number of samples. However, as \(S\) increases the bound is reached for an increasing number of observed samples.