Energy-Recycling Full-Duplex Radios for Next-Generation Networks
Marco Maso, Chen-Feng Liu, Chia-Han Lee, Tony Q. S. Quek, Leonardo S. Cardoso
aa r X i v : . [ c s . I T ] M a y Energy-Recycling Full-Duplex Radios forNext-Generation Networks
Marco Maso,
Member, IEEE,
Chen-Feng Liu, Chia-Han Lee,
Member, IEEE,
Tony Q. S. Quek,
Senior Member, IEEE, and Leonardo S. Cardoso,
Member, IEEE
Abstract —In this work, a novel energy-recycling single-antennafull-duplex (FD) radio is designed, in which a new 3-port elementincluding a power divider and an energy harvester is addedbetween the circulator and the receiver (RX) chain. The presenceof this new element brings advantages over the state of the artin terms of both spectral efficiency and energy consumption.In particular, it provides the means of performing both anarbitrary attenuation of the incoming signal, which in turnincreases the effectiveness of the state-of-the-art self-interferencecancellation strategies subsequently adopted in the RX chain,and the recycling of a non-negligible portion of the energyleaked through the non-ideal circulator. The performance ofthis architecture is analyzed in a practically relevant 4-nodescenario in which 2 nodes operate in FD and 2 nodes in half-duplex (HD). Analytical approximations are derived for boththe achievable rates of the transmissions performed by the FDand HD radios and the energy recycled by the FD radios.The accuracy of these derivations is confirmed by numericalsimulations. Quantitatively, achievable rate gains up to overthe state-of-the-art alternatives, in the considered scenario, arehighlighted. Furthermore, up to of the leaked energy at thecirculator, i.e., of the energy of the transmitted signal, canbe recycled. Index Terms —Full-duplex (FD) radios, self-interference can-cellation, energy harvesting, wireless backhaul, device-to-device(D2D) communications.
I. I
NTRODUCTION S INCE the introduction of personal mobile services, wire-less traffic demands have increased continuously. Thisincrease has picked up pace in the last decade due to thepopularity of mobile applications and the market penetrationof mobile devices with networking capabilities. In a momentwhen wireless research has continuously closed the gap be-tween the achievable and theoretical capacity bounds, innova-tive solutions are needed to further enhance the performance of
This research was supported in part by the A*STAR SERC under Grant1224104048, in part by the SUTD-ZJU Research Collaboration under GrantSUTD-ZJU/RES/01/2014, in part by the MOE ARF Tier 2 under GrantMOE2014-T2-2-002, and in part by the Ministry of Science and Technology(MOST), Taiwan under Grant MOST104-2221-E-001-013-MY2.M. Maso is with the Mathematical and Algorithmic Sciences Lab,Huawei France Research Center, Boulogne-Billancourt 92100, France (e-mail:[email protected]).C.-F. Liu and C.-H. Lee are with the Research Center for Informa-tion Technology Innovation, Academia Sinica, Taipei 115, Taiwan (e-mail:cfl[email protected]; [email protected]).T. Q. S. Quek is with the Singapore University of Technology and Design,Singapore 487372, and also with the Institute for Infocomm Research,A*STAR, Singapore 138632 (e-mail: [email protected]).L. S. Cardoso is with the Université de Lyon, INRIA, INSA-Lyon,CITI-INRIA, F-69621, Villeurbanne, France (e-mail: [email protected]). current networks. In this context, one very appealing approachthat has regained traction lately is the in-band full-duplex(FD) [1]. Devices adopting this approach can simultaneouslytransmit and receive signals in the same frequency band.FD radios can theoretically achieve a two-fold throughputimprovement without extra antennas or band over their half-duplex (HD) counterparts. Among the potential solutions toimplement FD radios for future networks, the single-antennaarchitectures have recently gained momentum [2]. This hap-pens for several reasons. These systems take significantlyless physical space on the devices than their multi-antennacounterparts, being less demanding in terms of the form factor.This is of utmost importance when the possible size of the FDdevice is limited either by physical constraints, e.g., a sensor,or by practical and commercial purposes, e.g., a portabledevice. Remarkably, these architectures have been shown tobe able to achieve the claimed theoretical performance bothin terms of academic prototypes and commercially availableproducts [3]. The further element in favor of single-antennaFD architectures is that they share the same radio-frequency(RF) circuitry basis with HD systems. Therefore, very littlehardware changes are needed to implement a single-antennaFD radio as compared to a HD one.The potential performance enhancement brought by the FDapproach depends on how much self-interference (SI) canbe subtracted from the received signal [4]. This aspect isespecially relevant in the context of single-antenna FD radios,where the self-interference cancellation (SIC) capability of thedevice is significantly hindered by practical hardware limita-tions, e.g., the non-ideality of the antenna circulator [5], [6].From a qualitative point of view, the SI in single-antenna FDradios has two main components: 1) signal leakage betweenthe transmit (TX) and receive (RX) ports of the circulator,and 2) signal received by the antenna after a propagation in amulti-path environment. In this regard, it is worth noting thatthe signal leakage from the circulator not only generates SIbut also reduces the energy efficiency of the FD radio sincepart of the power invested by the latter for the transmission iswasted due to the leakage.
