## 2 Moment-Based Estimators  
We assume that samples from a real sinusoidal signal in additive noise are available over an observation window of size \(K\), i.e.,  
\[y_{k}=Ax_{k}+w_{k},\qquad k=0,\ldots,K-1, \tag{1}\]  
where \(x_{k}=\cos\left(2\pi\
u k+\phi\right)\); the amplitude \(A\), the frequency \(\
u\) and phase \(\phi\) of the sinusoid are unknown; and the additive Gaussian noise \(w_{k}\) is a random variable with zero-mean and unknown variance \(\sigma^{2}\); i.e., \(w_{k}\sim\mathcal{N}(0;\sigma^{2})\). The signal-to-noise ratio (SNR), defined as \(\rho\triangleq A^{2}/2\sigma^{2}\), is then unknown and represents the parameter we wish to estimate when the other parameters of the sinusoid, frequency and phase, remain unknown. To that purpose we derive an estimator based on the method of moments. The method of moments is a well known statistical method used to derive estimators as functions of sample high-order moments [7]. The key idea is to express the parameter to be estimated as a function of the moments of \(y_{k}\) of a different order--i.e., \(\mu_{m}=E\left[y_{k}^{m}\right]\), where \(m\) is an integer--and use the natural estimators of the moments \(\hat{\mu}_{m}\triangleq(1/K)\sum_{k=0}^{K-1}y_{k}^{m}\) in place of the true moments \(\mu_{m}\) to get an estimate of the parameter from the observed samples. In many cases, such estimators are computationally not intensive and present good statistical performances. Though, in general, the optimality of an estimator derived with this method cannot be guaranteed, in many cases the estimator turns out to be consistent. The method relies on the possibility of expressing moments from the observed random signal as functions of the parameters to be estimated. However, when in the observed signal model, one term is deterministic, as is our case, so the derivation of the moments might be cumbersome, as shown, for example, in [6]. For these cases, [PERSON] suggests to assume a random nature for the \"deterministic\" sinusoid by assuming a random phase, and in this paper we follow such an approach [7] (p. 299). To derive moment-based estimators of the signal power \(S=A^{2}/2\), noise power \(\sigma^{2}\) and SNR \(\rho\) we use the second and fourth order moment, and therefore, we need to compute \(\mu_{2}\) and \(\mu_{4}\). Instead of deriving each moment separately, as done in [12], we show here that a general formula expressing any even-order moment of the signal (1) exists that is valid asymptotically as the number of observed samples grows to infinity. Equivalently, in the signal model (1), where the sinusoidal signal is deterministic, we assume that the sinusoidal signal's phase is a uniform random variable (r.v. ), i.e., \(\phi\sim\mathcal{U}\left[0,2\pi\right]\), independent of the noise \(w_{k}\). Consequently, we can treat the sequence \(y_{k}\) as the realization of a wide sense stationary (WSS) random sequence for which it is possible to calculate statistical moments as, under the given assumption,  
\[\frac{1}{K}\sum_{k=0}^{K-1}x_{k}^{2m}\simeq E\left[x_{k}^{2m}\right], \tag{2}\]  
for \(K\rightarrow\infty\). The general formula for even-order moments for the signal model (1) is given by the following expression. \[\mu_{2m}=\sum_{n=0}^{m}\left( \begin{pmatrix}2m\\ 2n\end{pmatrix} (2m-2n-1)!!\right)\frac{(2n-1)!!}{(2n)!! }A^{2n}\sigma^{2(m-n)}, \tag{3}\]  
where \(! !\) denotes the double factorial operator and its derivation is given in Appendix A.