## 4 Asymptotic Performances  
It is only possible to study the analytical performances of the proposed estimators asymptotically, i.e., for \(K\) that tends to infinity. In this section we derive the asymptotic variances of all proposed estimators and we will show in the next section that, though the analysis is only valid for \(K\rightarrow\infty\), in some cases we have found that the analytical results may predict performances for sufficiently high \(K\) as well. Unfortunately, it is not possible to predict the number of samples required to get the asymptotic results and we need to run Monte-Carlo simulations to have an idea of how large the number of samples needs to be. The mathematical method that we apply to derive the asymptotic analytical performances is outlined, for example, in [7], and makes it possible to overcome the mathematical intractability of the method that uses the transformation of random variables to get the pdf of the estimator.  
The variance of the estimator derived with the method of moments can be obtained from the first-order Taylor expansion of \(g(\mathbf{T})\) about the point \(\mathbf{T}=E\left(\mathbf{T}\right)=\mathbf{\mu}\), where \(\mathbf{T}\) is the vector of the sample moments and \(g\left(\mathbf{T}\right)\) is the function that maps the sample moments to the estimate. Under the first-order approximation the mean of the estimator is equal to \(g\  
where  
\[M_{m}=\sum_{p=0}^{m}\binom{2m}{2p}\frac{(2p-1)!!}{(2p)!!}\left(2m-2p-1\right)!!A^{2p }\sigma^{2\left(m-p\right)} \tag{27}\]  
and  
\[D_{i,j}=\sum_{p=0}^{i}\sum_{q=0}^{j}\binom{2i}{2p}\binom{2j}{2q}\frac{(2\left(i+j -p-q\right)-1)!!}{(2\left(i+j-p-q\right))!!}\left(2p-1\right)!!\left(2q-1\right)!!A^{2\left(i+j-p-q\right)}\sigma^{2\left(p+q\right)}. \tag{28}\]  
Note that the same covariance matrix can be used to study the asymptotic performances of all proposed estimators as the only difference is in the way the function \(g(T)\) is defined in each case. Furthermore, such a covariance matrix is useful to evaluate the performances of any moment based estimator given the signal model (1).  
To derive the asymptotic variance of all proposed estimators we also need the partial derivatives defined in (24). As we use the second and fourth moment of \(y_{k}\) we only need to consider the two sample moments \(T_{2}\) and \(T_{4}\). The derivatives of the function \(g\left(T_{2},T_{4}\right)\) are easily computed. First, consider the estimator \(\hat{S}\), that is defined by the following function \(g\left(T\right)\)  
\[g\left(T\right)=\sqrt{2T_{2}^{2}-\frac{2}{3}T_{4}}=g\left(T_{2},T_{4}\right). \tag{29}\]  
Hence, the partial derivatives are  
\[\frac{\partial g}{\partial T_{2}}=\frac{2T_{2}}{g\left(T_{2},T_{4}\right)} \tag{30}\]  
and  
\[\frac{\partial g}{\partial T_{4}}=-\frac{1}{3}\frac{1}{g\left(T_{2},T_{4} \right)}. \tag{31}\]  
The above results are useful for the estimator of the noise power \(\hat{N}\) defined by the function \(h\left(T_{2},T_{4}\right)\), as follows.  
\[h\left(T_{2},T_{4}\right)=T_{2}-g\left(T_{2},T_{4}\right). \tag{32}\]  
The corresponding derivatives are  
\[\frac{\partial h}{\partial T_{2}} =1-\frac{\partial g}{\partial T_{2}} \tag{33}\] \[\frac{\partial h}{\partial T_{4}} =-\frac{\partial g}{\partial T_{4}}. \tag{34}\]  
Finally, the derivatives of the function \(f\left(T_{2},T_{4}\right)=g\left(T_{2},T_{4}\right)/h\left(T_{2},T_{4}\right)\), that define the estimator of the SNR \(\hat{\rho}\), are  
\[\frac{\partial f}{\partial T_{2}} =\frac{\partial g}{\partial T_{2}}\frac{1}{h}-g\frac{\partial h} {\partial T_{2}}\frac{1}{h^{2}} \tag{35}\] \[\frac{\partial f}{\partial T_{4}} =\frac{\partial g}{\partial T_{4}}\frac{1}{h}-g\frac{\partial h} {\partial T_{4}}\frac{1}{h^{2}}. \tag{36}\]  
The general expression of the covariance matrix shows that the proposed estimators are consistent. However, it is difficult to know in advance when the asymptotic behavior describes the actual performances of the estimators well.