On the Achievable Rates of Multihop Virtual Full-Duplex Relay Channels

0.58 MB·English

by  Song-Nam Hong

On the Achievable Rates of Multihop Virtual
Full-Duplex Relay Channels
Song-Nam Hong∗, Ivana Maric´∗, Dennis Hui∗ and Giuseppe Caire†
∗ Ericsson Research, San Jose, CA,
email:(songnam.hong, ivana.maric, dennis.hui)@ericsson.com
† Technical University of Berlin, Germany,
email:[email protected]
Abstract—We study a multihop “virtual” full-duplex relay
5
channel as a special case of a general multiple multicast relay
1 path1
network. For such channel, quantize-map-and-forward (QMF)
0
(or noisy network coding (NNC)) achieves the cut-set upper
2
bound within a constant gap where the gap grows linearly with path2

r the number of relay stages K. However, this gap may not be
p
negligible for the systems with multihop transmissions (i.e., a
A
wireless backhaul operating at higher frequencies). We have
recently attained an improved result to the capacity scaling Fig. 1. Multihop virtual full-duplex relay channels when K = 3 (i.e., 4-
3
where the gap grows logarithmically as logK, by using an hop relay network). Black-solid lines are active for every odd time slot and
2
optimalquantizationatrelaysandbyexploitingrelays’messages red-dashedlinesareactiveforeveryeventimeslot.

(decoded in the previous time slot) as side-information. In this
]
T paper, we further improve the performance of this network by
presenting a mixed scheme where each relay can perform either
I
. decode-and-forward (DF) or QMF with possibly rate-splitting. while one relay transmits its signal to the next stage, the
s
c Wederiveanachievablerateandshowthattheproposedscheme other relay receives a signal from the previous stage. The
outperforms the optimized QMF. Furthermore, we demonstrate
[ role of the relays is swapped at the end of each time interval
that this performance improvement increases with K.
(see Fig. 1). This relaying operation is known as “successive
2 Index Terms—Multihop relay networks, wireless backhaul,
v quantize-map-and-forward relaying” [3]. In this way, the source can send a new message
0 tothedestinationateverytimeslotasiffull-duplexrelaysare
4 I. INTRODUCTION used. Every two consecutive source messages will travel via
4
Recent works have demonstrated the practical feasibility of two alternate disjoint paths of relays. In [4], 2-hop model has
6
0 full-duplex relays through the suppression of self-interference been studied, showing that dirty paper coding (DPC) achieves
. inamixedanalog-digitalfashioninordertoavoidtheproblem the performance of ideal full-duplex relay since the source
1
0 of receiver power saturation [1], [2]. These architectures are cancompletelyeliminatethe“known”interferenceatintended
5 based on some form of analog self-interference cancella- receiver. However, DPC is no longer applicable in a multihop
1 tion, followed by digital self-interference cancellation in the network model shown in Fig. 1 since a transmit relay has no
v: baseband domain. In some of these architectures, the self- knowledge on interference signals at other stages [5]. Thus,
i interference cancellation in the analog domain is obtained finding an optimal strategy for the multihop models is still an
X
by transmitting with multiple antennas such that the signals open problem.
r
a transmitted over different antennas superimpose in opposite Since the multihop model is a special case of a single-
phases and therefore cancel each other at the receiving anten- source single-destination (non-layered) network, QMF [6] (or
nas. Building on the idea of using multiple antennas to cope NNC [7], [8]) can be applied to this model. By setting the
withtheisolationofthereceiverfromthetransmitter,wemay quantization distortion levels to be at the background noise
consider a “distributed version” of such approach where the level,theseschemesachievethecapacitywithinaconstantgap
transmit and receive antennas belong to physically separated that scales linearly with the number of nodes in the network.
