# Uplink NOMA-MIMO Systems with Rayleigh Power Distribution
## 3 Capacity of Discrete-Continuous MU-MIMO Uplink Channel  
For a discrete-continuous channel, mutual information is described by the following Equation [9]:  
\[I(X;Y)=\sum\limits_{i=1}^{M}\int\limits_{Y}p(Y|X_{i})\,p(X_{i})\log\biggl{(} \frac{p(Y|X_{i})}{p(Y)}\biggr{)}dY \tag{8}\]  
Mutual information as for a continuous channel can be represented as an entropy difference:  
\[I(X;Y)=H(Y)-H(Y|X) \tag{9}\]  
where:  
\[ \begin{array}{l}H(Y)=-\int\limits_{Y}p(Y)\log(p(Y))dY\\ H(Y|X)=\sum\limits_{i=1}^{M}p(X_{i})H(Y|X_{i})\\ H(Y|X_{i})=-\int\limits_{Y}p(Y|X_{i})\log(p(Y|X_{i}))dY\\ p(Y)=\sum\limits_{i=1}^{M}p(Y|X_{i})p(X_{i})\end{array}  \tag{10}\]  
The bandwidth of a discrete-continuous channel is the maximum of mutual information according to an a priori discrete distribution:  
\[C=\max\limits_{p(X_{i}),j=1,M}I(X;Y) \tag{11}\]  
It is known that the maximum entropy has a source of discrete information with a uniform a priori distribution, i.e., \(p(X_{i})=\frac{1}{M}\). In this case we have:  
\[ \begin{array}{l}H(Y)=-\frac{1}{M}\sum\limits_{i=1}^{M}\int\limits_{Y}p(Y|X_{ i})\log\biggl{(}\frac{1}{M}\sum\limits_{j=1}^{M}p(Y|X_{j})\biggr{)}dY\\ H(Y|X_{i})=\frac{1}{M}\sum\limits_{i=1}^{M}H(Y|X_{i})\end{array}  \tag{12}\]  
From model (2) we have:  
\[p(Y|X_{i})=\frac{1}{(2\pi)^{N_{rx}}\sigma_{\eta}^{2N_{rx}}}\exp\biggl{(}-\frac {1}{2\sigma_{\eta}^{2}}\Bigl{(}Y-\widetilde{\mathbf{H}}X_{i}\Bigr{)}^{\prime }\Bigl{(}Y-\widetilde{\mathbf{H}}X_{i}\Bigr{)}\biggr{)} \tag{13}\]  
where \(X_{i}\triangleq X(\Theta_{i})\)-\(i\)-th value of extended QAM symbols vector of all users, \(i=\overline{1,M}\); \(\widetilde{\mathbf{H}}\triangleq\frac{\sqrt{p}}{\sqrt{N_{rx}}}\mathbf{H}\). Taking into account Equation (13), we have:  
\[ \begin{array}{l}H(Y|X_{i})=N_{rx}+N_{rx}\log\Bigl{(}2\pi\sigma_{\eta}^{2} \Bigr{)}\\ H(Y)=N_{rx}\log\Bigl{(}\Bigl{(}2\pi\sigma_{\eta}^{2}\Bigr{)}\Bigr{)}-\\ \frac{1}{M}\sum\limits_{i=1}^{M}\int\limits_{Y}\frac{1}{(2\pi\sigma_{\eta}^{2 })^{N_{rx}}}\exp\biggl{(}-\frac{\|Y-\widetilde{\mathbf{H}}X_{i}\|^{2}}{2\sigma _{\eta}^{2}}\biggr{)}\log\biggl{(}\frac{1}{M}\sum\limits_{j=1}^{M}\exp\biggl{(} -\frac{\|Y-\widetilde{\mathbf{H}}X_{j}\|^{2}}{2\sigma_{\eta}^{2}}\biggr{)} \biggr{)}dY\end{array}  \tag{14}\]  
As a result, the bandwidth of a discrete-continuous MIMO channel will be determined by the following equation:  
sian vectors \(Z_{i}\sim\mathbb{CN}\Big{(}0,2\sigma_{\eta}^{2}I_{N_{rx}}\Big{)}\) and random uniformly distributed discrete vectors \(X_{i}\in\Omega^{UN_{\rm{kr}}}\):  
\[ \begin{split}&\overline{C}=\log M-N_{rx}-E_{\mathbf{H},Z_{i},X_{i}} \Bigg{\{}\log\left(\sum\limits_{j=1}^{M}\exp\left(-\frac{\|Z_{i}-\widetilde{ \mathbf{H}}\left(X_{j}-X_{i}\right)\|^{2}}{2\sigma_{\eta}^{2}}\right)\right) \Bigg{\}}\approx\\ &\log M-N_{rx}-\frac{1}{N_{MK}}\sum\limits_{n=1}^{N_{MK}}\log \left(\sum\limits_{j=1}^{M}\exp\left(-\frac{\|Z^{(n)}-\widetilde{\mathbf{H}}^ {(n)}\left(X_{j}-X^{(n)}\right)\|^{2}}{2\sigma_{\eta}^{2}}\right)\right)=\\ &-N_{rx}-\frac{1}{N_{MK}}\sum\limits_{n=1}^{N_{MK}}\log\left( \frac{1}{M}\sum\limits_{j=1}^{M}\exp\left(-\frac{\|Z^{(n)}-\widetilde{ \mathbf{H}}^{(n)}\left(X_{j}-X^{(n)}\right)\|^{2}}{2\sigma_{\eta}^{2}}\right) \right)\end{split}  \tag{19}\]  
where \(\widetilde{\mathbf{H}}^{(n)},Z^{(n)},X^{(n)}\)-\(n\)-th implementations of random matrices and vectors used for averaging. The bandwidth of discrete channels is usually measured in bits per second per hertz ((bits/s)/Hz). To obtain the value in these units of measurement, it was necessary to use base 2 logarithm in the initial Equations (4) and (8). Considering \(\log_{2}x=\frac{\log x}{\log 2}\) and \(M=2^{UN_{\rm{kr}}k_{b}}\) we can finally write an expression for discrete-continuous MU-MIMO uplink channel capacity and its upper bound:  
\[