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4162 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 8, AUGUST 2013

Distributed Full-Duplex viaWireless Side-Channels: Bounds and Protocols

Jingwen Bai and Ashutosh Sabharwal

Abstract—In this paper, we study a three-node full-duplexnetwork, where a base station is engaged in simultaneous up-and downlink communication in the same frequency band withtwo half-duplex mobile nodes. To reduce the impact of inter-node interference between the two mobile nodes on the systemcapacity, we study how an orthogonal side-channel between thetwo mobile nodes can be leveraged to achieve full-duplex-likemultiplexing gains. We propose and characterize the achievablerates of four distributed full-duplex schemes, labeled bin-and-cancel, compress-and-cancel, estimate-and-cancel and decode-and-cancel. Of the four, bin-and-cancel is shown to achievewithin 1 bit/s/Hz of the capacity region for all values of channelparameters. In contrast, the other three schemes achieve thenear-optimal performance only in certain regimes of channelvalues. Asymptotic multiplexing gains of all proposed schemesare derived to show that the side-channel is extremely effectivein regimes where inter-node interference has the highest impact.

Index Terms—Full-duplex wireless, wireless side-channels, in-terference management, device-to-device communication, capac-ity region.

I. INTRODUCTION

CURRENT deployed wireless communication systems

adopt either time-division or frequency-division fortransmission and reception. Recently, many researchers have

implemented full-duplex wireless communication systemswhich support simultaneous transmission and reception in the

same frequency band. In these system level implementations,

different self-interference cancellation mechanisms have been

proposed, including passive self-interference suppression (e.g.,

[1], [2], [3]) and active self-interference cancellation (e.g.,

[4], [5], [6], [3]). While the first set of experiments were allshort-range, recent systems which employ antenna isolation at

infrastructure nodes [7] have reported full-duplex communica-tion ranges in the order of 100s of meters, which are suitable

for small cells [8]. The feasibility of full-duplex in short to

medium ranges opens new design opportunities for wireless

communication systems. We envision that the first adoption

of full-duplex will be in small cell infrastructure nodes, sinceinfrastructure nodes have more flexibility in their physical size

and shape compared to mobile nodes. Thus, we expect that

small form-factor mobile nodes will continue to be half-duplex

Manuscript received December 20, 2012; revised March 29, 2013; acceptedMay 25, 2013. The associate editor coordinating the review of this paper andapproving it for publication was S. Valaee.

The authors are with the Department of Electrical and Computer Engi-neering, Rice University, Houston, TX 77005, USA (e-mail: { jingwen.bai,ashu}@rice.edu).

This work was partially supported by NSF CNS-1012921 and NSF CNS-1161596.

Digital Object Identifier 10.1109/TWC.2013.071913.122015

BS

Rx Tx

M1 M2Inter-node Interference

Fig. 1. Three-node full-duplex network: inter-node interference becomes animportant factor when the infrastructure node communicates with uplink and downlink mobile nodes simultaneously.

in the near future. Within this context, we focus on studyingthe performance of a network of half-duplex mobiles being

served by a full-duplex base station (BS).

Consider the three-node network shown in Figure 1. A

full-duplex infrastructure node communicates with two half-

duplex mobiles simultaneously to support one uplink and one

downlink flow. The network has to deal with two forms of

interferences: self-interference at the full-duplex infrastructure

node, and inter-node interference (INI) from uplink mobile M1to downlink mobile M2. The two forms of interferences differ

in one important aspect: the self-interfering signal is knownat the interfered receiver, since the transmitter and receiver

are co-located at BS, but the INI is not known at the receiver

Node M2. In a small cell scenario, we assume that the self-

interference at the infrastructure has been suppressed to the

noise floor (see discussion in first para, in [7]) and hence focus

on INI. To the best of the authors’ knowledge, this paper is thefirst to study the impact of INI in the three-node full-duplex

network and the mitigation of INI using advanced interferencemanagement techniques.

In this paper, we study how a wireless side-channel between

Nodes M1 and M2 can be leveraged to manage INI. Conceptu-

ally, one can model the co-location of transmitter and receiver

on a full-duplex node as an infinite capacity side-channel.

Thus, our use of a wireless side-channel between M1 and M2mimics the inherent full-duplex side-channel but has finite

bandwidth and signal-to-noise ratio (SNR) like any practical

wireless channel. The wireless side-channel model is inspired

by the fact that most smartphones support simultaneous use

of multiple standards, and can thus access multiple orthogonalspectral bands. For example, the main network could be on a

cellular band while the M1→M2 wireless side-channel couldbe on an unlicensed ISM band, thus creating a novel use of

ISM-bands in licensed cellular networks. Based on the above

discussion, we label the protocols for communication in side-

channel assisted three-node network as distributed full-duplex .

1536-1276/13$31.00 c 2013 IEEE


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S 22

S 20

bin l

Y 2

Y 3

Y 1S 1 X 1

X 22

X 20

X 3

RxBS

RxM2TxBS

TxM1

Fig. 3. Depiction of bin-and-cancel.

The number of bin indexes is determined by the rate of theside-channel. At the interfered receiver M2, by decoding the

common message of the interfering signal from M1, part of the

interference can be subtracted out while treating the remaining

private message of M1 as noise. The bin index can first be

decoded from the side-channel. Then with the help of the bin

index, the uncertainty of decoding the common message of M1

can be resolved, allowing sending more common message of

M1 and mitigating INI.

In Gaussian channels, we first split the total transmit power

of Node M1 between side-channel and main-channel, then

split the power allocated to the main-channel between the

common and private message. Assuming inputs are i.i.d.

