impact of channel estimation errors and power allocation...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 7, SEPTEMBER 2012 3223

Impact of Channel Estimation Errors and PowerAllocation on Analog Network Coding and

Routing in Two-Way RelayingForoogh S. Tabataba, Student Member, IEEE, Parastoo Sadeghi, Senior Member, IEEE,

Charlotte Hucher, Member, IEEE, and Mohammad Reza Pakravan, Member, IEEE

Abstract—In this paper, we study two important transmis-sion strategies in full-duplex two-way relaying in the presenceof channel estimation errors. In analog network coding (ANC),the relay transmits the combined signals that were received fromboth sources, with the aim of achieving better spectral efficiency.However, due to imperfect channel-state information (CSI),sources cannot perfectly cancel their own data in the relayedsignal. We derive an achievable information rate for ANC inimperfect-CSI conditions and show how the ANC performancecan significantly be degraded as a result. Moreover, we derivecut-set bounds with channel estimation errors for traditional rout-ing (TR), in which time sharing is used at the relay. Although ithas been previously shown that ANC outperforms TR when theCSI is perfect, we find that it may not maintain its superiority inimperfect-CSI case at low signal-to-noise ratio (SNR) conditions.Next, we propose practical power allocation techniques that can beused in the sources and relay for both ANC and TR. The proposedpower allocation schemes are relatively simple to compute and relyonly on long-term channel statistics. Nevertheless, they are shownto be effective and close to optimal solutions for a wide range ofSNRs, to different positions of the relay, and for both perfect- andimperfect-CSI conditions. By using the proposed power allocationtechniques, it is possible to bring back advantages of ANC over TRfor a wide range of SNRs in imperfect-CSI conditions.

Index Terms—Analog network coding (ANC), capacity, channelestimation, channel-state information (CSI), power optimization,Rayleigh fading channels, routing.

Manuscript received August 1, 2011; revised January 23, 2012 andApril 5, 2012; accepted May 2, 2012. Date of publication June 1, 2012; dateof current version September 11, 2012. This work was supported in part byIran Telecommunications Research Center and the Australian Research Councilthrough the Discovery Projects Funding Scheme under Project DP0984950.The review of this paper was coordinated by Prof. O. B. Akan.

F. S. Tabataba and M. R. Pakravan are with the Data Networking ResearchLaboratory, Advanced Communications Research Institute, Department ofElectrical Engineering, Sharif University of Technology, Tehran, Iran (e-mail:[email protected]; [email protected]).

P. Sadeghi is with the Research School of Engineering, The AustralianNational University, Canberra ACT 0200, Australia (e-mail: [email protected]).

C. Hucher was with the Research School of Engineering, The AustralianNational University, Canberra ACT 0200, Australia. She is now with SequansCommunications, Paris, France (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2012.2201969

I. INTRODUCTION

R ELAY-BASED wireless communication has been iden-tified as an energy-efficient transmission scheme from

remote users to the desired destination through one or moreintermediate relays [1]–[3]. Well-known relaying techniquesinclude the amplify-and-forward (AF) scheme, which involvesthe analog retransmission of the noisy received signal at therelay, and the decode-and-forward (DF) scheme, which in-volves decoding the received signal at the relay, followed byreencoding and transmission. In multiway relaying, informationbetween multiple wireless sources is exchanged through a sin-gle relay. There exist different approaches to the transmission ofmultiple data streams by the relay. Traditional routing (TR) hasbeen the main strategy for multiway relaying, which is basedon time sharing between different data streams [4], [5].

Physical-layer or analog network coding (ANC) is a morerecent spectrally efficient technique for multiway relaying [4],[6]–[12]. For example, in ANC, for half-duplex two-way relaychannels, two sources simultaneously transmit to the relay inthe first time slot. In the second time slot, the relay eitheramplifies and forwards the physical superposition of two sourcesignals [4], [7], [8], [13] or computes a function of two sourcemessages and forwards it back to the sources [6], [12], [14],[15]. The advantage of ANC over TR and digital networkcoding [7] is due to embracing the interference and broadcastnature of wireless channels. In this paper, we are interested inthe AF strategy for ANC in two-way relaying (which we willcall ANC for short), because it is simpler to implement at therelay [16], [17].

One important requirement for ANC to properly work,however, is self-interference cancelation [4]. That is, to recoverthe message from the other source, each source needs to cancelits own transmission from the signal that was received from therelay. This approach can fully be accomplished if each sourcehas perfect knowledge about the channels that affect its ownsignal in the “combined” received signal from the relay.

In practical training-based wireless systems, however, onlyestimates of channels may be available [18], [19]. As a result,complete self-interference cancelation is not possible, and thiscondition can inevitably impact the performance of ANC. Manystudies of ANC assume that perfect channel-state information(CSI) is available at the receivers [4], [8], [16], [20], [21].Quantifying the impact of channel estimation errors on ANCis largely an unaddressed problem.

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3224 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 7, SEPTEMBER 2012

The first contribution of this paper is to derive achievableinformation rates for AF-based full-duplex ANC with channelestimation errors. The second contribution of this paper is toderive information rate cut-set bounds in TR with channel esti-mation errors. Both these derivations are nontrivial and have notbeen studied in the literature. We find that channel estimationerrors significantly affect the rates and existing derivations forperfect-CSI conditions cannot be extended to describe sucheffects. Therefore, we have derived new bounds and achievablerates for this problem, which are more general and can besimplified to be used for perfect-CSI conditions. Although thederived information rate lower and upper bounds for ANCand TR are not directly comparable, our analysis points to anew observation. Under perfect-CSI conditions, in [4], highercapacity for full-duplex ANC over TR for all ranges of thesignal-to-noise ratio (SNR) is reported. However, we find that,when the receivers do not have perfect CSI, TR cut-set boundscan noticeably be higher than achievable data rates in ANC atlow SNRs. Therefore, it might not be beneficial to use ANC inthis case. The possible advantage of TR is despite the fact thatANC requires only one time slot to complete the full-duplextransmission, whereas TR requires two time slots. This case isdue to the accumulation of many data-dependent noise terms inANC that are not present in perfect-CSI conditions.

The third contribution of this paper is to study and proposepractical power allocation techniques that can be used inthe sources and relay for both ANC and TR. For ANC, weintroduce a novel simple function based on the average receivedSNR that has a similar behavior to the average sum rate andcan be used for power optimization purposes. Analytical poweroptimizations are carried out based on perfect-CSI formulationsand high-SNR assumptions due to the complexity of problemformulation for imperfect-CSI cases. Moreover, the proposedpower allocation schemes do not impose any computation com-plexity to the systems due to having closed-form expressionsand depend only on long-term channel statistics. Nevertheless,they are shown to be effective and close to (numerically found)optimum values for a wide range of SNRs and for both perfect-and imperfect-CSI conditions. In addition, they can remedyadverse effects of channel estimation errors, particularlyfor ANC.

A. Related Work

Channel estimation for relay channels have been thetopic of recent research, predominantly for one-way relaying[22]–[24]. For AF one-way relaying (one source, one relay,one destination), lower bounds on the information rates withchannel estimation errors [25], [26], approximations of theoutage probabilities [27], asymptotical symbol error rate [22]and bit error rate [28] have been considered. Recently, [29] hasderived the outage probability, error probability, and ergodiccapacity for one-way multirelay systems in the presence ofchannel estimation errors. To the best of our knowledge, [30]and [31] are the only works that have studied information ratesfor two-way ANC with channel estimation errors. However,the bounds that were proposed in those works are for half-duplex channels, which makes the approaches and the final

Fig. 1. Two-way relay network that was considered in this paper.

results in [30] and [31] fundamentally different from this paper,and involve some approximations of the lower bound due tothe non-Gaussianity of source–relay–destination channels [26].The authors in [32] and [33] study information rates for otherhalf-duplex two-way relaying schemes, i.e., for DF relayingand superposition coding (SPC) in multiple antenna relays,respectively. In this paper, we consider full-duplex channels,for reasons which will shortly be clarified and provide aprovable achievable rate for the ANC with channel estimationerrors. Specific channel estimation or pilot design techniquesfor two-way relaying have recently been proposed in [34]–[37].However, in this paper, we are not directly concerned withoptimized pilot design for two-way relay systems. Instead,we are interested in understanding the fundamental effects ofresidual channel estimation errors, which are unavoidable inany training-based design, on the information rates of ANCand TR.

