## 4 The Joint Spatial-Code Clustered IA Scheme
### Error Performance of the JSCC-IA Scheme with Identical Propagation Delay  
In the JSCC-IA system, when all users share the same propagation delay, the reconstructed signal waveform vector at the \(k\)th receiver can be expressed as Equation (10). The demodulator employs \(g\)si\((t)\) to correlate with \(\tilde{\textbf{s}}_{k}(t)\), and the test statistic vector can be written as:  
\[\hat{\textbf{s}}_{k}=\int_{0}^{T_{b}}\sum_{i=1}^{k}\gamma_{k}\textbf{U}_{k}^ {\text{H}}\textbf{H}_{i}\textbf{V}_{i}\textbf{m}_{j}g_{i}(t)g_{k}(t)dt+\int_{ 0}^{T_{b}}\textbf{U}_{k}^{\text{H}}\textbf{n}_{k}(t)g_{k}(t)dt \tag{29}\]  
From Equations (24) and (26)-(28), the correlation between \(g\)(\(t\)) and \(g\)(\(t\)) can be calculated as:  
\[ \begin{split}\int_{0}^{T_{b}}g_{i}(t)g_{k}(t)dt&= \int_{0}^{T_{b}}c_{\underline{i}}(t)c_{\underline{k}}(t)dt\\ &=A^{2}\sum_{n=0}^{L-1}[C_{\underline{i}}(n)C_{\underline{k}}(n) \int_{nT_{c}}^{(n+1)T_{c}}\text{rect}_{T_{c}}(t-nT_{c})\text{rect}_{T_{c}}(t -nT_{c})dt]\\ &=A^{2}T_{c}\sum_{n=0}^{L-1}C_{\underline{k}}(n)C_{\underline{i} }(n)\cr&=E_{b}\rho_{\underline{k}k}(0)\,/\,L\end{split}  \tag{30}\]  
To attain the expected value of \(\mathbf{\hat{s}}_{k}\), all the possible transmitted messages have to be considered, and we introduce the transformation matrix \(\mathbf{T}_{\mathrm{JSCC1}}\), the overall bit vector \(\mathbf{M}\) and the overall test statistic vector \(\mathbf{\overline{s}}\)  as follows:  
\[\mathbf{T}_{\mathrm{JSCC1}}= \begin{bmatrix}\mathbf{F}_{11},\mathbf{F}_{12},..., \mathbf{F}_{1K}\\ \mathbf{F}_{21},\mathbf{F}_{22},...,\mathbf{F}_{2K}\\...\\ \mathbf{F}_{K1},\mathbf{F}_{K2},...,\mathbf{F}_{KK}\end{bmatrix} \epsilon\ \mathbf{C}^{k\ell d \cdot k\ell d},\ \mathbf{M}= \begin{bmatrix}\mathbf{m}_{1}\\ \mathbf{m}_{2}\\...\\ \mathbf{m}_{K}\end{bmatrix} \epsilon\ \mathbf{R}^{k\ell d\cdot 1},\ \mathbf{\overline{s}}= \begin{bmatrix}\mathbf{\hat{s}}_{1}\\ \mathbf{\hat{s}}_{2}\\...\\ \mathbf{\hat{s}}_{K}\end{bmatrix} \epsilon\ \mathbf{C}^{k\ell d\cdot 1} \tag{37}\]  
It is obvious that there are \(2^{kd}\) possible value of \(\mathbf{M}\), and we index them as \(\mathbf{M}_{1},\mathbf{M}_{2},...,\mathbf{M}_{2^{\ell\alpha}}\). From Equations (36) and (37), when \(\mathbf{M}_{j}\) (\(j=1\), \(2\), \(...\), \(2^{K\ell}\)) is transmitted, the expected value of \(\mathbf{\overline{s}}\) can be calculated as:  
\[\mathbf{E}[\mathbf{\overline{s}}\ |\ \mathbf{M}_{j}]=(E_{b}\ /\ L)\mathbf{T}_{ \mathrm{JSCC1}}\mathbf{M}_{j} \tag{38}\]  
Therefore, the expected values of \(\mathbf{\overline{s}}\)  under all the transmitted bit vectors can be written as:  
\[ \begin{split}\left[\mathbf{E}[\mathbf{\overline{s}}\ |\ \mathbf{M}_{1}],\mathbf{E}[\mathbf{\overline{s}}\ |\ \mathbf{M}_{2}],...,\mathbf{E}[\mathbf{\overline{s}}\ |\ \mathbf{M}_{2^{\ell\alpha}}\ ]\right]& =(E_{b}\ /\ L)\mathbf{T}_{\mathrm{JSCC1}}\mathbf{\overline{M}}_{ \mathrm{JSCC1}}\\ &=(E_{b}\ /\ L)\mathbf{D}_{\mathrm{JSCC1}}\end{split}  \tag{39}\]  
where:  
\[\mathbf{\overline{M}}_{\mathrm{JSCC1}}=\left[\mathbf{M}_{1},\mathbf{M}_{2},...,\mathbf{M}_{2^{\ell\alpha}}\right] \tag{40}\]  
\[\mathbf{D}_{\mathrm{JSCC1}}=\mathbf{T}_{\mathrm{JSCC1}}\mathbf{\overline{M}} _{\mathrm{JSCC1}} \tag{41}\]  
Based on the analysis above, the BER expression of the proposed JSCC-IA with identical propagation delay is straightforward and can be expressed as the following theorem. **Theorem 2**. : _The BER of the \(l\)-th data stream of the \(k\)-th user in the JSCC-IA system with identical propagation delay can be expressed as:_  
\[\mathrm{BER}_{\mathrm{JSCC1}}(k,l)=\frac{1}{2^{k\ell}}\sum_{q=1}^{2^{k\ell}} \left\{\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left[-\mathbf{\overline{M}}_{ \mathrm{JSCC1}}((k-1)d+l,q)\right]\frac{1}{L}\sqrt{\frac{2E_{b}}{N_{0}}}\right]\right\} \tag{42}\]  
_And the overall BER of the whole system is given by:_  
\[\overline{\mathrm{BER}}_{\mathrm{JSCC1}}=\frac{1}{Kd}\sum_{k=1}^{K}\sum_{l=1}^ {d}\mathrm{BER}_{\mathrm{JSCC1}}(k,l) \tag{43}\]  
**Theorem 2 of Proof.