CCapacity Atlas
gaussian-mimo-channel

Point-to-point Gaussian MIMO channel

A point-to-point Gaussian vector channel under a total covariance trace constraint has a log-determinant water-filling capacity formula.

Point-to-point Continuous alphabet Gaussian Power constraint Capacity Exact Single-letter characterization

Channel and question

Input
A real transmit vector \(X\in\mathbb R^{n_t}\) with covariance \(Q\succeq0\) and \(\operatorname{tr}Q\le P\).
Output
\(Y=HX+Z\in\mathbb R^{n_r}\), where \(Z\sim\mathcal N(0,\sigma^2I)\).
Law
A fixed real channel matrix \(H\) is known at encoder and decoder.
Quantity
Gaussian MIMO capacity \(C(H,P)\), measured in bits per channel use.

Criterion. Arbitrary blocklength vector codes under the total average power constraint.

  • Total average transmit power is at most \(P\).
  • Average decoding error vanishes.
  • Real-channel normalization contributes a factor one half.

Current status

\[C(H,P)=\max_{Q\succeq0,\,\operatorname{tr}Q\le P}\frac12\log_2\det\!\left(I+\sigma^{-2}HQH^{\mathsf T}\right)\]

The maximizing covariance is obtained by water-filling over the eigenmodes of \(H^{\mathsf T}H\).

ResultRelationMethodYear
Exact\(C(H,P)=\max_{Q\succeq0,\operatorname{tr}Q\le P}\frac12\log_2\det(I+\sigma^{-2}HQH^{\mathsf T})\)Gaussian extremality and eigenmode water-filling.1999

Lean formalization

Canonical statementStatement

Version 1 · Lean. The real-valued model fixes the one-half normalization; matrix optimization infrastructure remains an external proof task.

Substantial proofs0 linked

No external Lean proof is registered. Proofs longer than roughly 50 lines or requiring problem-specific infrastructure should live in a dedicated repository and link back to this statement version.

References

  1. Emre Telatar (1999). Capacity of Multi-antenna Gaussian Channels. European Transactions on Telecommunications. DOI 10.1002/ett.4460100604.

Discussion

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