Many existing works in two-way relaying schemes assumeequal power allocation between the nodes due to its simplicity.Some recent studies of power allocation in two-way relayingsystems can be found in [17], [33], [34], and [38]–[40]. Theauthors in [17] propose a rate adaptation scheme for ANCand a method of finding the optimal transmission power thatmaximizes the data rate. In [33], a numerical power allocationtechnique that jointly maximizes the achievable rate regionof SPC half-duplex multiple-antenna two-way relay systemis introduced, and in [38], the achievable sum rate of thehalf-duplex ANC scheme is maximized. In [39], the totaltransmit power is minimized, subject to some constraints onthe received SNRs at two sources, whereas [40] considers amultiuser two-way relay network and investigates the problemof optimal power allocation to maximize the weighted sumrate of all users. In [34], maximum-likelihood estimation atthe relay is adopted, and two power allocation schemes tomaximize the average effective SNR and minimize the meansquare error of the channel estimation is proposed. However,all the aforementioned works ([17], [33], [34], and [38]–[40])are based on half-duplex transmission, whereas we proposepower allocation for the full-duplex mode. In addition, [17] and[38]–[40] need instantaneous CSI for power allocation, whichis not available in imperfect-CSI conditions.

II. GENERAL SYSTEM MODEL AND ASSUMPTIONS

We consider a two-way relay channel with two source nodesA and B and one relay node R, as shown in Fig. 1. All channels

TABATABA et al.: IMPACT OF CHANNEL ESTIMATION ERRORS AND POWER ALLOCATION ON ANC AND ROUTING 3225

Fig. 2. Block transmission schemes of both strategies. (a) ANC. (b) TR.

are Rayleigh block-fading channels, i.e., the realization ofthe fading channel in each link follows a zero-mean complexGaussian distribution, which stays constant during the trans-mission of a block of symbols and changes to an independentvalue in the next block. In the mathematical equations, we usesubscripts a, b, and r to refer to nodes A, B, and R, respectively.The fading channel from node k to node j, (k, j) ∈ {a, b, r}2is denoted by hkj with variance σ2

hkj= 1/dνkj , where dkj is the

distance between nodes k and j, and ν is the path-loss exponent[41]. This channel model takes into account both long-term pathloss and short-term fading (a similar model has been used in [6],[38], [40], and [42]).

As depicted in Fig. 2, in full-duplex1 ANC, both sourcesbroadcast their data at the same time during the block timeT . Concurrently, the relay sends the amplified version of pre-viously received signals to nodes A and B. However, in full-duplex TR, time-division multiplexing is used, i.e., the blocktime T is shared between two sources. In the first half of eachblock, node A transmits to the relay and node B, whereas therelay sends node’s A previously received information to B atthe same time. The second half of each block is allocated toB−A transmission.

The main reasons for considering full-duplex systems in thispaper are described as follows. Based on [4, Fig. 2], we observethat, even in the presence of perfect CSI, the information rate ofhalf-duplex ANC may be slightly worse than that of half-duplexTR at a low SNR. Therefore, it is not unexpected if channelestimation errors can further degrade ANC information ratescompared to TR in such conditions. Based on [4, Fig. 3], onthe other hand, we observe that full-duplex ANC is better thanTR with perfect CSI for the entire range of the SNR considered.Therefore, it would be more interesting to see if channel estima-tion errors can bring ANC information rates below those of TRin full-duplex transmission. In addition, there has been a recentinterest in making full-duplex transmission practically possible.

1In half-duplex systems, devices transmit and receive in a single-frequencyband but at different times. In full-duplex systems, devices can transmit andreceive in a single-frequency band at the same time, with the aim of doublingthe spectral efficiency.

For example, if nodes have two directive antennas for receivingand transmitting [42] with enough spatial separation to reduceloop-back interference from the transmit antenna to the receiveantenna, they can work in the full-duplex mode. Loop-backinterference can further be reduced by employing other interfer-ence cancelation techniques such as time-domain cancellationor spatial suppression; see [43]–[46] and the references therein.Recently, in [47], a new technique called antenna cancellationhas been proposed to implement practical full-duplex radios,which provides about 30-dB interference reduction. Therefore,in our analysis, the loop-back interference residual power isassumed to be negligible compared to other noise such as chan-nel estimation errors.2 In addition, full-duplex transmission hasbeen considered in several papers related to one-way relay-assisted cooperative communications [42], [44]–[46] and intwo-way relay channels [4], [20].

We assume that neither the two sources nor the relay possessa priori instantaneous CSI but only the statistics of the chan-nels. That is, hkj for (k, j) ∈ {a, b, r}2 is not initially knownby any node, but the variances of the channel coefficients,σ2hkj

= 1/dνkj , are known to all nodes. These variances arelong-term statistical parameters of the channels that do notchange very often and, hence, can be estimated with somemodest processing at the receivers and sent to the transmittersusing low-rate feedback if needed [27], [48]. Pilot symboltransmission is used to estimate each channel link.3 We assumethat three pilot symbols are sent by nodes A, B, and R insequence to estimate the channels at the receivers before datatransmission begins. All nodes operate in the half-duplex modeduring pilot transmission in both the TR and ANC schemes (seeFig. 2). We model each channel as hkj = hkj + hkj , where hkj

is the channel estimate, and hkj is the estimation error with zeromean and variance σ2

hkj. If node k sends a pilot symbol to node

j, then the received signal at node j is given by

Yj =√

PkhkjXp + Zj , (k, j) ∈ {a, b, r}2, k �= j (1)

where Pk is the average transmit power of node k, and Xp isthe pilot symbol with unit power. Note that, without loss ofgenerality, we can assume that a pilot symbol with a constantunit value for pilot transmission [24], [25]. Zj is a zero-meancomplex-valued additive white Gaussian noise (AWGN) atnode j, and without loss of generality, we can assume that it hasunit variance. We assume that the receivers use the minimum-mean-square-error (MMSE) method for channel estimation.Note that pilot-symbol-based MMSE estimators are often usedfor optimal channel estimation in cellular channels [22]. There-fore, many papers [18], [19], [23], [24], [27]–[30], [34], [37]assume the MMSE method for performance evaluation underimperfect-CSI conditions. For a complex Gaussian channel hkj ,

2In cases where loop-back interference cannot be neglected, it should jointlybe considered along with channel estimation errors. This is an interesting openproblem but is beyond the scope of this paper and can be considered for futurework.

3The pilot transmission scheme is common between ANC and TR; hence, itis described first. We will describe the data transmission model for ANC andTR later in their corresponding sections.


MMSE estimation is identical to linear minimum mean squareerror (LMMSE) [49], and node j obtains

hkj =E{hkjY

∗j

}E−1

{|Yj |2

}=

σ2hkj

√PkXp

σ2hkj

Pk + 1Yj =

σ2hkj

√Pk

σ2hkj

Pk + 1Yj (2)

hkj =hkj − hkj σ2hkj

=σ2hkj

σ2hkj

Pk + 1. (3)

Note that Xp = 1 is known; thus, Yj and, as a result, hkj andhkj , are complex Gaussian random variables. At the end of thepilot transmission phase, nodes A, B, and R have hba, hra; hab,hrb; and har, hbr, respectively. Using reliable control channels,the estimates are sent from the receivers to the transmitters toassist them with decoding and input synchronization duringdata transmission, if and when required, as will be explainedin Section III and Appendix C.4

It is shown in [18] that, when the pilot and data powers areoptimized, the transmission of one pilot symbol (per unknownchannel per transmission block) is the optimal approach forachieving the highest information rate. When pilot and datasymbol powers are identical, it may be advantageous to sendmore than one pilot symbol for estimating each channel to re-duce the channel estimation error and increase the informationrate. However, the optimal number of pilot symbols under thissetting cannot be found in closed form. Furthermore, equalpower for pilot and data symbols leads to simpler implemen-tation of output amplifiers in practice. Based on this discussion,we assume that pilot and data symbol powers are identical andthat only one pilot symbol is sent from each node.

In the following sections, we provide a detailed descriptionof each scheme and derive an achievable rate for ANC and a ca-pacity upper bound for TR in the presence of channel estimationerrors. Then, the perfect-CSI formulation is shown as a specialcase of our results. Based on perfect-CSI formulations, weoptimize the power allocation of both the ANC and TR schemesand provide simple closed-form solutions that can be used inpractice. We write all information rates for A−B transmission.B−A rates can be found by changing all a indices to b, and viceversa. All the logarithms are taken in base 2, E{·} means thestatistical expectation, and Re{·} is the real part of a complexnumber.