A. Related Works
The design of SIC algorithms has been widely addressed be-fore. Notable examples in the literature are the solutions basedon either analog [5]–[9] or digital signal processing [10]–[13] SIC, and the so-called spatial SIC [14]–[16]. In general,performing part of the SIC in the analog domain reduces the problems of the saturation of the RX amplifiers and the lowdynamic range at the analog-to-digital converter (ADC) [5].Several approaches have been proposed in this context. Asolution based on a balanced feed network in which thetransmitted signal is fed to the TX antenna via two paths suchthat their corresponding SI signals are 180 degrees out of phaseis proposed in [6]. Alternative strategies are proposed in [8]and [9], where the adoption of a RF echo canceler and a phaseshifter to tune a reference signal in order to match it with the SIsignal are proposed, respectively. The best SIC capabilities arecertainly offered by hybrid solutions adopting both analog anddigital signal processing [5], [7]. A quantitatively remarkableresult in this sense is achieved in [5], where a novel single-antenna FD radio architecture able to achieve 110 dB ofSIC for a transmission of an orthogonal frequency-divisionmultiplexing (OFDM) signal over a bandwidth of 80 MHz isproposed. This solution, essentially based on a hybrid analogand digital cancellation algorithm, provides a very effectiveway of implementing a FD transceiver without significantlyincreasing both its size and cost. The market potential of suchsingle-antenna FD radio is certainly non-negligible. Motivatedby these achievements, prototypes and market-ready productshave already been showcased and proposed to demonstratethe feasibility of single-antenna FD transmissions in real-lifescenarioss [3], [17], confirming that real-time FD radios caneffectively operate in different environmental conditions.
B. Our Contribution
The effectiveness of state-of-the-art SIC algorithms aresubject to constraints on the TX power of the FD radio, andhence, the residual SI [5], [6]. As a matter of fact, the intensityof the SI can be brought down to the noise floor only if theTX power is below a certain threshold, which is determinedby the nature of the SIC algorithm. Conversely, residual SIappears in the RX chain and the signal-to-interference-plus-noise ratio (SINR) of the incoming received signal decreases.This can significantly degrade the throughput of the FD radio.In this work, we approach this problem, considering that therange of TX powers of the FD radio may span values abovethe threshold for complete SIC. To deal with this issue, anovel FD radio architecture is designed and proposed in thiswork. The core novelty of this architecture, as compared tothe state-of-the-art alternatives, is the introduction of a one-way 3-port composite element, composed of a variable gainpower divider and an RF energy harvester (EH), between thecirculator and the RX chain. In practice, the 3-port elementsplits the signal coming from the circulator into two parts,i.e., an information component (IC) to be decoded by theRX chain and an energy component (EC) to be harvested.This has a two-fold advantage on the performance of the FDdevice. Firstly, the intensity of the SI observed in the RXchain can be arbitrarily reduced, regardless of the TX powerof the FD radio, in turn both restoring the effectiveness ofthe state-of-the-art SIC algorithms and increasing the spectralefficiency of the incoming link. Secondly, the energy efficiencyof the FD radio can be increased by recycling a portion of theenergy leaked at the circulator, otherwise wasted. However, this performance gain may come at a price. In fact, on onehand the combined effect of the 3-port element and SICprovides an effective SINR enhancement for the useful signalcarried by the IC. On the other hand, the power of the lattermay significantly decrease as compared to the noise floorduring these operation, inducing an intrinsic signal-to-noiseratio (SNR) reduction. In practice, a trade-off, a function ofthe splitting ratio adopted by the power divider, exists. Asa consequence, a careful performance analysis needs to beperformed to optimize the relevant parameters of the proposedarchitecture and assess its merit as compared to the state ofthe art.A practically relevant case study is identified to achievethis goal. Accordingly, we consider a hybrid FD/HD four-node setting in which two FD nodes play the role of servernodes (SN) while the remaining two play the role of attachednodes (AN) operating in HD. In this context, we dividethe communication between the SNs and the ANs into twodirections: 1) forward (or “downlink”), and 2) backward (or“uplink”). Additionally, we assume that the two SNs exchangeinformation, regardless of the direction of the communication.The rationale behind this setting is that it perfectly captures themain elements of what is currently considered the best candi-date for future FD-enabled networks, i.e., a system in whichFD and HD devices operate side by side [18]. Interestingly,this rather general setting can model several different real-life scenarios. Possible relevant examples are: (i) a scenario inwhich the proposed architecture is used at two mobile devicesengaging in a device-to-device (D2D) communication, whilebeing served by two base stations (BSs), or (ii) a scenario inwhich the proposed architecture is used at two BSs exchangingsignaling via a wireless backhaul while serving two mobiledevices. In this regard, we would like to note that both D2Dcommunications and wireless backhauling are indeed two ofthe envisioned applications for the FD technology both inacademic [4], [19] and industrial [3] contexts. Furthermore,we assume that the SNs adopt orthogonal frequency-divisionmultiple access (OFDMA) to serve their ANs in the sametime slot and frequency band. This choice has a two-foldmotivation. Firstly, this waveform, together with its manyvariants (e.g., filtered OFDM [20]), is one of the most likelycandidates for the air interface design of future 5G networksfor its potential in terms of high data rates, mobility man-agement, transmission diversity exploitation potential, spectralefficiency, robustness against inter-symbol interference, andsimple receiver architecture requirements [21]. Secondly, manystate-of-the-art works on FD radios are actually based onOFDM, e.g., [5], due to the aforementioned reasons. As aconsequence, considering the same waveform as these worksgives us the possibility to compare our results against the stateof the art in a very direct and fair way. Finally, in order toframe a scenario in which the potential of the hybrid FD/HDis fully exploited, we impose that the FD communicationsbetween the two SNs do not affect the HD communicationsbetween SNs and ANs. In other words, we explicitly accountfor an intrinsic inter-node interference reduction at the PHYlayer, similarly to what is advocated in contributions suchas [22], [23]. In practice, we will impose that the signaling between the two FD SNs is exchanged an overlay techniquespecifically tailored for OFDM-based communications, the so-called cognitive interference alignment (CIA) [24].Subsequent to the definition of the reference setting, weanalytically derive the exact/approximated achievable rate ofeach link in the considered system. Additionally, we specif-ically analyze the aforementioned trade-off between SINRenhancement and SNR decrease by studying how the splittingratio of the power divider impacts the achievable rates. In thiscontext, we compute closed-form expressions for the sets ofvalues of the splitting ratio for which the proposed architectureoutperforms the state-of-the-art alternatives. In particular, weprovide the optimal value of such parameter as a functionof the TX power of the device. Finally, we validate ourderivations by means of numerical simulations and show thatfor the considered reference setting: 1) achievable rate gains ofup to over the state-of-the-art alternatives can be obtained,and 2) up to of the leaked energy at the circulator, i.e., of the energy of the transmitted signal, can be recycled.The remainder of the paper is organized as follows. Thenovel FD radio architecture is introduced in Sec. II. Thesystem setup is described in Sec. III. The forward and back-ward transmission modes are analyzed in Sec. IV and Sec. V,respectively. Finally, numerical results are presented in Sec. VIand conclusion in Sec. VII.Throughout this work, the mathematical notation adoptedis as follows. We denote matrices as boldface upper-caseletters, vectors as boldface lower-case letters, and we let ( · ) H be the conjugate transpose of a vector/matrix. All vectorsare columns, unless otherwise stated. In particular, I N is the N × N identity matrix, and L × N is the L × N zero matrix.We denote [ H ] mn and [ h ] n as the element in the m th row andthe n th column of a matrix and the n th element of a vector,respectively. Finally, diag ( h ) is a diagonal matrix constructedfrom the vector h .II. E NERGY - RECYCLING
FD A
RCHITECTURE