nodes.Thishastheadvantagethateachofsuchnodesoperates In [9], we improved this result by using the principle of
in conventional half-duplex mode. Furthermore, by allowing QMF(orNNC)andbyoptimizingthequantizationlevels.We
a large physical separation between nodes, the problem of showed that the gap from the capacity scales logarithmically
receiver power saturation is eliminated. with the number of nodes. The same rate-scaling was also
Motivatedbythedistributedapproach,weintroduceacom- attainedin[10]bychoosingquantizationlevelsataresolution
munication scheme that utilizes “virtual” full-duplex relays, decreasing with the number of relay stages (in short, stage-
eachconsistingoftwohalf-duplexrelays.Inthisconfiguration, depth quantization). Furthermore, the optimized QMF has
each relay stage is formed of at least two half-duplex relays, a lower decoding complexity because it deploys successive
used alternatively in transmit and receive modes, such that decoding (SD) instead of joint decoding (JD) used in [6]–[8],
[10]. discrete memoryless channel is described by the conditional
However,innetworkconsistingofmanyrelays,constraining probabilities given by (cid:81)(cid:100)kK=/12(cid:101)p(y¯i,2k−1|xi,2k−1,x¯i,2k−2)
all relays to perform the same scheme might not be optimal (cid:81)(cid:98)kK=/12(cid:99)p(yi,2k|xi,2k−1,x¯i,2k) p(yD|xi,K), where i = 1 for
since they observe signals of different strengths. Relays in oddtandi=2forevent,andwherex andy denotethe
i,k i,k
favorable positions can perform DF thereby eliminating the respective output and input at relay R , and x and x
i,k 1,0 2,0
noise otherwise partially propagated via quantization based denote the source outputs.
schemes[8],[11].Ontheotherhand,decodingrequirementat
arelaycanseverelylimitthetransmissionrateifthereception III. MAINRESULTS
link is weak. Motivated by this, we presented in [9] a mixed We present a mixed scheme in which each relay performs
strategy using both DF and QMF (with optimal quantization) either QMF or DF depending on channel coefficients. Each
for Gaussian multihop virtual full-duplex channels. We con- DF relay decodes its incoming message which can be either a
sidered a special case of this scheme restricted to a symmetric source message or a quantization index sent by a QMF relay.
relaying configuration in which all relays on one transmission The destination explicitly decodes relays messages as well as
path perform DF and others in the second path perform QMF the source message and hence it can use these messages as a
with rate-splitting. side-informationinthenexttimeslot.Furthermore,eachrelay
In this paper, we generalize our previous work of [9] in can incorporate rate-splitting into its encoding scheme (QMF
three ways: 1) we consider a general discrete memoryless or DF) to reduce interference it creates to another relay. To
channelbeyondaGaussianchannel;2)weanalyzeanarbitrary be specific, relay Ri,k uses a rate-splitting if R¯i,k performs
relaying configuration in which each relay can perform either DF. This enables DF relays to partially eliminate the inter-
DF or QMF to optimize the overall rate performance; 3) relayinterference.WhenR¯i,k performsQMF,therate-splitting
each relay (either DF or QMF) can employ rate-splitting is not used because a QMF relay does not need to decode
that enables interference cancellation at DF relays thereby any message (unlike a DF relay) and because the destination
increasingtheachievablerate.Wederiveanachievablerateof decodes a message with full-knowledge of the interference.
the proposed scheme. Via numerical evaluation, we show that Detaileddescriptionoftheencoding/decodingschemeisgiven
the proposed mixed scheme outperforms the optimized QMF in Section III-A.
in[9]aswellastheQMFwithnoise-level[6]andstage-depth In order to state the achievable rate, we next introduce the
quantization[10],andthattheperformancegainincreaseswith following notation. Let V ={k ,...,k }⊆{1,...,K}
i i,1 i,|Vi|
the number of hops. This result implies that using DF relays denote the index subset containing the indices of QMF relays
infavorablepositionsreducesthegapfromcapacitytologK(cid:48) in the path i, where k < k < ··· < k . For a given
where K(cid:48) ≤ K denotes the number of stages that contain a V , let I = {k ,...i,,1k i,2−1} for (cid:96) =i,|V0i,|...,|V | with
i i,(cid:96) i,(cid:96) i,(cid:96)+1 i
QMF relay. Our results indicate that deployment of the mixed k = 0 and k = K +1. Notice that I includes all
i,0 i,|Vi|+1 i,(cid:96)
strategy for a general multiple multicast relay network can DF relays that transmit message sent by QMF relay R .
