When is Noisy State Information at the Encoder as Useless as No Information or as Good as Noise-Free State?

TL;DR

This paper investigates conditions under which noisy state information at the encoder becomes effectively useless or as beneficial as perfect information, impacting channel capacity in information theory.

Contribution

It establishes a threshold-based criterion for when noisy encoder state information does not improve capacity, extending results beyond binary-input channels.

Findings

Capacity equals no side information when mutual information is below threshold.

Threshold depends solely on the state distribution.

Extensions to non-binary input channels are provided.

Abstract

For any binary-input channel with perfect state information at the decoder, if the mutual information between the noisy state observation at the encoder and the true channel state is below a positive threshold determined solely by the state distribution, then the capacity is the same as that with no encoder side information. A complementary phenomenon is revealed for the generalized probing capacity. Extensions beyond binary-input channels are developed.