Gaussian distributed satisfying the per-node power constraint,

we have the following achievability result.

Proposition 1. For Gaussian side-channel assisted three-node

full-duplex network, in the weak interference regime defined

by INR ≤ SNR2 , 3 the following rate region is achievable for bin-and-cancel,

R1 ≤C

SNR1

1 + β ̄λINR

R2 ≤ minC λ̄SNR2 , C β ̄λSNR2+C β̄ ̄λINR

1 + β ̄λINR+W C λSNRside

W

R1 + R2 ≤C

β ̄λSNR2

+ C

SNR1 + β̄ ̄λINR

1 + β ̄λINR

+ W C

λSNRside

W

,

(1)

where β, λ ∈ [0, 1], β + β̄ = 1, λ + λ̄ = 1 and C (X ) =W mlog (1 + X ).

In the strong interference regime defined by SNR2 ≤INR ≤ SNR2 (1 + SNR1) , the achievable rate region is

R1

≤C (SNR1)

R2 ≤C λ̄SNR2R1 + R2 ≤C

SNR1 + λ̄INR

+ W C

λSNRside

W

.

(2)

In the very strong interference regime defined by INR ≥SNR2 (1 + SNR1) , the capacity region is

R1 ≤C (SNR1) C 1R2 ≤C (SNR2) C 2

(3)

Proof: The encoding procedure is depicted in Figure 3.

Using the Han-Kobayashi common-private message splitting

3When W = 0, we define W log

1 + xW

0.

strategy, the common message can be decoded at both re-

ceivers, while private message can only be decoded at the

intended receiver. Message S 1 is of size 2nR1 , and X 1 is

intended to be decoded at Y 1 only. Message S 2 is divided

into common part S 20 of size 2nR20, and private part S 22 of

size 2nR22. Superpose the codewords of both S 22 and S 20such that X 2 = X 20 + X 22. Then partition the set [1 : 2nR20]into 2nR3 equal size bins, B(l) = [(l

−1)2n(R20−R3) + 1 :

l2n(R20−R3)], l ∈ [1 : 2nR3], and X 3(l) is sent through theside-channel over n blocks. In the Gaussian channels, we

assume all inputs are i.i.d. Gaussian distributed satisfying

X 1 ∼ N (0, P 1), X 20 ∼ N (0, β̄ ̄λP 2), X 22 ∼ N (0, β ̄λP 2), andX 3 ∼ N (0, λP 2), respectively, where β, λ ∈ [0, 1], β̄ + β =1, λ̄ + λ = 1. The parameter λ denotes the fraction of powerallocated to the side-channel and β denotes the fraction of

power split for the private message of M1.Decoding occurs in two steps. Upon receiving

Y 2, (S 20, S 22) are first decoded. The achievable rateregion of (R20, R22) is the capacity region of a Gaussianmultiple-access channel denoted as C1, where

R20 ≤ C β̄ ̄λSNR2R22 ≤ C

β ̄λSNR2

R20 + R22 ≤ C

λ̄SNR2

.

(4)

Then upon receiving Y 1, Y 3, with the help of the side-

channel (S 1, S 20) are decoded treating S 22 as noise. This isa multiple-access channel with side-channel. The capacity re-

gion of such channel denoted as C2 is given in Appendix VI-A.Thus we have

R1 ≤ C

SNR11 + β ̄λINR

R20

≤C

β̄ ̄λINR

1 + β ̄λINR+W C λSNRside

W R1 + R20 ≤ C

SNR1 + β̄ ̄λINR

1 + β ̄λINR

+W C

λSNRside

W

.

(5)

The achievable rate region of Gaussian side-channel assistedthree-node network is the set of all (R1, R2) such thatR1, R2 = R20 + R22 satisfying that (R20, R22) ∈ C1 and(R1, R20) ∈ C2. Using Fourier-Motzkin elimination, we canderive the results in Proposition 1.

In the strong interference regime where SNR2 ≤ INR ≤SNR2 (1 + SNR1), it is not difficult to find out the achievablerate region is a decreasing function of β (see e.g. [10]).

Therefore, we can obtain the achievable region by substitutingβ = 0.When the interference is very strong where INR ≥

SNR2 (1 + SNR1), Carleial ([12]) shows that the communi-cation is not impaired by the very strong interference at all.Because the interfered receiver can first decode the interfering

signal, then subtract it out from the received signal and finally

decode its own message. Thus the capacity region is contained

by the point-to-point capacity of uplink and downlink.

Remark 1. In the weak interference regime, BC scheme

contributes to improving the downlink rate R1 which is limited

by the interference from M1. While in the strong interference

regime where INR

≥SNR2 , M1 sends all common message,


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BAI and SABHARWAL: DISTRIBUTED FULL-DUPLEX VIA WIRELESS SIDE-CHANNELS: BOUNDS AND PROTOCOLS 4165

R1 ≤ C SNR1

1 + SNR1 + (1 + SNR1 + λ̄INR)

1 + λSNRside

W

W − 1(1 + SNR1 + λ̄INR)

1 + λSNRside

W

W − 1 + (1 + SNR1)(1 + λ̄INR))

R2 ≤ C

λ̄SNR2

.

(7)

and there is no interference at the receiver of M2. In this

case, BC scheme enables M1 to deliver more common message

which is restricted by the interference link, thus increasing the

uplink rate R2.

