# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Detection  
Let us consider the hypothesis test  
\[ \begin{array}{ll}\mathrm{H}_{0}\ :&r(t)=n(t)\\ \mathrm{H}_{1}\ :&r(t)=x(t)+n(t)\end{array} \qquad t\in[-T/2,T/2] \tag{37}\]  
where \(x(t)\) is the signal-of-interest (SOI) and \(n(t)\) is noise. The detection problem consists in deciding which hypothesis holds true on the basis of the observation of the (unique) received signal \(r(t)\) observed in the finite time interval \(t\in[-T/2,T/2]\). If this unique observed signal is modeled as a segment of an infinitely long time series, the natural framework for its statistical characterization is the FOT probability framework. If the joint FOT probability density functions of \(x(t+\tau_{i})\) and \(n(t+\tau_{i})\) for \(t+\tau_{i}\in[0,T]\), \(i=1,\ldots,N\), is unknown or complicated, then the likelihood ratio test (LRT) or the generalized likelihood ratio test (GLRT) cannot be derived. However, sub-optimal receivers can be adopted which use cyclic statistical function measurements as front end data. In this section, several cyclostationarity-based detectors; that is, that adopt cyclic statistical function measurements as front end data, are presented by a unified notation. Specifically, let us denote by \(z_{\tau}^{\alpha}(\theta)\) any of the cyclic statistical functions introduced in Section 2.2. Thus, \(\theta\) represents any of the parameters \(\tau\), \(f\), or \(\xi\). In the case of the 4 th-order cyclic moment, \(\theta\) is the three-dimensional vector of lags \((\tau_{1},\tau_{2},\tau_{3})\). Let \(\widehat{z}_{r}(\alpha,\theta;t_{0})\) be an estimator of \(z_{\tau}^{\alpha}(\theta)\) obtained by observing \(r(t)\) for \(t\in[t_{0}-T/2,t_{0}+T/2]\) (see Section 2.3). In order to simplify the notation, the dependence on \(t_{0}\) in \(\widehat{z}_{r}(\alpha,\theta;t_{0})\) is omitted when unnecessary, if this does not create ambiguity. As discussed in Section 2.3, under mild assumptions, the normalized estimation error  
\[\
u_{T}[\widehat{z}_{r}(\alpha,\theta;t_{0})-z_{\tau}^{\alpha}(\theta)] \tag{38}\]  
is asymptotically (\(T\to\infty\)) complex normal. In (38), the normalizing factor \(\
u_{T}\) is given by \(\
u_{T}=\sqrt{T}\) for the estimators of the cyclic CDF (Section 2.3.1), cyclic autocorrelation (Section 2.3.3), and cyclic 4 th-order moment (Section 2.3.5), by \(\
u_{T}=\sqrt{T\Delta f_{T}}\) for the estimators of the cyclic spectrum (Section 2.3.4), and by \(\
u_{T}=\sqrt{Tb_{T}}\) for the kernel-based estimators of the cyclic CDF and pdf with \(\gamma=0\) (Section 2.3.2). Furthermore, the normalized estimation errors  
\[\
u_{T}\big{[}\widehat{z}_{r}(\alpha_{k},\theta_{k};t_{0})-z_{\tau}^{\alpha_{ k}}(\theta_{k})\big{]} \tag{39}\]  
at pairs \((\alpha_{k},\theta_{k})\), \(k=1,\ldots,K\) can be shown to be asymptotically jointly complex normal (see [2] (Sections 5.2 and 5.7), [45] (Sections 2.4 and 4.7) and [47; 48] for results with the stochastic approach). All the considered cyclostationarity-based detectors have the following structure:  
\[\mathcal{T}\underset{\mathrm{H}_{0}}{\overset{\mathrm{H}_{1}}{\gtrless}}\lambda \tag{40}\]  
where the decision statistics \(\mathcal{T}\) are built from cyclic statistical function measurements \(\widehat{z}_{r}(\alpha,\theta)\) and possibly their FOT statistics as the autocovariance matrix. The thresholds \(\lambda\) are derived analytically when the FOT distribution of \(\mathcal{T}\mid\mathrm{H}_{0}\) can be derived from the FOT asymptotic properties of \(\widehat{z}_{r}(\alpha,\theta;t_{0})\). Otherwise, the thresholds are derived by Monte Carlo trials using side data under \(\mathrm{H}_{0}\). Although the structure and parameters of the proposed detectors are derived in the FOT approach--that is, considering the received signal as a unique time series, without interpreting this time series as a realization of a stochastic process--the detector performances are defined in terms of classical probabilities computed over an ensemble of realizations.