Let us now focus on the architecture of the FD SN. Westart by recalling that state-of-the-art SIC algorithms for FDradios are limited in the amount of SI they can cancel. Inpractice, the full effectiveness of the adopted SIC algorithms,which aim at reducing the intensity of the SI at least at thesame level of the noise floor, is guaranteed only if the TXpower is below a certain threshold [5]. We denote this upperbound as P th . Now, let us assume that the FD device isable to induce an arbitrary attenuation of the SI such thatthe power of the latter is lower than α c P th , with α c definedas the ratio between the power of the signal leakage and theTX power. In this case, the SIC technique would provide itsbest cancellation performance. As a consequence, significantSINR gains w.r.t. state-of-the-art solutions could be achieved,thanks to the joint effect of the arbitrary attenuation and theSIC. The design of the novel FD architecture proposed inthis work starts from these considerations. Consider the FDradio architecture depicted in Fig. 1. As seen in Fig. 1, a 3-port composite element is introduced in the new architecturebetween the port B of the circulator and the RX chain, and TX chain RX chainAnalog SIDigital SIDigital reference signalAnalog reference signal Decoded signalEH Battery √ ρ √ − ρ Circulator 3-port elementPower divider cancellationcancellation
C BA
Figure 1. Novel FD architecture. hybrid SIC approach as in [5] is assumed. Interestingly, thiselement can be built using off-the-shelf components easilyavailable on the market. In particular, it is composed of: • Power Divider:
The input signal to the 3-port elementis first fed to an adjustable gain power divider andsplit into two with the √ ρ and √ − ρ power ratios.We refer to ρ as the power splitting ratio. This com-ponent is commonly considered at the core of manywireless power transfer applications, as the enabler ofthe so-called power splitting approach [25]. In particular,it can be implemented by means of Wilkinson powerdividers and balanced-unbalanced (balun) transformers,commonly developed and studied components in the RFcircuitry domain. In this context, variable gain variantsof such power dividers/balun have been the subject ofintense research in last years. Relevant examples of theoutcome of these efforts are the variable gain Wilkinsonpower divider proposed and implemented in microstripfor microwave applications in [26] or, more recently, thevariable power division balun proposed and implementedin microstrip in [27]. In the proposed architecture, theoutputs of the divider are two attenuated replicas of theinput signal, i.e., the IC and the EC. After the powerdivision, the IC is fed to the RX chain connected to thesecond port of the 3-port element, whereas the EC isfed to the second component of the 3-port element, i.e.,the EH circuitry. In this regard, we note that no powerdivision occurs whenever ρ = 1 , i.e., the entirety of theinput signal to the power divider is fed to the RX chain ofthe FD radio. This corresponds to a situation in which theproposed FD radio operates exactly as the state-of-the-artsolution [5], as discussed in the following. • Energy Harvester:
The EH circuitry converts a timevarying signal, i.e., the EC, into a direct current (DC)signal suitable for battery recharging or powering circuits.The structure and operations performed by a state-of-the-art EH [28], [29] are:1) The input signal is rectified by means of a devicecommonly referred to as rectifier . In general, this operation is performed by means of suitable diodes.In particular, p-n junction diodes are adopted whenthe signal has frequencies in the kHz-MHz range,whereas devices with shorter transit times and lowerintrinsic capacitances, such as the gallium arsenide(GaAs) Schottky diodes, are adopted when the sig-nal has frequencies in the GHz-THz range [29];2) The rectified signal is filtered by means of a second-order low-pass filter to obtain a DC voltage;3) Finally, a DC-to-DC converter, e.g., an unregulatedbuck-boost converter operating in the discontinuousconduction mode [28], is usually adopted to adaptthe rectified voltage to the level required by theapplication load, e.g., a storage device, connectedto the third port of the 3-port element in Fig. 1.The obtained voltage by means of this procedure cancharge a battery within a range of few Volts. The effi-ciency of the overall RF-to-DC conversion provided bythe EH can be modeled by a factor β ∈ [0 , , obtainedas the ratio of the DC output energy over the RF inputenergy [28], [30]. It is worth noting that the state-of-the-art RF EH is already capable of delivering remarkableconversion efficiencies, i.e., η ≥ , and is ready forcommercial usage [28], [31], [32]. Naturally, the conver-sion efficiency strongly depends on how much the EHcan be tailored to the specific architecture/application. Inthis context, an