On Gaussian Half-Duplex Relay Networks

1.4 MB·English

by  Martina Cardone

1
On Gaussian Half-Duplex Relay Networks
Martina Cardone, Daniela Tuninetti, Raymond Knopp and Umer Salim
Abstract—ThispaperconsidersGaussianrelaynetworkswhere asself-interference,whichmaketheimplementationofdecod-
a source transmits a message to a sink terminal with the help of ing algorithm challenging. As a result HD relaying proves to
oneormorerelaynodes.Therelaysworkinhalf-duplexmode,in
beamorepracticaltechnologywithitsrelativelysimplesignal
thesensethattheycannottransmitandreceiveatthesametime.
processing. Thus it is more realistic to assume that the relay
For the case of one relay, the generalized Degrees-of-Freedom
is characterized first and then it is shown that capacity can operates in HD mode either in Frequency Division Duplexing
be achieved to within a constant gap regardless of the actual (FDD) or Time Division Duplexing (TDD). In FDD, the
value of the channel parameters. Different achievable schemes relay uses one frequency band to transmit and another one
arepresentedwitheitherdeterministicorrandomswitchforthe
to receive; in TDD, the relay listens for a fraction γ ∈ [0,1]
3 relay node. It is shown that random switch in general achieves
1 higher rates than deterministic switch. For the case of K relays, oftimeandthentransmitsintheremaining1−γ timefraction.
0 it is shown that the generalized Degrees-of-Freedom can be From an application point of view, the HD model fits future
2 obtained by solving a linear program and that capacity can be 4G network with relays [1], where the relay communicates
n achievedtowithinaconstantgapofK/2log(4K).Thisgapmay over-the-air with the source, which is called the Donor-eNB.
a be further decreased by considering more structured networks We keep our focus on deployment scenarios where the relay
J such as, for example, the diamond network.
works in TDD HD mode.

3 Index Terms—Relay Channel, Generalized Degrees-of- In this work we concentrate on the HD relay networks,
2 Freedom, Capacity to within a Constant Gap, Inner bound,
where the relays transmit and receive in different time slots.
] Outer bound, Half-duplex. HD relaying has received considerable attention lately, as
T summarized next.
I
. I. INTRODUCTION
s
c A. Related Work
The performance of wireless systems can be enhanced by
[
enablingcooperationbetweenthewirelessnodes.Thesimplest a) Single Relay Networks: The RC was first introduced
1
form of cooperation is modeled by the Relay Channel (RC) by van der Meulen [2] and then thoroughly studied by Cover
v
where a source terminal communicates to a destination with and El Gamal [3]. In [3] the authors study the general memo-
2
2 the help of a relay node. In this multi-hop system the relay ryless RC, derive inner and outer bounds on the capacity and
5 helps to increase the coverage and the throughput of the establish the capacity for some classes of RCs. The proposed
5 network. outer bound is now known as the max-flow min-cut outer
.
1 Relays employed in practical wireless networks can be bound, or cut-set for short, which can be extended to more
0 classified into two categories: Full-Duplex (FD) and Half- general memoryless networks [4]. Two achievable relaying
3 Duplex (HD). In the former case the relays transmit and strategies were proposed in [3], whose combination is still
1
receive simultaneously; in the latter case the relays can either the largest known achievable rate for a general RC, namely
:
v transmit or receive at any given time, but not both. There are Decode-and-Forward (DF) and Compress-and-Forward (CF).
Xi some relatively expensive relay devices which work in FD In DF, the relay fully decodes the message sent by the
mode, normally used in military communications. However source and then coherently cooperates with the source to
r
a FDrelayinginwirelessnetworkshaspracticalrestrictionssuch communicate this information to the destination. In CF, the
relay does not attempt to recover the source message, but it
M. Cardone and R. Knopp are with the Mobile Communications just compresses the information received and then sends it to
Department at Eurecom, Sophia Antipolis, 06560, France (e-mail: car-
the destination. The capacity of the general memoryless RC
[email protected];[email protected]).D.TuninettiiswiththeElectricaland
Computer Engineering Department of the University of Illinois at Chicago, is known for some special classes, in particular for degraded
Chicago, IL 60607 USA (e-mail: [email protected]). U. Salim is with Al- RCs, reversely degraded RCs and semi-deterministic RC [3].
gorithm Design group of Intel Mobile Communications, Sophia Antipolis,
The HD-RC was studied by Host-Madsen in [5]. Here
06560,France(e-mail:[email protected]).
TheworkofD.TuninettiwaspartiallyfundedbyNSFunderawardnumber the author derives both an upper and a lower bound on the
0643954;thecontentsofthisarticlearesolelytheresponsibilityoftheauthor capacity. The former is based on the cut-set arguments, the
and do not necessarily represent the official views of the NSF. The work
latterexploitsthePartial-Decode-and-Forward(PDF)strategy
of D. Tuninetti was possible thanks to the generous support of Telecom-
ParisTech,ParisFrance,whiletheauthorwasonasabbaticalleaveatthesame where the relay only decodes part of the message sent by the
institution.Eurecom’sresearchispartiallysupportedbyitsindustrialpartners: source. Host-Madsen considers the transmit/listen state of the
BMW, Cisco Systems, Monaco Telecom, Orange, SFR, ST Ericsson, SAP,
relay as fixed and therefore known a priori to all nodes.
SwisscomandSymantech.TheresearchworkcarriedoutatIntelbyU.Salim
has received funding from the European Community’s Seventh Framework In [6], Kramer shows that larger rates can be achieved by
Program(FP7/2007-2013)SACRAproject(grantagreementnumber249060) using a random transmit/listen switch strategy at the relay. In
The results in this paper have been submitted in part to the 2013 IEEE
this way, the source and the relay can harness the randomness
International Conference on Communications (ICC) and to the 2013 IEEE
InternationalSymposiumonInformationTheory(ISIT). thatliesintheswitchinordertotransmitextrainformation.An
2
importantobservationof[6]isthatthereisnoneedtodevelop Their achievable scheme is a clever extension of the two-hop
a separate theory for memoryless networks with HD nodes as DF strategy of [16]. It is interesting to note that [15] derived
the HD constraints can be incorporated into the memoryless closed-formexpressionsforthefractionsoftimetherelaysare
FD framework. In this work we shall adopt this approach in active and tight outer bounds based on the dual of the linear
deriving outer and inner bounds for HD relay networks. program (LP) associated with the classical cut-set bound.
b) Multiple Relay Networks: The pioneering work of Extensions of these ideas to more than two relays appear
Cover and El Gamal [3] has been extended to networks with difficult due to the combinatorial structure of the problem.
