# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Fraction-of-Time Probability
The second-order characterization of a time series \(x(t)\) is based on the almost-periodic joint FOT CDF of \(x(t+\tau)\) and \(x(t)\)[2] (Theorem 2.34), [31]  
\[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]3 \[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]4  
where \(\Gamma_{2}\) is a countable set of possibly incommensurate cycle frequencies, and the Fourier coefficients  
\[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]5  
are referred to as _cyclic joint CDFs_. If the joint FOT CDF is differentiable with respect to both \(\xi_{1}\) and \(\xi_{2}\), the almost-periodic joint FOT pdf of \(x(t+\tau)\) and \(x(t)\) can be defined. This can be formally expressed as [2] (Equation (2.50))  
\[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]6 \[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]7 (17b) \[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]8  
where the Fourier coefficients  
\[f_{x}^{\{\alpha\}}(\xi;t) \triangleq\frac{\mathrm{d}}{\mathrm{d}\xi}F_{x}^{\{\alpha\}}(\xi;t) \tag{10a}\]9 \[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]0  
are referred to as _cyclic joint pdfs_. The following fundamental theorem of expectation holds [2] (Theorem 2.35), [31]:  
\[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]1 \[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]2 (19b) \[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]3where the function \(\mathrm{E}^{\{\alpha\}}\{x(t+\tau)\ x(t)\}\) has all the properties of an _autocorrelation function_ and the Fourier coefficient  
\[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]4 \[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]5 (20b) \[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]6  
is referred to as _cyclic autocorrelation function_ at _cycle frequency_\(\alpha\). It results that  
\[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]7  
The second-order characterization of a time series \(x(t)\) in the spectral domain is made in terms of the _cyclic spectrum_\(S^{\alpha}_{x}(f)\). This is defined as the correlation between the spectral components of the signal at frequencies \(f\) and \(f-\alpha\), where \(\alpha\in\mathrm{A}_{2}\) is one of the cycle frequencies of the signal \(x(t)\)[1] (Chapter 11), [2] (Chapter 2). For this reason, \(S^{\alpha}_{x}(f)\) is also referred to as _spectral correlation density_. It is linked to the cyclic autocorrelation by [PERSON]'s relation [2] (Section 2.3.1.10)  
\[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]8  
also referred to as the cyclic Wiener-Khinchin relation. Higher-order statistics of almost-cyclostationary signals in the FOT probability framework were introduced in [7; 32]. In this paper, only the 4 th-order reduced-dimension cyclic temporal moment function  
\[=\mathrm{E}^{\{\alpha\}}\{\delta(\xi-x(t))\}\]9  
referred to as, in short, 4 th-order cyclic moment, is considered to build a higher-order cyclostationarity (HOCS)-based detector.