# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Single Cycle (SC) Detectors
#### 2.6.2 QF Detectors Based on Measurements of the Cyclic CDF and pdf  
Let  
\[f_{x}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{x}\in\Gamma_{x}}f_{x}^{\gamma_{x}}(\xi)\,e^{|2\pi \gamma_{x}t} \tag{57}\] \[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]  
be the 1 st-order almost-periodic FOT pdf of \(x(t)\) and \(n(t)\), respectively. With reference to the SOI-noise additive model in the hypothesis test (37), under hypothesis \(\mathrm{H}_{0}\), one has  
\[f_{r|\mathrm{H}_{0}}^{\{\alpha\}}(\xi,t)=f_{n}^{\{\alpha\}}(\xi,t) \tag{59}\]  
and under hypothesis \(\mathrm{H}_{1}\), one obtains  
\[f_{r|\mathrm{H}_{1}}^{\{\alpha\}}(\xi,t) =f_{x}^{\{\alpha\}}(\xi,t)\,\underset{\xi}{\otimes}\,f_{n}^{\{ \alpha\}}(\xi,t) \tag{60a}\] \[=\sum_{\gamma_{x}\in\Gamma_{x}}\sum_{\gamma_{n}\in\Gamma_{x}}\! \left[f_{x}^{\gamma_{x}}(\xi)\,\underset{\xi}{\otimes}\,f_{n}^{\gamma_{n}}( \xi)\right]e^{|2\pi|\gamma_{x}+\gamma_{n}|t} \tag{60b}\]where \(\otimes_{\xi}\) denotes convolution with respect to the variable \(\xi\). Therefore, from (59) it follows that  
\[f^{\beta}_{r|\mathrm{H}_{0}}(\xi) \triangleq\left\langle f^{\{a\}}_{r|\mathrm{H}_{0}}(\xi,t)\,e^{-j2 \pi\beta t}\right\rangle_{t} \tag{61a}\] \[=f^{\beta}_{n}(\xi) \tag{61b}\]  
and from (60b), one has  
\[f^{\beta}_{r|\mathrm{H}_{1}}(\xi) \triangleq\left\langle f^{\{a\}}_{r|\mathrm{H}_{1}}(\xi,t)\,e^{- j2\pi\beta t}\right\rangle_{t} \tag{62a}\] \[=\sum_{\gamma_{x}\in\Gamma_{x}}\left[f^{\gamma_{x}}_{x}(s)\ \otimes\limits_{\xi}\ f^{\beta-\gamma_{x}}_{n}(\xi)\right] \tag{62b}\]  
Let the following _Assumptions_ be satisfied:  
1. \(x(t)\) exhibits 1 st-order cyclostationarity at \((\beta,\xi)\): \(\beta\in\Gamma_{x}\) and \(f^{\beta}_{x}(\xi)\
eq 0\);
2. \(\beta-\gamma_{x}\
ot\in\Gamma_{n}/\{0\}\ \forall\gamma_{x}\in\Gamma_{x}\). Condition 2 is satisfied, for example, if \(n(t)\) is a stationary time series. Thus, for \(\beta\in\Gamma_{x}\), it results that  
\[\mathrm{H}_{0}\ :\quad f^{\beta}_{r|\mathrm{H}_{0}}(\xi) = f^{\beta}_{n}(\xi)=0\] \[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]0  
or, equivalently,  
\[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]1 \[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]2  
Equations (37) and (63) suggest the following ad hoc (non-optimum and not equivalent to (37)) binary hypothesis test to discriminate between hypotheses \(\mathrm{H}_{0}\) and \(\mathrm{H}_{1}\) when \(r(t)\) is observed for \(t\in[-T/2,T/2]\):  
\[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]3 \[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]4  
or, equivalently,  
\[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]5 \[f_{n}^{\{\alpha\}}(\xi,t) =\sum_{\gamma_{n}\in\Gamma_{n}}f_{n}^{\gamma_{n}}(\xi)\,e^{|2\pi \gamma_{n}t} \tag{58}\]6  
where \(\widehat{f}^{\beta}_{r|\mathrm{H}_{1}}(\xi)\) and \(\widehat{f}^{\beta}_{r|\mathrm{H}_{1}}(\xi)\) denote estimates of \(f^{\beta}_{r|\mathrm{H}_{1}}(\xi)\) and \(F^{\beta}_{r|\mathrm{H}_{1}}(\xi)\), respectively, for \(i=0,1\) (see Sections 2.3.1 and 2.3.2). In (66) and (65), \(\epsilon^{(T)}_{i}(\beta,\xi)\) and \(\eta^{(T)}_{i}(\beta,\xi)\), \(i=0,1\), represent estimation errors. If \(f^{0}_{n}(\xi)\) is unimodal around \(\xi=0\) (e.g., zero-mean Gaussian), then the magnitude of the convolution product \(f^{\beta}_{x}(\xi)\ \otimes\limits_{\xi}\ f^{0}_{n}(\xi)\) has the same point of maximum as the magnitude of \(f^{\beta}_{x}(\xi)\). This fact is exploited for the choice of the value of \(\xi\) to be used in the quadratic-form detector.