# Cyclic Detectors in the Fraction-of-Time Probability Framework
## 2 Materials and Methods
### Single Cycle (SC) Detectors
\begin{array}{rclrcl}\mathrm{H}_{0}&:&z_{r}^{\mathsf{z}}(\theta)&=&z_{n}^{ \mathsf{z}}(\theta)=0\\ \mathrm{H}_{1}&:&z_{r}^{\mathsf{z}}(\theta)&=&z_{x}^{\mathsf{z}}(\theta)+z_{nx}^ {\mathsf{z}}(\theta)+z_{xn}^{\mathsf{z}}(\theta)+z_{n}^{\mathsf{z}}(\theta)\\ &=&z_{x}^{\mathsf{z}}(\theta)\end{array}  \tag{45}\]  
Equations (37) and (45) suggest the following ad hoc (non-optimum and not equivalent to (37)) binary hypothesis test for discriminating between the hypotheses \(\mathrm{H}_{0}\) and \(\mathrm{H}_{1}\) when \(r(t)\) is observed for \(t\in[-T/2,T/2]\)[2] (Section 8.4):  
\[ \begin{array}{rclrcl}\mathrm{H}_{0}&:&\widehat{z}_{r}(\alpha,\theta)&=& \epsilon_{0}^{(T)}(\alpha,\theta)\\ \mathrm{H}_{1}&:&\widehat{z}_{r}(\alpha,\theta)&=&z_{x}^{\mathsf{z}}(\theta)+ \epsilon_{1}^{(T)}(\alpha,\theta)\end{array}  \tag{46}\]  
In (46), \(\epsilon_{i}^{(T)}(\alpha,\theta)\) denotes the estimation error on \(z_{r}^{\mathsf{z}}(\theta)\) using the measurement \(r(t)\) observed for \(t\in[-T/2,T/2]\), under hypothesis \(\mathrm{H}_{i}\). The decision test (46) can be generalized considering measurements in multiple pairs \((\alpha_{k},\theta_{k})\), \(k=1,\ldots,K\). Under the _Assumptions_,  
1. \(x(t)\) exhibits cyclostationarity at \((\alpha_{k},\theta_{k})\): \(z_{x}^{\alpha_{k}}(\theta_{k})\
ot\equiv 0\);
2. \(n(t)\) does not exhibit cyclostationarity at \((\alpha_{k},\theta_{k})\): \(z_{n}^{\alpha_{k}}(\theta_{k})\equiv 0\);
3. \(x(t)\) and \(n(t)\) do not exhibit joint cyclostationarity at \((\alpha_{k},\theta_{k})\): \(z_{xn}^{\alpha_{k}}(\theta_{k})\equiv z_{nx}^{\alpha_{k}}(\theta_{k})\equiv 0\). The test (46) for several pairs \((\alpha_{k},\theta_{k})\), \(k=1,\ldots,K\) is  
\[ \begin{array}{rclrcl}\mathrm{H}_{0}&:&\widehat{z}_{r}(\alpha_{k},\theta_{k} )&=&\epsilon_{0}^{(T)}(\alpha_{k},\theta_{k})\\ \mathrm{H}_{1}&:&\widehat{z}_{r}(\alpha_{k},\theta_{k})&=&z_{x}(\alpha_{k}, \theta_{k})+\epsilon_{1}^{(T)}(\alpha_{k},\theta_{k})\end{array} \qquad k=1, \ldots,K \tag{47}\]  
Let us define the normalized measurement column vector  
\[\mathbf{Z}\triangleq\ \{\
u_{T}\,\widehat{z}_{r}(\alpha_{k},\theta_{k});\,k=1, \ldots,K\}\;. \tag{48}\]  
Due to the asymptotic joint complex normality of the normalized measurement errors (39), for \(T\) sufficiently large the column random vector \(\mathbf{Z}\) is (practically) complex normal under both hypotheses  
\[\mathbf{Z}\,|\,\mathrm{H}_{i}\sim\mathcal{N}(\boldsymbol{\mu}_{i},\boldsymbol{ \Sigma}_{i},\boldsymbol{\Sigma}_{i}^{(c)})\qquad i=0,1 \tag{49}\]  
with mean vectors  
\[ \begin{array}{rl}\boldsymbol{\mu}_{0}&=\mathbb{E}\{\mathbf{Z}\,|\,\mathrm{H} _{0}\}=\mathbf{0}\\ \boldsymbol{\mu}_{1}&=\mathbb{E}\{\mathbf{Z}\,|\,\mathrm{H}_{1}\}=\{\
u_{T}\,z _{r}^{\alpha_{k}}(\theta_{k});\,k=1,\ldots,K\}\end{array}  \tag{50}\]  
asymptotic covariance matrix \(\boldsymbol{\Sigma}_{i}\) with entries  
\[ \begin{array}{rl}\boldsymbol{\Sigma}_{i}(k_{1},k_{2})=&\lim_{T\to\infty} \mathrm{cov}\big{\{}Z_{k_{1}},Z_{k_{2}}\,|\,\mathrm{H}_{i}\big{\}}\\ =&\lim_{T\to\infty}\
u_{T}^{2}\,\mathrm{cov}\Big{\{}\epsilon_{i}^{(T)}(\alpha_ {k_{1}},\theta_{k_{1}}),\,\epsilon_{i}^{(T)}(\alpha_{k_{2}},\theta_{k_{2}}) \Big{\}}\qquad i=0,1\end{array}  \tag{51}\]  
and asymptotic conjugate covariance matrix \(\boldsymbol{\Sigma}_{i}^{(c)}\) with entries  
\[ \begin{array}{rl}\boldsymbol{\Sigma}_{i}^{(c)}(k_{1},k_{2})=&\lim_{T\to\infty }\mathrm{cov}\big{\{}Z_{k_{1}},Z_{k_{2}}^{*}\,|\,\mathrm{H}_{i}\big{\}}\\ =&\lim_{T\to\infty}\
u_{T}^{2}\,\mathrm{cov}\Big{\{}\epsilon_{i}^{(T)}(\alpha_ {k_{1}},\theta_{k_{1}}),\,\epsilon_{i}^{(T)*}(\alpha_{k_{2}},\theta_{k_{2}}) \Big{\}}\qquad i=0,1\,,\end{array}  \tag{52}\]  
where superscript \(*\) denotes complex conjugation. Complex-valued random vectors can be suitably described and characterized by introducing the _augmented random vector_  
\[\boldsymbol{\zeta}\triangleq\left[ \begin{array}{c}\boldsymbol{Z}\\ \boldsymbol{Z}^{*}\end{array} \right] \tag{53}\]  
the _augmented mean vector_  
\[\boldsymbol{\mu}_{\zeta|\mathrm{H}_{i}}\triangleq\left[