multiple relays. In [7] Kramer et al. proposed several inner Inspired by [15], the authors in [17] showed that for a very
and outer bounds as a generalization of DF, CF and the cut- specific HD diamond network with (K−2)=3 relays, (K−
setbound.ItwasshownthatDFachievestheergodiccapacity 1) = 4 states out of 2K−2 = 8 are active in the cut-set outer
of a wireless Gaussian network with phase fading if phase bound.Inthesamework,itwasverifiednumericallythatfora
information is available only locally and the relays are close generalHDdiamondnetworkwith(K−2)≤7relays,(K−1)
to the source node. states suffice for the cut-set upper bound and it is conjectured
The exact characterization of the capacity region of a that the same holds for any number of relays. We remark that
general memoryless network is challenging. Recently it has in[17]onlythecut-setupperboundwasconsidered;moreover
been advocated that progress can be made towards under- only the case of deterministic switch and per-symbol power
standing the capacity by showing that achievable strategies constraint was considered.
are provably “close” to (easily computable) outer bounds [8]. Multi relay networks were also studied in [18] where the
As an example, Etkin et al. characterized the capacity of the authors determine numerically the optimal fractions of time
GaussianInterferenceChanneltowithin1bitregardlessofthe each relay transmits/receives with DF through an iterative
system parameters [9]. In [10], the authors study unicast and algorithm.Alsointhiscasetherelaysusedeterministicswitch.
multicastGaussianrelaynetworkswithK nodesandshowthat
capacitycanbeachievedtowithin(cid:80)K 5min{M ,N }bits
k=1 k k B. Contributions
with quantize-remap-and-forward (QMF), where M and N
k k
are the number of transmit and receive antennas, respectively, In this work we focus on the HD relay networks. The exact
of node k ∈[1:K]. Interestingly, the result is valid for static capacity of this channel is unknown. In this paper we make
and ergodic fading networks where the nodes operate either progress toward determining its capacity by giving a constant
in FD mode or in HD mode with deterministic listen/transmit gap result for any Gaussian network with random switch. Our
schedulefortherelays.Forsingleantennasystems,Limetal. main contribution can be summarized as follows:
in [11] recently showed that this 5K bits gap can be reduced 1) We determine the generalized Degrees-of-Freedom
to0.6K bitsforFDrelaynetworkswithnoisynetworkcoding (gDoF) of the HD RC with a single relay. We identify
(NNC). Both QMF and NNC are network extensions of CF. three schemes that achieve the gDoF upper bound. The
Thegapcharacterizationof[11]isvalidforanymultirelay simplest one is inspired by the Linear Deterministic
network but linear in the number of nodes in the network, Approximation (LDA) of the Gaussian noise channel
which could be a too coarse capacity characterization for at high SNR [10]; it uses superposition coding at the
networks with a large number of nodes. Tighter gaps can source, DF at the relay and stripping decoding at both
be obtained for more structured networks. For example, the the relay and destination; we note that neither power al-
diamondnetworkmodelwasfirstproposedin[12].Adiamond locationnorbackwarddecodingisrequiredatthenodes.
network consists of a source, a destination and K−2 relays. The second and third schemes use more sophisticated
The source and the destination cannot communicate directly coding techniques and are based on PDF and NNC
and the relays cannot communicate among themselves. In strategies [4].
other words, a general Gaussian multi relay network with K 2) We prove that the three schemes above achieve the
nodes is characterized by K(K −1) channel gains, while a capacity to within a constant gap regardless of the
diamond network only has 2(K−2) non zero channel gains. channel parameters. We consider both deterministic and
In[12]thecaseoftworelayswasstudiedforwhichanachiev- randomswitchfortherelay.Thusinthesecondcasethe
able region based on time sharing between DF and amplify- relayharnessestherandomnessthatliesintheswitchto
and-forward(AF)wasproposed.ThecapacityofageneralFD achieve larger rates and therefore smaller gaps from the
diamond network is known to within 2log(K −1) bits [13]. cut-set upper bound.
If in addition the FD diamond network is symmetric, that is, 3) We prove that PDF with random switch is optimal for
all source-relay links are equal and all relay-destination links a Gaussian diamond network with one relay, i.e., it
are equal, the gap is less than 2 bits for any K [14]. HD achieves the capacity, even though we were not able
diamond networks have been studied as well, albeit only for to determine the capacity achieving input distribution.
deterministic switch for the relays. In a HD diamond network 4) WedeterminethecapacityofthenoiselessLDAchannel.
with (K −2) relays, there are 2K−2 possible combinations In particular we show that random switch and non-
of listening and transmit states, since each relay can either uniform inputs at the relay are optimal.
transmit or receive. For the case of (K−2)=2 relays, [15] 5) For HD networks with K nodes (HD-MRC), all
shows that out of 2K−2 =4 possible states only (K−1)=3 of which employ random switch, we prove that
states suffice to achieve capacity to within less than 4 bits. NNC achieves the cut-set outer bound to within
3
K/2log(4K) bits. For diamond relay networks assum- B. The Gaussian Half-Duplex RC
ing that the conjecture in [17] holds for any K, the gap We consider a single-antenna complex-valued Gaussian
can be reduced to 5log(K). Half-Duplex Relay Channel (G-HD-RC), shown in Fig. 1,
where the inputs are subject to an average power constraint,
C. Paper Organization
described by the input/output relationship
The rest of the paper is organized as follows. Section √
Y = CX (1−S )+Z , (1a)
II describes the channel model and states our main result. r s r r
√ √
Section III derives the gDoF upper-bound based on the cut- Y = SX +ejθ IX S +Z , θ ∈R, (1b)
d s r r d
set bound. Section IV provides a lower bound based on the
where the channel gains C,S,I are constant and therefore
PDFstrategy.SectionVhighlightsamotivatingexamplebased
known to all terminals. Without loss of generality we can
on the LDA and provides a simple achievable based on this
assume that the relay output Y does not contain the relay
approach.SectionsVIandVIIaredevotedtotheanalyticaland r
input X because the relay node can always subtract X
numericalproofsthatthecapacityforasinglerelaynetworkis r r
from Y . Moreover, since a node can compensate for the
achievable to within a constant gap, respectively. Section VIII r
phase of one of its channel gains, we can assume without
considers networks with multiple relays and it shows that the
loss of generality that the channel gains from the source to
gDoF can be computed by solving a linear program and that
the other two terminals are real-valued and nonnegative. The
NNC achieves the capacity to within a constant gap. Section
channelinputsaresubjecttounitaryaveragepowerconstraints
IX concludes the paper.
