# Capacity Outer Bound and Degrees of Freedom of Wiener Phase Noise Channels with Oversampling

## TL;DR

This paper derives an outer capacity bound for Wiener phase noise channels with oversampling, revealing that phase modulation's maximum degrees of freedom at high SNR is limited to 1/4, even with increasing oversampling.

## Contribution

It introduces a novel outer bound on capacity for Wiener phase noise channels considering oversampling, and characterizes the degrees of freedom as the oversampling factor grows with transmit power.

## Key findings

- Maximum pre-log with phase modulation is at most 1/4 at high SNR.
- Outer capacity bound depends on the oversampling factor.
- Degrees of freedom are limited despite increased oversampling.

## Abstract

The discrete-time Wiener phase noise channel with an integrate-and-dump multi-sample receiver is studied. A novel outer bound on the capacity with an average input power constraint is derived as a function of the oversampling factor. This outer bound yields the degrees of freedom for the scenario in which the oversampling factor grows with the transmit power $P$ as $P^{\alpha}$. The result shows, perhaps surprisingly, that the largest pre-log that can be attained with phase modulation at high signal-to-noise ratio is at most $1/4$.