Remark 2. In [10], the authors adopted a similar approach

for Z-interference channel with a digital relay link which

is error-free between the receivers. Our system model and

the system model in [10] will reduce to the Gaussian Z-

interference channel when there is no side-channel between

the interfered transceiver or there is no relaying link between

the two receivers. However, in our system model, the side-

channel is a wireless channel which exists between the two

mobile nodes. This is motivated by the prevalence of multi-

radio interfaces on current mobile devices. Also, we consider

the power allocation between main-channel and side-channel

for the proposed scheme to meet the per-node power con-

straint. In contrast, in [10] there is no such power constraint

since the relay schemes are employed between the receivers

over a noiseless relay link.

BC scheme adopts Han-Kobayashi strategy which allows

arbitrary splits of total transmit power between common mes-

sage and private message in the weak interference regime in

addition to the power split between the side-channel and main-

channel. Therefore, the optimization among such myriads of

possibilities is hard in general. We can simplify BC scheme by

setting power splitting of private message of M1 at the levelof Gaussian noise at the receiver M2 in the weak interferenceregime. This particular simple common-private power splitting

is used in [13] and is shown to achieve to within one bit

of the capacity region of the general two-user interference

channel. We will use the simplified BC scheme with fixed

power splitting for the capacity analysis in Section III-D.

B. Three Simpler Schemes

We propose three simpler schemes compared to BC scheme

for encoding X 3 and sequentially post-processing (Y 1, Y 3) atreceiver M2. The first is compress-and-cancel, which sim-

plifies the transmitter design of Node M1. Then we giveanother two schemes: decode-and-cancel and estimate-and-

cancel, both of which not only simplify the transmitter design

of M1, but also the receiver design of M2.

1) Compress-and-cancel: In compress-and-cancel scheme,

the transmitter design of M1 is simplified compared to BCscheme. Node M1 sends a compressed version of the inter-

fering message over the side-channel using source coding.

Noticing that the correlated information with the interfering

message can be observed at receiver M2, Wyner-Ziv strat-

egy [14] can be adopted for compression. We first compress

X 2 into

X 2, then encode

X 2 as X 3 and transmit it over

the side-channel. At the receiver M2, with the knowledge of

the distribution of Y 1, we can recover the interference under

certain distortion and subtract it from the main-channel. The

capacity of the side-channel should allow reliable transmission

of the compressed signal X 2.In the Gaussian channels, all inputs are assumed to be

i.i.d. Gaussian distributed, satisfying X 1 ∼ N (0, P 1), X 2 ∼N (0, λ̄P 2) and X 3 ∼ N (0, λP 2), respectively, where λ ∈[0, 1], λ̄ + λ = 1. For the ease of analysis, we also assume X 2 to be a Gaussian quantized version of X 2: X 2 = X 2 + Z q, (6)where Z q is the quantization noise with Gaussian distribution

N (0, σ2q ). We can obtain the following inner bound, with

calculation of rate in Appendix VI-B.


full-duplex network,the achievable rate region for compress-

and-cancel is given in (7) at the top of the page.

2) Decode-and-cancel: Decode-and-cancel can lead to a

simpler transceiver design for M1 compared to BC scheme

because no common-private message splitting is adopted at

the transmitter M1, and only single-user decoders are involved

at the receiver M2. In DC scheme [9], the interfering messageS 2 of Node M1 is encoded into two codewords X 2 and X 3using two independent Gaussian codebooks. Then X 2 is sentthrough the main-channel and X 3 is sent through the side-

channel. The message S 2 is required to be decoded at boththe intended receiver BS and the side-channel receiver at M2.

After decoding S 2 from the side-channel, the interference can

be cancelled out at the main-channel receiver at M2.

In the Gaussian channels, all inputs are assumed to be

i.i.d. Gaussian distributed, satisfying X 1 ∼ N (0, P 1), X 2 ∼N (0, λ̄P 2) and X 3 ∼ N (0, λP 2), respectively, where λ ∈[0, 1], λ̄+ λ = 1. The parameter λ denotes the power allocatedto the side-channel. We obtain the following inner bound

achieved by DC.

Proposition 3. (from [9]) For Gaussian side-channel assisted

three-node full-duplex network, the following rate region is

achievable for decode-and-cancel

R1 ≤ C 1R2 ≤ min

W C

λSNRside

W

, C

λ̄SNR2

(8)

3) Estimate-and-cancel: We also propose an estimate-and-

cancel scheme which is even simpler than DC scheme. In

estimate-and-cancel, M1 sends a scaled version of the wave-

form of the interfering signal over the side-channel. At the

receiver M2, we first estimate the interfering signal from the

side-channel received signal Y 3 , then rescale it according toX 2 and cancel it out from Y 1. Finally decode the signal-of-

interest at the main-channel receiver at M2 using single-user


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decoder. Note that in EC scheme, we have less flexibility in

using the side-channel since the bandwidth of the side-channel

is required to be equal or larger than that of the main-channel.

Let X 3 = K X 2, where K is a scalar (K ≥ 0) satisfying theper-node power constraint, i.e., E(|X 2 + X 3|2) ≤ P 2. ThusE(|X 2|2) ≤ P 2(1+K )2 .

We adopt independently Gaussian codebooks for X 1 and X 2

with X 1 ∼N

(0, P 1), X 2 ∼N 0, P 2(1+K )2. Side-channel sig-

nal X 3 is correlated with X 2 such that X 3 ∼ N

0, K 2P 2

(1+K )2

,

where K ≥ 0. The resulting rate of EC is given as follows,with calculation of rate shown in Appendix VI-B.


full-duplex network, the following rate region is achievable for

estimate-and-cancel

R1 ≤C SNR1

1 + K

2SNRside(1+K )2

1 + K

2SNRside(1+K )2

+ INR(1+K )2

R2

≤C

SNR2

(1 + K )2 ,(9)

where K ≥ 0.