i,ki,(cid:96)
result in performance gains. Define a mapping: g (k) = k if k ∈ I , (cid:96) = 0,...,|V |.
i i,(cid:96) i,(cid:96) i
Notice that {I }|Vi| forms a partition of {1,...,K}.
II. NETWORKMODEL i,(cid:96) (cid:96)=0
Definition 1: According to the mode of a receiving relay,
Weconsideravirtualfull-duplexrelaychannelwithK relay
we define:
stages illustrated in Fig. 1. Encoding/decoding operations are
performed over time slots consisting of n channel uses of a I =∆ (cid:26) I(Xi,k;Yi,k+1|U¯i,k+1), k+1∈Vic
discrete memoryless channel. Successive relaying [9] is as- i,k I(Xi,k;Yˆi,k+1|X¯i,k+1), k+1∈Vi
sumedsuchthat,ateachtimeslott,thesourcetransmitsanew I =∆ (cid:26) I(Xi,k;Yi,k+1|U¯i,k+1,Ui,k), k+1∈Vic ,
message wt ∈ {1,...,2nri} where i = 1 for odd time slot i,k1 I(Xi,k;Yˆi,k+1|X¯i,k+1,Ui,k), k+1∈Vi
t and i = 2 for even time slot t, and the destination decodes
where Y = Y , X = φ, and U denotes an
a new message w . We define two message rates r and i,K+1 D i,K+1 i,k
t−K 1 auxiliaryrandomvariabletobeusedforsuperpositioncoding.
r since the odd-indexed and even-indexed messages are con-
2 By letting r denote the rate of R , we have:
veyedtothedestinationviatwodisjointpaths,namely,path1: i,k i,k
Theorem 1: For a (K + 1)-hop virtual full-duplex relay
(S,R ,...,R ,D)andpath2:(S,R ,...,R ,D).The
1,1 1,K 2,1 2,K
channel, the achievable rate-region of the mixed strategy with
role of relays is alternatively reversed in successive time slots
SD is the set of all rate pairs (r /2,r /2) that satisfy:
(seeFig.1).DuringN+K timeslots,thedestinationdecodes 1 2
the N/2 messages from each path. Thus, the achievable rate (cid:26) (cid:27)
of the messages via path i is given by r N/2(N +K). By ri ≤min min Ii,k, min I(Ui,k;Y¯i,k)+Ii,k1 ,
i k∈Ii,0 k∈Ii,0∩V¯ic
letting N → ∞, the rate r /2 is achievable, provided that
i
and for k ∈V with g (k)=k ,
the error probability vanishes with n. As in standard relay i i i,(cid:96)
channels (see for example [6], [7]), we take first the limit for I(Yˆi,k;Yi,k|X¯i,k)
n → ∞ and then for N → ∞, and focus on the achievable
(cid:26) (cid:27)
rate ri. Throughout, we use the notation ¯i to indicate the =min min Ii,k(cid:48), min I(Ui,k(cid:48);Y¯i,k(cid:48))+Ii,k(cid:48)1 ,
complement of i, i.e.,¯i=2 if i=1 and¯i=1 if i=2. The k(cid:48)∈Ii,(cid:96) k(cid:48)∈Ii,(cid:96)∩V¯ic
for any index subset V ⊆ {1,...,K},i = 1,2, side-information:
i
and any joint distributions that factors as (cid:81)2i=1p(xi,0) decoded messages:
(cid:81) (cid:81) (cid:81)
k∈VP¯iropo(xf:i,Sk)ee Sk∈ecVt¯iciopn(uIiI,Ik-)Ap.(xi,k|ui,k) k∈Vip(yˆi,k|yi,k). Destination
Theorem 2: For a (K + 1)-hop virtual full-duplex relay 2-stage relays
channel, the achievable rate region of the mixed strategy with
JD (at DF-only stages) is the set of all rate pairs (r /2,r /2)
1 2
1-stage relays
to satisfy:
r ≤min{I :k ∈I },
i i,k i,0
Source
and for k ∈Vc∩Vc,
1 2
r +r ≤min{I(U ,X ;Y )+I ,
1,g1(k) 2,g2(k) 1,k 2,k−1 2,k 1,k1 Fig.2. Timeexpanded3-hopnetworkwhererelaysR1,2 andR2,1 perform
I(U2,k,X1,k−1;Y1,k)+I2,k1}, DFandothersperformQMF.li denotesthequantizationindexobtainedata
QMFrelayintimesloti.
