CCapacity Atlas
gaussian-mimo-broadcast-channel

Gaussian MIMO broadcast channel

A Gaussian vector transmitter sends independent private messages to multiple receivers, and dirty-paper coding achieves the full capacity region.

Broadcast Continuous alphabet Gaussian Additive noise Power constraint Capacity region Exact Single-letter characterization

Channel and question

Input
A real vector \(X\in\mathbb R^t\) carrying independent uniform private messages \(M_1,\ldots,M_K\).
Output
Receiver i observes \(Y_i\in\mathbb R^{r_i}\) and decodes only \(M_i\).
Law
\(Y_i=H_iX+Z_i\), with fixed \(H_i\in\mathbb R^{r_i\times t}\) and \(Z_i\sim\mathcal N(0,N_i)\), where \(N_i\succ0\).
Quantity
Private-message capacity region \(\mathcal C(P)\), measured in rate tuples in bits per real vector channel use.

Criterion. Closure of achievable nonnegative rate tuples, with \(R_i=\liminf_{n\to\infty}\frac1n\log_2|\mathcal M_i^{(n)}|\) and the average probability that any receiver decodes incorrectly tending to zero.

  • The matrices H_i and N_i are known to the encoder and all receivers.
  • Noise vectors are independent across uses and receivers and independent of the messages.
  • One encoder knows all messages. There is no feedback, receiver cooperation, or common message.
  • Arbitrary deterministic block encoders and measurable decoders are allowed. For each blocklength n, \(\frac1n\sum_{s=1}^n\mathbb E[\|X_s(M_1,\ldots,M_K)\|_2^2]\le P\), where the expectation is over the uniform message tuple. Power is shared across transmit antennas and averaged over time and messages.

Parameters

\(K,t,r_i\)
Number of receivers, transmit antennas, and receive antennas at receiver i. Range: \(K\ge2,\quad t,r_i\ge1\).
\(P\)
Total average transmit power per channel use. Range: \(P\ge0\).

Current status

\[\begin{gathered} \mathcal C(P)=\overline{\operatorname{conv}}\! \bigcup_{\substack{\pi,\ Q_1,\ldots,Q_K\succeq0\\\sum_i\operatorname{tr}Q_i\le P}} \{R\in\mathbb R_+^K:R_{\pi(k)}\le r_{\pi(k)},\ k=1,\ldots,K\},\\ r_{\pi(k)}=\frac12\log_2 \frac{\det(N_{\pi(k)}+H_{\pi(k)}S_kH_{\pi(k)}^{\mathsf T})} {\det(N_{\pi(k)}+H_{\pi(k)}S_{k-1}H_{\pi(k)}^{\mathsf T})} \end{gathered}\]

Conditions. Here \(\pi\) ranges over permutations of the receivers, each \(Q_i\) is a real symmetric transmit covariance matrix, \(S_0=0\), and \(S_k=\sum_{j=1}^kQ_{\pi(j)}\).

Write the displayed region as \(\mathcal R_{\rm DPC}(P)\). With this indexing, encode in order \(\pi(K),\ldots,\pi(1)\); dirty-paper coding cancels the already encoded layers, leaving interference with receive covariance \(H_{\pi(k)}S_{k-1}H_{\pi(k)}^{\mathsf T}\). Convex closure allows time sharing. The factor one half is for real signals. The source's Definition 6 and Corollary 4 give the dirty-paper region and its equality to capacity under total average power.

Known results and bounds for Gaussian MIMO broadcast channel
ResultRelationMethodYear
Inner Region\(\mathcal R_{\rm DPC}(P)\subseteq\mathcal C(P)\)The fixed real Gaussian model and message-averaged total-power constraint specified above.Gaussian dirty-paper coding over all covariance allocations and encoding orders, with time sharing.2006
Outer Region\(\mathcal C(P)\subseteq\mathcal R_{\rm DPC}(P)\)The fixed real Gaussian model and message-averaged total-power constraint specified above.Channel enhancement and the entropy-power inequality, followed by the reduction from covariance constraints to total power.2006

Formal verification

Lean coverageFormally stated

The general capacity equality has a complete local Lean proof under the original model assumptions. The operational model uses mathlib multivariate Gaussian noise, joint decoding error, and power averaged over time and uniform message tuples. Achievability constructs deterministic layered random-binning codes and proves time sharing and boundary approximation. The converse combines finite-message Fano bounds, channel enhancement, Gaussian entropy power, and a compact alignment limit. The canonical theorem and its dependencies are kernel checked with only the standard Lean axioms.

Claims

  • The real Gaussian MIMO broadcast private-message capacity region equals the closed convex dirty-paper coding region under message-averaged total power.
    operational-capacity · exact capacity · solved · Formally proved · v1
Lean declarations (2)

References

  1. Hanan Weingarten, Yossef Steinberg, and Shlomo Shamai (Shitz) (2006). The Capacity Region of the Gaussian Multiple-Input Multiple-Output Broadcast Channel. IEEE Transactions on Information Theory 52(9), 3936–3964. DOI 10.1109/TIT.2006.880064.

Discussion

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Related problems

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