## 3 Capacity Analysis of the IWMSNs
### The Optimal Joint Decoding Capacity
In order to get the approximations of \(\mathbf{C}_{\text{OPT}}\), therefore, the MGF of \(\text{logdet}(\mathbf{I}_{N}+\gamma\mathbf{\Sigma}\mathbf{W}\mathbf{\Theta} \mathbf{W}^{\dagger})\) can be rewritten as:  
\[ \begin{array}{l}\mathbf{F}(u)\approx\mathbb{E}\left[\exp\left(u\cdot\text{ logdet}(\mathbf{I}_{N}+\gamma\mathbf{\Sigma}\mathbf{W}\mathbf{\Theta}\mathbf{W}^{ \dagger})\right)\right]\\ =\mathbb{E}\left[\text{det}(\mathbf{I}_{N}+\gamma\mathbf{\Sigma}\mathbf{W} \mathbf{\Theta}\mathbf{W}^{\dagger})^{u}\right]\\ \approx\int_{\mathbf{W}}\text{det}(\mathbf{I}_{N}+\gamma\mathbf{\Sigma} \mathbf{W}\mathbf{\Theta}\mathbf{W}^{\dagger})^{u}d\mathbf{W}\end{array}  \tag{17}\]  
\(d\mathbf{W}\) is defined as the Haar measure from [52]. In [53], the integral can be written as a hypergeometric function and expressed in terms of a scalar argument,\[ \begin{array}{l}\mathbf{F}(u)=\int_{\mathbf{W}}\det(\mathbf{I}_{N}+\gamma\Sigma \mathbf{W}\mathbf{\Theta}\mathbf{W}^{\dagger})^{u}d\mathbf{W}\\ =_{1}F_{0}(-u;-\gamma\boldsymbol{\Sigma},\mathbf{\Theta})\\ =\frac{\prod_{n=1}^{N}(n-1)!}{\prod_{n=1}^{N}(-u-N+1)_{N-n}}\cdot\frac{\det \left(_{1}\mathcal{F}_{0}(-u-N+1;-\gamma\zeta_{i}\theta_{i})\right)}{V(- \gamma\boldsymbol{\Sigma})V(\mathbf{\Theta})}\\ =\frac{\prod_{n=1}^{N}(n-1)!}{\prod_{n=1}^{N}(-u-N+1)_{N-n}}\cdot\frac{\det \left((1+\gamma\theta_{i}\zeta_{j})^{u+N-1}\right)_{ij}}{V(-\gamma\boldsymbol {\Sigma})V(\mathbf{\Theta})}\end{array}  \tag{18}\]  
where \(V(\mathbf{X})\) denotes the Vandermonde determinant and \(\lambda_{s}\) are the eigenvalues of \(\mathbf{X}\). \(V(\mathbf{X})=\prod_{s>t}\left(\lambda_{s}-\lambda_{t}\right)\), and \(V(b\mathbf{X})=b^{\sum_{n=1}^{N}u}V(\mathbf{X})\). \(\zeta_{i}\) and \(\theta_{j}\) are the diagonal element of diagonal matrices  
Since \(\log(\prod_{n=1}^{N-1}{(-u-n)})=\sum_{n=1}^{N-1}{n\log(-u-n)}\), then:  
\[ \begin{array}{l}\frac{d\log(\mathbf{G}(u))}{du}=\frac{d(\mathbf{G}(u))}{du} \frac{1}{\mathbf{G}(u)}\\ =\frac{d\binom{N-1}{n-1}{du}}{\left.\begin{array}{c}\\ \\ \\ \\ \\ \\ \end{array}\right.}\\ =\sum\limits_{n=1}^{N-1}{n\frac{d\log(-u-n)}{du}}\\ =\sum\limits_{n=1}^{N-1}{\frac{n}{u+n}}\\ \end{array}  \tag{24}\]  
Then, we can get that:  
\[\frac{d\left(\mathbf{G}(u)\right)}{du}=\mathbf{G}(u)\sum\limits_{n=1}^{N-1}{ \frac{n}{u+n}} \tag{25}\]  
We substitute the Equations (22) and (25) into Equation (21), then:  
\[ \begin{array}{l}\frac{\frac{d\det\mathbf{Y}(u)}{du}\cdot\mathbf{G}(u)-\det \mathbf{Y}(u)\cdot\frac{\mathbf{G}(u)}{du}}{\mathbf{G}^{2}(u)}\\ =\frac{\det\mathbf{Y}(u)\cdot\text{tr}\left(\left(\left(\left(1+\gamma\theta_{i }\zeta_{j}\right)^{+N-1}\right)_{i,j}\right)^{-1}\cdot\left(\log(1+\gamma\theta _{i}\zeta_{j})\cdot\left(1+\gamma\theta_{i}\zeta_{j}\right)^{+N-1}\right)_{i,j }\right)\cdot\mathbf{G}(u)-\det\mathbf{Y}(u)\cdot\mathbf{G}(u)\sum\limits_{n= 1}^{N-1}{\frac{n}{u+n}}}\\ =\frac{\det\mathbf{Y}(u)}{\mathbf{G}(u)}\left(\text{tr}\left(\mathbf{Z}(u) \cdot\left(\log(1+\gamma\theta_{i}\zeta_{j})\cdot\left(1+\gamma\theta_{i} \zeta_{j}\right)^{n+N-1}\right)_{i,j}\right)-\sum\limits_{n=1}^{N-1}{\frac{n} {u+n}}\right)\end{array}  \tag{26}\]  
After substituting Equations (20) and (26) into Equation (19), we derive the optimal capacity of the return link for the IWSMSNs as:  
\[ \begin{array}{l}\lim\limits_{N\rightarrow\infty}\mathbf{C}_{\text{OPT}} \left(\left(\mathbf{T}\otimes\mathbf{1}_{K}\right)\mathbf{R};\alpha,\beta, \gamma\right)\\ \approx\frac{1}{N}\frac{\prod_{n=1}^{N-1}{(k-1)!}}{V(-\gamma\mathbf{\Sigma})V( \mathbf{\Theta})}\frac{\det\mathbf{Y}(u)}{\mathbf{G}(u)}\cdot\left(\text{tr} \left(\mathbf{Z}(u)\cdot\left(\log(1+\gamma\theta_{i}\zeta_{j})\cdot\left(1+ \gamma\theta_{i}\zeta_{j}\right)^{n+N-1}\right)_{i,j}\right)-\sum\limits_{n=1}^ {N-1}{\frac{n}{u+n}}\right)\bigg{|}_{u=0}\end{array}