and for k ∈V with g (k)=k ,
i i i,(cid:96)
I(Yˆi,k;Yi,k|X¯i,k)=ri,k side-information
r ≤min{I :k(cid:48) ∈I }
i,k i,k(cid:48) i,(cid:96)
ri,k ≤I(Ui,k;Y¯i,k)+Ii,k1,k ∈V¯ic, (a) Equivalent model for path 1
for any subset Vi ⊆ {1,...,K} and any joint distribution side-information
given in Theorem 1, where r =r .
i,0 i
Proof: The proof is omitted due to the lack of space.
Remark 1: Recall that JD is a key component in the Han- (b) Equivalent model for path 2
Kobayashicodingscheme[12]fortwo-userinterferencechan- Fig.3. Equivalentmodelfor3-hopnetworkwhereredcirclesperformQMF
nel, that achieves a higher rate than SD. In other words, andwhitecirclesperformDF.
JD attains a higher rate than SD when both messages of
two transmitters should be decoded at two receivers, i.e., the
corresponding rates should be chosen in the intersection of message (cid:96)p at rate r with r =r +r . Randomly
i,k i,k1 i,k i,k1 i,k0
two MAC regions. In our network, this case occurs when and independently generate 2nri,k0 codewords u ((cid:96)c ) of
i,k i,k
both relays at the same stage perform DF. Thus, we applied length n indexed by (cid:96)c ∈ {1,...,2nri,k0} with i.i.d. com-
i,k
JD for such stages to obtain an improved achievable rate in ponents ∼ p(u ). For each (cid:96)c , randomly and condition-
i,k i,k
Corollary 2. ally independently generate 2nri,k1 codewords x ((cid:96)c ,(cid:96)p )
i,k i,k i,k
of length n indexed by (cid:96)p ∈ {1,...,2nri,k1} with i.i.d.
A. Proof of Theorem 1 i,k
components ∼ p(x |u ). Randomly and independently
i,k i,k
Fig. 2 shows a time-expanded graph of a 3-hop virtual full- generate 2nrˆi,k codewords yˆ (µ) of length n indexed by
i,k
duplex relay channel in which relays R1,2 and R2,1 perform µ ∈ {1,...,2nrˆi,k} with i.i.d. components ∼ p(yˆi,k). The
DF and R and R perform QMF. As pointed out earlier,
1,1 2,2 quantization codewords are randomly and independently as-
in the proposed scheme, the destination explicitly decodes signedwithuniformprobabilityto2nri,k bins.Denotethe(cid:96)i,k-
relays’messagesandhence,itcanusethesemessagesasside- thbinbyB((cid:96)i,k)with(cid:96)i,k ∈{1,...,2nri,k}.Here,Wyner-Ziv
information in the next time slot thereby completely knowing
quantization is assumed, such that the quantization distortion
the inter-relay interference (see Fig. 2). From this, we can level is chosen by imposing I(Yˆi,k;Yi,k|X¯i,k) = ri,k. This
produce a simplified network model shown in Fig. 3, which
enablesthedestinationtofindanuniquequantizationsequence
can be straightforwardly extended to a (K +1)-hop network yˆ from the bin index (cid:96) and the side-information x .