appropriate tuning of the EH correspondsto the adoption of a suitable diode able to operate withsignals at the frequencies of interest. As a consequence,every RF EH can be appropriately tuned as long as itscomponents are suitably chosen [29].The proposed architecture brings two main advantages overthe state of the art: first, it allows to increase the upper boundon the TX power for the FD radio while guaranteeing theeffectiveness of the preexisting SIC algorithms; second, itrecycles a portion of the wasted energy at the circulator. Itshould be noted that such architecture also offers an implicitadditional advantage in the form of general RF energy har-vesting capabilities. In particular, and similar to what is donein classic wireless power transfer settings [30], [33], [34], RFenergy could always be harvested from the environment bythe proposed FD architecture by setting ρ = 0 , whenever noinformation signal is received. Concerning the latter aspect, wewould like to note that differently from other FD architecturesthat resort to RF energy harvesting to increase their energyefficiency [35], no restrictions on the usage that may bedone of the recycled energy are enforced in this contribution.Indeed, several reasonable strategies to make use of suchenergy could be envisioned for the proposed architecture, e.g.,the one proposed in [35]. However any choice in this sensewould be arbitrary and out of the scope of this work. Thus,the identification of possible strategies for the proposed FDarchitecture to use the recycled energy is deferred to a furthercontribution. A. Impact on the Performance of the FD Radio
The operating scenario in which the proposed architectureunveils its potential is when the TX power of the FD radio,denoted by P , is larger than P th . In this case, the residual SIsuffered by the state-of-the-art solution after the SIC wouldhave power ρP N P th . Hence, the choice of an appropriate valueof ρ , i.e., ρ ≤ P th P , could reduce the power of the signalcoming from the circulator such that the full effectivenessof the subsequent SIC algorithms is restored. Conversely, noadvantage would be brought by the proposed FD architectureover the state of the art when P ≤ P th , i.e., ρ = 1 isoptimal. Naturally, as previously discussed, the power dividertargets the entirety of its input signal. This results in a non-negligible SNR reduction w.r.t. the state-of-the-art solution,potentially affecting the performance of the device. To providethe best operating point, the choice of an optimal ρ must bemade accounting for both gains and losses associated to SINRenhancement and SNR reduction. This aspect will be furtherdiscussed in the following.Finally, it is worth noting that the performance of the FDradio is constrained not only by the limitation of the state-of-the-art SIC solutions but also by practical hardware limitationsthat affect the RX chain of every radio device. In practice, thecircuitry performing the digital signal processing in the RXchain can easily saturate if the power of its input signal is toolarge. Let P sat ≥ P th be the TX power of the FD device forwhich such saturation occurs. Thus, in general, the decoding ofthe IC can be performed by the FD radio only when ρ ≤ P sat P .III. S YSTEM S ETUP
Consider the hybrid FD/HD four-node scenario described inSec. I, in which two FD SNs serve two HD ANs, while sharinginformation/signaling over a wireless in-band link. In particu-lar, and without loss of generality, we assume that the commu-nication between the FD SNs and the HD ANs occurs in thetime-division duplexing (TDD) mode. The two phases of thiscommunication are illustrated in Fig. 2. During the forward : Undesired signal: Forward signal + signaling
SN 1SN 2 AN 2AN 1 : Self-interference signal (a) Forward phase. : Backward signal: Undesired signal: Signaling
SN 1SN 2 AN 2AN 1 : Self-interference signal (b) Backward phase.Figure 2. Communication phases between the SNs and the ANs. phase, the i th SN simultaneously communicates with both the i th AN and the j th SN. In the meanwhile, the i th AN onlyreceives the signals from the two SNs, without transmitting.Conversely, in the backward phase, the i th AN communicateswith its corresponding SN, without receiving information fromthe latter. At the same time, the i th SN communicates with the j th SN while receiving signals from all the other nodes. We assume frequency-selective block-fading channels with l + 1 taps between all the devices, distributed as independent com-plex Gaussian random variables with zero mean and variancegiven by ξ n ∈ R , depending on the power delay profile (PDP)of the considered