The Capacity Region of the Two-Receiver Gaussian Vector Broadcast Channel With Private and Common Messages

Abstract

We develop a new method for showing the optimality of the Gaussian distribution in multiterminal information theory problems. As an application of this method we show that Marton's inner bound achieves the capacity of the vector Gaussian broadcast channels with common message.Proof. The proof is a trivial consequence of the fact that h(Remark 3. An interesting consequence of Gaussian noise is thatZ and Z ′ are again independent and distributed according to N (0, I). HenceỸ, Y ′ can be regarded as the outputs of the Gaussian channel when the inputs are distributed according toX, X ′ . This observation is peculiar to additive Gaussian channels.Claim 2. In vector additive Gaussian product broadcast channels with invertible channel gain matrices, the random variables Y 11 and Y 22 are independent if and only if X 1 and X 2 are independent.Proof. Here we prove the non-trivial direction. Suppose Y 11 and Y 22 are independent. We know that Y 11 = G 11 X 1 + Z 11 and Y 22 = G 22 X 2 + Z 22 where Z 11 , Z 22 are mutually independent and independent of the pair X 1 , X 2 . Taking characteristic functions we see thatOn the other handSince E e it1·Z11 , E e it2·Z22 > 0 ∀t 1 , t 2 we have that E e i(t1·G11X1+t2·G22X2) = E e it1·G11X1 E e it2·G22X2 , ∀t 1 , t 2 .Hence G 11 X 1 and G 22 X 2 are independent; since G 11 and G 22 are invertible, X 1 and X 2 are independent.X:E(XX T ) K

Superposition coding is almost always optimal for the Poisson broadcast channel

“…We first show that L 1 (q) is concave in [0, t]. Recall that I 1 (q) − I 2 (q) is concave in [0, κ] for κ in (28). Thus it suffices to show that t ≤ κ. Equivalently we show that f …”

Feedback-capacity of degraded Gaussian Vector BC using directed information and concave envelopes

“…This is done by extending the technique of concave envelopes [5]. However this extension is not very straightforward.…”

“…Recall that the upper concave envelope of a function f (x) is the smallest concave function g(x) such that g(x) ≥ f (x) throughout the domain of f (x). Equivalently, g(x) can be expressed as [5] g(x) = sup…”

“…The key technique of [5] was to show that the concave envelope defined over the joint distribution p(…”

“…But the concave envelopes in [5] do not factorize in the presence of feedback. We reformulate the factorization of concave envelopes in terms of directed information to yield the capacity region of a PD-GVBC with feedback.…”