# A Lower Bound on the Expected Distortion of Joint Source-Channel Coding

## TL;DR

This paper establishes a fundamental lower bound on how quickly the expected distortion in joint source-channel coding approaches its limit, showing it cannot converge faster than order n^{-1/2} in general.

## Contribution

The work provides the first general lower bound on the convergence rate of expected distortion in joint source-channel coding, resolving a long-standing open problem.

## Key findings

- Convergence rate is at least proportional to n^{-1/2} in general.
- For transmitting uniform bits over BSC with bandwidth ratio > 1, distortion gap is at least Omega(n^{-1/2}).
- The result applies broadly to memoryless source-channel pairs.

## Abstract

We consider the classic joint source-channel coding problem of transmitting a memoryless source over a memoryless channel. The focus of this work is on the long-standing open problem of finding the rate of convergence of the smallest attainable expected distortion to its asymptotic value, as a function of blocklength $n$. Our main result is that in general the convergence rate is not faster than $n^{-1/2}$. In particular, we show that for the problem of transmitting i.i.d uniform bits over a binary symmetric channels with Hamming distortion, the smallest attainable distortion (bit error rate) is at least $\Omega(n^{-1/2})$ above the asymptotic value, if the ``bandwidth expansion ratio'' is above $1$.