C. New Outer Bound

In this section, we derive a new outer bound to char-

acterize the performance of the four schemes discussed in

Sections III-A and III-B. When there is no INI in the side-

channel assisted three-node full-duplex network, we can obtain

the no-interference outer bound,

R1 ≤ C 1R2

≤C 2.

(10)

However, the no-interference outer bound can be arbitrarily

loose. We notice that one of the transmitter and receiver pairs

in Figure 2 is co-located at the base station, thus causal

information of M1 is available at the transmitter BS. Hence we

derive a new outer bound on the capacity region of Gaussian

side-channel assisted three-node network as follows.

Theorem 1. For Gaussian side-channel assisted three-node

full-duplex network under per-node power constraint, in the

weak interference regime, i.e., INR ≤ SNR2 , the capacityregion is outer bounded by

R1 ≤

C 1

R2 ≤C λ̄SNR2R1 + R2 ≤C

ᾱλ̄INR + SNR1 + 2

̄αλ̄SNR1INR

1 + αλ̄INR

+ C

αλ̄SNR2

+ W C

λSNRside

W

.

(11)

In the strong interference regime, i.e., INR ≥ SNR2 , thecapacity region is outer bounded by

R1 ≤C 1R2

≤C λ̄SNR2

R1 + R2 ≤C

SNR1 + λ̄INR + 2 ̄

λSNR1INR

+ W C

λSNRside

W

,

(12)

where α, λ ∈ [0, 1], α + ᾱ = 1, λ + λ̄ = 1.Proof: The first two bounds on R1 and R2 come from the

point-to-point capacity of AWGN channel. The third bound on

R1+R2 is the sum-capacity upper bound. Messages S 1 and S 2are chosen uniformly and independently at random, and X 1iis a function of (S 1, Y

i−12 ). We define U i = (S 1, Y

i−12 , X

i−11 )

for i ∈ [1, n]. From Fano’s inequality, we have H (S i|Y in) ≤n for i = 1, 2, where n goes to infinity as n goes to zero.Thus for any codebook of block length n, we have

nR1 ≤ I (S 1; Y n1 , Y n3 ) + n1 (13)= I (S 1; Y 1

n) + I (S 1; Y n

3 |Y n1 ) + n1 (14)= I (S 1; Y

n1 ) + H (Y

n3 |Y n1 ) − H (Y n3 |Y n1 , S 1) + n1 (15)

≤ I (S 1; Y n1 ) + I (X n3 ; Y n3 ) + n1 (16)

≤

n

i=1 I (S 1, Y i−11 ; Y 1i) +n

i=1 I (X 3i; Y 3i) + n1 (17)≤

ni=1

I (S 1, Y i−1

2 , Y i−1

1 ; Y 1i)+

ni=1

I (X 3i; Y 3i)+ n1 (18)

=

ni=1

I (S 1, Y i−1

2 , X i1, Y

i−11 ; Y 1i) +

ni=1

I (X 3i; Y 3i)

+n1 (19)

=ni=1

I (S 1, Y i−1

2 , X i1; Y 1i) +

ni=1

I (X 3i; Y 3i) + n1 (20)

=n

i=1I (U i, X 1i; Y 1i) +

n

i=1I (X 3i; Y 3i) + n1, (21)

where (16) follows from H (Y n3 |Y n1 ) ≤ H (Y n3 ) and−H (Y n3 |Y n1 , S 1) ≤ −H (Y n3 |X n3 ); (19) and (21) follow be-cause X 1i is a function of (S 1, Y

i−12 ); (20) follows because

Y i−11 → (X i1, Y i−12 ) → Y 1i forms a Markov chain. And forany codebook of block length n , we have

nR2 = H (S 2|S 1) (22)= I (S 2; Y 2

n|S 1) + H (S 2|Y n2 , S 1) (23)≤ I (S 2; Y 2n|S 1) + n2 (24)

=ni=1

I (S 2; Y 2i|Y i−12 , S 1) + n2 (25)

=ni=1

I (S 2, Y 2i|Y i−12 , S 1, X i1) + n2 (26)

≤ni=1

I (X 2i, Y 2i|U i, X 1i) + n2, (27)

where (24) follows from Fano’s inequality; (26) follows be-

cause X 1i is a function of (S 1, Y i−1

2 ).Now by applying the standard time sharing argument, we

can obtain

R1 + R2 ≤ I (U, X 1; Y 1) + I (X 2, Y 2|U, X 1) + I (X 3; Y 3), (28)for product distribution p(u) p(x1x2x3

|u) p(y1y2

|x1x2x3).


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Similarly, in the strong interference regime where

I (X 2; Y 2|X 1, U ) ≤ I (X 2; Y 1|X 1, U ), the sum-capacity isupper bounded by

R1 + R2 ≤ I (U, X 1; Y 1) + I (X 2, Y 2|U, X 1) + I (X 3; Y 3)≤ I (U, X 1; Y 1) + I (X 2, Y 1|U, X 1) + I (X 3; Y 3)= I (U, X 1, X 2; Y 1) + I (X 3; Y 3).

In the Gaussian channels, the capacity region is outerbounded by Gaussian inputs. The auxiliary random variable U

captures the correlation between the codewords X 1 and X 2.

