# Feedback-Capacity of Degraded Gaussian Vector BC using Directed Information and Concave Envelopes

## TL;DR

This paper extends the understanding of feedback capacity in physically degraded Gaussian vector broadcast channels by employing directed information and concave envelope factorization, showing feedback does not enlarge capacity regions.

## Contribution

It introduces a novel approach using directed information and concave envelope factorization to determine feedback capacity in Gaussian vector BCs, beyond EPI-based methods.

## Key findings

- Feedback does not enlarge capacity in physically degraded Gaussian vector BCs.
- Directed information and concave envelope factorization effectively characterize feedback capacity.
- Reversely physically degraded BCs are unaffected by feedback in terms of capacity.

## Abstract

It is known that the capacity region of a two user physically degraded discrete memoryless (DM) broadcast channel (BC) is not enlarged by feedback. An identical result holds true for a physically degraded Gaussian BC, established later using a variant of the Entropy Power Inequality (EPI). In this paper, we extend the latter result to a physically degraded Gaussian Vector BC (PD-GVBC). However, the extension is not EPI based, but employs a recent result on the factorization of concave envelopes. While the existing concave envelope factorization results do not hold in the presence of feedback, we show that factorizing the corresponding directed information quantities suffice to attain the feedback capacity region of a PD-GVBC. Our work demonstrates that factorizing concave envelopes of directed information can handle situations involving feedback. We further show that the capacity region of a discrete memoryless reversely physically degraded BC is not enlarged by feedback.