without loss of generality, i.e., E[|X |2]≤1, u∈{s,r}. The
u
II. SINGLERELAYNETWORKS:SYSTEMMODEL ‘switch’ random variable Sr is binary. The noises Zd,Zr are
A. General Memoryless Relay Channel assumedtobezero-meanjointlyGaussianandwithunitpower
without loss of generality. In particular (but not without loss
A RC consists of two input alphabets (X ,X ), two output
s r of generality) in this work we assume that Z is independent
alphabets (Y ,Y ) and a transition probability P . d
r d Yr,Yd|Xs,Xr of Z . In the following we will only consider G-HD-RC for
ThesourcesendssymbolsfromX ,therelayreceivessymbols r
s which C > 0 and I > 0 in (2), since for either C = 0 or
inY andsendssymbolsfromX ,andthedestinationreceives
r r I =0 the relay is disconnected from either the source or the
symbols in Y . The source has a message W ∈[1:2NR] for
d destination, respectively, so the channel reduces to a point-to-
the destination where N denotes the codeword length and R
point channel with capacity log(1+S).
the transmission rate in bits per channel use 1. At time i,
The capacity of the channel in (1) is unknown. Here we
i ∈ [1 : N], the source maps its message W into a channel
makeprogresstowarddeterminingitscapacitybyestablishing
input symbol X (W) and the relay maps its past channel
s,i its gDoF, i.e., an exact capacity characterization in the limit
observations into a channel input symbol X (Yi−1). The
r,i r for infinite SNR [9], and its capacity to within a constant gap
channel is assumed to be memoryless, that is, the following
atanyfiniteSNR.ConsiderSNR>0andtheparameterization
Markov chain holds for all i∈[1:N]
(W,Yi−1,Yi−1,Xi−1,Xi−1)→(X ,X )→(Y ,Y ).
r d s r s,i r,i r,i d,i S :=SNRβsd,source-destination link, (2a)
At time N, the destination makes an estimate W(cid:99)(YdN) of the I :=SNRβrd,relay-destination link, (2b)
message W based on all its channel observations YN. A rate
R is said to be (cid:15)-achievable if P[W(cid:99) (cid:54)= W] ≤ (cid:15)dfor some C :=SNRβsr,source-relay link, (2c)
(cid:15) ∈ [0,1]. The capacity is the largest nonnegative rate that is for some (β ,β ,β )∈R3. We define:
sd rd sr +
(cid:15)-achievable for any (cid:15)>0.
Definition 1. The gDoF is
We note that half-duplex channels are a special case of the
memoryless full-duplexframework in the followingsense [6]: C(HD−RC)
d(HD−RC) := lim ,
let the channel input of the relay be the pair (Xr,Sr), where SNR→+∞log(1+SNR)
X ∈ X as before and S ∈ {0,1} is the state random
r r r where C(HD−RC) is the capacity of the G-HD-RC.
variable that indicates whether the relay is in receive-mode
(Sr = 0) or in transmit-mode (Sr = 1). The memoryless Definition 2. The capacity C(HD−RC) is said to be known to
channel transition probability is defined as within b bits if one can show rates R(in) and R(out) such that
PYr,Yd|Xs,Xr,Sr=0 =PY(0r),Yd|Xs,Sr=0 R(in) ≤C(HD−RC) ≤R(out) ≤R(in)+blog(2).
P =P(1) P(1) , Our main result for single relay networks can be summa-
Yr,Yd|Xs,Xr,Sr=1 Yd|Xs,Xr,Sr=1 Yr|Sr=1
rized as
that is, when the relay is in receive-mode (S = 0) the
r
outputsYr,Yd areindependentofXr andwhentherelayisin Theorem 1. ThegDoFoftheG-HD-RCisgivenby(3)atthe
transmit-mode (Sr =1) the relay output Yr is independent of top of next page and the cut-set upper bound is achieved to
everythingelse.Inotherwords,the(stillmemoryless)channel within the following number of bits
isnowspecifiedbythetwotransitionprobabilitiesoneforeach
Achievable scheme LDA NNC PDF
mode of operation [6].
analytical gap 3 1.61 1
1Logarithmsareinbase2. numerical gap 1.59 1.52 1
4
(cid:40)
β + (βrd−βsd)(βsr−βsd) for β >β , β >β
d(HD−RC) = sd (βrd−βsd)+(βsr−βsd) sr sd rd sd (3)
β otherwise.
sd
whereLDAisaverysimpleachievableschemeinspiredbythe • in (7): the terms b1 and b2 are defined as
linear deterministic approximation of the G-HD-RC at high
(cid:16) √ √ (cid:17)
SNR, PDF is partial-decode-and-forward and NNC is noisy- log 1+( I+ S)2
network-coding, or compress-and-forward. b1 := log(1+S) >1 since I >0, (20)
Sections III-VII are devoted to the proof of Theorem 1. log(1+C+S)
b := >1 since C >0. (21)
2 log(1+S)
Remark 1. The gDoF of the Gaussian Full-Duplex Relay
Channel (G-FD-RC) is Proof: The proof and the definitions of the above quan-
tities can be found in Appendix A.
d(FD−RC) =β +min{[β −β ]+,[β −β ]+}, (4)
sd sr sd rd sd Theupperboundin(5)willbeusedtoprovethatPDFwith
and its capacity C(FD−RC) is achievable to within 1 bit randomswitchachievescapacitytowithin1bit,theonein(6)
to prove that PDF with deterministic switch also achieves
per channel use [10]. We notice that HD achieves the same
capacity to within 1 bit and for numerical evaluations (since
gDoF of FD if min{β ,β } ≤ β , in which case the RC
rd sr sd
wedonotknowthedistributionP∗ thatmaximizesthe
behaves gDoF-wise like a point-to-point channel from TX to Xs,Xr,Sr
cut-set upper bound in (5)), and the one in (7) for analytical
RX with gDoF given by β . In either FD or HD the gDoF
sd
computations such as the derivation of the gDoF.
has a ‘routing’ interpretation [10]: if the weakest link from
the source to the destination through the relay is smaller Proposition 2. The gDoF of the G-HD-RC is upper bounded
than the direct link from the source to the destination then by the right hand side of (3).
directtransmissionisoptimalandtherelaycanbekeptsilent,
otherwise it is optimal to communicate with the help of the Proof: The proof can be found in Appendix B.
relay.
IV. SINGLERELAYNETWORKS:LOWERBOUNDSBASED
III. SINGLERELAYNETWORKS:UPPERBOUND ONPARTIAL-DECODE-AND-FORWARD
This section is devoted to the proof of a number of upper This section is devoted to the proof of a number of lower
bounds that we shall use for the converse part of Theorem 1. bounds that we shall use for the direct part of Theorem 1.
