## 3 Cross-Correlation Functions for Different Modulation Schemes  
In this part, the cross-correlation functions for several modulation schemes are analyzed in an effort to pinpoint distinctive traits that may be utilized to identify these schemes. Across the whole of the exploration, we are going to operate on the following presumptions.  
* We employ the widely held premise in the literature that data symbols are unrelated to each other [30; 31]. We write \[\mathrm{E}[u(k_{1})u(k_{2})]=\begin{cases}\delta(k_{1}-k_{2})&\text{for $ \alpha=\text{BPSK}$}\\ 0&\text{Otherwise,}\end{cases}\] (7) where \(\mathrm{E}[\cdot]\) is a random variable's statistical expectation, \(\delta(\cdot)\) is the function of Kronecker delta, and BPSK refers to a binary phase shift keying modulation. Furthermore, we compose \[\mathrm{E}[u(k_{1})u^{*}(k_{2})]=\delta(k_{1}-k_{2}),\] (8) for all potential values of \(\alpha\) pertain to the configuration choices of \(M-\)PSK and \(M-\)QAM. Here, \(M\) is the modulation order, and
* denotes the complex conjugate operator. We suppose that the transmit power is one without sacrificing generality.
* The information symbols have no relationship with the noise values: \[\mathrm{E}[u(k_{1})n_{R}(k_{2})] = \mathrm{E}[u(k_{1})n_{R}^{*}(k_{2})]=0,\] \[\mathrm{E}\Big{[}u(k_{1})n_{D}^{(1)}(k_{2})\Big{]} = \mathrm{E}\Big{[}u(k_{1})n_{D}^{(1)*}(k_{2})\Big{]}=0,\] \[\mathrm{E}\Big{[}u(k_{1})n_{D}^{(2)}(k_{2})\Big{]} = \mathrm{E}\Big{[}u(k_{1})n_{D}^{(2)*}(k_{2})\Big{]}=0.\] (9)* The noise values are unrelated to one another: \[\mathrm{E}\Big{[}n_{R}(k_{1})n_{D}^{(1)}(k_{2})\Big{]} = \mathrm{E}\Big{[}n_{R}(k_{1})n_{D}^{(2)}(k_{2})\Big{]}=0,\] \[\mathrm{E}\Big{[}n_{R}(k_{1})n_{D}^{(1)*}(k_{2})\Big{]} = \mathrm{E}\Big{[}n_{R}(k_{1})n_{D}^{(2)*}(k_{2})\Big{]}=0,\] \[\mathrm{E}\Big{[}n_{D}^{(1)}(k_{1})n_{D}^{(2)}(k_{2})\Big{]} = \mathrm{E}\Big{[}n_{D}^{(1)}(k_{1})n_{D}^{(2)*}(k_{2})\Big{]}=0.\] (10)
* The parameters of the channel are initialized at random, and they remain constant throughout the frame. We describe the following correlation functions: \[F_{1}(k_{1},k_{2})=\mathrm{E}\Big{[}z_{D}^{(1)}(k_{1})z_{D}^{(2)}(k_{2})\Big{]},\] (11a) \[F_{2}(k_{1},k_{2})=\mathrm{E}\Big{[}z_{D}^{(1)}(k_{1})\Big{(}z_{D}^{(2)}(k_{2}) \Big{)}^{3}\Big{]},\] (11b) \[F_{3}(k_{1},k_{2})=\mathrm{E}\Big{[}z_{D}^{(1)}(k_{1})\Big{(}z_{D}^{(2)}(k_{2}) \Big{)}^{7}\Big{]},\] (11c) and \[F_{4}(k_{1},k_{2})=\mathrm{E}\Big{[}\Big{(}z_{D}^{(1)}(k_{1})\Big{)}^{4}\Big{(} z_{D}^{(2)*}(k_{2})\Big{)}^{2}\Big{]}.\] (11d)  
We develop the following expressions with the assistance of Table 1 and the aformentioned assumptions.  
\[F_{1}(k_{1},k_{2})=\left\{\begin{array}{ll}C\sum\limits_{l_{1},l_{2},l_{3}= 0}^{L-1}h_{SD}(l_{1})h_{RD}(l_{2})h_{SR}(l_{3})\delta(k_{1}-k_{2}-l_{1}+l_{2}+ l_{3})&\text{for $\alpha=\mathrm{BPSK}$}\\ 0&\text{otherwise},\end{array}\right.\] (12a) \[F_{2}(k_{1},k_{2})=\left\{\begin{array}{ll}C\sum\limits_{l_{1},l_{2},\cdots,l_{ \eta}=0}^{L-1}h_{SD}(l_{1})\prod\limits_{i=2}^{4}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!  
Figure 3: \(|F_{2}(k_{1},k_{2})|\) for \(16-\)QAM and \(16-\)PSK modulations.