In a bi-directional relay channel, two nodes wish to exchange independent messages over a shared half-duplex channel with the help of relays. In this thesis, we derive performance bounds for three temporal protocols of the single relay bi-directional channel. Then we extend the protocols to the multiple relay bi-directional channel. The relays may forward information in one of four manners: Amplify and Forward (AF), Decode and Forward (DF), Compress and Forward (CF) and Mixed Forward. The last scheme is a combination of CF in one direction and DF in the other.;In the first part of the thesis, we derive achievable rate regions and outer bounds for three temporal protocols with four possible relaying schemes. The first protocol is a two phase protocol where both users simultaneously transmit during the first phase and the relay alone transmits during the second. The second protocol considers sequential transmissions from the two users followed by a transmission from the relay while the third protocol is a hybrid of the first two protocols and has four phases. We provide a comprehensive treatment of protocols in Gaussian noise, obtaining their respective achievable rate regions, outer bounds, and their relative performance under different SNR and relay geometries.;The second part of the thesis considers bi-directional communications with multiple relays. We derive achievable rate regions and outer bounds for half-duplex protocols with multiple decode and forward relays and compare these to the same protocols with amplify and forward relays in a Gaussian noise channel. We consider three new classes of half-duplex protocols: the (m, 2) 2 phase protocol with m relays, the (m, 3) 3 phase protocol with m relays, and general (m, t) Multiple Hops and Multiple Relays (MHMR) protocols, where m is the total number of relays and 3 t ≤ m + 2 is the number of temporal phases in the protocol. The (m, 2) and (m, 3) protocols extend previous bi-directional relaying protocols for a single m = 1 relay, while the (m, t) protocol efficiently combines multi-hop routing with network coding.;In summary, the main contribution and goal of this thesis is the comprehensive treatment of a communication channel of recent interest: the bi-directional relay channel. Through the derivation of general inner and outer bounds on the capacity region of such channels and numerical analysis in the corresponding Gaussian noise channels we are able to quantitatively compare the merits of different temporal protocols and relaying schemes.
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