We define α = 1 − ρ2, where ρ is the correlation coefficientbetween X 1 and X 2 and α ∈ [0, 1]. With per-node powerconstraint, i.e., E(|X 1|2) ≤ P 1, E(|X 2 + X 3|2) ≤ P 2, letpower allocated to the side-channel be E(|X 3|2) = λP 2, thuswe have E(|X 2|2) ≤ λ̄P 2, where λ ∈ [0, 1], λ̄ + λ = 1. Hencethe capacity region outer bound can be expressed in terms

of parameters α, λ, channel coefficient and per-node power

constraints.

D. Within One Bit of the Capacity Region

In this section, we show that BC is within one-bit of thenew outer bound derived in Section III-C. Furthermore, we

also characterize the capacity gap of DC and EC schemes,

and show that they are also approximately optimal in specific

regimes.Let RBC =

β,λRBC(β, λ) denote the achievable rate

region for bin-and-cancel including all possible power split

at Node M1, where RBC(β, λ) denote the achievable rateregion for a fixed power split between common and private

information of M1, as well as side-channel and main-channel

at M1. The main result is given in the following theorem.

Theorem 2. The achievable region RBC is within 1 bit/s/Hz

of the capacity region of Gaussian side-channel assisted three-node network, for all values of channel parameters and

bandwidth ratio.

Proof: Let δ BCR1 denote the difference between the upper

bound on R1 and achievable rate R1 in RBC. Likewise, we

have δ BCR2 and δ BCR1+R2

, where all rates are divided by the

total bandwidth W m + W s. In order to prove the rate pair(R1 − 1, R2 − 1) in RBC is achievable for any (R1, R2) inthe capacity region of Gaussian three-node full-duplex withside-channel, we just need to show that

δ BCR1


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δ DCR1

= 0

δ DCR2

≤ W m

W m + W slog (1 + SNR2)

− W m

W m + W slog

SNR2SNRsideSNR2 + SNRside

= W m

W m+W slog1+ SNR

22

SNR2+SNRside+SNR2SNRside

≤ 1

1 + W ≤

1

2(32)

2) For EC scheme, the bandwidth ratio between side-

channel and main-channel is required to be greater than1, i.e., W ≥ 1. Comparing the rate region of EC inPropostion 4 with the no-interference outer bound we

have

δ ECR1

≤ W m


− W m

W m

+ W s

log1 +SNR1

1 + K

2SNRside(1+K )2

1 +

K 2SNRside(1+K )2 +

INR(1+K )2

=

W m

W m + W slog (1+

SNR1INR(1+K )2

1 + K 2SNRside(1+K )2

+ INR(1+K )2

+ SNR1

1 + K 2SNRside(1+K )2

δ ECR2

≤ W m


− W m

W m + W slog

1 +

SNR2(1 + K )2

= W m

W m + W slog

1 +

K (K +2)SNR2(1+K )2

1 + SNR2(1+K )2

.

(33)

The achievable rate region of EC is the union of rate

pairs for all possible values of K , where K ≥ 0.Therefore, when SNRside ≥

1 + 2√

2−1

(INR−2), and

we chose K =√

2 − 1, the following inequalities willhold simultaneously

δ ECR1 ≤ 1

1 + W ≤ 1

2

δ ECR2 ≤ 1

1 + W ≤ 1

2.

(34)

E. Discussion of capacity analysis results

Practical communication systems usually operate in the

inference-limited regime, so the strength of the signal and

interference are much larger than the thermal noise. Our one-

bit capacity gap result shows that for all channel parameters

and bandwidth ratio, BC scheme is asymptotic optimal in high

SNR, INR limit, because one bit is relatively small compared

to the rate of the users. For the two simple schemes, DC and

EC can achieve the near-optimal (half-bit gap) performance

only in certain regimes of channel values. From the capacityanalysis in previous section, we also notice that the capacity

gap for each scheme is inversely proportional to the bandwidth

ratio W , where W ∈ [0, W max]. Often, in practical wirelesscommunication, the bandwidth of side-channel is different

from the main-channel. For example, ISM band radio can

operate on a bandwidth up to 80 MHz in 2.4 GHz. In contrast,

generally the bandwidth of current 3G/4G radio is 20 MHz.

Motivated by the application of leveraging ISM band as theside-channel in a cellular network, the bandwidth of side-

channel usually is much larger than that of main-channel.If W = W max, for all values of SNR and INR, BC isnear-optimal which achieves within 11+W max bits/s/Hz of the

capacity region.

F. No side-channel Special Case

We can obtain the system model without side-channel when

bandwidth ratio W = 0 and Y 3 = 0 in Equation (1), which canbe modeled as causal cognitive Z-interference channel [15]. In

the cognitive Z-interference channel, we assume that at time i,

the encoder of base station will have access to the information

sequence sent by M1 up to time i − 1.When we do not use the causal information at BS, the

resulting channel is classic Z-channel. The rate region of classic Z-channel is the capacity inner bound of cognitive Z-

interference channel. And the rate region of classic Z-channel

is a special case of the achievable rate region of BC scheme

in Proposition 1 when W = 0. The sum-rate can be achievedby treating interference as noise in the weak regime and joint

decoding in the strong interference regime.The state-of-art approaches based on block Markov codes

[15], [16] do not seem to enhance the achievable rate region

over Z-channel (to the best of our knowledge), a counter-

intuitive result which may be due to the causality of infor-

mation at BS. At time i, the way that the transmitter BS canobtain the causal message of M1 is by decoding the message

through the direct link between M1 and BS before time i , sowe can not take advantage of the decoding schemes for block

Markov codes which require delay. Thus we conjecture that

this causal information will not help improve the achievable

sum-rate over classic Z-channel. Nevertheless, we can show

that the sum-capacity is established in the high SNR, INRlimit. As SNR1, SNR2, INR → ∞, with their ratios beingkept constant, we obtain the following theorem.