i,k i,k i,k
withanarbitraryrelayconfiguration.Thisnetworkmodelwill
Encoding: Source transmits a message w by sending the
i
be used for the proof.
codeword x (w ) where i is either 1 or 2 depending on
Fix relay modes V , i = 1,2. For given V , fix input i,0 i
i i time slot. For QMF relay with k ∈ V , R observes y
distributions as defined in Theorem 1. i i,k i,k
and finds µ such that (y ,yˆ (µ))∈T(n)(Y ,Yˆ ). If no
Codebook generation: Randomly and independently gen- i,k i,k (cid:15) i,k i,k
erate 2nri codewords xi,0(wi) of length n indexed by wi ∈ quantization codeword satisfies the joint typicality condition,
({i1.e,..,..Q,M2nFri}inwteirtfherie.di.d.recloaym)p,ornaenndtosm∼lyp(axnid,0)i.ndFeopreknd∈entVly¯i tthhaetryˆelia,ky(µch)o∈osBes((cid:96)µi,k=).1T.oTsheennd, itthefinmdesstshaegebi(cid:96)ni,kin=dex((cid:96)(cid:96)ci,ik,k,(cid:96)spiu,kc)h,
generate 2nri,k codewords xi,k((cid:96)i,k) of length n indexed by it transmits the downstream codeword xi,k((cid:96)ci,k,(cid:96)pi,k) using
(cid:96)i,k ∈ {1,...,2nri,k} with i.i.d. components ∼ p(xi,k). For superposition coding. If rate-splitting is not used, then the
k ∈ V¯ic (i.e., DF interfered relay), split message (cid:96)i,k into codeword xi,k((cid:96)i,k) is sent. For DF relay with k ∈ Vic, Ri,k
independent“common”message(cid:96)c atrater and“private” decodes the incoming message (cid:96)ˆ . To send the message
i,k i,k0 i,k−1
"rate-splitting" "rate-splitting"
x¯i,k+1 if
ri,k ≤I(Xi,k;Yˆi,k+1|X¯i,k+1). (4)
Type-IV: With the same argument in Type-II, R uses
i,k
rate-splitting and hence, the common message (cid:96)c should be
i,k
(a)Type-I (b)Type-II (c)Type-III (d)Type-IV decoded at R¯i,k, yielding
Fig.4. Fourscenariosdeterminedbythenext-hopandinterferedrelays.Red ri,k0 ≤I(Ui,k;Y¯i,k). (5)
circles perform QMF, white circles perform DF, and black circles perform
The destination can reliably decode the (cid:96) =((cid:96)c ,(cid:96)p ) if
eitherQMForDF. i,k i,k i,k
ri,k0 ≤I(Ui,k;Yˆi,k+1|X¯i,(k+1))
(cid:96)ˆi,k−1 = ((cid:96)ˆci,k−1,(cid:96)ˆpi,k−1), it transmits the downstream code- ri,k1 ≤I(Xi,k;Yˆi,k+1|Ui,k,X¯i,(k+1)).
