# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Fraction-of-Time Probability  
In this section, a brief review of the FOT approach is provided, in order to introduce the essential definitions and notation. In the FOT approach, signals are modeled as single functions of time rather than as realizations of an ensemble of functions, namely a stochastic process. All time-invariant statistical functions such as the distribution, autocorrelation, moments, and cumulants can be constructed starting from the unique available signal by resorting to the concept of relative measure [29; 30], which plays a role analogous to the probability measure in the classical stochastic approach [25]. In such a case, the infinite-time average turns out to be the expectation operator that allows one to construct time-invariant statistical functions. The time-invariant FOT approach can be extended to accommodate periodically or almost-periodically time-variant statistical functions, in order to provide a suitable FOT probabilistic description for time series that exhibit cyclostationarity. In such a case, the expectation operator is the almost-periodic component extraction operator (6). For comprehensive treatments, see [1], [2] (Chapter 2), [31; 32]. For extensions, see [33; 34]. Let \(x(t)\) be a continuous-time real-valued signal. The almost-periodic FOT cumulative distribution function (CDF) of \(x(t)\) is defined as [2] (Equation (33))  
\[F_{x}^{\{\alpha\}}(\xi;t) \triangleq\mathrm{E}^{\{\alpha\}}\{\mathrm{u}(\xi-x(t))\} \tag{7a}\] \[=\sum_{\gamma\in\Gamma_{1}}F_{x}^{\gamma}(\xi)\ e^{2\pi\gamma t} \tag{7b}\]  
where  
\[\mathrm{u}(\xi)\triangleq\left\{ \begin{array}{ll}1&\xi\geq 0\\ 0&\xi<0\end{array} \right. \tag{8}\]  
is the unit-step function, \(\Gamma_{1}\) is a countable set of possibly incommensurate cycle frequencies, and the Fourier coefficients [2] (Equation (2.34))  
\[F_{x}^{\gamma}(\xi)\triangleq\left\langle\mathrm{u}(\xi-x(t))\,e^{-j2\pi \gamma t}\right\rangle_{t}\quad\gamma\in\Gamma_{1} \tag{9}\]  
are referred to as _cyclic CDFs_. If the CDF is differentiable with respect to \(\xi\), the almost-periodic FOT probability density function (pdf) of the real-valued signal \(x(t)\) can be defined. This can be formally expressed (in the sense of generalized functions [35]) as [2] (Equation (2.37))  
\[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\] \[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\] (10b) \[=\sum_{\gamma\in\Gamma_{1}}f_{x}^{\gamma}(\xi)\ e^{2\pi\gamma t} \tag{10c}\]  
where the Fourier coefficients are formally expressed as  
\[f_{x}^{\gamma}(\xi) \triangleq\left\langle\delta(\xi-x(t))\,e^{-j2\pi\gamma t}\right\rangle _{t} \tag{11a}\] \[=\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\gamma}(\xi)\quad\gamma \in\Gamma_{1} \tag{11b}\]  
and are referred to as _cyclic pdfs_. In (10b) and (11a), \(\delta(\cdot)\) denotes Dirac delta. The following fundamental theorem of expectation holds [31]:  
\[\mathrm{E}^{\{\alpha\}}\{x(t)\} \triangleq\int_{\mathbb{R}}\xi\,\mathrm{d}F_{x}^{\{\alpha\}}(\xi;t) \tag{12a}\] \[=\int_{\mathbb{R}}\xi\,f_{x}^{\{\alpha\}}(\xi;t)\,\mathrm{d}\xi\] (12b) \[=\sum_{\eta\in\mathrm{A}_{1}}x^{\eta}\ e^{2\pi\gamma t} \tag{12c}\]where the Fourier coefficients of the almost-periodic component of \(x(t)\) can be expressed as  
\[x^{\eta} \triangleq\left\langle x(t)\,e^{-j2\pi\eta t}\right\rangle_{t} \tag{13a}\] \[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]0 (13b) \[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]1  
with  
\[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]2  
The function defined in (12a) is the almost-periodic mean (in the FOT sense) of the signal \(x(t)\) and (12b) is the FOT counterpart of the fundamental theorem of expectation in the stochastic approach. The Fourier coefficients defined in (13a) are referred to as _cyclic means_.