From the cut-set bound we have: From the achievable rate with PDF we have:
Proposition 1. The capacity of the G-HD-RC is upper Proposition 3. The capacity of the G-HD-RC is lower
bounded as in (5), (6) and (7) at the top of next page where bounded as in (8), (9) and (10) at the top of next page where
• in (5): the distribution PX∗s,Xr,Sr is the one that maxi- • in (8): we fix the input PU,Xs,Xr,Sr to evaluate the PDF
mizes the cut-set upper bound, lower bound; in particular we set P to be the
• in (6): the parameter γ := P[Sr = 0] ∈ [0,1] represents samedistributionthatmaximizesthecXuts-,sXert,Surpperbound
the fraction of time the relay node listens, H(γ) is the in (5) and we choose either U = X or U = X S +
r r r
binary entropy function defined as X (1−S ).
s r
• in (9): the parameter γ := P[Sr = 0] ∈ [0,1] represents
H(γ):=−γlog(γ)−(1−γ)log(1−γ), (12)
the fraction of time the relay node listens, the maximiza-
tion is over the set (13)-(15) as for the cut-set upper
the maximization is over the set
bound in (6), the mutual informations I ,...,I are
5 8
γ ∈[0,1], (13)
I :=I in (16), (22)
|α |≤1, (14) 5 1
1
I :=I in (17), (23)
(P ,P ,P ,P )∈R4 6 2
s,0 s,1 r,0 r,1 +
I :=log(1+max{C,S}P )≤I in (18), (24)
:γP +(1−γ)P ≤1, u∈{s,r}, (15) 7 s,0 3
u,0 u,1
I :=I in (19), (25)
8 4
and the mutual informations I ,...,I are defined as
1 4
and I(PDF) :=I(S ;Y ) is computed from the density
0 r d
I :=log(1+S P ), (16)
1 s,0
I2 :=log(cid:16)1+SPs,1+IPr,1+2|α1|(cid:112)SPs,1IPr,1(cid:17), (17) fYd(t)=πγv exp(−|t|2/v0)+1π−vγexp(−|t|2/v1),
0 1
I :=log(1+(C+S)P ), (18) (26)
3 s,0
I4 :=log(cid:0)1+(1−|α1|2)S Ps,1(cid:1), (19) with t∈C,v =exp(I ),v =exp(I ).
0 5 1 6
5
(cid:110) (cid:111)(cid:12)
C(HD−RC) ≤min I(X ,X ,S ;Y ),I(X ;Y ,Y |X ,S ) (cid:12) (5)
s r r d s r d r r (cid:12)
(Xs,Xr,Sr)∼PX∗s,Xr,Sr
(cid:110) (cid:111)
≤maxmin H(γ)+γI +(1−γ)I ,γI +(1−γ)I =:r(CS−HD) (6)
1 2 3 4
(cid:18) (cid:19)
(b −1)(b −1)
≤2log(2)+log(1+S) 1+ 1 2 , (7)
(b −1)+(b −1)
1 2
(cid:110)
C(HD−RC) ≥min I(X ,X ,S ;Y ),
s r r d
(cid:111)(cid:12)
I(U;Y |X ,S )+I(X ;Y |X ,S ,U) (cid:12) , (8)
r r r s d r r (cid:12)
(Xs,Xr,Sr)∼PX∗s,Xr,Sr andU =Xr orU =XrSr+Xs(1−Sr)
(cid:110) (cid:111)
C(HD−RC) ≥maxmin I(PDF)+γI +(1−γ)I ,γI +(1−γ)I =:r(PDF−HD) (9)
0 5 6 7 8
(cid:18) (cid:19)
(c −1)(c −1)
≥log(1+S) 1+ 1 2 , (10)
(c −1)+(c −1)
1 2
(cid:16) (cid:17)
r(LDA−HD):=log(1+S)+ log 1+ 1+IS (cid:20)log(cid:18)1+ C (cid:19)−log(cid:18)1+ S (cid:19)(cid:21)+. (11)
(cid:16) (cid:17) (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105)+ 1+S 1+S
log 1+ I + log 1+ C −log 1+ S
1+S 1+S 1+S
• in (10): the terms c1 and c2 are The rest of the section is devoted to the proof of Proposi-
tion 5. Before we provide the details of the scheme, we point
log(1+I+S)
c := >1 since I >0, (27) out three important practical aspects of this scheme that are
1 log(1+S)
worth noticing:
log(1+max{C,S})
c2 := log(1+S) >1 since C >0. (28) 1) the destination does not use backward decoding, which
simplifies the decoding procedure and incurs no delay,
Proof: The proof can be found in Appendix C. 2) the destination uses successive decoding, which is sim-
The lower bound in (8) will be compared to the upper pler than joint decoding, and
bound in (5) to prove that PDF with random switch achieves 3) no power allocation is applied at the source or at the
capacity to within 1 bit, the one in (9) with the one in (6) relay, which simplifies the encoding procedure and can
to prove that PDF with deterministic switch also achieves be used for time-varying channel as well. The source
capacity to within 1 bit and for numerical evaluations, and uses superposition coding to ‘route’ part of its data
the one in (10) for analytical computations such as evaluation through the relay.
of the achievable gDoF.
These aspects will be clear from the actual description of the
Proposition 4. The gDoF of the G-HD-RC is lower bounded scheme. Moreover we can show that
by the right hand side of (3).
Proposition 6. The LDA strategy achieves the gDoF upper
Proof: The proof can be found in Appendix D. bound in (3).
Propositions 2 and 4 show that the gDoF for the G-HD-RC
Proof: The proof can be found in Appendix E.
is given by (3).
V. SINGLERELAYNETWORKS:ASIMPLEACHIEVABLE A. A Motivating Example
STRATEGY The LDA of the G-HD-RC in (1) is a deterministic channel
Inthissectionweproposeaverysimpleachievablescheme with input-output relationship
that is gDoF optimal, that achieves capacity to within 3 bits
and that can be implemented in practical HD relay networks. Yr =Sn−βsrXs (1−Sr), (29a)
In Section V-A we describe a deterministic-switch achievable Yd =Sn−βsdXs+Sn−βrdXr Sr, (29b)
strategyfortheLinearDeterministicApproximation(LDA)of
the G-HD-RC at high SNR which we mimic in Section V-B for some nonnegative integers β ,β ,β , where the inputs
sr sd rd
to derive an achievable rate for the G-HD-RC at any SNR. andoutputsarevectorsoflengthn:=max{β ,β ,β }and
sr sd rd
This achievable scheme is referred to as the LDA-strategy, or S is the n×n shift matrix [10].
LDA for short. The main result of this section is: The capacity of a deterministic RC is given by the cut-set
upperbound[10].FortheLDAin(29)thecut-setupper-bound
Proposition 5. The capacity of the G-HD-RC is lower
evaluates to
bounded as in (11) at the top of this page.