word x ((cid:96)ˆc ,(cid:96)ˆp ) using superposition coding. If the
i,k i,k−1 i,k−1 From the above, we obtain:
rate-splitting is not used, then the codeword x ((cid:96)ˆ ) is
sent. i,k i,k−1 ri,k ≤min{I(Xi,k;Yˆi,k+1|X¯i,(k+1)), (6)
Decoding: We first observe that rate ri,k at relay Ri,k is I(Ui,k;Y¯i,k)+I(Xi,k;Yˆi,k+1|Ui,k,X¯i,(k+1))}
determined according to the modes of its neighboring relays,
Wearenowreadytoderiveanachievablerateofthemixed
i.e., the next-hop relay Ri,k+1 and the interfered relay R¯i,k. scheme. From Fig. 4, we can classify the types of each relay
WhenR performsQMF,thedestinationdecodes(cid:96)ˆ from
i,k+1 i,k R . Using (1)-(6) and from Definition 1, we have:
a quantized observation yˆ with the full-knowledge of i,k
i,k+1 (cid:26)
the interference x¯i,k+1. In the other case, Ri,k+1 decodes the r ≤ Ii,k, Types I and III
(cid:96)ˆi,k from its observation yi,k+1 with the partial-knowledge of i,k min{Ii,k,I(Ui,k;Y¯i,k)+Ii,k1}, Types II and IV
interference u¯i,k+1. In addition, the mode of R¯i,k determines which can be represented as
the use of rate-splitting at R , yielding an additional rate-
i,k r ≤I (7)
constraint of r since the common message (cid:96)c should i,k i,k
be decoded ati,Rk0¯i,k. Namely, the modes of thei,nkeighbor- ri,k ≤I(Ui,k;Y¯i,k)+Ii,k1,k ∈V¯ic. (8)
ing relays determines the types of observations (unquantized
Then, we have:
vs quantized), side-information (full-knowledge vs partial-
• ThesourceandDFrelaysRi,k fork ∈Ii,0 sendasource
knowledge),andtheuseofrate-splitting.Basedonthis,wecan
message. This message can be reliably decoded at those
define the four scenarios given in Fig. 4. The corresponding
DF relays and the destination if
rate-constraints of r are derived as follows.
i,k
Type-I: Since the interfered relay R¯i,k performs QMF, Ri,k ri ≤min{ri,k :k ∈Ii,0}. (9)
does not use rate-splitting as explained before. Due to the
• Letgi(k)=ki,(cid:96).Therelay’smessage(cid:96)i,k canbedecoded
rate-splitting at R¯i,k+1, the next-hop relay Ri,k+1 can reliably at DF relays R for k(cid:48) ∈I and the destination if
i,k(cid:48) i,(cid:96)
decode the (cid:96) with the partial-knowledge of interference as
i,k
u¯i,k+1, yielding: ri,k ≤min{ri,k(cid:48) :k(cid:48) ∈Ii,(cid:96)}. (10)
ri,k ≤I(Xi,k;Yi,k+1|U¯i,k+1). (1) • Due to the use of Wyner-Ziv quantization, we have:
Type-II: In this case, Ri,k uses rate-splitting so that the I(Yˆi,k;Yi,k|X¯i,k)=ri,k,k ∈Vi.
interfered relay R¯i,k can partially eliminate the interference. Substituting (7) and (8) into (9) and (10) completes the proof.
Thus, the common message (cid:96)i,k0 should be decoded at R¯i,k,
yielding IV. ACHIEVABLERATESFORGAUSSIANCHANNELS
ri,k0 ≤I(Ui,k;Y¯i,k). (2) In order to evaluate the performance of the proposed
scheme, we consider a Gaussian channel where both paths
The next-hop relay R can reliably decode the (cid:96) =
i,k+1 i,k experience the same channel gains, i.e., SNR denotes the
((cid:96)c ,(cid:96)p ) if k
i,k i,k direct channel gain from R to R and INR denotes
i,k−1 i,k k
ri,k0 ≤I(Ui,k;Yi,k+1|U¯i,k+1) the interference channel gain from R¯i,k to Ri,k. Due to the
symmetric structure of each stage, we naturally assume that
ri,k1 ≤I(Xi,k;Yi,k+1|Ui,k,U¯i,k+1). ∆
each stage uses the same relaying scheme, i.e., V = V =
1 2
From the above, we get: V ={k ,...,k }. From Definition 1, we obtain:
1 |V|
(cid:18) (cid:19)
ri,k ≤min{I(Xi,k;Yi,k+1|U¯i,k+1), (3) I =log 1+ SNRk+1
I(Ui,k;Y¯i,k)+I(Xi,k;Yi,k+1|Ui,k,U¯i,k+1)}. k 1+(1−θk+1)INRk+1+σˆk2+1
(cid:18) (cid:19)
(1−θ )SNR
Type-III: The destination can reliably decode the (cid:96)i,k from Ik1 =log 1+ 1+(1−θ k)INR k+1+σˆ2 ,
a quantized observation yˆi,k+1 using the side-information k+1 k+1 k+1
6 6
useQ5 useQ5
perchannel4 perchannel4
metricrateSbits3 metricrateSbits3
m m
sy2 sy2
Achievable1 MMQMiixxFeeddSossccphhtieemmmaeelQSSJSDDQQ Achievable1 MMQMiixxFeeddSossccphhtieemmmaeelQSSJSDDQQ
QMFSstage−depthQ QMFSstage−depthQ
QMFSnoise−levelQ QMFSnoiselevelQ
0 0
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
NumberofstagesSKQ NumberofstagesSKQ
Fig.5. SNR=20dBandINRk=SNRαk whereαk∼Unif[1,2]. Fig.6. SNR=20dBandINRk=SNRαk whereαk∼Unif[0,1].
