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

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