# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Monte Carlo \(P_{\text{md}}\) versus \(T\)  
In this experiment, the Monte Carlo probability of missed detection \(P_{\text{md}}\) versus the data-record length \(T\) is computed for the energy detector, quadratic-form detectors, and single-cycle detectors. The results are reported in Figure 1.  
The results show that the best performance was obtained by the SC detectors based on estimates of the cyclic pdf (kernel-based estimator) and the cyclic CDF (Fourier coefficient of the indicator). A good performance was obtained by the QF detectors based on estimates of the cyclic CDF (kernel-based estimator) and the cyclic CDF (Fourier coefficient of the indicator). The poor performance of the energy detector was the consequence of the severe noise and interference environment. In fact, due to the very low values of SNR and SIR and the fluctuation of the noise power from one Monte Carlo realization to the other,  
Figure 1: Monte Carlo \(P_{\text{md}}\) versus \(T\). Energy detector (black): (\(\ast\)). Quadratic form detectors (blue) and single-cycle detectors (green): (\(\bigtriangledown\)) cyclic pdf (kernel-based estimator); (\(\square\)) cyclic CDF (Fourier coefficient of the indicator); (\(\bigcirc\)) cyclic autocorrelation (cyclic correlogram); (\(\lozenge\)) cyclic CDF (kernel-based estimator); (\(\ast\)) cyclic 4 th-order moment. the estimated threshold of this detector, which was obtained from measurements under H0, was not reliable. Detectors based on cyclic statistic measurements benefit from the signal selectivity property of cyclostationarity-based algorithms and are intrinsically immune to the effect of noise and interference, regardless of the temporal and spectral overlap of SOI and disturbance, provided that a sufficiently long data-record length is adopted for the measurements [1] (Chapter 14), [2] (Section 9.2). The stationary noise only affects the cyclic statistic measurements due to the leakage phenomenon, and the performance degradation decreases as the data record increases [2] (Section 9.7). The narrow-band interference can degrade the cyclic detector performance only in the rare case in which the double of the carrier frequency of one of the three tones (which randomly changes from one realization to the other) is equal to the cycle frequency \(1/T_{p}\) within an error of the order of \(1/T\). Therefore, all the cyclostationarity-based detectors are potentially able to counteract the effects of noise and interference, provided that a sufficiently long observation interval is available. For the FSK signal considered here, the strength of the sine wave regenerated by computing the cyclic CDF and cyclic pdf, compared with the corresponding continuous component, is higher than that obtained by computing the second- and fourth-order cyclic statistics, and this constitutes an advantage at very low values of SNR and SIR. In particular, for the considered values of \(T\), these detectors provide satisfactory performance, while larger values of \(T\) are needed to obtain the same performance with SC and QF detectors based on second- and fourth-order statistic measurements. The CS detectors exhibit better performance than the QF detectors, since the former exploit side information for the threshold computation. The results obtained for moderate or high values of SNR and SIR are not presented here, since they showed that all cyclostationarity-based methods exhibit comparable performance. Receiver Operating Characteristic (ROC): Monte Carlo \(P_{\text{d}}\) versus Nominal \(P_{\text{fa}}\)  
In this experiment, the receiver operating characteristic (Monte Carlo \(P_{\text{d}}\) versus nominal \(P_{\text{fa}}\), in short MC-\(P_{\text{d}}\) vs. nominal-\(P_{\text{fa}}\)) is computed for the energy detector, quadratic-form detectors, and single-cycle detectors. The results are reported in Figure 2.  
These results confirm those of Section 3.4. Figure 2: ROC: Monte Carlo \(P_{\text{d}}\) versus nominal \(P_{\text{fa}}\). (**Left**) \(T=2^{15}T_{s}\); (**Right**) \(T=2^{19}T_{s}\).