Theorem 4. For the Gaussian three-node full-duplex network,

the asymptotic sum-capacity is established

C No−SCsum =

C (SNR2) + C SNR11+INR INR ≤ SNR2,C (SNR1 + INR) SNR2 ≤ INR ≤ SNR2(1 + SNR1),C (SNR1) + C(SNR2) INR ≥ SNR2(1 + SNR1).

(35)

Proof: We only need to prove that the upper bound onthe sum-capacity of Gaussian three-node full-duplex network

is tight as SNR1, SNR2, INR → ∞. We can compute thesum-capacity upper bound directly from Theorem 1 and

lower bound from Proposition 1 by substituting W = 0

and λ = 0. We use C No−SCsum and C

No−SCsum to denote upper

bound and lower bound on the no-side-channel sum-capacity,

respectively.


8/12


limSNR1,SNR2,

INR→∞

C No−SCsum

C No−SCsum= lim

SNR1,SNR2,INR→∞

log(1 + αSNR2)+log

1+INR+SNR1+2√ ̄αSNR1INR

1+αINR

log(1 + β SNR2)+log

1+INR+SNR1

1+βINR

= lim

SNR1,SNR2,INR→∞

log(αSNR2)+log(INR+SNR1 +2√ ̄

αSNR1INR)−log(αINR)log(β SNR2)+log(SNR1 +INR)−log(β INR)

= limSNR1,SNR2,

INR→∞

log(SNR2) + log(INR + SNR1) − log(INR) + f 1(α)log(SNR2) + log(SNR1 + INR) − log(INR) + f 2(β )

= 1, (36)

where f 1(α) and f 2(β ) are constant for given α and β , respectively.

limSNR1,SNR2,

INR→∞

INR∗

SNR2(1 + SNR1) = lim

SNR1,SNR2,INR→∞

2SNR1 + SNR2(1 + SNR1) − 2

SNR1(SNR1 + SNR2 + SNR1SNR2)

SNR2(1 + SNR1) = 1.

(37)

In the weak interference regime where INR ≤

SNR2,

when SNR1, SNR2, INR → ∞, the lower bound matches withupper bound as shown in (36) at the top of the page.

The strong interference regime for the sum-capacity lower

bound is defined as SNR2 ≤ INR ≤ SNR2(1 + SNR1),and for the sum-capacity upper bound is defined as SNR2 ≤INR ≤ INR∗, where INR∗ = 2SNR1 + SNR2(1 + SNR1) −2

SNR1(SNR1 + SNR2 + SNR1SNR2). Thus in the stronginterference regime, using the result obtained in (37) at the

top of the page, we have

limSNR1,SNR2,INR→∞

C No−SCsum

C No−SCsum=

limSNR1,SNR2,INR→∞

C SNR1 + INR + 2√ SNR1INRC (SNR1 + INR)

= 1.

If INR ≥ INR∗ or INR ≥ SNR2(1+SNR1), the constrainton R1 + R2 in (12) becomes redundant and the sum-capacityupper bound reduces to (3) when W = 0, λ = 0.

Therefore, in the high SNR and INR limit, for the Gaussian

three-node full-duplex network, treating interference as noise

and joint decoding are asymptotically sum-capacity achievingin weak and strong interference regimes, respectively.

IV. ASYMPTOTIC AND FINITE SNR MULTIPLEXING GAIN

COMPARISONS A. High SNR Multiplexing Gain

For simplicity, we assume SNR1 = SNR2 = SNR and wedefine µ = log INRlog SNR , ν =

log SNRsidelog SNR . Parameter µ captures

the interference level, parameters ν and ν µ

capture the side-

channel level compared with main-channel.In this paper, we use multiplexing gain as the metric to

characterize the rate improvement by leveraging the side-

channel which is given as

M = Rsum(SNR, INR)

C single−user(SNR), (38)

where C single−

user = W mlog (1 + SNR) .

The single-user capacity is the maximum uplink/downlink

rate of the three-node network in the main-channel. Ideally,a perfect full-duplex can achieve the maximum multiplexing

gain of 2 when there is no interference, which corresponds tothe largest sum-capacity of 2W m log (1 + SNR). AssumingSNR 1, INR 1 and SNRside W , we can computethe asymptotic multiplexing gain of our proposed schemes as

SNR, SNRside, INR → ∞ and µ ≥ 0, ν ≥ 0, W ∈ [0, W max].First we derive the asymptotic multiplexing gain of the

no-side-channel sum-capacity over single-user capacity in the

high SNR and INR limit, which is given by

M No−SC = C No−SCsumC single−user

= 2 − µ 0 ≤ µ


9/12


RBCsumC single−user

=W m min

log

1 + SNR2

+ log(1 + λ̄SNR), log

1 + SNR+λ̄INR−12

+ log

1 + SNRINR

+ Wlog

1 + λSNRsideW

W mlog(1 + SNR)

≈ min {2log(SNR) , 2log (SNR) − log(INR) + Wlog (SNRside)}log(SNR)

≈ min{2, 2 + W ν − µ}.(41)

As SNR, SNRside, INR → ∞, in the weak interferenceregime (i.e., 0 ≤ µ < 1), set the power for the private messageof M1 at the level of Gaussian noise at the receiver M2,

namely, let β = 1λ̄INR

. Thus for a fixed λ (λ = 0), we derivethe expression in (41) at the top of the page.