III. ANALOG NETWORK CODING ACHIEVABLE RATE

AND POWER ALLOCATION

A. Achievable Rate

In the ANC scheme, both source nodes A and B simultane-ously broadcast their own data, whereas the relay concurrentlysends the amplified version of the previously received signal

4Similar assumptions have been made in [25] and [28] for one-way AFrelaying and in [36] for the ANC scheme. Here, we are interested in the impactof channel estimation errors on ANC. The practical aspect of this assumption isa separate issue, which is out of the scope of this paper. See some analysis anddiscussions provided in [28] and [35].

from A and B [4]. The received signals at the relay and node Bfor symbol time i are

Yr[i] =√

Pa(har + har)Xa[i]

+√

Pb(hbr + hbr)Xb[i] + Zr[i] (4)

Yb[i] =√

Pa(hab + hab)Xa[i]

+ α(hrb + hrb)Yr[i− 1] + Zb[i] (5)

where Xk is the transmitted information symbol of node k (k ∈{a, b}) with zero mean and unit variance. Zr and Zb are zero-mean complex-valued AWGN with unit variance at the relayand node B, respectively. Because perfect CSI is not availableat the receiving nodes, we have replaced the channel coeffi-cients with the sum of their estimates and estimation errors.We assume that hab and hrb have been calculated during pilottransmission at node B, and similarly, hbr and har have beencalculated at the relay. α is the relay amplification factor thatwas chosen such that the relay power constraint is satisfied, i.e.,

α =

√√√√ Pr

1 + Pa

(|har|2 + σ2

har

)+ Pb

(|hbr|2 + σ2

hbr

) . (6)

Node B knows its own symbol Xb. We assume that channelestimates hbr and har, which are calculated at the relay, areforwarded to the sources. Therefore, node B can calculate αduring data transmission and attempt to delete its own contribu-tion from the received signal. Inserting (4) and (5) and deletingthe terms known to node B in Yb[i] results in

Yb[i] =√

PahabXa[i] + αhrbhar

√PaXa[i− 1] + Zanc[i]

(7)

Zanc[i] �√


√PaXa[i− 1]

+ αhrbhar

√PaXa[i− 1]+αhrbhar

√PaXa[i− 1]

+ αhrbhbr

√PbXb[i− 1] + αhrbhbr

√PbXb[i− 1]

+ αhrbhbr

√PbXb[i− 1] + αhrbZr[i− 1]

+ αhrbZr[i− 1] + Zb[i]. (8)

It is evident that node B cannot completely delete its owncontribution because of the channel estimation errors. There-fore, many data-dependent noise terms that are not presentin the perfect-CSI case have been accumulated in Zanc[i]. Inparticular, for perfect CSI, we let hab = hrb = har = hbr ≡ 0to obtain

Yb,p[i] =√PahabXa[i] + αphrbhar

√PaXa[i− 1]

+ Zanc,p[i] (9)

Zanc,p[i] �αphrbZr[i− 1] + Zb[i] (10)

where

αp =

√Pr

1 + Pa|har|2 + Pb|hbr|2. (11)


As a result, our approach in deriving a lower bound on theinformation rate of ANC with channel estimation errors isdifferent from the approach in [4]. In the following theorem, weprove that the lower bounding techniques in [18] and [19] canbe adopted to replace Zanc[i] by a worst-case Gaussian noiseand help us solve the problem.

Theorem 1: Let hΔ= {hab, har, hrb} denote a given realiza-

tion of the channel estimates. An achievable rate of ANC forA−B transmission is given by

Ranc(h) = log

⎛⎜⎜⎜⎝A1

⎛⎜⎜⎝

1 +

√1 −(

A2

A1

)22

⎞⎟⎟⎠

2⎞⎟⎟⎟⎠ (12)

where

A1 � 1 +Pa

Nanc

(|hab|2 + α2|hrb|2|har|2

)

A2 �α2Pa

Nanc|hab||hrb||har| (13)

and Nanc is the variance of Zanc, which is given by

Nanc =σ2hab

Pa+α2Pa

(|hrb|2σ2

har+|har|2σ2

hrb+σ2

harσ2hrb

)+ α2Pb

(|hrb|2σ2

hbr+ |hbr|2σ2

hrb+ σ2

hbrσ2hrb

)+ α2σ2

hrb+ α2|hrb|2 + 1. (14)

Proof: See Appendix A. �Because h is random, Ranc(h) is also a random variable,

and the average achievable rate Ranc = Eh{Ranc(h)} dependson the distribution of h. This expectation cannot be found inclosed form, and as a result, we numerically calculate it usingthe Monte Carlo method. The achievable rate for perfect CSI isa special case of our results. Substituting all h’s with h’s andassuming zero variances for the estimation errors in (12), wearrive at

Ranc,p(h) = log

⎛⎜⎜⎜⎝A1p

⎛⎜⎜⎝

1 +

√1 −(

A2p

A1p

)22

⎞⎟⎟⎠

2⎞⎟⎟⎟⎠ (15)

A1p � 1 +Pa

Nanc,p

(|hab|2 + α2

p|hrb|2|har|2)

A2p �αp2Pa

Nanc,p|hab||hrb||har| (16)

Nanc,p =α2p|hrb|2 + 1. (17)

This result is a new achievable rate that is derived for ANCin perfect-CSI conditions. Another achievable rate was derivedin [4], with somewhat different assumptions in derivations (seeAppendix A for more details).

B. ANC Power Optimization

Under the same total power constraint PT for both ANC andTR, we wish to optimize the power allocation of the sourcenodes A and B and the relay node to maximize the averageachievable rate for ANC and the cut-set bound for TR. Inthis section, we study power allocation for ANC. Due to thecomplexity of the lower bound expressions in the imperfect-CSI case in (12), the analysis has been done for perfect-CSIconditions. In the following discussion, we introduce the powerconstraint for ANC and define some power ratios to allocatepower to different nodes. We provide an analytical poweroptimization technique that leads to closed-form expressionsfor power allocation. We compare our analytical power allo-cation with equal power assignment and with the numericaloptimization of power ratios in Section V. We will also applythe analytical solution on both systems in imperfect-CSI condi-tions and investigate the efficiency of the power distribution inthis case.

The power constraint for the ANC scheme is given by

Pa + Pr + Pb = PT (18)

where all powers are constant during pilot and data transmis-sions. Subject to the aforementioned power constraint, we aimat maximizing the sum of the average achievable rates of A−Band B−A transmissions, i.e.,

maxPa,Pb,Pr

Rabanc,p + Rba

anc,p

s.t. Pa + Pr + Pb = PT (19)

where Rabanc,p is given in (15), and Rba

anc,p is derived byappropriate changes in the indices of Rab

anc,p. We denote by γrthe fraction of the total power that is allocated to the relay andγa the fraction of the sources’ power that is allocated to nodeA, i.e.,

Pr = γrPT (20)

Pa =(1 − γr)γaPT Pb = (1 − γr)(1 − γa)PT . (21)

As aforementioned, the average rate (Ranc,p) cannot becomputed in closed form. In the first step of simplification,we can maximize the sum of instantaneous achievable rates(Rab

anc,p +Rbaanc,p). However, replacing Pr and Pa in (15) with

(20) and (21) results in a complicated function of power ratiosthat does not lead to any closed-form solution for the powerallocation. Therefore, we need to find a simpler function thatdoes not depend on instantaneous channel coefficients whileproviding close-to-optimal power ratios.

Proposition 1: Instead of directly maximizing the sum of theaverage information rates in (19), we can solve the followingoptimization problem, which yields close-to-optimal solutionsfor the original problem:

maxPa,Pb,Pr

g(Pa, Pb, Pr)

s.t. Pa + Pr + Pb = PT (22)


where g(Pa, Pb, Pr) is defined in (23), shown at the bottom ofthe page.

Appendix B provides the details of how the objective func-tion is derived as well as some supporting evidence for itseffectiveness in solving the original problem.

We substitute (20) and (21) in (23) and set the derivatives of gwith respect to (w.r.t.) γr and γa to zero. It yields the followingequation set: { ∂g

∂γr|(γopt

r ,γopta ) = 0

∂g∂γa

|(γoptr ,γopt

a ) = 0.(24)

The partial derivatives of g are rather complicated, particularlyfor ∂g/∂γr, which do not result in closed-form expressions forpower ratios. Therefore, we need to do more simplifications andsolve the problem in a suboptimal way. First, we assume high-SNR conditions, in which α2

p in (11) has a simpler form (α2p �

Pr/(Paσ2har

+ Pbσ2hrb

)). This assumption leads to solutionsthat are independent of the total power PT . In addition, withoutloss of generality, we assume normalized distances between thetwo source nodes. That is, σ2

ab = 1 (in practice, the physicaldistances can easily be incorporated into the model, whichaffects the average SNR). In the following section, we providea suboptimal solution for this problem.

1) Calculating γoptr for Equal Distances: First, we find the

solution for the case in which the relay is located at equaldistances from nodes A and B, i.e., σ2

har= σ2

hrb. In this case,

γopta = 0.5, and we just need to find the optimal γr, which is

given by

∂g

∂γr= 0 ⇒

4 − 2σ2har

+ 4γr(σ2har

− 1)

ln(2)(γr − 1)(1 + γr

(σ2har

− 1)) = 0

(25)

⇒ γoptr =

{0, for σ2

har≤ 2

σ2har

−2

2σ2har

−2, for σ2

har> 2.

