## 4 The Joint Spatial-Code Clustered IA Scheme
### Error Performance of the Conventional IA scheme  
To make a comparison between the proposed JSCC-IA and the conventional IA schemes, the error performance of the latter one is investigated in this subsection. The equivalent baseband waveform function and the bit energy \(E_{b}\) of the BPSK signal in the conventional IA system can be respectively expressed as:  
\[g_{k}(t)=A\ \text{rect}_{T_{b}}(t) \tag{67}\]  
\[E_{b}=\int_{0}^{T_{b}}g_{k}(t)g_{k}(t)dt=A^{2}T_{b} \tag{68}\]  
Substitute Equations (1), (10), (67) and (68) into Equation (14), and the test statistic vector can be rewritten as:  
\[\mathbf{\hat{s}}_{k}=E_{b}\gamma_{kk}\mathbf{U}_{k}^{\mathrm{H}}\mathbf{H}_{kk} \mathbf{V}_{k}\mathbf{m}_{k}+\sum_{i=1,j=k}^{K}E_{b}\gamma_{ik}\mathbf{U}_{k}^ {\mathrm{H}}\mathbf{H}_{ik}\mathbf{V}_{i}\mathbf{m}_{i}+\mathbf{\hat{n}}_{k} \tag{69}\]  
Therefore, the SINR of the \(l\)-th data stream of the \(k\)-th user can be calculated as:  
\[\text{SINR}_{\mathrm{LA}}(k,l)=\frac{\mid E_{b}\gamma_{lk}\mathbf{U}_{k,l}^{ \mathrm{H}}\mathbf{H}_{ik}\mathbf{V}_{k,l}m_{k,l}\mid^{2}}{\sum_{i=1,j=k}^{K} \mid E_{b}\gamma_{ik}\mathbf{U}_{k,l}^{\mathrm{H}}\mathbf{H}_{ik}\mathbf{V}_{ i,j}m_{i,j}\mid^{2}+\sum_{j=1,j=l,j=l}^{d}\mid E_{b}\gamma_{ik}\mathbf{U}_{k,l}^{ \mathrm{H}}\mathbf{H}_{ik}\mathbf{V}_{k,j}m_{k,j}\mid^{2}+N_{0}E_{b}\mid 2} \tag{70}\]  
Similar as the JSCC-IA scheme with identical propagation delay, we introduce the matrix \(\mathbf{\overline{M}}\) as Equation (40) and the the transformation matrix \(\mathbf{T}_{\mathrm{IA}}\) as follows:  
\[\mathbf{T}_{\mathrm{IA}}=\begin{bmatrix}\mathbf{B}_{11},\mathbf{B}_{12},..., \mathbf{B}_{1K}\\ \mathbf{B}_{21},\mathbf{B}_{22},...,\mathbf{B}_{2K}\\...\\ \mathbf{B}_{K1},\mathbf{B}_{K2},...,\mathbf{B}_{KK}\end{bmatrix}\epsilon \mathbf{\overline{C}}^{k\delta\delta\delta\delta\delta} \tag{71}\]  
where \(\mathbf{B}_{\delta}=\gamma_{\delta}\mathbf{U}_{i}^{\mathrm{H}}\mathbf{H}_{ \delta}\mathbf{V}_{j}\). Then the BER expression of the conventional IA system can be expressed as the following theorem.  
**Theorem 4**.: _The BER of the l-th data stream of the k-th user in the conventional IA system can be expressed as:_  
Figure 3: The propagation delay between two users.  
\[\mathrm{BER}_{\mathrm{IA}}(k,l)=\frac{1}{2^{k\ell}}\sum_{q=1}^{2^{k\ell}}\left\{ \frac{1}{2}+\frac{1}{2}\mathrm{erf}\Bigg{[}-\mathrm{sign}[\overline{\mathbf{M}}( (k-1)d+l,q)]\mathrm{real}\big{[}\mathbf{D}_{\mathrm{IA}}((k-1)d+l,q)\big{]} \sqrt{\frac{2E_{b}}{N_{0}}}\Bigg{]}\right\} \tag{72}\]  
_And the overall averaged BER of the whole system can be expressed as:_  
\[\overline{\mathrm{BER}}_{\mathrm{IA}}=\frac{1}{Kd}\sum_{k=1}^{K}\sum_{l=1}^{d} \mathrm{BER}_{\mathrm{IA}}(k,l) \tag{73}\]  
**Theorem 4 of Proof.** The proof for Theorem 4 is similar as that of Theorem 2.  
In this section, we have introduced the JSCC-IA scheme and formulated the associated BER expressions analytically. We have clustered the users randomly, without further considering the effect of different grouping combinations. In the next section, we turn to more practical situations of asymmetric networks, and propose a grouping algorithm to select more appropriate clustering combinations with better performance.