What is the sum capacity of a scalar gaussian broadcast channel?

Question

"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')

Answer 1

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}