(26)

It can easily be verified that, for σ2har

> 2, the aforementionedsolution is the maximizing argument of the function g. Forσ2har

≤ 2, which corresponds to large A−R distances (dar >

0.7937 for ν = 3), (σ2har

− 2)/(2σ2har

− 2) is not in the validrange [0, 1], and the maximum of the simplified function g isobtained for γr = 0. This result means that the relay shouldnot be used in the system.5 Note that this result is reasonable.The larger the distance of the relay from the nodes, the lowerthe quality of the relay links compared with the direct link.

5Note that, even in the absence of relay, there is direct link between thesources.

Therefore, it is more efficient to use the total power budget forthe source nodes only.

2) Calculating γopta for a Known γopt

r : Now, we assume thatwe know γopt

r and wish to find γopta , which is the solution to the

following equation:

∂g

∂γa= 0 ⇒ C4γ

4a + C3γ

3a + C2γ

2a + C1γa + C0 = 0 (27)

where

C4 � − 2(σ2har

− σ2hbr

)3(γr − 1)3

C3 � −(σ2har

−σ2hbr

)2 (σ2har

− 7σ2hbr

− 4σ2har

γr + 4σ2hbr

γr)

× (γr − 1)2

C2 � −(σ2har

− σ2hbr

)(γr − 1)

×(σ4har

σ2hbr

γr + 11σ2har

σ2hbr

γr − 2σ4har

γr

+ σ2har

σ4hbr

γr − 4σ2har

σ2hbr

γ2r + 2σ4

harγ2r

−3σ2har

σ2hbr

+ 2σ4hbr

γ2r − 11σ4

hbrγr + 9σ4

hbr

)C1 �σ2

hbr

(σ2hbr

+ σ2har

γr − σ2hbr

γr)

×(5σ2

hbr− 3σ2

har+3σ2

harγr − 5σ2

hbrγr+2σ2

harσ2hbr

γr)

C0 � − σ4hbr

(σ2hbr

+ σ2har

γr − σ2hbr

γr) (

σ2har

γr − γr + 1).

(28)

Therefore, γopta is the real root of a quadratic equation that can

be solved in the general case. Note that the aforementionedconstants are related to the mean of the channel coefficientsand do not require knowledge of instantaneous channel real-izations. They also depend on the optimum value of γr, whichis not available in the general case but can be approximated.For example, we can assume that γr = 1/3, which is used inequal power allocation or other reasonable constants. Anothersuggestion is to define σ2 as the average of σ2

harand σ2

hbrand

use it in (26), which has been derived for equal distances, i.e.,

γ∗r �

{0, for σ2 < 2σ2−2

2σ2−2, for σ2 > 2

σ2 �σ2har

+ σ2hbr

2. (29)

We use (27) and (29) to calculate analytical power ratiosfor the ANC scheme. Although the aforementioned solutionfor the ANC scheme is suboptimal, it performs remarkablyclose to optimum solutions for both perfect- and imperfect-CSIconditions, as shown by numerical analysis in Section V.

g(Pa, Pb, Pr) � log

⎛⎜⎜⎜⎝

PaPb

(σ2hab

+ Pr

1+Paσ2har

+Pbσ2hrb

σ2hrb

σ2har

)2

(1 + Pr

1+Paσ2har

+Pbσ2hrb

σ2hrb

)(1 + Pr

1+Paσ2har

+Pbσ2hrb

σ2har

)⎞⎟⎟⎟⎠ (23)


IV. TRADITIONAL ROUTING CUT-SET BOUND

AND POWER ALLOCATION

A. Cut-Set Bound

In the TR scheme, the block time T is shared betweentwo sources. In the first half of each block, node A transmitsto the relay and node B, whereas the relay sends node’s Ainformation to B at the same time. The second half of eachblock is allocated to B−A transmission.6 Unlike ANC, keepingthe symbol index i is not necessary for our derivations. Forexample, it is understood that the received signal at node Bat symbol index i corresponds to the transmission from nodeA, Xa[i], and the relay Xr[i] (where Xr[i], in fact, containsinformation about Xa[i− 1]). To simplify the notation, we dropthe symbol index from the received signals at the relay and nodeB and write

Yr =√

PaharXa +√

PaharXa + Zr (30)

Yb =√

PahabXa +√

PrhrbXr +√PahabXa

+√

PrhrbXr + Zb. (31)

Because our aim is to derive a capacity cut-set bound for TR,we have not assumed any specific type of data processing andforwarding at the relay, such as AF or DF. That is, a generaltransmitted signal from the relay in (31) should be considered.Similar formulation has been used in [5], [42], and [50].

For A−B communication, let CTR(h) denote the cut-setcapacity upper bound for a given realization of the channel esti-mates with h = {hab, har, hrb}. Because h is random, CTR(h)is also a random variable, and the average cut-set bound CTR =Eh{CTR(h)} depends on the distribution of h.

Theorem 2: For a given realization of channel estimates h, acut-set capacity upper bound for TR for A−B communications

6Note that unequal time sharing between nodes A and B can also be incor-porated into the TR scheme. To put our results in perspective with [4], however,we adhere to a basic TR scheme with equal time sharing. The optimization ofthe time sharing can potentially be an advantage of TR compared with ANC toachieve higher rates in the presence of channel estimation errors. Understandingthe effects of this additional degree of freedom in TR is an interesting problemfor future research.

with jointly Gaussian inputs is given by

CTR(h) = max0≤|ρ|≤1

min

{12I(Xa, Xr;Yb|Xb, h),

12I(Xa;Yr, Yb|Xr, Xb, h)

}(32)

where ρ is the correlation between inputs from source A andthe relay, i.e.,

ρ = E {XaX∗r} . (33)

Upper bounds for I(Xa, Xr;Yb|Xb, h) and I(Xa;Yr, Yb|Xr,

Xb, h) are given in (34) and (35), shown at the bottom of thepage.

Proof: See Appendix C. �As evident from the aforementioned bounds, the presence

of channel estimation errors has resulted in far more involvedformulas compared with the perfect-CSI case given as follows.The aforementioned results cannot simply be inferred fromexisting cut-set bounds for perfect CSI. Therefore, the boundshave carefully been recalculated in the proof for imperfect-CSIconditions.

In fact, the cut-set bound for perfect CSI is a specialcase of our results. In this case, there is no estimation error(hkj = hkj), and by changing all h’s to h’s and assuming zerovariances for estimation errors (σ2

hkj= 0), we obtain

CpTR(h)

≤ max0≤|ρ|≤1

min

{12log(

1 + Pa|hab|2 + Pr|hrb|2

+ 2|ρ||hab||hrb|√

PaPr

),

12log(1+Pa

(1−|ρ|2

) (|hab|2+|har|2

))}(36)

which is in complete accordance with those provided in [4], [5],and [42].

B. TR Power Optimization

Similar to Section III-B, we introduce the power constraintand power ratios for the TR scheme. We provide analytical

I(Xa, Xr;Yb|Xb, h)

≤EXa,Xr

{log

(Pa|hab|2 + Pr|hrb|2 + 2|ρ||hab||hrb|

√PaPr + Paσ

2hab

+ Prσ2hrb

+ 1

Paσ2hab

|Xa|2 + Prσ2hrb

|Xr|2 + 1

)}(34)

I(Xa;Yr, Yb|Xr, Xb, h)

≤EXa,Xr

⎧⎨⎩log

⎛⎝(Paσ

2har

+1)(

Pa|hab|2(1−|ρ|2

)+Paσ

2hab

+Prσ2hrb

+1)+ Pa|har|2

(1 − |ρ|2

)(Paσ

2hab

+ Prσ2hrb

+1)

(Paσ2

har|Xa|2 + 1

)(Paσ2

hab|Xa|2 + Prσ2

hrb|Xr|2 + 1

)⎞⎠⎫⎬⎭(35)


power optimization with closed-form expressions for powerallocation. The analysis has been done for perfect-CSI condi-tions. However, we apply the analytical solution in both perfect-and imperfect-CSI conditions, which is shown to be close tothe optimal results in Section V. Moreover, we compare ouranalytical power allocation with equal power assignment andwith the numerical optimization of power ratios.

The total power constraint for the TR scheme is

Pa + P abr + Pb + P ba

r = PT (37)

where P abr and P ba

r are the relay powers during A−B andB−A data transmissions, respectively. Assuming differentpowers for the relay during two transmissions enables moredegrees of freedom for power optimization. Because we needa single pilot symbol to be sent by the relay to estimate hra andhrb at A and B, we can assume Pr = P ab

r + P bar for the relay

during pilot transmission. For the source nodes, we assumeequal power for pilot and data symbols.

