# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 1 Introduction
This approach has the methodological advantage that the statistical characterization of the measurements of the cyclic statistical functions of the received signal can be performed without resorting to any abstract stochastic process. The advantage is not only methodological but also in the detector implementation. In fact, a class of detectors is presented here such that the decision statistic is derived exclusively from the received signal, with no aid from side data under the null hypothesis, and the threshold is analytically determined. Two classes of cyclostationarity-based detectors are considered: single-cycle (SC) detectors, and quadratic-form (QF) detectors. For each class, the decision statistic is constructed using measurements of the Fourier coefficients of the almost-periodically time-variant CDF and pdf. Moreover, for comparison purposes, analogous detectors based on measurements of second- and higher-order cyclic statistic are also considered. Performance analysis in a very severe noise and interference environment is carried out via Monte Carlo simulations. The novel contributions of this paper are (i) the proposal of new cyclostationarity-based detection tests that exploit the almost-periodicity of the first-order FOT CDF and pdf; (ii) the proposal of new kernel-based cyclic CDF and pdf estimators; (iii) the exploitation of the FOT approach to statistically characterize the decision statistic, in order to analytically derive the detection threshold; and (iv) the presentation of the cyclostationarity-based detectors in a unified framework that accommodates both existing and new detectors. The paper is organized as follows: The materials and methods are presented in Section 2. Specifically, in Section 2.1, the signal decomposition into an almost-periodic component and a residual term is considered; in Section 2.2, the definitions and notation for signal analysis in the FOT approach for almost-cyclostationary signals are briefly reviewed. In Section 2.3, estimators for FOT cyclic statistical functions are presented. Specifically, the estimator for the cyclic CDF (Section 2.3.1), the new proposed kernel-based estimators for the cyclic CDF and pdf (Section 2.3.2), estimators for cyclic autocorrelation (Section 2.3.3), the cyclic spectrum (Section 2.3.4), and the 4 th-order cyclic moment (Section 2.3.5) are considered. The detection problem is stated in Section 2.4 and cyclostationarity-based detectors are presented in a unified general formalism that accommodates several detection strategies. Single-cycle detectors are treated in Section 2.5, and quadratic-form detectors in Section 2.6. QF detectors based on measurements of the cyclic autocorrelation, spectrum, and moment are addressed in Section 2.6.1. A new class of QF detectors based on measurements of the cyclic CDF and pdf is presented in Section 2.6.2. For QF detectors, the detector structure is presented in Section 2.6.3. A statistical test for the presence of cyclostationarity is described in Section 2.6.4. The numerical results are reported in Section 3. The simulation setup is described in Section 3.1. The cases of threshold determined using Monte Carlo simulation and of analytically derived threshold are treated in Sections 3.2 and 3.3, respectively. Performance analysis is carried out in Section 3.4, in terms of the probability of missed detection versus data-record length, and in Sections 3.5 and 3.6, in terms of the receiver operating characteristic (ROC). For the QF detectors, the match or mismatch between the Monte Carlo probability of false alarm and the nominal or design probability of false alarm is analyzed in Section 3.7. A conclusive discussion is given in Section 4.