Analysis of the Second Moment of the LT Decoder

TL;DR

This paper provides a detailed analysis of the second moment of the ripple size in LT decoding, establishing bounds on the error probability and offering new analytic expressions for variance and expectation at finite lengths.

Contribution

It introduces the first finite-length analytic expressions for the ripple size variance and refines existing expectation bounds, advancing understanding of LT decoding performance.

Findings

Standard deviation of ripple size is of order √k.

Provides an analytic expression for ripple size variance.

Refines expectation bounds for ripple size at finite lengths.

Abstract

We analyze the second moment of the ripple size during the LT decoding process and prove that the standard deviation of the ripple size for an LT-code with length $k$ is of the order of $\sqrt k.$ Together with a result by Karp et. al stating that the expectation of the ripple size is of the order of $k$ [3], this gives bounds on the error probability of the LT decoder. We also give an analytic expression for the variance of the ripple size up to terms of constant order, and refine the expression in [3] for the expectation of the ripple size up to terms of the order of $1/k$, thus providing a first step towards an analytic finite-length analysis of LT decoding.