In a bi-directional relay channel, two nodes wish to exchange independent messages over a shared wireless half-duplex channel with the help of relays. Recent work has considered information theoretic limits of the bi-directional relay channel with a single relay. In this work we consider bi-directional relaying 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 an additive white Gaussian noise channel. We consider three novel 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+3 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 new
arXiv:0810.1268v2 [cs.IT] 9 Sep 2010
1
Bi-directional half-duplex protocols with
multiple relays
Sang Joon Kim, Natasha Devroye, and Vahid Tarokh
Abstract
In a bi-directional relay channel, two nodes wish to exchange independent messages over a shared
wireless half-duplex channel with the help of relays. Recent work has considered information theoretic
limits of the bi-directional relay channel with a single relay. In this work we consider bi-directional
relaying 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 an additive white Gaussian noise channel. We consider three novel 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 new (m, t)
protocol efficiently combines multi-hop routing with message-level network coding. Finally, we provide
a comprehensive treatment of the MHMR protocols with decode and forward relaying and amplify and
forward relaying in the Gaussian noise, obtaining their respective achievable rate regions, outer bounds
and relative performance under different SNRs and relay geometries, including an analytical comparison
on the protocols at low and high SNR.
Index Terms
bi-directional communication, achievable rate regions, decode and forward, amplify and forward,
multiple relays
Sang Joon Kim and Natasha Devroye were, and Vahid Tarokh is with the School of Engineering and Applied Sciences,
Harvard University, Cambridge, MA 02138. Natasha Devroye is currently with the University of Illinois at Chicago, Chicago,
IL 60607. Emails: adella0919@gmail.com, devroye@ece.uic.edu, vahid@seas.harvard.edu. This research is supported in part
by NSF grant number ACI-0330244 and ARO MURI grant number W911NF-07-1-0376. This work was supported in part by
the Army Research Office, under the MURI award No. N00014-01-1-0859. The views expressed in this paper are those of the
authors alone and not of the sponsor.
November 10, 2018
DRAFT
2
I. INTRODUCTION
In bi-directional channels, two terminal nodes (a and b) wish to exchange independent messages.
The “two-way channel” was first considered in [37], where full-duplex operation in which nodes may
transmit and receive simultaneously was assumed. While the capacity region is known for “restricted” and
additive white Gaussian noise two-way channels, it remains unknown in general. In wireless channels or
mesh networks, two-way or bi-directional communication may take place with the help of m other nodes
ri, i ∈{1, 2, · · · m} termed relays. Since full-duplex operation is, with current technology, of limited
practical significance, in this work we assume that the nodes are half-duplex, i.e. at each point in time, a
node can either transmit or receive symbols, but not both. Consequently, nodes communicate according
to pre-defined “protocols” which indicate which node transmits when.
Our main goal is to determine the limits of bi-directional communication with multiple relays. To do
so, we propose and determine the achievable rate regions, as well as outer bounds obtained using several
protocols. The protocols we propose for the multiple-relay bi-directional channel may be described in
terms of two parameters: the number of relays, m, and the number of temporal phases t, called hops.
Throughout this work, phases and hops are used interchangeably. We also define an intermediate hop as a
hop in which only relays transmit (and not the terminal nodes). Note that our protocols are all composed
of a number of temporal phases/hops due to the half-duplex nature of the channel.
Protocols. We denote our proposed protocols as (m, t) MHMR (Multiple Hops and Multiple Relays)
protocols, for general positive integers m ≥2 and t ≥2. For the special case of two hops (t = 2), the
terminal nodes may simultaneously transmit in phase 1 as in the MABC (Multiple Access Broadcast
Channel) protocol of [15], while the relays transmit the decoded messages to the terminal nodes in phase
2. For the special case of three hops (t = 3) the terminal nodes may sequentially transmit in the first
two phases as in the TDBC (Time Division Broadcast Channel) protocol of [15], after which the relays
transmit in phase 3. However, the (m, t) MHMR (Multiple Hops and Multiple Relays) protocol for t > 3
is not an immediate extension/generalization of the (m, 2) and (m, 3) protocols.
Relaying scheme. While a protocol in this work defines the temporal aspect (phases) of bi-directional
communication, it does not specify the type of relaying a node may perform, or relaying scheme.
That is, for each of the
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