Assuming the aforementioned power constraint, we aim atmaximizing the sum of the instantaneous capacity upper boundsof A−B and B−A transmissions subject to the aforementionedpower constraint, i.e.,

maxPa,Pb,Pab

r ,P bar

Cp,abTR (h) + Cp,ba

TR (h)

s.t. Pa + P abr + Pb + P ba

r = PT (38)

where Cp,abTR (h) is given in (36), and Cp,ba

TR (h) is derived byappropriate changes in the indices of Cp,ab

TR (h). We denote byγab the fraction of the total power that was allocated to theA−B transmission. βab is the ratio of relay power to node Apower, i.e.,

Pa + P abr = γabPT , P ab

r = βabPa. (39)

For A−B transmission, we have

Pb + P bar = (1 − γab)PT , P ba

r = βbaPb. (40)

Proposition 2: The optimum value of γab is 0.5, and theoptimum βab, which is denoted by βopt

ab , is given by

βoptab =

|har|2|hrb|2

(|hrb|2 + |hab|2)2 + |har|2|hab|2. (41)

The optimum value of βba, which is denoted by βoptba , can easily

be found by suitable change of indices.Proof: See Appendix D. �

The optimized power ratios depend on the instantaneouschannel values, which are not available during pilot trans-mission. In addition, it is more convenient to fix the powerallocation for a long time rather than to change it for each blocktransmission. Therefore, in (41), we replace the square of themagnitude of channels with their means to obtain

β∗ab �

σ2har

σ2hrb(

σ2hrb

+ σ2hab

)2+ σ2

harσ2hab

. (42)

Fig. 3. Average achievable rate of ANC compared with the average capacityupper bound of TR against SNR using equal power allocation in Pos-1 (dar =0.5, drb = 0.5). ANC outperforms TR in the perfect-CSI case; however, it mayhave a weaker performance in low SNRs in imperfect-CSI conditions..

We use the aforementioned power allocation for both perfect-and imperfect-CSI cases and for all SNRs and compare theresults with optimum numerical results found by global search.Numerical analysis in Section V (for example, see Fig. 9) willshow that, for perfect CSI, cut-set bounds using our analyticalpower allocation are very close to those found through nu-merical optimization, even for low SNRs. In addition, it hasgood performance in imperfect-CSI situations. Note that usingsimilar power ratios for perfect- and imperfect-CSI conditionsdoes not mean that the rates are the same. It means that powerallocation is insensitive to SNR and channel estimation errorsand mainly depends on statistical properties of channels.

V. NUMERICAL RESULTS

In this section, we provide numerical results for the analysisderived in the paper. We have used the Monte Carlo method tocalculate average rates over different realizations of the chan-nels. In all figures, we plot the mean of A−B and B−A rates.Without loss of generality, we assume a normalized distancebetween the two sources, i.e., dab = 1. The path-loss exponentis ν = 3, and the total power PT for both schemes varies from0 dBW to 20 dBW, which is equivalent to 0–20 dB for the totalSNR due to the normalized noise power assumption. Similarsystem parameters have been used in [27], [38], [42], and thereferences therein. We define the following two main positionsfor the relay: 1) Pos-1, in which the relay is located at an equaldistance from both source nodes (dar = 0.5, drb = 0.5), and2) Pos-2, in which the relay is far from one source and closeto the other source (dar = 0.9, drb = 0.1).

In Fig. 3, we have compared the average achievable rate ofANC with the average capacity upper bound of TR in Pos-1 us-ing equal power allocation. That is, the total power is uniformlydistributed among all nodes, i.e., Pa = Pb = Pr = PT /3 forANC and Pa = Pb = P ab

r = P bar = PT /4 for TR. It can be

observed that the achievable rate of ANC is higher than the


Fig. 4. Power-optimized average rates of ANC and TR compared with theaverage rates using equal power allocation for imperfect-CSI conditions inPos-2 (dar = 0.9, drb = 0.1). With proper power optimization, ANC canmaintain its superiority against TR for a larger range of SNRs. For example,at the SNR of 6 dB, ANC cannot outperform TR using equal power allocation.However, ANC outperforms TR using optimum power allocation.

capacity upper bound of TR for the perfect-CSI case (this resulthas been reported in [4]). However, for imperfect CSI at lowSNRs, ANC has poorer performance compared with TR. Thiscase is due to the accumulation of many data-dependent noiseterms that are not present in the perfect-CSI case. Note that,because we study achievable rates for ANC and capacity upperbounds for TR, we cannot definitely decide which scheme hasbetter performance at low SNRs. However, this observation isdifferent from the perfect-CSI case and should be taken intoaccount in practice.7 Similar phenomena have been observedfor Pos-2 and other positions of the relay between two sourcenodes.

Now, we investigate the effect of power optimization on ANCand TR. Fig. 4 represents the average achievable rate of ANCand the capacity upper bound of TR in Pos-2 for imperfect-CSIconditions. We have compared equal and optimum power allo-cations, where optimum power allocation is numerically foundby global search. It can be observed that power optimizationhas significantly improved the rates of both systems. However,the improvement for ANC is more than for TR. As an example,at the SNR of 14 dB, the rate improvement for ANC is about0.46 b/s/Hz, and for TR, it is 0.21 b/s/Hz. It also affects theprevious conclusions at low SNRs. For example, at the SNR of6 dB, ANC cannot outperform TR using equal power allocation.However, ANC outperforms TR using optimum power alloca-tion. Note that this result is not seen in perfect-CSI conditions.In other words, in the perfect-CSI case, the rate improvementthat results from power optimization is almost the same forboth systems (as can be seen in Figs. 7 and 9). Therefore,

7Note that, in the derivation of the TR cut-set bound in (56), we assumedjointly Gaussian input distributions. Therefore, although cut-set bounds canbe loose in general, the choice of a specific input distribution makes ourobservation stronger.

Fig. 5. Average achievable rate of ANC against a normalized distance of therelay from node A (dar/dab). The relay has been moved from the centerto node B on the straight line between two nodes for SNR = 10 dB. Poweroptimization has a more significant impact when the relay is close to one ofthe source nodes. Analytical results are almost identical to optimal solutions inperfect-CSI conditions. They also have a good performance in the imperfect-CSI case.

power optimization is an important issue, particularly for ANCin imperfect-CSI conditions.

Furthermore, if the relay is closer to one of the sources,power optimization affects the system performance even more.Fig. 5 shows the average rate of ANC against dar at the SNRof 10 dB. We can see that the impact of power optimizationgrows as the relay is moved from the center to one of theend points. This result is in accordance with intuition, becausewhen the relay is at the center, Pa = Pb is optimal. This figurealso includes performance obtained with the analytical powerallocation for ANC. Although our analysis in Section III-B forANC was suboptimal, it matches the optimal solution for allpositions of the relay in the perfect-CSI case. Moreover, it per-forms close to the optimum in imperfect-CSI conditions. Thereason for this result can be explained as follows. Because thedifference between the perfect- and imperfect-CSI cases comesfrom channel estimation errors, we expect that, at high SNRs(where the channel estimation error decreases), the perfect-CSIassumption becomes a very good approximation in the system.As a result, our derived power allocation scheme for perfectCSI is somewhat a high-SNR approximation of optimal powerallocation for imperfect CSI conditions. Therefore, it is notsurprising that it has a very good performance for both casesat a high SNR.

Optimal power ratios for ANC have been plotted in Fig. 6 forSNR = 10 dB. Analytical γa and γr have been calculated from(27) and (29) and compared with optimal ratios found throughnumerical search. It can be observed that analytical power ratiosare close to optimal values.

Fig. 7 shows the average rate of ANC against SNR usingdifferent power allocations in Pos-2. Optimum power allocationhas been found by global numerical search of power ratios tosolve the optimization problem in (19), which cannot be usedin practice. Whereas the analytical power allocation employs


Fig. 6. Optimum power ratios for ANC (found through numerical search)compared with analytical power ratios against dar/dab for SNR = 10 dB.Analytical power ratios are close to optimal values.

Fig. 7. Average achievable rate of the ANC scheme against SNR usingdifferent power allocations in Pos-2 (dar = 0.9, drb = 0.1). For the perfect-CSI case, the analytical power allocation results are almost identical to theoptimal solution in all SNRs. Moreover, it performs close to the optimal inimperfect-CSI conditions. We can save about 2 dB of power compared withthe equal power scheme to achieve the average rate of 2 b/s/Hz. The impact ofusing optimal power ratios is more noticeable in the imperfect-CSI case.