6
(cid:40) (cid:110) (cid:111)
β +max min (1−θ∗(γ))log 1 +θ∗(γ)log L−1,γ[β −β ]+ for β >β , β >β
C(HD) = sd γ∈[0,1] 1−θ∗(γ) θ∗(γ) sr sd sr sd rd sd
β otherwise.
sd
(30)
(cid:110) (cid:111)
max{R}= max min I(X ,X ,S ;Y ),I(X ;Y ,Y |X ,S )
s r r d s r d r r
PXs,Xr,Sr
(cid:110) (cid:111)
= max min H(Y ),H(Y ,Y |X ,S )
d r d r r
PXs,Xr,Sr
(cid:110) (cid:111)
≤ max min H(Y |S ),H(Y ,Y |X ,S ) +H(S )
d r r d r r r
PXs,Xr,Sr
≤ max min{γβ +(1−γ)max{β ,β },γmax{β ,β }+(1−γ)β }+log(2)
sd sd rd sd sr sd
γ∈[0,1]
=β +γ∗ [β −β ]++log(2), (31)
sd LDA sr sd
Theorem 2. The capacity of the deterministic HD RC in (29) Remark 2. The HD optimal strategy in Figs. 2(a) and 2(b)
is given by (30) at the top of this page where θ∗(γ) = 1− shouldbecomparedwiththeFDoptimalstrategyinFig.2(c).
max{1/L,γ} and L:=2[βrd−βsd]+. In Fig. 2(c), in a given time slot t, the source sends b1[t] (of
length β bits) directly to the destination and b [t+1] (of
Proof: The proof can be found in Appendix K. sd 2
length at most β −β bits) to the relay; the relay decodes
Next we further upper bound the capacity in (30) because sr sd
bothb [t]andb [t+1]andforwardsb [t+1]inthenextslot;
ourgoalistogetinsightsintoasymptoticallyoptimalstrategies 1 2 2
in slot t the relay sends b [t] (of length at most β −β
for the G-HD-RC. For the channel in (29) we have that (31) 2 rd sd
bits) to the destination; the number of bits the relay forwards
at the top of this page holds, where γ∗ is the optimal γ :=
LDA must be the minimum among the number of bits the relay
P[S = 0] ∈ [0,1] obtained by equating the two arguments
r can decode (given by β −β ) and the number of bits that
within the min and is given by sr sd
can be decoded at the destination without harming the direct
(cid:40)
γL∗DA := 0(βrd−(ββsrdd)−+β(βsdsr)−βsd) foothreβrwrdis>e.βsd, βsr >βsd tthraentsomtaislsniounmfbreormotfhbeitssoudrecceod(gediveant tbhyeβdredst−inβatsido)n. Tisherefore,
Next we show that the upper bound in (31) is achievable βsd+min{βrd−βsd,βsr−βsd},
to within log(2) = 1 bit. This 1 bit represents the maximum
which gives precisely the optimal gDoF for the full-duplex
amount of information I(S ;Y ) that could be conveyed to
r d channel in (4)
the destination by a random switch at the relay. If we neglect
the term log(2) we can achieve the upper bound in (31) Remark 3. Fig.3comparesthecapacitiesoftheFDandHD
with the scheme shown in Figs. 2(a) and 2(b) for the case LDAchannels;italsoshowssomeachievableratesfortheHD
min{β ,β } > β , which is the case where the upper LDA channel. In particular, the capacity of the FD channel is
sr rd sd
bound differs from direct transmission, i.e., X = 0. In given by (4) (dotted black curve labeled “FD”), the capacity
r
Phase I/Fig. 2(a) the relay listens and the source sends b (of of the HD channel is given by (30) (solid black curve labeled
1
length βsd bits) directly to the destination and b2 (of length “HD” obtained with the optimal p∗0 in Appendix K) and its
β −β bits) to the relay; note that b is below the noise upper bound by (31) (red curve labeled “HDlda upper”). For
sr sd 2
floor at the destination; the duration of Phase I is γ, hence comparison we also show the performance when the source
the relay has accumulated γ(β −β ) bits to forward to the uses i.i.d. Bernoulli(1/2) bits and the relay uses one of the
sr sd
destination. In Phase II/Fig. 2(b) the relay forwards the bits followingstrategies:i.i.d.Bernoulli(q)bitsandrandomswitch
learntinPhaseItothedestinationby‘repackaging’theminto (bluecurvelabeled“HDiidq+rand”obtainedbynumerically
a (of length β −β bits); the source keeps sending a new optimizing q ∈ [0,1]), i.i.d. Bernoulli(1/2) bits and random
rd sd
b (of length β bits) directly to the destination; note that a switch (green curve labeled “HDiid 1/2+rand” obtained with
1 sd
does not interfere at the destination with b2; the duration of p0 = 1/L in Appendix K), and i.i.d. Bernoulli(1/2) bits and
Phase II is such that all the bits accumulated in Phase I can deterministicswitch(magentacurvelabeled“HDiid1/2+det”
be delivered to the destination, that is and given by βsd+min{γ[βsr−βsd]+,(1−γ)[βrd−βsd]+}).
We can draw conclusions from Fig. 3:
γ(β −β )=(1−γ)(β −β ),
sr sd rd sd • With deterministic switch: i.i.d. Bernoulli(1/2) bits for
which gives precisely the optimal γ∗ . The total number of the relay are optimal but this choice is quite far from
LDA
bits decoded at the destination is capacity (magenta curve vs. solid black curve); this
choice however is at most one bit from optimal (magenta
1·β +γ∗ ·(β −β ),
sd LDA sr sd curve vs. red curve).
which gives precisely the optimal gDoF for the half-duplex • Withrandomswitch:theoptimalinputdistributionforthe
channel in (3). relayisnoti.i.d.bits;i.i.d.inputsincursarateloss(blue
7
curve vs. solid black curve); if in addition we insist on Note that γ∗ → γ∗ as SNR increases. Moreover
LDA
i.i.d. Bernoulli(1/2) bits for the relay we incur a further we give here an explicit closed form expression for the
loss (green curve vs. blue curve). optimal duration of the time the relay listens to the
This shows that for optimal performance the relay inputs are channel.
correlated and that random switch should be used. The rate sent directly from the source to the destination,
that is, the sum of (33) and (35), is
B. An achievable strategy inspired by the LDA (cid:18) (cid:19)
S
R +R =log(1+S)−γ∗log 1+ .
We can mimic the LDA strategy in Section V-A for the G- b1[1] b1[2] 1+S
HD-RCasfollows.WeassumeS <C,otherwiseweusedirect (cid:124) (cid:123)(cid:122) (cid:125)
transmission to achieve R = log(1+S). The transmission is ∈[0,log(2)]
divided into two phases: Thereforethetotalratedecodedatthedestinationthrough
• Phase I of duration γ: the transmit signals are the two phases is
√ √ 1
X [1]= 1−δX + δX , δ = , R +R +R =r(LDA−HD) in (11).
s b1[1] b2 1+S b1[1] b1[2] b2
Xr[1]=0. Wenoticethattherateexpressionforr(LDA−HD) in(11),
The relay applies successive decoding of X followed which was derived under the assumption C >S, is valid
by X from b1[1] forallC sinceforC <S itreducestodirecttransmission
b2 √ √ √ √ from the source to the destination.
