# The Velocity of the Propagating Wave for General Coupled Scalar Systems

## TL;DR

This paper derives a formula for the propagation velocity of wave-like profiles in spatially coupled scalar systems, with applications to coding and compressive sensing, enhancing understanding of message-passing algorithms.

## Contribution

It introduces a novel formula for wave velocity in coupled scalar systems, applicable to various algorithms like LDPC decoding and compressive sensing.

## Key findings

- Derived a general velocity formula for solitonic wave profiles
- Validated the formula with applications to LDPC codes and compressive sensing
- Provided insights into the dynamics of message-passing algorithms

## Abstract

We consider spatially coupled systems governed by a set of scalar density evolution equations. Such equations track the behavior of message-passing algorithms used, for example, in coding, sparse sensing, or constraint-satisfaction problems. Assuming that the "profile" describing the average state of the algorithm exhibits a solitonic wave-like behavior after initial transient iterations, we derive a formula for the propagation velocity of the wave. We illustrate the formula with two applications, namely Generalized LDPC codes and compressive sensing.