# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 1 Introduction  
Cyclostationarity is a property exhibited by almost all modulated signals adopted in communications, radar/sonar, and telemetry [1; 2]. Cyclostationary signals exhibit statistical functions, such as the cumulative distribution, probability density, autocorrelation, moments, and cumulants that are periodic functions of time. More generally, these statistical functions can be almost periodic [3]; that is, they can be expressed by the superposition of sine waves with possibly incommensurate frequencies. In such a case, the signals are referred to as almost cyclostationary (ACS). The statistical properties of ACS signals are suitably described by the Fourier coefficients, referred to as cyclic statistics, of these almost-periodic functions. The frequencies of the (generalized) Fourier series expansions are related to signal parameters such as the carrier frequency, band rate, and sampling or scanning frequency. Thus, the frequencies of the harmonics, referred to as cycle frequencies, are a characteristic of a signal. In particular, signals having different abovementioned parameters exhibit different cycle frequencies. This fact has been exploited for the design of detection and estimation algorithms that are signal selective. In fact, in the presence of a mixture of signal of interest (SOI) and disturbance, characteristics of the SOI alone can be estimated, provided that a cycle frequency of the SOI exists, which is not shared with the disturbance signal, regardless of the temporal and spectral overlap of the SOI and disturbance [1] (Chapter 14), [2] (Section 9.2). Several cyclostationarity-based detectors have been presented in the literature. Single-cycle and multi-cycle detectors are proposed in [4; 5; 6], with reference to second-order statistics, and in [7], with reference to higher-order statistics. A statistical test for the presence of cyclostationarity is proposed in [8] and its application to signal detection isdiscussed in [2] (Section 8.4), [9]. A statistical test for the presence of spectral coherence is proposed in [10]. Spectral coherence is also exploited in [11; 12]. Detection strategies based on the estimated confidence interval of estimated cyclic statistics are presented in [2] (Section 8.7). They are based on the results in [13]. Applications to cognitive radio are presented in [14; 15; 16; 17; 18]. The case of an unknown cyclostationarity period is addressed in [19; 20]. The case of irregular cyclicity is treated in [21; 22; 23; 24]. All the of abovementioned cyclostationarity-based detectors exploit measurements of second- or higher-order cyclic statistics. In this paper, a new class of cyclic detectors is proposed. Specifically, detectors based on measurements of the Fourier coefficients of the almost-periodically time-variant cumulative distribution function (CDF) and probability density function (pdf) are introduced. Numerical results presented here show that performance gains can be obtained by constructing cyclostationarity-based detectors that exploit these cyclic statistical functions. In practical detection problems, a single realization of the received signal is available and a decision on the hypothesis \"SOI not present\" (null hypothesis) or \"SOI present\" (alternative hypothesis) must be taken on the basis of this unique observed signal. That is, no ensemble of realizations is available for making the decision. For this reason, in this paper, the statistical characterization of the received signal and of its cyclic statistical function measurements is made using the fraction-of-time (FOT) probability approach. In such an approach, a signal is modeled as a single function of time, rather than a realization of a stochastic process [1; 25; 26; 27; 28]. Then, statistical functions such as distribution, autocorrelation, moments, and cumulants are constructed, starting from this unique function of time.