Y [1]= C 1−δX + C δX +Z [1],
r b1[1] b2 r
which is possible if (rates are normalized by the total VI. SINGLERELAYNETWORKS:ANALYTICALGAPS
duration of the two phases)
In the previous sections we described upper and lower
(cid:18) (cid:19)
1
R ≤γlog(1+C)−γlog 1+C bounds to determine the gDoF of the G-HD-RC. Here we
b1[1] 1+S
show that the same upper and lower bounds are to within a
(cid:18) (cid:19)
1 constant gap of one another thereby concluding the proof of
R ≤γlog 1+C . (32)
b2 1+S Theorem 1. We consider both the case of random switch and
ThedestinationdecodesX treatingX asnoisefrom of deterministic switch for the relay.
b1[1] b2
√ √ √ √
Proposition 7. [PDF and random switch] PDF with random
Y [1]= S 1−δX + S δX +Z [1],
d b1[1] b2 d switch is optimal to within 1 bit.
which is possible if
Proof: The proof can be found in Appendix G.
(cid:18) (cid:19)
1
R ≤γlog(1+S)−γlog 1+S . (33)
b1[1] 1+S Proposition 8. [PDF and deterministic switch] PDF with
deterministic switch is optimal to within 1 bit.
Finally, since we assume S < C, Phase I is successful
if (32) and (33) are satisfied. Proof: The proof can be found in Appendix H.
• Phase II of duration 1−γ: the transmit signals are
Proposition 9. [LDA (deterministic switch)] LDA is optimal
X [2]=X to within 3 bits.
s b1[2]
X [2]=X
r b2 Proof: The proof can be found in Appendix I.
The destination applies successive decoding of X (by
b2
exploiting also the information about b2 that it gathered We conclude this section with a discussion on the gap that
in the first phase) followed by Xb1[2] from can be obtained with NNC. The NNC strategy is a network
√ √
generalization of the CF. It has been proposed for general
Y [2]= SX +e+jθ IX +Z [2],
d b1[2] b2 d memorylessnetworksanditisoptimaltowithinaconstantgap
which is possible if for full-duplex multicast networks with an arbitrary number
(cid:18) (cid:19) (cid:18) (cid:19) of relays, where the gap grows linearly with the number of
I S
Rb2≤(1−γ)log 1+1+S +γlog 1+1+S (34) relays [11]. In the case with only one relay, NNC reduces
to the classical CF [4, Remark 18.6] and represents a good
R ≤(1−γ)log(1+S). (35)
b1[2] alternative to the PDF especially in the case when the link
• ByimposingthattherateRb2 isthesameinbothphases, between the source and the relay is weaker than the direct
that is, that (32) and (34) are equal, we get that γ should link. The NNC rate is presented in Appendix F. By using
due chosen equal to γ∗ Remark 5 in Appendix F we have
(cid:16) (cid:17)
log 1+ I Proposition 10. [NNC and deterministic switch] NNC with
γ∗= 1+S . deterministic switch is optimal to within 1.61 bits.
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
log 1+ I +log 1+ C −log 1+ S
1+S 1+S 1+S Proof: The proof can be found in Appendix J.
8
VII. SINGLERELAYNETWORKS:NUMERICALGAPS The LDA achievable rate in (11) with S =0 is given by
In this section we show that the gap results obtained in r(LDA−HD)| = max min{γlog(1+C)
S=0
Section VI are pessimistic and are due to crude bounding in γ∈[0,1]
boththeupperandlowerbounds,whichwasnecessaryinorder (1−γ)log(1+I)}.
toobtainrateexpressionsthatcouldbehandledanalytically.In
and its gap from the outer bound can be reduced from 3 bits
order to illustrate our point, we first consider a relay network
to about 1.5 bits since
without the source-destination link, that is, with S = 0, in
Section VII-A and then we show that the same observations GAP≤r(CS−HD)|S=0−r(LDA−HD)|S=0
(cid:26) (cid:18) (cid:19)
are valid for any network in Section VII-B. C
≤ max γlog 1+ −γlog(1+C),
γ∈[0,1] γ
(cid:18) (cid:19) (cid:27)
I
A. Single Relay Networks without a Source-Destination Link, H(γ)+(1−γ)log 1+ −(1−γ)log(1+I)
1−γ
a.k.a. Diamond Networks with One Relay
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
1 1
c) Upper Bound: We start by showing that the (upper ≤ max γlog ,H(γ)+(1−γ)log
bound on the) cut-set upper bound in (6) can be improved γ∈[0,1] γ 1−γ
(cid:26) (cid:18) (cid:19)(cid:27)
1
upon. Note that we were not able to evaluate the actual cut-
= max H(γ)+(1−γ)log =1.5112 bits.
set upper bound in (5) so we further bounded it as in (6), γ∈[0,1] 1−γ
which for S =0 reduces to Note that the actual gap is even less than 1.5 bits.
(cid:26) (cid:18) I (cid:19) By numerically evaluating the difference between
r(CS−HD)|S=0 =γm∈[a0,x1]min H(γ)+(1−γ)log 1+1−γ , min{C(FD),r(CS−HD)}|S=0 and r(LDA−HD)|S=0 we found
(cid:18) (cid:19)(cid:27) that the gap is at most 1.11 bits.
C
γlog 1+ . f) Numerical gaps with deterministic switch: Similarly,
γ
by numerical evaluations one can find that the PDF strat-
ThecapacityoftheG-FD-RCforS =0isknownexactlyand egy with deterministic switch in Remark 4-Appendix C and
is given by the cut-set upper bound the NNC strategy with deterministic switch in Remark 5-
Appendix F are to within 0.80 bits and 1.01 bits, respectively,
C(FD)|S=0 =log(1+min{C,I}). of the improved upper bound. Notice that in these cases there
is no information conveyed by the relay to the destination
C(FD) is a trivial upper bound for the capacity of the G-HD-
throughtheswitch.Furtherreductionsinthegapwithrandom
RC. Now we show that our upper bound r(CS−HD)| can
S=0 switch are discussed next for a general network.
be larger than C(FD)| . For the case C =15/2>I =3/2
S=0 Fig. 4 shows different upper an lower bounds for the G-
we have HD-RC for S = 0, C = 15, I = 3 vs γ = P[S = 0].
r
(cid:26) (cid:18)1(cid:19) 1 1 (cid:27) We see that the cut-set upper bound exceeds the capacity
r(CS−HD)| ≥min H + log(1+2I), log(1+2C)
S=0 2 2 2 of the G-FD-RC (maximum of the solid black curve vs.
dashed black curve). Different achievable strategies are also
=log(4)>C(FD)| =log(2.5).
