## 4 The Joint Spatial-Code Clustered IA Scheme
### Error Performance of the JSCC-IA Scheme with Identical Propagation Delay
** For the case where the priori probabilities of the transmitted data are equal, the probability of the transmitted bit vector can be expressed as:  
\[P(\mathbf{M}_{q})=1/2^{k\ell},\ q=1,2,...,2^{k\ell} \tag{44}\]  
As shown in Equation (39), when \(\mathbf{M}_{q}\) is transmitted, the expected value of \(\mathbf{\overline{s}}\) can be given by:  
\[\mathbf{E}[\mathbf{\overline{s}}\ |\ \mathbf{M}_{q}]=(E_{b}\ /\ L)\mathbf{D}_{ \mathrm{JSCC1},q} \tag{45}\]where \(\mathbf{D}_{\text{JSCC1},q}\) represents the \(q\)-th column of \(\mathbf{D}_{\text{JSCC1}}\). As we make the decision according to the real part of the test statistic, the expected value and variance of \(\text{real}(\hat{s}_{k,l})\) when \(\mathbf{M}_{q}\) is transmitted can be respectively given as:  
\[\mathbf{E}[\text{real}(\hat{s}_{k,l})\,|\,\mathbf{M}_{q}]=(E_{b}\,/\,L)\,\text{ real}\big{[}\mathbf{D}_{\text{JSCC1}}((k-1)d+l,q)\big{]} \tag{46}\]  
\[\text{var}[\text{real}(\hat{s}_{k,l})\,|\,\mathbf{M}_{q}]=\sigma_{R}^{2}=N_{0}E _{b}\,/\,4 \tag{47}\]  
Therefore, the probability density function of \(\text{real}(\hat{s}_{k,l})\) can be formulated as:  
\[ \begin{split} P(z\,|\,\mathbf{M}_{q})=&\frac{1}{ \sqrt{2\pi\sigma_{R}^{2}}}\exp\left\{-\frac{\left\{z-\mathbf{E}[\text{real}( \hat{s}_{k,l})\,|\,\mathbf{M}_{q}]\right\}^{2}}{2\sigma_{R}^{2}}\right\}\\ =&\frac{1}{\sqrt{2\pi\sigma_{R}^{2}}}\exp\left\{- \frac{\left\{z-(E_{b}\,/\,L)\,\text{real}\big{[}\mathbf{D}_{\text{JSCC1}}((k- 1)d+l,q)\big{]}\right\}^{2}}{2\sigma_{R}^{2}}\right\}\end{split}  \tag{48}\]  
When \(m_{k,l}=1\), the probability of the error occurrence is:  
\[ \begin{split} P(e\,|\,\mathbf{M}_{q},m_{k,l}=1)&= P(z<0\,|\,\mathbf{M}_{q},m_{k,l}=1)\\ =&\int_{-\infty}^{0}P(z\,|\,\mathbf{M}_{q})dz\\ =&\frac{1}{2}+\frac{1}{2}\,\text{erf}\left\{\frac{- (E_{b}\,/\,L)\,\text{real}\big{[}\mathbf{D}_{\text{JSCC1}}((k-1)d+l,q)\big{]} }{\sqrt{2\sigma_{R}^{2}}}\right\}\\ =&\frac{1}{2}+\frac{1}{2}\,\text{erf}\left\{\frac{- m_{k,l}(E_{b}\,/\,L)\,\text{real}\big{[}\mathbf{D}_{\text{JSCC1}}((k-1)d+l,q) \big{]}}{\sqrt{2\sigma_{R}^{2}}}\right\}\end{split}  \tag{49}\]  
When \(m_{k,l}=-1\), the probability of the error occurrence is:  
\[ \begin{split} P(e\,|\,\mathbf{M}_{q},m_{k,l}=&-1) =P(z>0\,|\,\mathbf{M}_{q},m_{k,l}=-1)\\ =&\int_{0}^{+\infty}P(z\,|\,\mathbf{M}_{q})dz\\ =&\frac{1}{2}+\frac{1}{2}\,\text{erf}\left\{\frac{(E_ {b}\,/\,L)\,\text{real}\big{[}\mathbf{D}_{\text{JSCC1}}((k-1)d+l,q)\big{]}}{ \sqrt{2\sigma_{R}^{2}}}\right\}\\ =&\frac{1}{2}+\frac{1}{2}\,\text{erf}\left\{\frac{- m_{k,l}(E_{b}\,/\,L)\,\text{real}\big{[}\mathbf{D}_{\text{JSCC1}}((k-1)d+l,q) \big{]}}{\sqrt{2\sigma_{R}^{2}}}\right\}\end{split}  \tag{50}\]  
As \(m_{k,l}=\overline{\mathbf{M}}_{\text{JSCC1}}((k-1)d+l,q)\), we substitute it into Equations (49) and (50) to attain the error probability of \(m_{k,l}\), _i.e._,  
\[P\big{(}e\,|\,\mathbf{M}_{q}\big{)}=\frac{1}{2}+\frac{1}{2}\,\text{erf}\left\{ \frac{-(E_{b}\,/\,L)\overline{\mathbf{M}}_{\text{JSCC1}}((k-1)d+l,q)\text{ real}\big{[}\mathbf{D}_{\text{JSCC1}}((k-1)d+l,q)\big{]}}{\sqrt{2\sigma_{R}^{2}}}\right\} \tag{51}\]  
Substitute Equation (47) into Equation (51), and the error probability of \(m_{k,l}\) can be expressed as:  
\[P\big{(}e\,|\,\mathbf{M}_{q}\big{)}=\frac{1}{2}+\frac{1}{2}\,\text{erf}\left\{ -\overline{\mathbf{M}}_{\text{JSCC1}}((k-1)d+l,q)\,\text{real}\big{[}\mathbf{ D}_{\text{JSCC1}}((k-1)d+l,q)\big{]}\frac{1}{L}\sqrt{\frac{2E_{b}}{N_{0}}}\right\} \tag{52}\]When all the possible transmitted bit vectors are considered, the overall averaged BER of \(m_{k,l}\) can be formulated as:  
\[\mathrm{BER}_{\mathrm{JSCC1}}(k,l)=P(\mathbf{M}_{\mathrm{q}})\sum_{q=1}^{2\omega }P(e\,|\,\mathbf{M}_{\mathrm{q}}) \tag{53}\]  
Substitute Equations (44) and (52) into Equation (53), and Equation (42) can be verified.