Broadcast Channels and Their Capacity Under Noisy Feedback

Broadcast channels model the downlink in a wireless network, i.e., the link from a base station transmitter to several receivers. The broadcast channel model was proposed by Tom Cover in 1972. While characterizing the capacity region of the broadcast channel remains an important open problem in Information Theory, the capacity of the Gaussian version of this channel has been resolved (single antenna version in 1974 and multi-antenna version in 2005).

Feeding signals received from the channel by the receiver(s) back to the transmitter(s) may, in general, lead to an enlargement of the capacity region of multi-terminal channels. A famous exception to this rule is the point-to-point channel (i.e., a single transmitter & a single receiver). For a limited class of broadcast channels, called the physically degraded broadcast channels, feedback does not enlarge the capacity region (El Gamal, '78,'81). However, this class does not contain wireless channels which arise in practice.

Results available in the literature show that feedback of even a limited form can result in an enlargement of the capacity region of broadcast channels. Full feedback (Dueck '80, Ozarow-Leung '84), feedback from only one receiver (Pillai '08), noisy feedback (Shayevitz-Wigger '13, Venkataramanan-Pradhan '13), rate-limited feedback (Yu-Wigger '14) have all been shown to lead to an enlargement of the capacity region in channels of practical interest. For multiple access channels (the uplink analogue of broadcast channels), it is known that even partial noisy feedback always enlarges the capacity region. A duality which is known to exist between multiple access and broadcast channels has led to the speculation that a similar result may hold for broadcast channels. We show that this is not the case in a class of channels which includes channels which arise in practice. In particular, we identify a threshold for the noise power in the feedback link above which feedback from a single receiver does not enlarge the capacity region (joint work with Sibiraj B Pillai (IIT Bombay).