S=0
shown, whose order from the most performing to the least
The reason why the capacity of the FD channel can be performing is: PDF with random switch (red curve, 1.913
smaller than our upper bound r(CS−HD)| is the crude bits/ch.use), PDF with deterministic switch (blue curve, 1.702
S=0
bound I(S ;Y )≤H(S )=H(γ). As mentioned earlier, we bits/ch.use), NNC with random switch (cyan curve, 1.446
r d r
neededthisboundinordertohaveananalyticalexpressionfor bits/ch.use), NNC with deterministic switch (magenta curve,
the upper bound. Actually for S =0 the cut-set upper bound 1.402 bits/ch.use), and LDA (green curve, 1.333 bits/ch.use).
in (5) is tight, as we show next. In this particular setting, the maximum rate using the NNC
d) Exact capacity with PDF: strategy with random switch (cyan curve, 1.446 bits/ch.use)
is achieved for P[Q = 0,S = 0] = 0,P[Q = 0,S = 1] =
r r
Theorem 3. In absence of direct link between the source and 0.33,P[Q = 1,S = 0] = 0.45,P[Q = 1,S = 1] = 0.22.
r r
the destination PDF with random switch achieves the cut-set This is due to the absence of the direct link (S =0) between
upper bound.
the source and the destination. Actually, since the source can
communicate with the destination only through the relay, it
Proof:WithS =0,thecut-setupperboundin(5)andthe
is necessary a coordination between the transmissions of the
PDF lower bound in (8) are the same (see also Appendix G
source and those of the relay. This coordination is possible
with S =0).
thankstothetime-sharingrandomvariableQ,i.e.whenQ=0
e) Improved gap for the LDA Lower Bound: Despite
thesourcestayssilent,whilewhenQ=1thesourcetransmits.
knowing the capacity expression for S = 0, its actual eval-
uation is elusive as it is not clear what the optimal input
distribution P∗ in (5) is. For this reason we next B. Single Relay Network with a Source-Destination Link
Xs,Xr,Sr
specialized the LDA strategy to the case S = 0 and evaluate Although the considerations in Section VII-A were for a
its gap from the (upper bound on the) cut-set upper bound relay channel without a source-destination link, they are valid
in (6). in general.
9
Fig. 5 and Fig. 6 show the rates achieved by using the VIII. NETWORKSWITHMULTIPLERELAYS
differentachievableschemespresentedintheprevioussections
In this section we extend our gDoF and gap results to
with S > 0. In Fig. 5 the channel conditions are such that
general HD-MRC. Similarly to the the full-duplex case [11],
the PDF strategy outperforms the NNC, while in Fig. 6 the
ourmainresultisthatNNCisoptimaltowithinaconstantgap
oppositeholds.InFig.5thePDFstrategywithrandomswitch
for the HD-MRC where the gap is a function of the number
(red curve, 11.66 bits/ch.use) outperforms both the NNC
of relays.
with random switch (cyan curve, 11.11 bits/ch.use) and the
PDF with deterministic switch (blue curve, 11.4 bits/ch.use);
then the PDF with deterministic switch outperforms the NNC A. Network Model
with deterministic switch (magenta curve, 10.94 bits/ch.use),
In this model we have K ≥ 3 nodes, i.e., one transmit-
which is also encompassed by the NNC with random switch.
ter/node1,onereceiver/nodeK andK−2relayswithindices
Differently from the case without direct link, we observe that
2,...,K −1. Each node is equipped with a single antenna
the maximum NNC rates both in Fig. 5 and in Fig. 6 are
and is subject to an average power constraint, which we set
achieved with the choice Q=∅, i.e. the time-sharing random to 1 without loss of generality, i.e. E(cid:2)|X |2(cid:3) ≤ 1 ∈ R+,
k
variable Q is a constant. This is due to the fact that the
k ∈ [1 : K −1]. The system is described by the input/output
source is always heard by the destination even when the relay
relationship
transmits so there is no need for the source to remain silent
when the relay sends. Y =(I−S)HS X+Z (36a)
InFig.7weconsiderthecaseofdeterministicswitch.Fig.7 Z=[Z1,...,ZK]T ∼N(0,I) (36b)
shows, as a function of SNR for βsd =1,(βrd,βsr)∈[0,2.4], Y =[Y1,...,YK]T ∈CK (36c)
the maximum gap between the cut-set upper bound r(CS−HD)
X=[X ,...,X ]T ∈CK :
in (6) and the following lower bounds with deterministic 1 K
switch:thePDFlowerboundobtainedfromr(PDF−HD) in(9) E[|Xk|2]≤1 for k ∈[1:K−1], (36d)
with I(PDF) = 0, the NNC lower bound in Remark 5 in H∈CK×K (36e)
0
Appendix F, and the LDA lower bound in (11). From Fig. 7
S=diag{S ,...,S }:
1 K
we observe that the maximum gap with PDF is of 1 bit as
S =1, S ∈{0,1} for k ∈[2:K−1], S =0, (36f)
in Proposition 8, but with NNC is around 1.52 bits and with 1 k K
LDA is around 1.59 bits, which are lower than the analytical where the vector S represents the state of the nodes, either
gaps we found in Propositions 10 and 9, respectively. receive (S = 0) or transmit (S = 1). The channel matrix
The lower bounds can be improved upon by considering H∈CK×K is constant and therefore known to all terminals.
that information can be transmitted through a random switch The entry hij with (i,j) ∈ [1 : K]2 represents the channel
for the relay. However, this improvement depends on the between source j and destination i. Without loss of generality
channel gains. If the information cannot be routed through we assume that the noises Z are zero-mean jointly Gaussian
the relay because min{C,I} ≤ S, then the system cannot and with unit power. Furthermore we assume that the noises
exploit the randomness of the switch, and so IPDF = 0 and are iid N(0,1), this is however not without loss of general-
0
INNC = 0 are approximately optimal (in this case the relay ity [19].
0
can remain silent). For this reason the maximum numerical The capacity of the channel described in (36) is not known
gap obtained with a random switch coincides with the one in general. Here we show that a scheme based on the NNC
obtained with a deterministic switch, as there are channel strategy achieves the capacity within a constant gap for any
conditions for which random switch is not necessary. This number of relays and for any choice of channel parameters.
behavior for the PDF strategy is represented in Fig. 8. In As for the FD case, the gap is found to be a function of the
thisfigurewenumericallyevaluatethedifferencebetweenthe numberofrelays.Weproposeadifferentandsimplerbounding
analytical gap, i.e., the one computed with IPDF = 0, and technique than that of [11], which might overestimate the
0
the numerical one, i.e., computed with Iopt (actual value of actual gap between inner and outer bound.
