# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Single Cycle (SC) Detectors
#### 2.6.3 Qf Detector Structure  
If in (56) the augmented mean vectors and covariance matrix are unknown, the hypothesis test can be performed using the _generalized log-likelihood ratio test (GLLRT)_[51] (Section 2.5)  
\[\ln\,\frac{f_{\rm\,]H_{1}}(\zeta_{a};\widehat{\mathbf{\eta}}_{\zeta[{\rm H_{1}}}, \widehat{\mathbf{\Gamma}}_{1})}{f_{\rm\,]H_{0}}(\zeta_{a};\widehat{\mathbf{\eta}}_{ \zeta[{\rm H_{0}}},\widehat{\mathbf{\Gamma}}_{0})}\gtrapprox \tag{67}\]  
where the estimated parameters are maximum likelihood estimates. If the parameter estimates happen to not be maximum likelihood estimates, then the test is no longer GLLRT. However, if such estimates are consistent, the test can still be applied as an ad hoc test for the considered problem [52]. In (67), \(\zeta_{a}\) is the measurement of \(\mathbf{Z}\) made on the whole observation interval \([-T/2,+T/2]\), \(\widehat{\mathbf{\eta}}_{\zeta[{\rm H_{0}}}=\mathbf{0}\), \(\widehat{\mathbf{\eta}}_{\zeta[{\rm H_{1}}}=\mathbf{\zeta}_{a}\), and the estimates \(\widehat{\mathbf{\Gamma}}_{1}\), and \(\widehat{\mathbf{\Gamma}}_{0}\) can be obtained using an FOT-subsampling procedure. For example, if \(z_{r}^{a}(\theta)\) is the cyclic autocorrelation function, then by defining  
\[R_{rr^{(*)}}^{(b,u)}(\alpha,\tau) \triangleq \frac{1}{b}\int_{u-b/2}^{u+b/2}r(t+\tau)\,r^{(*)}(t)\,e^{-j2\pi at }\,\mathrm{d}t \tag{68}\] \[= \frac{1}{b}\int_{-b/2}^{b/2}r(t^{\prime}+u+\tau)\,r^{(*)}(t^{ \prime}+u)\,e^{-j2\pi a(t^{\prime}+u)}\,\mathrm{d}t^{\prime}\]  
with \(b\ll T\), one has  
\[\widehat{\Sigma}_{i}(k_{1},k_{2}) = \left\langle\sqrt{b}\,R_{rr^{(*)}\!{}_{k_{1}}}^{(b,u)}(\alpha_{k_ {1}},\tau_{k_{1}})\,\sqrt{b}\,R_{rr^{(*)}\!{}_{k_{2}}}^{(b,u)*}(\alpha_{k_{2}}, \tau_{k_{2}})\,\left|\,{\rm H}_{i}\right.\right\rangle_{u} \tag{69}\] \[- \left\langle\sqrt{b}\,R_{rr^{(*)}\!{}_{k_{1}}}^{(b,u)}(\alpha_{k_ {1}},\tau_{k_{1}})\,\left|\,{\rm H}_{i}\right.\right\rangle_{u}\!\left\langle \sqrt{b}\,R_{rr^{(*)}\!{}_{k_{2}}}^{(b,u)*}(\alpha_{k_{2}},\tau_{k_{2}})\, \left|\,{\rm H}_{i}\right.\right\rangle_{u}\]  
where \(\langle\cdot\rangle_{u}\) is the time average made on the whole observation interval \(u\in[-T/2+b/2,+T/2-b/2]\) and the convergence conditions are \(T\to\infty\), \(b\to\infty\), with \(T/b\to\infty\). Therefore, with these estimates substituted into, the decision test (67) reduces to comparing a quadratic form \(\widehat{Q}(\zeta_{a})\) with a threshold [2] (Sections 8.4-8.5), [9]:  
\[\widehat{Q}(\zeta_{a})\triangleq\zeta_{a}^{\rm H}\widehat{\Gamma}_{0}^{-1} \zeta_{a}\gtrapprox \tag{70}\]  
The detector (70) is not optimum. However, if one assumes as front-end data the cyclic statistic measurements \(\mathbf{Z}\), then such a detector is optimum in the sense of Neyman-Pearson. In (70), the estimate \(\widehat{\mathbf{\Gamma}}_{0}^{-1}\) of the inverse of the covariance matrix \(\mathbf{\Gamma}_{0}\) must be obtained using the available side data under H\({}_{0}\). If these data are not available, one should resort to the statistical test for presence of cyclostationarity (Section 2.6.4). When such a test is adopted for signal detection, this results in a sub-optimum detector with respect to (70) [2] (Section 8.4.3), [9]. The quadratic forms \(Q(\xi)\) and \(\widehat{Q}(\zeta)\) have asymptotically the same central \(\chi_{2K}^{2}\) distribution \(F_{\chi_{2K}^{2}}(\cdot)\) (over the ensemble of realizations) [39] (Section 4.1.2, p. 140). Therefore, for a desired false-alarm rate  
\[P_{\rm fa}=\mathrm{P}[\widehat{Q}>\lambda\,\left|\,{\rm H}_{0}\right|=1-F_{ \chi_{2K}^{2}}(\lambda) \tag{71}\]  
the threshold \(\lambda\) can be analytically derived  
\[\lambda=F_{\chi_{2K}^{2}}^{-1}(1-P_{\rm fa})\,. \tag{72}\]  
(Section 3.6.2). The continuous-time FSK SOI is cyclostationary with period \(T_{p}\) and the sampled SOI is cyclostationary with period \(T_{p}/T_{s}=8\). FSK modulation is adopted in satellite communications due to its constant-envelope property, which minimizes the effect of nonlinear amplification in the high-power amplifier.