(27) and (29) to calculate the power ratios. It can be observedthat analytical power allocation results are almost identicalto optimal solutions for the perfect-CSI case in all SNRs.In addition, they have a good performance in imperfect-CSIconditions. For example, analytical power allocation can save2-dB power compared with equal power distribution at the rateof 2 b/s/Hz.

All power ratios from different power allocations have beenplotted in Fig. 8. It is shown that analytical power ratios are veryclose to the optimal ratios found by numerical optimization ofthe average rate, particularly at a high SNR.

Fig. 9 shows the average capacity upper bound of TR as afunction of SNR using different power allocations in Pos-2.

Fig. 8. Optimum power ratios of the ANC scheme (found through nu-merical search) compared with analytical power ratios in Pos-2 (dar = 0.9,drb = 0.1)) for perfect CSI. The analytical power ratios are very close to theoptimal power ratios, particularly at high SNRs.

Fig. 9. Average capacity upper bound of TR against SNR using differentpower allocations in Pos-2 (dar = 0.9, drb = 0.1). The capacities that wereachieved with the analytical power allocation are very close to the optimalsolution. Using optimum power allocation results in noticeable power savingscompared with using equal power allocation.

Optimum power allocation has been found by global numericalsearch using power constraints and maximizing the averagesum rate for all channel realizations. This power optimization iscomplicated and time consuming and cannot be implemented inpractice. However, analytical power allocation simply assumesthat γab = 0.5 and uses (42) to calculate βab, and the resultingrates are very close to the optimum solution for both perfect-and imperfect-CSI conditions (the difference in rates based onoptimal and analytical solutions is less than 3%). In addition,optimal power allocation allows significant power savings com-pared with equal power allocation. For example, at the rate of2 b/s/Hz, analytical power allocation provides 1.13-dB powersavings.


VI. CONCLUSION

In this paper, we have derived achievable information ratesin ANC and cut-set upper bounds in TR when the CSI at thereceiving nodes is imperfect. Although there is inevitably somerate loss due to imperfect CSI in both schemes, our studiesindicated that the performance of ANC is more sensitive tosuch errors, because self-inference cancelation in ANC dependson the availability and accuracy of the CSI, and many data-dependent noise terms accumulate at the receiver when theCSI is not perfect. More specifically, we observed that theinformation rate of ANC fall below the TR upper bound in lowSNR and imperfect-CSI conditions (this result is in contrastto perfect-CSI observations in the literature). To remedy thiscondition and bring back the advantages of ANC over TR, westudied and proposed simple power allocation techniques thatare well suited for both perfect- and imperfect-CSI conditionsand high and low SNRs and rely only on channel statistics.Using the proposed power allocation methods, we showed thatANC can outperform TR for most ranges of SNRs, even inimperfect-CSI conditions. Our derivations provide a frameworkto study capacity bounds and power allocation methods forsimilar schemes. One interesting problem for future work is togeneralize our analysis to the case that loop-back interferencecancellation is not perfect and affects the performance.

APPENDIX APROOF OF THEOREM 1

Referring to (7), Yb[i] can be modeled as the output of afrequency-selective channel [20], with the channel transformfunction given as follows:

Hanc(f) =√

Pahab + αhrbhar

√Pae

−j2πfTs , Ts �1W

(43)

where Ts is the symbol period, and W is the bandwidth,which will be considered as unity for the calculation of therates per hertz. It can be observed in (8) that the spectrum ofZanc[i] depends on the spectrum of the channel inputs Xa andXb, which makes the optimization on the input spectrum verycomplicated. Therefore, we cannot use standard water-fillingapproaches that were used in [4] to arrive at the capacity lowerbound in ANC under the imperfect-CSI case. However, undersome mild conditions on the input (input symbols need to beuncorrelated) and because all Gaussian channels are estimatedusing the MMSE technique, we can obtain an achievable infor-mation rate. We first prove the following lemma.

Lemma 1: Assume that MMSE estimates of Gaussianchannels hkj , (k, j) ∈ {a, b, r}2, k �= j are given. The ANCnoise terms (Zanc[i]) and the data terms (the first two terms)in Yb[i] in (7) are uncorrelated, and the variance of Zanc[i] isgiven by

Nanc =σ2hab

Pa+α2Pa

(|hrb|2σ2

har+|har|2σ2

hrb+σ2

harσ2hrb

)+ α2Pb

(|hrb|2σ2

hbr+ |hbr|2σ2

hrb+ σ2

hbrσ2hrb

)+ α2σ2

hrb+ α2|hrb|2 + 1. (44)

Proof: First, we note that, for Gaussian channels, MMSE isidentical to LMMSE, and through (2), there is an equivalencebetween hkj and the received signal Yj during pilot transmis-sion. To verify that the signal and noise terms in Yb[i] areuncorrelated, we write

E{(√


√PaXa[i− 1]

)Z∗anc[i]

∣∣∣h}= E

{(√PahabXa[i] + αhrbhar

√PaXa[i− 1]

)×(√

Pah∗abX

∗a[i] + αh∗

rbh∗ar

√PaX

∗a[i− 1]

+ αh∗rbh

∗ar

√PaX

∗a[i− 1]+αh∗

rbh∗ar

√PaX

∗a[i−1]

+ αh∗rbh

∗br

√PbX

∗b [i− 1]+αh∗

rbh∗br

√PbX

∗b [i−1]

+ αh∗rbh

∗br

√PbX

∗b [i− 1] + αh∗

rbZ∗r [i− 1]

+ αh∗rbZ

∗r [i− 1] + Z∗

b [i]) ∣∣∣h} (45)

where the expectation is understood as expectation over channelestimation errors, input symbols, and noises conditioned ongiven channel estimates or, equivalently, on given received pilotsignals. Using the property of MMSE estimation, we knowthat E{h∗

kj |hkj} = 0 for (k, j) ∈ {a, b, r}2, k �= j. In addition,the mean of AWGN terms is zero, i.e., E{Zr} = E{Zb} = 0.Hence, all terms in the aforementioned expectation will bezero. Using this result, it is easy to verify that noise varianceE{Zanc[i]Z

∗anc[i]} is given in (44). �

Now that the uncorrelatedness of noise and signal terms isestablished, we can use the results in [18] and [19] to replaceZanc[i] by a worst-case Gaussian noise with the same noisevariance as Zanc[i], which is given in (44). In addition, due toassuming uncorrelated input symbols (E{Xa[i]X

∗a[i− 1]} =

0), the input spectrum is flat in its bandwidth. Therefore, we canassert that an achievable rate for ANC is obtained as follows:

Ranc =

W/2∫−W/2

log

(1 +

1W |Hanc(f)|2

Nanc

)df (46)

=

1/2∫−1/2

log (A1 +A2 cos(2πfTs)) df (47)

where

A1 � 1 +Pa

Nanc

(|hab|2 + α2|hrb|2|har|2

)

A2 �α2Pa

Nanc|hab||hrb||har|. (48)

Based on [51, eq. (4.292.3)] and because 0 ≤ |A2/A1| ≤ 1 andTsW = 1, (46) can be found in closed form, we have

Ranc = log

⎛⎜⎜⎜⎝A1

⎛⎜⎜⎝

1 +

√1 −(

A2

A1

)22

⎞⎟⎟⎠

2⎞⎟⎟⎟⎠ . (49)


We remark the differences between our derivations and those in[30]. In [30], the concatenate non-Gaussian channel harhrb isjointly estimated (denoted by harhrb) using LMMSE, becausethe optimum MMSE for non-Gaussian channels is not math-ematically tractable. However, the lower bounding technique,which is based on [18], requires MMSE estimation for allchannels involved (see [26] for more details). As a result,some approximation is inherent in the derivations in [30]. Here,we have avoided this approximation and provided a provablelower bound by requiring that individual channels are estimatedusing MMSE (which is tractable, because each channel isGaussian) and communicated from the relay to nodes A andB, if needed.8 Moreover, the approach and the final rates in[30] are different from (49) due to the full-duplex assumptionin this paper, which makes the problem and its derivation rathercomplicated.

APPENDIX BDERIVATION AND VERIFICATION OF THE OBJECTIVE

FUNCTION IN PROPOSITION 1

Due to the complexity of the derived rates for ANC, weaim at finding a simple novel function that has a similarbehavior to the sum of average rates for power optimization.We build our function based on the average received SNR.Note that we do not use this function to approximate therate; we only use it for power allocation. We also highlightthat the average SNR is an important determining factor inthe performance of many wireless communication systems.For many well-known cases, this dependence is explicit. Forexample, under perfect-CSI conditions, the average BER andoutage probability in Rayleigh fading channels are explicitfunctions of the average SNR. Other performance metrics suchas average capacity and achievable rate are increasing functions(albeit sometimes unknown or complicated) of the averageSNR. Hence, maximizing the average SNR is justified in manysituations of interest. Similar SNR-based techniques have beenused in [52] and [39] for one- and two-way AF relayingschemes, respectively, and in [18] and [19] for multiple-antennasystems.