0
IPDF), at a fix SNR = 20dB and by varying (β ,β ). We
0 rd sr
observethatwhentheinformationcannotbeconveyedthrough
B. Capacity to within a constant gap
the relay, i.e., min{β ,β }≤1, then IPDF =0 is optimal,
rd sr 0
since the information only flows through the direct link. In Our main result is
Fig. 9 the channel channel gains are set such that the use of
Theorem4. Thecut-setupperboundforthehalf-duplexmulti-
the relay increases the gDoF of the channel (β = 1 and
sd relay network is achievable to within
(β ,β ) ∈ [1,2.4]). Here the relay uses PDF. We observe
rd sr
that we have a further improvement in terms of gap by using GAP≤ max {min{1+(cid:96),K−1−(cid:96)}log(1+(cid:96))
a random switch (blue curve) instead of using a deterministic (cid:96)∈[0:K−2]
switch (red curve). We notice that at high SNR, where the +min{1+3(cid:96),(cid:96)+K−1}} (37)
gap is maximum, this improvement is around 0.1 bits. As
bits per channel use.
mentioned earlier, the rate advantage of random switch over
deterministic switch depends on the channel gains. Proof: The proof can be found in Appendix L.
10
For high value of K, i.e. K >> 1, the optimal value of (cid:96) channel gains as
in (37) is (cid:96)∼= K−2 and the corresponding limit gap becomes
2  ∗ ∗ ∗ ∗ 
GAP∼= K log(4K). (38) (cid:20)log(|hij|2)(cid:21) =αs1 ∗ β1 ∗ 
2 log(SNR) (i,j)∈[1:4]2 αs2 β2 ∗ ∗ 
1 α α ∗
Fig. 10 shows the trend of the gap in (37) as a function of the 1d 2d
numberofnodesK (bluecurve).Theredcurverepresentsthe where ∗ denotes entries that do not matter; this is so because
limit behaviour of the gap in (37). the source node never listens to the channel (first row), the
Notethatbyapplyingtheabovegap-resulttothecaseK = destination node never transmits (last column), and a relay
3 we obtain GAP ≤ 4 bits which is a much larger gap than can remove the ‘self-interference’ (main diagonal). Note that
1.61 bits we found for the case of one relay. thedirectlinkfromthesourcetothedestinationhasgainSNR
A smaller gap may be obtained by computing tighter and all other channel gains are expressed with reference to it.
bounds. This can be accomplished by several means. For g) Full Duplex: In the following the set A indicates the
example: relaysthatlieonthesource/node1sideofthecut.Thecut-set
• Byusingmorecomplexandsophisticatedboundingtech- upper bound for the FD-MRC [11] gives
niques:in[11]anupperboundonthewater-fillingpower
A=∅: I :=I(X ;Y ,Y ,Y |X ,X )
allocationforageneralMIMOchannelwasderived.This 1 1 4 2 3 2 3
moreinvolvedupperboundcouldbeusedheretoobtaina ≤log(1+|h |2+|h |2+|h |2)
41 21 31
smallergap.Wenotethatourboundingtechniqueapplied A={2}: I :=I(X ,X ;Y ,Y |X )
2 1 2 4 3 3
to the FD-MRC gives a gap of order K log(2K), which
2 ≤log|I +H HH|+log(2)
is larger that 0.63K found in [11]. 2 1 1
=log(1+A)+2log(2)
• ByusinganachievablestrategybasedonPDF,whichina
singlerelaycasegivesasmallergapthanNNC.However, A={3}: I :=I(X ,X ;Y ,Y |X )
3 1 3 4 2 2
PDFseemsnottobeeasilyextendedtonetworkswithan ≤log|I +H HH|+log(2)
2 2 2
arbitrary number of relays, which is the main motivation
=log(1+B)+2log(2)
we consider NNC here.
• By deriving tighter bounds on specific network topolo- A={2,3}: I4 :=I(X1,X2,X3;Y4|∅)
gies: in [20] it is found that for a FD diamond network ≤log(1+(|h |+|h |+|h |)2),
41 42 43
with K relaysthe gapis ofthe order log(K),rather than
linearinK [11].Moreover,forasymmetricFDdiamond where
network with K relays the gap does not depend on the
A=|h |2+|h |2+|h |2+|h |2+|h h −h h |2,
number of relays and is upper bounded by 2 bits. The 41 42 31 32 31 42 32 41
keydifferencebetweenageneralmulti-relaynetworkand B =|h41|2+|h43|2+|h21|2+|h23|2+|h21h43−h23h41|2,
a diamond network is that for each subset A we have (cid:20)1 0(cid:21) (cid:20)h h (cid:21) (cid:20)h h (cid:21)
I = , H = 31 32 and H = 21 23 .
Rank[HA,s]=2, i.e., the rank of a generic channel sub- 2 0 1 1 h41 h42 2 h41 h43
matrix does no longer depend on the cardinality of A.
Based on this observation we have Thus for the full-duplex case, the cut-set bound is given by
Proposition11. Thecut-setupperboundforthegaussian r(CS−FD) :=min{I ,I ,I ,I }
1 2 3 4
half-duplex diamond network with (K−2) relays is
achievable to within which at high SNR gives the following gDoF (achievable to
within a constant gap [11])
GAP≤(K−2)log(2)+4log(K)+2log(e/2).
r(CS−FD)
Moreover, if the conjecture in [17] holds then the gap d(FD) = lim
K=4 SNR→+∞log(1+SNR)
above could be decreased to
(cid:110)
=min max{1,α ,α },max{α +α ,β +1},
s1 s2 s2 1d 2
GAP≤5log(K)+2log(e/2).
(cid:111)
max{α +α ,β +1},max{1,α ,α } .
s1 2d 1 1d 2d
Proof: The proof can be found in Appendix M.
In the following we are interested in identifying cases for
which
C. Example: Fully connected network with K =4
d(FD) >d(FD) :=max(cid:8)1,min{α ,α },
To gain insights into how relays are best utilized, we K=4 K=4,bestrelay s1 1d
(cid:9)
consider a network with two relays, i.e. K = 4 nodes. In min{α ,α } ,
s2 2d
particular we highlight under which channel conditions the
gDoFperformanceisenhancedbyexploitingbothrelaysrather where d(FD) is the gDoF that can be obtained by
K=4,bestrelay
than using only the best one. We also compare the loss that selecting the relay which achieves the highest gDoF in (4)
incurs by using HD with respect to FD. Let parameterize the while leaving the other silent. From the expression for d(FD)
K=4