channel. We denote the vectors representingthe channel between the i th-SN-to- j th-AN link, the channelbetween the SNs, and the multi-path channel experienced bythe SI signal at the j th SN as h ij = [ h ij (0) , · · · , h ij ( l )] T ∈ C l +1 , h s = [ h s (0) , · · · , h s ( l )] T ∈ C l +1 , and h mj = (cid:2) h mj (0) , · · · , h mj ( l ) (cid:3) T ∈ C l +1 , ∀ i, j ∈ { , } , respectively,with h ij ( n ) , h s ( n ) , and h mj ( n ) ∈ CN (0 , ξ n ) , ∀ n ∈ [0 , l ] .Assuming no RF impairments and perfect synchronization atthe receiver, the channel matrix which models the convolutionof the signal transmitted by the i th SN and h ij can be writtenas H ij = h ij (0) 0 · · · h ij ( l ) · · · h ij (1) h ij (1) . . . .. . . . . . . . ...... . . . .. . . . . . . . h ij ( l ) h ij ( l ) . . . .. . h ij (0) . . . ... . . . .. . . . . . . . · · · h ij ( l ) · · · · · · h ij (0) . (1)Analogously, H s and H mj are constructed from h s and h mj ,respectively. At this stage, we assume that perfect channel stateinformation (CSI) is available at each device. As discussedin Sec. I, we assume that the communication between SNsand ANs is based on OFDMA with N sub-carriers and L cyclic prefix (CP) symbols. This assumption guarantees threeconditions: 1) the frequency-selectivity of the channels can beeffectively dealt with at the receivers with simple equalizationprocedures, 2) the considered waveform is consistent with thetrends envisioned for the 5G air interface design [20], and 3)the resulting scenario is compliant with what is considered inthe reference state-of-the-art case [5]. We denote N i as theset of sub-carriers allocated to the i th SN-to-AN link, where N ∪ N = { , · · · , N } and N ∩ N = ∅ . In this context,the signaling is performed by means of a block transmissionwith block size N + L symbols, as detailed in the following.IV. T HE F ORWARD P HASE
A. Signal Model
According to the sub-carrier allocation for the OFDMAtransmission, the information transmitted by the i th SN in the t th OFDM block, i.e., u i, o [ t ] ∈ C N , can be expressed as (cid:2) u i, o [ t ] (cid:3) n = ((cid:2) u i, o [ t ] (cid:3) n ∈ C , n ∈ N i , , otherwise , (2)with covariance matrix P F i, o ∈ C N × N . Let F be the normal-ized N -point discrete Fourier transform (DFT) matrix, i.e., [ F ] mn = √ N exp (cid:16) − j π ( m − n − N (cid:17) , ∀ m, n ∈ { , · · · , N } ,and A = (cid:20) L × ( N − L ) I L I N (cid:21) a CP insertion matrix, then the signal transmitted by the i th SNto the i th AN in the forward phase is obtained as x i, o [ t ] = AF − u i, o [ t ] ∈ C N + L , such that Tr (cid:0) E [ x i, o [ t ] (cid:0) x i, o [ t ] (cid:1) H ] (cid:1) =( N + L ) P o . Let x i, b [ t ] be the signaling transmitted from the i th to the j th SN in the t th block, detailed in the following forsimplicity. Thus, we can write the overall transmitted signalby the i th SN in the t th block as x i [ t ] = x i, o [ t ]+ x i, b [ t ] . Now,if we define α ij as the path loss attenuation between the i thSN and the j th AN, then the t th block is received by the i thAN as y F i, o [ t ] = √ α ii (cid:16) H ∇ ii x i [ t ] + H △ ii x i [ t − (cid:17) + √ α ji (cid:16) H ∇ ji x j [ t ] + H △ ji x j [ t − (cid:17) + w i, o [ t ] , (3)with w i, o [ t ] ∈ CN ( , N I N + L ) additive white Gaussian noise(AWGN), and where H ∇ ij and H △ ij are the lower and theupper triangular part of H ij responsible for the inter-symboland inter-block interference (ISI and IBI), respectively. In thisregard, we note that all the received signals by the FD radioat time t appear in (3) for the sake of completeness, i.e.,the signals transmitted by the serving and the interfering SN,including both the t th OFDM block and the IBI componentresulting from the multi-path propagation of the ( t − thOFDM block, and received with the t h block. In order todecode the received signal, the i th AN performs a legacyOFDMA demodulation and get y F i, o [ t ] = D i FB (cid:16) √ α ii (cid:16) H ∇ ii x i [ t ] + H △ ii x i [ t − (cid:17) + √ α ji (cid:16) H ∇ ji x j [ t ] + H △ ji x j [ t − (cid:17) + w i, o [ t ] (cid:17) , (4)where B = [ N × L I N ] is the CP removal matrix, and D i isthe N × N sub-carrier selection matrix for N i , i.e., [ D i ] mn = ( , m = n ∈ N i , , otherwise . In practice, the signal obtained after this process can be writtenas y F i, o [ t ] = √ α ii D i FBH ∇ ii (cid:0) AF − u i, o [ t ] + x i, b [ t ] (cid:1) + √ α ji D i FBH ∇ ji x j, b [ t ] + D i FBw i, o [ t ] . (5)Switching our focus to the signaling between the two SNs, wenote that it should not generate undesired interference duringthe OFDM decoding of u i, o [ t ] at the i th