---
abstract: 'In this paper, we investigate the additive Gaussian noise channel with noisy feedback.  We consider the setup of linear coding of the feedback information and Gaussian signaling of the message (i.e. Cover-Pombra Scheme).  Then, we derive the upper and lower bounds on the largest achievable rate for this setup.  We show that these two bounds can be obtained by solving two convex optimization problems.  Finally, we present some simulations and discussion.' 
author:
- 
bibliography:
- 'ref.bib'
title: '**Bounds on the Achievable Rate of Noisy feedback Gaussian Channels under Linear Feedback Coding Scheme**'
---

Introduction
============

The study of the additive Gaussian noise channel with feedback has been a hot research topic for decades.  So far, a large body of work has looked at the ideal feedback case and obtained many notable results [@Schalkwijk66; @Omura68; @Massey1990; @cover89; @Kim10].  As an illustration, it is known that noiseless feedback improves the error exponent and reduces the coding complexity.  However, only few papers have studied channels with noisy feedback and many open problems still exist.  Literature on noisy feedback problems can be largely classified into two categories.  The first category studies the usefulness of noisy feedback by investigating reliability functions and error exponents [@Kim07],[@Love_arxiv].  The second focuses on the derivation of coding schemes based on the well-known Schalkwijk-Kailath scheme.  We refer interested readers to [@Omura68; @Lavenberg71; @Martins08; @Chance10; @Kumar] for details.\
In this paper, we investigate the behavior of the largest achievable rate of the additive Gaussian noise channel with noisy feedback, under the restriction of linear feedback coding scheme (i.e.Cover-Pombra Scheme).  We derive informative upper and lower bounds on the largest achievable rate.  This upper bound outperforms the bound presented in [@Love10], especially, in the case of having small feedback noise.  The lower bound shows the enhancement, in terms of the achievable rate, by exploiting the noisy feedback link.  Additionally, the derived bounds provide insight on how the noisy feedback channel behaves with respect to the feedback noise.\
The paper is organized as follows.  In Section \[sec\_pre\], we introduce some important definitions and lemmas, which are used throughout the paper.  In Section III, we introduce the signal model of the noisy feedback channel and then derive the formula of the achievable rate.  In Section \[sec\_upper\] and \[sec\_lower\], we derive an upper bound and a lower bound on the largest achievable rate, respectively.  We present some simulation results and discussion in Section \[sec\_simu\] and conclude the paper in Section \[sec\_conclude\].\
Notations: Uppercase and corresponding lowercase letters $(e.g. Y,Z,y,z)$ denotes random variables and realizations, respectively.  $x^n$ represents the vector $[x_1,x_2,\cdots,x_n]^T$ and $x^0=\emptyset$. $\mathbf{I}_n$ represents an $n\times n$ identity matrix.  $\mathbf{K}_n>0$ ($\mathbf{K}_n\geq 0$) denotes that the $n\times n$ matrix $\mathbf{K}_n$ is positive definite (semi-definite).  $log$ denotes the logarithm base $2$ and $0\log0=0$. The expectation operator over $X$ is presented as $\mathbb{E}(X)$.

Technical Preliminaries
=======================

\[sec\_pre\]

Modeling and Achievable Rate
============================

\[sec\_m\]

An Upper Bound on $R_{n,max}^{noisy}(P)$
========================================

\[sec\_upper\]

A Lower Bound on $R_{n,max}^{noisy}(P)$
=======================================

\[sec\_lower\]

Simulations and Discussion
==========================

\[sec\_simu\]

Conclusion
==========

We have derived the upper and lower bounds on the largest achievable rate for a linear feedback coding setup.  It is shown that these two bounds can be obtained as the optimal values of two convex optimization problems.  Furthermore, these bounds provide us the following insight: 1.  The achievable rate is very sensitive to the feedback noise.  2.  The achievable rate of channels with more correlated channel noise is less sensitive to the feedback noise.  \[sec\_conclude\]