In the strong interference regime where µ ≥ 1, β = 0. AsSNR, SNRside, INR → ∞, we have

RBCsumC single−user

≈

min {2log (SNR) , log(INR) + Wlog (SNRside)}log(SNR)

≈ min{2, µ + W ν }.(42)

From Theorem 2, we have proved that the inner bound

achieved by BC is within one bit of the outer bound by a

simple common-private power split when the power splitting

of private message of M1 is set at the level of Gaussian

noise at the receiver M2. Hence in the high SNR and INRlimit, BC scheme is asymptotic optimal in that it is capacity-

achieving for the Gaussian side-channel assisted three-nodenetwork. Therefore, the asymptotic multiplexing gain of the

sum-capacity of the Gaussian side-channel assisted three-node

full-duplex network over single-user capacity is

M side−channel = C side−channelsum

C single−user

= M BC =

min {2, 2 + W ν − µ} 0 ≤ µ


10/12


0 0.5 1 1.5 2 2.51

1.2

1.4

1.6

1.8

2

µ =log INRlog SNR

R s u m

C s i n g l e − u s e r

Bin−and−cancel

Decode−and−cancel

Compress−and−cancel

No−side−channel

No−side−channelsum−capacity

W=0.5, 2 ν=3µ

Side−channelsum−capacity

(a) W


11/12


Y 1

Y 3

X 1

X 2

X 3

M 2: [1 : 2nR2]

M 1: [1 : 2nR1]

Fig. 7. Multiple access channel with side-channel.

cancel scheme can achieve within one bit of the capacity

region for all values of channel parameters and is asymptoti-cally optimal. The other three schemes are also analyzed and

shown to be optimal in specific regimes. Both analytical andnumerical results demonstrate the factors that dominate the

performance of the proposed schemes, which contributes to

guiding the design of transceiver architecture.

VI. APPENDIX

A. Capacity Region of a Multiple Access Channel with Side-

Channel

Lemma 1. The capacity region of a multiple-access channelwith side-channel in Figure 7 is

R1 ≤ I (X 1;Y 1|X 2)

R2 ≤ I (X 2;Y 1|X 1) + I (X 3;Y 3)

R1 + R2 ≤ I (X 1,X 2;Y 1) + I (X 3;Y 3),

for some p(x1) p(x2, x3).

Proof: We first give the outline of achievability.

1) Codebook generation: Fix p(x1) and p(x2, x3) thatachieves the lower bound. Randomly and independently

generate 2nR1 sequence xn1 (m1), m1 ∈ [ 1 : 2nR1 ],each i.i.d. according to distribution

n

i=1 pX1(x1i).

Randomly and independently generate 2nR3

sequencexn3 (l), l ∈ [1 : 2nR3 ], each i.i.d. according to distributionn

i=1 pX3(x3i). For each l, randomly and condition-ally independently generate 2nR2 sequence xn2 (m2|l),m2 ∈ [ 1 : 2nR2 ], each i.i.d. according to distributionn

i=1 pX2|X3(x2i|x3i). Partition the set [1 : 2nR2 ] into2nR3 equal size bins, B(l) = [(l − 1)2n(R2−R3) + 1 :l2n(R2−R3)], l ∈ [ 1 : 2nR3 ]. The codebook and binassignments are revealed to all parties.

2) Encoding: Upon observing m2 ∈ [1 : 2nR2 ], assign m2to bin B(l). The encoders send xn1 (m1), x

n2 (m2|l) over

the main-channel, and xn3 (l) over the side-channel in nblocks.

3) Decoding: Upon receiving Y 3, the receiver for the side-channel declares that l̂ is sent if it is the unique messagesuch that (xn3 (l̂), Y

n3 ) ∈ T n ; otherwise, it declares an

error. Then upon receiving Y 1, the receiver for the main-

channel declares that ( m̂1, m̂2) are sent if it is the uniquepair such that (xn1 ( m̂1), x

n2 ( m̂2|l̂), Y n1 ) ∈ T n and m̂2 ∈

B(l̂); otherwise, it declares an error.Therefore, the following rate region is achievable

R1 ≤ I (X 1;Y 1|X 2)

R2 ≤ I (X 2;Y 1|X 1) + I (X 3;Y 3)

R1 + R2 ≤ I (X 1,X 2;Y 1) + I (X 3;Y 3),

for some p(x1) p(x2, x3).

The converse comes from cut-set upper bound.

n(R2) ≤ I (X n

2 ,X n

3 ; Y n

1 , Y n

3 |X n

1 ) (48)

= I (X n2 ;Y n

1 , Y n

3 |X n

1 ) + I (X n

3 ; Y n

1 , Y n

3 |X n

1 , X n

2 ) (49)

= I (X n2 ;Y n

1 |X n

1 ) + I (X n

2 ; Y n

3 |X n

1 , Y n

1 ) (50)

+H (X n3 |X n

1 , X n

2 ) −H (X n

3 |X n

1 , X n

2 , Y n

1 , Y n

3 ) (51)

= I (X n2 ;Y n

1 |X n

1 ) + I (X n

2 ; Y n

3 |X n

1 , Y n

1 ) (52)

≤ I (X n2 ;Y n

1 |X n

1 ) + I (X n

3 ; Y n

3 ), (53)

where (52) follows that X n3 is a function of X n2 , thus

H (X n3 |X n

1 , X n

2 ) = H (X n

3 |X n

1 , X n

2 , Y n

1 , Y n

3 ),

and (53) follows that

I (X n2 ; Y n

3 |X n

1 , Y n

1 ) = H (Y n

3 |X n

1 , Y n

1 ) −H (Y n

3 |X n

1 , X n

2 , Y n

1 )

≤ H (Y n3 ) −H (Y n

3 |X n

3 )

≤ I (X n3 |Y n

3 ).(54)

Similarly, we can also prove that

R1 ≤ I (X 1;Y 1|X 2)

R1 + R2 ≤ I (X 1,X 2;Y 1) + I (X 3;Y 3).

