# Randomized vs. orthogonal spectrum allocation in decentralized networks: Outage Analysis

## TL;DR

This paper compares randomized frequency hopping and fixed frequency division in decentralized wireless networks, showing that FH can significantly improve outage capacity and spectral efficiency under uncertain user activity and channel conditions.

## Contribution

It introduces new bounds on user rates using entropy power inequality and demonstrates FH's advantages over FD in outage capacity for decentralized networks.

## Key findings

- Outage capacity is higher with frequency hopping than with frequency division.
- Frequency hopping achieves better spectral efficiency in decentralized networks.
- New bounds on achievable rates are derived using entropy power inequality.

## Abstract

We address a decentralized wireless communication network with a fixed number $u$ of frequency sub-bands to be shared among $N$ transmitter-receiver pairs. It is assumed that the number of users $N$ is a random variable with a given distribution and the channel gains are quasi-static Rayleigh fading. The transmitters are assumed to be unaware of the number of active users in the network as well as the channel gains and not capable of detecting the presence of other users in a given frequency sub-band. Moreover, the users are unaware of each other's codebooks and hence, no multiuser detection is possible. We consider a randomized Frequency Hopping (FH) scheme in which each transmitter randomly hops over a subset of the $u$ sub-bands from transmission to transmission. Developing a new upper bound on the differential entropy of a mixed Gaussian random vector and using entropy power inequality, we offer a series of lower bounds on the achievable rate of each user. Thereafter, we obtain lower bounds on the maximum transmission rate per user to ensure a specified outage probability at a given Signal-to-Noise Ratio (SNR) level. We demonstrate that the so-called outage capacity can be considerably higher in the FH scheme than in the Frequency Division (FD) scenario for reasonable distributions on the number of active users. This guarantees a higher spectral efficiency in FH compared to FD.