When does Partial Noisy Feedback Enlarge the Capacity of a Gaussian Broadcast Channel?

Feedback is known to enlarge the capacity region of a Gaussian Broadcast Channel (GBC) with independent noise realizations at the receivers, and an average power constraint at the transmitter. The capacity enlargement may occur even when there is noisy feedback from only one of the two receivers. However, recent results show the existence of a feedback noise threshold, beyond which one-sided feedback from only the stronger receiver is futile in enlarging the capacity region. The current paper presents a tight characterization of the feedback noise threshold, which separates the regimes where feedback from only the stronger receiver enlarges the capacity or leaves it unchanged. The scheme used to prove this result also leads to some interesting observations on noisy feedback from only the weak receiver.