As explained in Appendix A, the received signal at nodeB can be modeled as the output of a frequency-selectivechannel with Hanc,p =

√Pahab + αphrbhar

√Pae

−j2πfTs

channel transform and Nanc,p noise variance [given in(17)] for perfect-CSI conditions. Because the inputs areassumed to be uncorrelated, the input spectrum is flat in itsbandwidth. Therefore, similar to (46), the received SNR can bewritten as

SNR(f) =1W |Hanc,p(f)|2

Nanc,p

=A1p − 1 +A2p cos(2πfTs). (50)

8In practical systems where the intention is not the analysis of informationrates, we can adopt any channel estimation technique that is simple and efficientand requires minimum amount of overhead.

Fig. 10. Sum of average achievable rates of ANC and the simplified function(g) in (53) used for power optimization against power ratios, dar = 0.7, drb =0.3, and SNR = 10 dB. They both achieve their maximum at around the samevalues of γa and γr .

First, we calculate the average SNR over the frequencyrange, and as a result, the second term disappears, i.e.,

SNR =

W/2∫−W/2

SNR(f)df = A1p − 1. (51)

In addition, we replace the square of magnitude of all thechannel gains in A1p in (16) with their mean and define anew simple function based on the average SNR for poweroptimization purposes, which is given as

gab � log(SNR)||hkj |2←σ2hkj

= log

⎛⎜⎜⎝ Pa(

1 + Pr

1+Paσ2har

+Pbσ2hrb

σ2hrb

)

×(σ2hab

+Pr

1 + Paσ2har

+ Pbσ2hrb

σ2hrb

σ2har

)⎞⎟⎟⎠ .

(52)

For B−A transmission, we can define a similar function calledgba. Considering the sum of gab and gba results in the functiong given in (53), shown at the bottom of the next page, for poweroptimization.

We have investigated the behavior of g as a function ofthe power ratios and compared it with that of the sum ratein Fig. 10 for dab = 1, dar = 0.7, and drb = 0.3. We remarkthat the value of function g does not need to be close to theinformation sum rate. It is only important that the power ratiosthat maximize g and those that maximize the sum rate are close.This condition is confirmed by visual and numerical evaluation.For example, the maximum value of the sum of average ratesfor perfect CSI is obtained for γr = 0.45, γa = 0.731, and


the maximum value of our simplified function is obtainedfor γr = 0.44, γa = 0.741.9 Therefore, we can approximatethe optimum power allocation by the one resulting from themaximization of g in (53). Note that we have observed identicalpatterns for different relay positions and various SNRs, whichis omitted here for brevity. The confirmation of the efficacyof the proposed objective function is verified in Figs. 6 and8, in which analytical power ratios that were derived from themaximization of g are close to optimal ratios that were obtainedby numerically optimizing (19).

APPENDIX CPROOF OF THEOREM 2

We start with a general joint probability distribution func-tion for the channel inputs p(Xa, Xr) to write the cut-setbound as

CTR(h) = maxp(Xa,Xr)

min



}(54)

with hΔ= {hab, har, hrb}. The 1/2 factor comes from the time-

division duplexing in TR. Because of the additional noise termsin (31), the maximization over p(Xa, Xr) is not tractable ingeneral. Therefore, we assume jointly Gaussian inputs with thefollowing correlation10:

ρ = E {XaX∗r} . (55)

Therefore, the cut-set capacity upper bound should be maxi-mized over ρ as

CTR(h) = maxρ

min



}. (56)

9The maximizing argument of the sum rate for the imperfect CSI case islocated at γr = 0.47, γa = 0.79, which is, again, very close to the simplifiedfunction.

10Similar assumptions have been used in the literature [30], [42], [53],[54] for situations where obtaining a general cut-set bound was not possible.The Gaussian distribution may not be optimal for maximizing the mutualinformation; however, it provides a lower bound for it. In addition, becausewe assume block-fading channels, the capacity bound should be averagedover all realizations of the channels, and if the number of channel samplesis large enough and the channel input is coded, then the input distribution isapproximately Gaussian [54].

The first term in (56) is

I(Xa, Xr;Yb|Xb, h) = h(Yb|Xb, h)− h(Yb|Xb, Xa, Xr, h)(57)

where h(·) is the differential entropy function. Based on (31),for a given Xa, Xr, and h, Yb is a Gaussian random variable.Therefore, its entropy is given by

h(Yb|Xb, Xa, Xr, h)

= EXa,Xr

{log(

2πe(

Var(Yb|Xb, Xa, Xr, h)))}

= EXa,Xr

{log(

2πe(Paσ

2hab

|Xa|2+Prσ2hrb

|Xr|2+1))}(58)

where Var(X) denotes the variance of random variable X . Theentropy h(Yb|Xb, h) is maximized if Yb, given h, is Gaussian[50], i.e.,

h(Yb|Xb, h)

≤ log(

2πe(

Var(Yb|Xb, h)))

= log(

2πe(Pa|hab|2+ Pr|hrb|2+ Paσ

2hab+ Prσ

2hrb

+ 2Re{ρhabh

∗rb

}√PaPr + 1

)). (59)

Because we assume slow fading, the receivers can sendthe estimated channels to the transmitters to be used duringdata transmission. Therefore, node A and the relay know therealizations for hab and hrb and can use it to synchronize theiroutputs by appropriately adjusting the phase of ρ. Therefore,only the absolute value of ρ (which ranges from 0 to 1) appearsin the entropy expression, i.e.,

h(Yb|Xb, h) ≤ log(

2πe(Pa|hab|2+ Pr|hrb|2

+Paσ2hab+ Prσ

2hrb

+ 2|ρ||hab||hrb|√

PaPr + 1))

. (60)

Based on (58) and (60), we have (61), shown at the bottom ofthe next page.

For the second term in (56), we can write


= h(Yr, Yb|Xr, Xb, h)− h(Yr, Yb|Xb, Xa, Xr, h)

= h(Yb|Xr, Xb, h) + h(Yr|Yb, Xr, Xb, h)

− h(Yr, Yb|Xb, Xa, Xr, h). (62)

g(Pa, Pb, Pr) = log

⎛⎜⎜⎜⎝

PaPb

(σ2hab

+ Pr

1+Paσ2har

+Pbσ2hrb

σ2hrb

σ2har

)2

(1 + Pr

1+Paσ2har

+Pbσ2hrb

σ2hrb

)(1 + Pr

1+Paσ2har

+Pbσ2hrb

σ2har

)⎞⎟⎟⎟⎠ (53)


Based on (30) and (31), for a given Xa, Xr, and h, Yr and Yb

are independent Gaussian random variables. Hence

h(Yr, Yb|Xb, Xa, Xr, h)

= EXa,Xr

{log(

2πe(

Var(Yr, Yb|Xa, Xr, Xb, h)))}

= EXa,Xr

{log(

2πe(Paσ

2har

|Xa|2 + 1))

+ log(

2πe(Paσ

2hab

|Xa|2

+Prσ2hrb

|Xr|2 + 1))}

.

The upper bounds for h(Yb|Xr, Xb, h) and h(Yr|Yb, Xr,

Xb, h) are

h(Yb|Xr, Xb, h) ≤ log(

2πeVar(Yb|Xr, Xb, h))

(63)

h(Yr|Yb, Xr, Xb, h) ≤ log(

2πeVar(Yr|Yb, Xr, Xb, h)).

(64)

Var(Yb|Xr, Xb, h) can be calculated using the LMMSE esti-mate of Yb, given (Xr, Xb), i.e.,

Yb = αbXr αb =√

Pahabρ+√

Prhrb (65)

Var(Yb|Xr, Xb, h) ≤ Var(Yb − Yb) = E{(Yb − αbXr)

2}

= Pa|hab|2(1 − |ρ|2

)+ Paσ

2hab

+ Prσ2hrb

+ 1. (66)

Similarly, Var(Yr|Yb, Xr, Xb, h) can be calculated using theLMMSE estimate of Yr, given (Xr, Xb, Yb), in (67)–(70),shown at the bottom of the page. Inserting (66) and (70) in (63)and (64) and using the results in (62) yields (71), shown at thebottom of the page.

Based on (61) and (71), we can calculate a capacity upperbound on CTR(h) for the current realization of the chan-nel estimates. Note that the expectation over Xa and Xr

should numerically be calculated (assuming that they are jointlyGaussian with correlation 0 ≤ ρ ≤ 1) for each realization ofchannel estimates using the Monte Carlo method. Then, we cal-culate the average bound CTR = Eh{CTR(h)} over differentrealizations of channel estimates.

