# DoF Region of the MISO BC with Partial CSIT: Proof by Inductive Fourier-Motzkin Elimination

## TL;DR

This paper introduces an alternative proof for the DoF region of the K-user MISO BC with partial CSIT, using Fourier-Motzkin elimination to simplify the achievable region and showing that tuning a single power variable suffices.

## Contribution

It presents a new proof method employing Fourier-Motzkin elimination, reducing the complexity of power tuning needed to achieve the entire DoF region.

## Key findings

- Single power variable tuning suffices for full DoF region
- Fourier-Motzkin elimination simplifies the proof process
- Achieves known outer bound directly

## Abstract

We provide a fresh perspective on the problem of characterizing the DoF region of the $K$-user MISO BC with arbitrary levels of partial CSIT. In a previous achievability proof, Piovano and Clerckx characterized all faces describing a polyhedral outer bound region, and then with the aid of mathematical induction, prescribed a scheme based on rate-splitting with flexible assignment of common DoF and power levels to achieve each such face. We propose an alternative approach in which we deal directly with the region achievable through rate-splitting and employ a Fourier-Motzkin procedure to eliminate all auxiliary variables, hence reducing the achievable region to the known outer bound. A key insight emerging from our proof is that tuning only one power variable, as well as assigning the common DoF, is sufficient to achieve the entire DoF region, as opposed to $K$ power variables previously employed.