"*On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel*" by Giuseppe Carie and Shlomo Shamai talks, in part, about the following type of link (paraphrasing):



> 
> A transmitter with $t$ antennas broadcasts to $r$ independent receivers through a flat-fading channel (modeled by a matrix $H\in \mathbb{C}^{r\times t}$) with iid additive white gaussian noise. i.e. each reception looks like: 
>  $$\vec{y}=H\vec{x}+\vec{z}; \quad \vec{z}\sim \mathcal{N}(\vec{0}, I\_{r\times r})$$ Each receiver seeks to recieve a different message.
> 
> 
> 


The **sum capacity** of such a channel is: $\displaystyle \underset{\vec{r}\in R}{\sup} \textstyle\sum\_{i}r\_i$ where $R$ is the achievable-rate region. It's just the maximum overall rate.


Given $H$, what is the **sum capacity** of a channel where a single transmit antenna is talking to $r$ independent (non-communicating) receive antennas? 


Note that finding the actual region $R$ is difficult, but the sum capacity is a known result.




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The paper says that the capacity region of the system in question is well-known. It cites Cover/Thomas' information theory textbook. However, Cover/Thomas only characterizes the $r=2$ case (relevant details start at pg515), and doesn't focus on the *sum capacity*


"*Sum Capacity of the Vector Gaussian Broadcast Channel and Uplink–Downlink Duality*" by Viswanath and David finds the sum capacity for the more complicated case where $t>1$, and says that the $t=1$ case is well-known. (but also alludes to $t=1$ systems being 'degenerate')



What is the sum capacity of a scalar gaussian broadcast channel?
The sum capacity result in both of the linked papers. For any Gaussian broadcast channel of this form:
$$\vec{y}= H^\dagger \vec{x} + \vec{z}; \quad \vec{z}\sim\mathcal{N}(0,I\_{r\times r})$$
 ($\vec{y}\in \mathbb{C}^{r\times 1}$ reception, $\vec{x}\in\mathbb{C}^{t\times 1}$ transmission, $H\in \mathbb{C}^{t\times r}$ channel, $\vec{z}\in\mathbb{C}^{r\times 1}$ noise)


It's capacity is as follows:


$$C^{(I)}= \underset{D}{\sup} \log \det (I+HDH^{\dagger})$$


Where $D$ is an $r\times r$ nonnegative diagonal matrix with $\mathrm{Tr}(D)\leq P$, where $P$ is the average power in $x$.




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For the scalar case, $t=1$ so the 'determinant' is really just of a 1x1 matrix and the quadratic is easy to expand:
\begin{align} 
& \sup\_D \log \det ( 1 + \textstyle \sum\_n H\_{1,n} D\_{n,n} \overline{H\_{1,n}}) \\
=& \sup\_D \log ( 1 + \textstyle \sum\_n |H\_{1,n}|^2 D\_{n,n}) \\
=&\max\_n \log( 1 + |H\_{1,n}|^2 P) \\ 
\end{align}


