# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 4 Discussion  
The problem of cyclostationary signal detection is addressed in the fraction-of-time probability approach. In this approach, signals are modeled as single functions of time, rather than realizations of a stochastic process. Two classes of detectors are considered: the single-cycle detectors, and the quadratic-form detectors. The single-cycle detectors have a decision statistic that only depends on the received signal. Such a decision statistic is compared with a threshold that is derived from the available side data in the null hypothesis. In contrast, quadratic-form detectors have a decision statistic that depends on the received signal and side data in the null hypothesis and the threshold that can be derived analytically, provided that the decision statistic has a chi-squared fraction-of-time distribution under the null hypothesis. However, at the price of a modest performance degradation, the decision statistic can be computed using the received signal only, with no need of side data. For quadratic-form detectors, the covariance matrix is estimated with a subsampling procedure that is naturally derived in the fraction-of-time approach. For both single-cycle and quadratic-form detectors, a new class of cyclic statistical function measurements are considered: the cyclic cumulative distribution function, and the cyclic probability density function. The simulation results showed that these measurements can allow one to gain in performance with respect to measurements of cyclic autocorrelation, cyclic spectrum, and cyclic 4 th-order moment. The advantage is particularly evident when the almost-periodically time-variant cumulative distribution and probability density functions exhibit harmonics whose strength is comparable with that of the continuous component. The effectiveness of the proposed detectors was tested on a FSK signal-of-interest in a severe noise and interference environment. Specifically, both additive stationary noise and narrow-band interference were present, with a power spectral density that  
Figure 4: Monte Carlo (effective) \(P_{\text{fa}}\) is versus nominal or design \(P_{\text{fa}}\). (**Left**) \(T=2^{15}T_{s}\); (**Right**) \(T=2^{19}T_{s}\). Quadratic form detectors (blue): (\(\bigtriangledown\)) cyclic pdf (kernel-based estimator); (\(\square\)) cyclic CDF (Fourier coefficient of the indicator); (\(\bigcirc\)) cyclic autocorrelation (cyclic correlogram); (\(\lozenge\)) cyclic CDF (kernel-based estimator); (\(\star\)) cyclic 4 th-order moment. Black line: Monte Carlo \(P_{\text{fa}}\) = nominal \(P_{\text{fa}}\). completely overlapped that of the signal-of-interest. The SNR and SIR were both \(-20\) dB. By operating at a cycle frequency equal to the symbol rate, a performance analysis was carried out using Monte Carlo simulations for the new proposed single-cycle and quadratic-form detectors based on measurements of theylic CDF and pdf. Specifically, the probability of missed detection versus the data-record length and the receiver operating characteristics were estimated. Moreover, the performances were compared with those of analogous detectors based on measurements of cyclic autocorrelation, cyclic spectrum, and cyclic 4 th-order moment measurements. The simulation results showed the better performance of the new proposed detectors. A further analysis was carried out to check the assumption made on the quadratic-form distribution that was exploited to analytically derive the detector threshold. Specifically, for all considered quadratic-form detectors, the nominal or design probability of false alarm (adopted to derive the threshold) was compared with the effective probability of false alarm obtained using Monte Carlo simulations. By increasing the data-record length, the values of the two probabilities of false alarm tended to be equal, except for the case of quadratic forms of measurements of cyclic CDF and pdf obtained using kernel-based estimators.