where θk+1 =1 if k+1∈V and σˆk2+1 =0 if k+1∈Vc. In schemeoutperformstheoptimizedQMFand,furthermore,that
this section, we focus on the symmetric achievable rate of r this improvement increases with the number of hops. This
with r =r1 =r2. We obtain: implies that using DF relays in favorable positions can reduce
Corollary 1: Fora(K+1)-hopGaussianvirtualfull-duplex the gap from the capacity to logK(cid:48) where K(cid:48) ≤ K denotes
relay channel, the achievable symmetric rate of the mixed the number of stages containing a QMF relay. Based on the
strategy with SD (or JD) is given by obtained results, we expect that the proposed mixed scheme
r =min{min{I :k ∈I },min{I(cid:48) :k ∈I \{0}}} can bring performance gains in a general multiple multicast
k 0 k 0 relay network. This is a subject of our future work.
r =min{min{I :k ∈I },min{I(cid:48) :k ∈I \{k }}}
k(cid:96) k (cid:96) k (cid:96) (cid:96)
σˆk2(cid:96) =(1+SNRk(cid:96))/(2rk(cid:96) −1),(cid:96)=1,...,|V|, REFERENCES
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k
 (cid:16) (cid:17) [2] J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, “Achieving
Ik(cid:48) = l12oglog1(cid:16)+1+θ1k++S1NS1+NIRN(RkR1+k−k+1+θ+11kθ+k1++)1ININIRRkk1k++,11(cid:17)+ 12Ik1, SJDD. [3] S1Csi6.hntSighcl.aeACgnco.nh,RuaUaenzlSnaIAeenli,,t,.SFCSeu.oplOnlt.f,..DGo2un0hp1aMl0erxa.onbW,ilaeinrCedloeAmss.pKuCt.oinKmghmaannuddnaiNcnaeit,tiwo“Cno,ro”koiipnnegrP(aMtriovoceb.iSCotfroamtthe)e-,
Proof: The proof follows the proof of Theorem 2 by giesfortheHalf-DuplexGaussianParallelRelayChannel:Simultaneous
RelayingversusSuccessiveRelaying,”inProc.oftheAllertonConf.on
choosing Gaussian input distributions with the conventional
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k =1,...,K. The detailed proof is omitted. Half-DuplexDiamondChannel,”inProc.ofIEEEInt.Symp.Inf.Theory
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In Figs. 5 and 6, we numerically evaluate the achievable
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k
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results show that the proposed mixed scheme outperforms the Decode-ForwardOption,”[Online]http://arxiv.org/abs/1304.1692.
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in Fig. 5, that JD significantly improves the performance GaussianRelayNetworks,”[Online]http://arxiv.org/abs/1407.3841.
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V. CONCLUSION
In this work, we proposed a mixed scheme for multihop
“virtual” full-duplex relay channels. We showed that our