AN, as discussedin Sec. I. Interestingly, the structure of the OFDMA signalinherently offers a means to achieve this goal [24]. As a matterof fact, x i, b [ t ] does not interfere with u i, o [ t ] at the i th AN onlyif ( D i FBH ∇ ii x i, b [ t ] = , D j FBH ∇ ij x i, b [ t ] = , ∀ i, j ∈ { , } and i = j. (6)In practice, (6) is satisfied if x i, b [ t ] ∈ null (cid:0) D i FBH ∇ ii + D j FBH ∇ ij (cid:1) with dim (cid:0) null (cid:0) D i FBH ∇ ii + D j FBH ∇ ij (cid:1)(cid:1) = L .We know from [24] that (6) can always be satisfied if perfectCSI is available at the transmitter. This means that the i thSN can transmit up to L independent information streams byprecoding them with a matrix whose column space is equal to null (cid:0) D i FBH ∇ ii + D j FBH ∇ ij (cid:1) , i.e., u i, b [ t ] ∈ C L withcovariance matrix P F i, b ∈ C L × L . Now, given an arbitrary semi-unitary matrix Γ F i ∈ C ( N + L ) × L that satisfies ( D i FBH ∇ ii Γ F i = , D j FBH ∇ ij Γ F i = , (7)obtainable by means of matrix operations such as the LQfactorization, we can express the signaling of the i th SN as x i, b [ t ] = Γ F i C F i u i, b [ t ] ∈ C N + L , (8)such that Tr (cid:0) E [ x i, b [ t ] (cid:0) x i, b [ t ] (cid:1) H ] (cid:1) = ( N + L ) P b and with C F i ∈ C L × L as a matrix offering auxiliary degrees of freedomto the i th SN, to be designed according to criteria of interestas detailed in the following. Additionally, we note that P b ischosen such that the TX power budget per symbol at the SN,i.e., P , is not exceeded, that is P b = P − P o . B. Performance of the Transmission in the Forward Phase
We first consider the rates achieved by the transmission inthe forward phase. Given that the signaling between the SNssatisfy (6), we can rewrite (5) as y F i, o [ t ] = √ α ii D i diag (cid:0) ˜h ii (cid:1) u i, o [ t ] + D i FBw i, o [ t ] , (9)where ˜h ii = √ N F (cid:2) h T ii , × ( N − l − (cid:3) T . Like the conven-tional OFDMA scheme, the i th SN transmits | N i | independentinformation streams to the i th AN via | N i | parallel Gaussianchannels. Thus, the achievable rate over the link between the i th SN to the i th AN in the forward phase is given by R F i, o = 1 N + L X n ∈ N i log α ii [ P F i, o ] nn | [ ˜h ii ] n | N ! , (10)where (cid:2) P F i, o (cid:3) nn = (cid:16) κ ,i − N α ii | [ ˜h ii ] n | (cid:17) + , ∀ n ∈ N i , is thewater-filling (WF) power allocation solution, with κ ,i chosensuch that Tr (cid:0) P F i, o (cid:1) = N P o [36]. C. Performance of the Signaling between SNs
We now focus on the achievable rate over the link betweenthe two SNs in the considered scenario. The received signalat the j th SN, including both the SI and the signal from theother SN, in the t th block is y F j, b [ t ] = √ ρα b H ∇ s x i [ t ] + √ ρα b H △ s x i [ t −
1] + √ ρα c x j [ t ]+ √ ρα m H m ∇ j x j [ t ] + √ ρα m H m △ j x j [ t −
1] + w j, b [ t ] , (11)where √ α c x j represents the signal leakage from the circulatorof the FD radio, α m is the path loss factor modeling theattenuation affecting the component of the SI that experiencesa multi-path propagation, and α b is the path loss factorcharacterizing the link between the two SNs. Given the natureof the communication, we can safely assume a block-wisedecoding of the signaling at each SN. In this context, asuccessive interference cancellation approach could be adoptedto correctly subtract the signal decoded in one block fromthe signal present in the subsequent block. Thus, if a perfect cancellation can be performed, the obtained signal after theSIC can be written as y SIC j, b [ t ] = √ ρα b H ∇ s Γ F i C F i u i, b [ t ] + w j, b [ t ]+ √ ρα b (cid:0) H ∇ s AF − u i, o [ t ] + H △ s AF − u i, o [ t − (cid:1) + √ ρα eq (cid:0) AF − u j, o [ t ] + Γ F j C F j u j, b [ t ] (cid:1) , (12)with w j, b ∈ CN ( , N I N + L ) an AWGN and where α eq = ( , if ρ ≤ P th P , N P th , if P th P < ρ ≤ min (cid:8) , P sat P (cid:9) . (13)The power of the multi-path SI signal in practical implemen-tations is typically at least dB lower than the power ofthe SI signal leaking from the circulator [6]. In practice, itcan be safely assumed that the contribution of the formerto the overall SI signal in (12) is quantitatively negligible.Then, only the SI signal leaking from the circulator will beconsidered in the following, for simplicity. At this stage, it isconvenient to divide the analysis into two parts, for the sakeof completeness. First, we will focus on the case ρ ≤ P th P ,in which no residual SI affects the decoding of the signaling.Then, we will study the case P th P < ρ ≤ min (cid:8) , P sat P (cid:9) , inwhich residual SI remains after the SIC.