APPENDIX DPROOF OF PROPOSITION 2

First, we find the optimal ρ that maximizes the capacity upperbound and then solve (38) to find the optimum power allocation.

The first term of the minimization in (36) is an increasingfunction of |ρ|, whereas the second term is a decreasing func-tion. Therefore, the optimal |ρ| is zero if βab > |har|2/|hrb|2.For βab ≤ |har|2/|hrb|2, the optimal |ρ| can be found by equat-ing the two terms as

|ρ|opt=−|hab||hrb|

√βab+|har|

√|hab|2−βab|hrb|2+|har|2

|hab|2 + |har|2.

(72)

I(Xa, Xr;Yb|Xb, h) ≤ EXa,Xr

{log

(Pa|hab|2 + Pr|hrb|2 + 2|ρ||hab||hrb|

√PaPr + Paσ

2hab

+ Prσ2hrb

+ 1

Paσ2hab

|Xa|2 + Prσ2hrb

|Xr|2 + 1

)}(61)

Yr =αr1Xr + αr2Yb (67)

αr1 =

√Paharρ

(Paσ

2hab

+ Prσ2hrb

+ 1)− Pa

√Prh

∗abharhrb

(1 − |ρ|2

)Pa|hab|2 (1 − |ρ|2) + Paσ2

hab+ Prσ2

hrb+ 1

(68)

αr2 =Pah

∗abhar

(1 − |ρ|2

)Pa|hab|2 (1 − |ρ|2) + Paσ2

hab+ Prσ2

hrb+ 1

(69)

Var(Yr|Yb, Xr, Xb, h) ≤Pa|har|2

(1 − |ρ|2

) (Paσ

2hab

+ Prσ2hrb

+ 1)

Pa|hab|2 (1 − |ρ|2) + Paσ2hab

+ Prσ2hrb

+ 1+ Paσ

2har

+ 1 (70)


≤EXa,Xr

⎧⎨⎩log

⎛⎝(Paσ

2har

+1)(

Pa|hab|2(1−|ρ|2)+Paσ2hab

+Prσ2hrb

+1)+Pa|har|2

(1−|ρ|2

) (Paσ

2hab

+Prσ2hrb

+1)

(Paσ2

har|Xa|2+1

)(Paσ2

hab|Xa|2+Prσ2

hrb|Xr|2+1

)⎞⎠⎫⎬⎭

(71)


Cp,abTR (h) ≤

⎧⎪⎪⎨⎪⎪⎩

12 log

(1 + Pa

(|har|2 + |hab|2

)), for βab >

|har |2|hrb|2

12 log

(1 +

Pa

(|har ||hrb|

√βab+|hab|

√|hab|2−βab|hrb|2+|har |2

)2|hab|2+|har |2

), for βab ≤ |har |2

|hrb|2(73)

fab(βab) �

⎧⎨⎩

1βab+1

(|har|2 + |hab|2

), for βab >

|har |2|hrb|2

1βab+1

(|har ||hrb|

√βab+|hab|

√|hab|2−βab|hrb|2+|har |2

)2|hab|2+|har |2 , for βab ≤ |har |2

|hrb|2(75)

Therefore, the capacity upper bound is as given in (73),shown at top of the page.

Assuming high-SNR conditions, we can approximatelog(1 + x) with log(x). It simplifies the problem and leadsto an optimal power distribution independent of the SNR,which is more practical. Using this approximation, (73), andthe definitions of power ratios in (39) and (40), our optimizationproblem in (38) is equivalent to

maxγab,βab,βba

γab(1 − γab)fab(βab)fba(βba) (74)

where fab is defined in (75), shown at top of the page, and fba issimilarly defined by changing the a indices in fab to b, and viceversa. Based on (74), it is obvious that the optimum γab equals0.5. It means that the two-way relaying power optimizationproblem is equivalent to two independent one-way relayingpower allocation problems. In other words, knowing γopt

ab , wecan independently maximize fab and fba.

The maximum value of the first term in (75) is obtained forthe minimum value of βab, which is the threshold |har|2/|hrb|2.Taking the derivative of the second term w.r.t. βab and setting itto zero results in the following optimal value:

βoptab =

|har|2|hrb|2

(|hrb|2 + |hab|2)2 + |har|2|hab|2(76)

which is always smaller than |har|2/|hrb|2. It can easily beverified that βopt

ab is the maximizing argument of fab. Themaximization procedure for fba is identical, and βopt

ba can befound by suitable index changes.

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Foroogh S. Tabataba (S’09) received the B.S. de-gree (with honors) in electrical engineering fromIsfahan University of Technology, Isfahan, Iran, in2004 and the M.S. and Ph.D. degrees in electricalengineering from Sharif University of Technology,Tehran, Iran, in 2006 and 2011, respectively.

From March 2004 to December 2011, she wasa Research Assistant with the Wireless ResearchLaboratory and was also with the Data NetworkResearch Laboratory, Advanced Communication Re-search Institute, Department of Electrical Engineer-

ing, Sharif University. In 2008, she received a visiting fellowship from theAustralian National University, Canberra, ACT, Australia, where she workedwith Dr. Sadeghi’s group for six months. In 2010, she was a Visiting Fellowwith the Signal Processing and Wireless Communication Laboratory, Uni-versity of Western Australia, Perth, WA, Australia, for eight months. She iscurrently an Assistant Professor with the Department of Electrical and Com-puter Engineering, Isfahan University of Technology. Her research interestsinclude communication systems theory, wireless communications, cooperativenetworks, information theory applications, and optical networks.

Parastoo Sadeghi (S’02–M’06–SM’07) receivedthe B.E. and M.E. degrees in electrical engineeringfrom Sharif University of Technology, Tehran, Iran,in 1995 and 1997, respectively, and the Ph.D. degreein electrical engineering from the University of NewSouth Wales, Sydney, NSW, Australia, in 2006.

From 1997 to 2002, she was a Research Engi-neer and then a Senior Research Engineer with IranCommunication Industries, Tehran, and with Deqx(formerly as Clarity Eq), Sydney, NSW, Australia.She is currently a Fellow (Senior Lecturer) with the

Research School of Engineering, Australian National University, Canberra,ACT, Australia. She has visited various research institutes, including theInstitute for Communications Engineering, Technical University of Munich,Munich, Germany, from April to June 2008 and the Massachusetts Instituteof Technology, Cambridge, from February to May 2009. She is a coauthorof more than 70 refereed journal papers or conference proceedings and is aChief Investigator with a number of Australian Research Council Discoveryand Linkage Projects. Her research interests include wireless communicationssystems and signal processing.

Dr. Sadeghi received IEEE Region 10 Student Paper Awards in 2003 and2005 for her research in the information theory of time-varying fading channels.


Charlotte Hucher (S’06–M’10) received the B.Eng.degree in digital communications from the ÉcoleNationale Supérieure des Télécommunications(ENST), Paris, France, in 2005, the M.S. degree indigital communications from the Université de Nice,Sophia Antipolis, France, and the Ph.D. degree incommunications and electronics from ENST, in2009, for her work on cooperative networks.

From 2009 to 2011, she was a Research Fellowwith the Research School of Engineering, AustralianNational University, Canberra, ACT, Australia,

where she pursued her research on cooperative communications. She is cur-rently a DSP Engineer with Sequans Communications, Paris, France. Herresearch interests include cooperative networks, physical network coding,multiuser communications, and interference channels.

Mohammad Reza Pakravan (M’89) received theB.S. degree (with honors) in electrical engineeringfrom the University of Tehran (UT), Tehran, Iran, in1990 and the M.S. and Ph.D. degrees in electricalengineering from the University of Ottawa, Ottawa,ON, Canada, in 1992 and 2000, respectively.

From 1997 to 2001, he was a Member of Tech-nical Staff with Nortel Networks and helped withthe development of a broadband access systems andits associated algorithms, software, and hardware.Based on his activities, several patents were obtained

by Nortel Networks in the U.S., Canada, and Europe. In 2001, he joinedthe Department of Electrical Engineering, Sharif University of Technology,Tehran, where he is currently an Associate Professor. He is the Director ofthe Data Networks Research Laboratory and the Networking Group, AdvancedCommunications Research Institute. His research interests include optical com-munication systems and networks, data networking algorithms and protocols,and cognitive and cooperative communication systems.

Dr. Pakravan received several awards and scholarships from the University ofOttawa and the Canadian Government for his performance and achievements.He received the IEEE Neal Shepherd Memorial Best Propagation Paper Awardfrom the IEEE Vehicular Technology Society in 2001.

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