## 3 Performance Analysis
### Erroneous Index Detection Probability \(P_{ed}\)  
Because the proposed system does not need to superimpose pilot signal and channel estimation over an AWGN channel, the output of the matched filter is given by  
\[ \begin{split} v(m)&=\left[\mathbf{T}^{m(L+1)} \boldsymbol{c}\right]^{T}r\\ &=\sum_{j=1}^{J}\boldsymbol{c}^{T}\mathbf{T}^{M-m(L+1)}\left(S_ {j,R}\mathbf{T}^{k_{Ij}(L+1)}\boldsymbol{c}+\sqrt{-1}S_{j,I}\mathbf{T}^{k_{Sj }(L+1)}\boldsymbol{c}\right)+\boldsymbol{c}^{T}\mathbf{T}^{M-m(L+1)} \boldsymbol{n}.\end{split}  \tag{12}\]  
Since channel estimation is not required, the matched filter output \(v(m)\) is equal to the decision variable \(z_{m}\). Because the detection processes of the two branches are independent and consistent, we take the in-phase branch as an example to analyze the erroneous index detection probability. Therefore, the mean and variance of decision variable \(z_{m}\) are obtained as  
\[ \begin{split}\mathbb{E}[\mathcal{R}(z_{m})]&= \mathbb{E}\left\{\sum_{j=1}^{J}S_{j,R}\boldsymbol{c}^{T}\mathbf{T}^{M-m(L+1)+k_ {Ij}(L+1)}\boldsymbol{c}\right\}\\ &=\left\{\begin{array}{l}\left(\sum_{j=1,m\
eq k_{Ij_{1}}}^{l} -\frac{S_{j,R}}{M}\right)+S_{j_{1},R}=\mu_{11},\quad m\in\left\{k_{I,1}, \cdots,k_{I,J}\right\}\\ \\ \sum_{j=1}^{J}-\frac{S_{j,R}}{M}=\mu_{12},\quad m\
otin\left\{k_{I,1}, \cdots,k_{I,J}\right\}\end{array}\right.,\end{split}  \tag{13}\]  
\[ \begin{split}\text{Var}[\mathcal{R}(\Delta_{m})]&= \text{Var}\Big{[}\mathcal{R}\Big{(}\boldsymbol{c}^{T}\mathbf{T}^{M-m(L+1)} \boldsymbol{n}\Big{)}\Big{]}\\ &=\frac{N_{0}}{2}\mathbb{E}\Big{(}\boldsymbol{c}^{2}\Big{)}= \sigma_{1}^{2},\end{split}  \tag{14}\]  
where \(\mathbb{E}\big{(}\boldsymbol{c}^{2}\big{)}\) denotes the energy of spreading sequence \(\boldsymbol{c}\). As shown in Figure 2, to obtain the indices of the transmitted sequences, the index detection block is used to find \(J\) maximums \(|\mathcal{R}\Big{(}z_{u}^{{}^{\prime}}\Big{)}|,u\in\left\{k_{I,1},\cdots,k_{I,J}\right\}\) from the variables \(|\mathcal{R}(z_{m})|,m=1,\ldots,J_{max}-1\). The remaining absolute values are given as \(|\mathcal{R}\Big{(}z_{i}^{{}^{\prime}}\Big{)}|,i=1,\ldots,J_{max}-1-J\). Let \(X=\min_{u\in\left\{k_{I,1}\cdots k_{I,J}\right\}}\Big{(}|\mathcal{R}\Big{(}z_ {u}^{{}^{\prime}}\Big{)}|\Big{)}\) and \(Y=\max_{i\
otin\left\{k_{I,1}\cdots k_{I,J}\right\}}\Big{(}|\mathcal{R}\Big{(} z_{i}^{{}^{\prime}}\Big{)}|\Big{)}\); when \(Y\leq X\), index detection is correct, so the error probability of in-phase branch index detection can be expressed as  
\[ \begin{split} P_{edl}&=1-\mathrm{P_{r}}\{Y\leq X\} =\int_{0}^{\infty}[1-\mathrm{P}\{Y\leq X\}]f_{X}(x)\mathrm{d}\mathrm{x}\\ &=\int_{0}^{\infty}\!\!\left[1-\Big{(}F_{\mathcal{R}\big{(}z_{i}^ {{}^{\prime}}\big{)}}(\boldsymbol{x})\Big{)}^{J_{max}-J-1}\right]f_{X}(x) \mathrm{d}\mathrm{x},\end{split}  \tag{15}\]where \(F_{\mathcal{R}\left(z_{i}^{{}^{\prime}}\right)}(x)\) is the cumulative distribution function (CDF) and \(f_{X}(x)\) is the probability density function (PDF). When the index detection is correct, the real part of the decision variable approximately follows the normal distribution and the absolute value of the real part follows a folded normal distribution, so their PDF and CDF are respectively expressed  
\[f_{\left|\mathcal{R}\left(z_{u}^{{}^{\prime}}\right)\right|}(x)=\frac{1}{\sqrt{ 2\pi\sigma_{1}^{2}}}\left(e^{\frac{-(x-\mu_{I1})^{2}}{2\sigma_{1}^{2}}}+e^{ \frac{-(x+\mu_{I1})^{2}}{2\sigma_{1}^{2}}}\right), \tag{16}\]  
\[F_{\left|\mathcal{R}\left(z_{u}^{{}^{\prime}}\right)\right|}(x)=\frac{1}{2} \left[\text{erf}\left(\frac{x-\mu_{I1}}{\sqrt{2\sigma_{1}^{2}}}\right)+\text{ erf}\left(\frac{x+\mu_{I1}}{\sqrt{2\sigma_{1}^{2}}}\right)\right], \tag{17}\]  
where \(\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{x}^{x}e^{-t^{2}}\text{dt}\) and \(x\geq 0\) is error function.