For Gaussian MAC with side-channel, the above constraintson R1, R2 and R1 + R2 will be simultaneously maximized byGaussian inputs.

B. Calculation of Achievable Rates of Compress-, estimate-

and-cancel schemes

For compress-and-cancel scheme, with Wyner-Ziv strat-

egy [14], the achievable rate region is given by

R1 ≤I (X 1; Y 1, X 2)R2 ≤I (X 2; Y 2)s.t. I (X 2; X 2|Y 1) ≤ I (X 3; Y 3),

(55)

for distribution product p(x1) p(x2) p(x3) p( x2|x2, y1).In the Gaussian case, all inputs are assumed to be i.i.d.

Gaussian distributed, satisfying X 1 ∼ N (0, P 1), X 2 ∼N (0, λ̄P 2) and X 3 ∼ N (0, λP 2), respectively, where λ ∈[0, 1], λ̄+λ = 1. Assuming X 2 is a Gaussian quantized versionof X 2: X 2 = X 2 + Z q, (56)where Z q is the quantization noise with Gaussian distribution

N (0, σ2q ). We have

RX2|Y 1 = I (X 2; X 2|Y 1)

= h( X 2|Y 1) − h( X 2|Y 1, X 2)= W mlog

1 +

λ̄P 2(γ 1P 1 + σ2)

(γ 1P 1 + γ 21λ̄P 2 + σ2)σ2q

,

(57)

and

I (X 3; Y 3) ≤ W slog

1 + γ 3λP 2

W σ2

. (58)

Thus subjecting to the constraint, we have

σ2q ≥λ̄P 2(γ 1P 1 + σ

2)

(γ 1P 1 + γ 21λ̄P 2 + σ2)

1 + γ 3λP 2

Wσ2

W

− 1

. (59)


12/12


And

R1 ≤ I (X 1; Y 1, X̂ 2)

= W mlog

1 +

γ 1P 1(λ̄P 2 + σ2q )

γ 21λ̄P 2 + σ2q + σ2(λ̄P 2 + σ2q )

.

(60)

Since R1 is a decreasing function of σ2q , now we can calculate

R1 by substituting (59) into (60). And R2 is given as

R2 ≤ I (X 2; Y 2)= C

λ̄SNR2

.

(61)

For estimate-and-cancel, since E(|X 2|2) ≤ P 2(1+K )2 , wechoose X 1 and X 2 from independent Gaussian codebooks

with X 1 ∼ N (0, P 1), X 2 ∼ N

0, P 2(1+K )2

. The correla-

tion coefficient between X 2 and X 3 is one, and X 3 ∼N

0, K

2P 2(1+K )2

. For Gaussian channels, by substituting the

channel parameters and power constraint, the achievable rate

region can be calculated

R1 ≤ I (X 1; Y 1, Y 3)= h(Y 1, Y 3)

−h(Y 1, Y 3

|X 1)

= C

SNR1 1 + K 2SNRside(1+K )2 1 + K

2SNRside(1+K )2 +

INR(1+K )2

R2 ≤ I (X 2; Y 2)

= C

SNR2(1 + K )2

.

(62)

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[8] Small cell forum. Available: http://www.smallcellforum.org/.[9] J. Bai and A. Sabharwal, “Decode-and-cancel for interference cancel-

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[11] T. Han and K. Kobayashi, “A new achievable rate region for theinterference channel,” IEEE Trans. Inf. Theory, Jan. 1981.

[12] A. Carleial, “A case where interference does not reduce capacity(correspondence),” IEEE Trans. Inf. Theory, vol. 21, Sept. 1975.

[13] R. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacityto within one bit,” IEEE Trans. Inf. Theory, vol. 54, Dec. 2008.

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[15] S. Seyedmehdi, J. Jiang, Y. Xin, and X. Wang, “An improved achievablerate region for causal cognitive radio,” in Proc. 2009 IEEE ISIT .

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Jingwen Bai received her B.E degree in ElectronicScience and Technology from Beijing University of Posts and Telecommunications, Beijing, China, in2011, and the M.S degree in Electrical and Com-puter Engineering from Rice University, Houston,TX, in 2013. She is currently working toward herPh.D. at Rice University, Houston, TX under theguidance of Dr. Ashutosh Sabharwal. Her researchinterest includes information theory and wirelesscommunication.

Ashutosh Sabharwal (S91-M99-SM04) received

the B.Tech. degree from the Indian Institute of Tech-nology, New Delhi, in 1993 and the M.S. and Ph.D.degrees from The Ohio State University, Columbus,in 1995 and 1999, respectively. He is currentlya Professor in the Department of Electrical andComputer Engineering, Rice University, Houston,TX. His research interests include the areas of information theory and communication algorithmsfor wireless systems.

Dr. Sabharwal was the recipient of PresidentialDissertation Fellowship Award in 1998, and the founder of WARP project(http://warp.rice.edu).

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