1) Absence of Residual Self-interference:
From (13) wehave that α eq = 0 in this case, then (12) can be rewrittenas y SIC j, b [ t ] = √ ρα b H ∇ s Γ F i C F i u i, b [ t ] + z j, b [ t ] , (14)where z j, b [ t ] = w j, b [ t ] + √ ρα b (cid:0) H ∇ s AF − u i, o [ t ] + H △ s AF − u i, o [ t − (cid:1) is the equivalent noise. In this regard,we note that it is reasonable to assume that each SN mayhave an estimation of the covariance matrix of the OFDMsignal transmitted by the other SN in this phase [37], [38].Accordingly, letcov (cid:0) z j, b (cid:1) = N I N + L + ρα b H ∇ s AF − P F i, o FA H (cid:0) H ∇ s (cid:1) H + ρα b H △ s AF − P F i, o FA H (cid:0) H △ s (cid:1) H (15)be the corresponding noise-whitening matrix. The whitenedversion of (14) can be written as y SIC j, b [ t ] = √ ρα b cov − (cid:0) z j, b (cid:1) H ∇ s Γ F i C F i u i, b [ t ]+ cov − (cid:0) z j, b (cid:1) z j, b [ t ] . (16)Now, we focus on the first right-hand side elementof (16) and let cov − (cid:0) z j, b (cid:1) H ∇ s Γ F i = U F i Σ F i (cid:0) Q F i (cid:1) H be a singular value decomposition (SVD), with U F i ∈ C ( N + L ) × ( N + L ) and Q F i ∈ C L × L unitary matrices, and Σ F i = [ diag ( σ F1 , z , . . . , σ F L, z ) L × N ] T where σ F i, z is the i thsingular value of cov − (cid:0) z j, b (cid:1) H ∇ s Γ F i . The link between thetwo SNs can then be decomposed into L parallel indepen-dent Gaussian channels by letting C F i = Q F i and defining (cid:0) U F i (cid:1) H cov − (cid:0) z j, b (cid:1) as the decoding matrix for the i th signal-ing. The achievable rate over the link between the two SNsin the forward phase, from the point of view of the i th SN, is then given by R F i, b ( ρ ) = 1 N + L L X n =1 log (cid:16) ρα b (cid:2) P F i, b (cid:3) nn (cid:2) Σ F i (cid:3) nn (cid:17) , (17)with (cid:2) P F i, b (cid:3) nn = (cid:0) κ ,i − (cid:0) ρα b (cid:2) Σ F i (cid:3) nn (cid:1) − (cid:1) + , ∀ n ∈{ , · · · , L } being the WF solution, where κ ,i is chosen suchthat Tr (cid:0) P F i, b (cid:1) = ( N + L ) P b . In practice, the informationrequired at both SNs to perform these computations could beexchanged in a preliminary phase of the communication, atthe beginning of each new coherence time of the channel.
2) Presence of Residual Self-interference:
From (13) wehave that α eq = N P th in this case, then (12) can be rewrittenas y SIC j, b [ t ] = √ ρα b H ∇ s Γ F i C F i u i, b [ t ] + z j, b [ t ] , (18)where z j, b [ t ] = w j, b [ t ] + r ρN P th (cid:0) AF − u j, o [ t ] + Γ F j C F j u j, b [ t ] (cid:1) + √ ρα b (cid:0) H ∇ s AF − u i, o [ t ] + H △ s AF − u i, o [ t − (cid:1) (19)is the equivalent noise. Following the same proceduresas for the previous case we write cov − (cid:0) z j, b (cid:1) H ∇ s Γ F i = U F i Σ F i (cid:0) Q F i (cid:1) H , where U F i , Σ F i , and Q F i are defined analo-gously. Computing cov (cid:0) z j, b (cid:1) is rather complex in this case.Differently from before, herein the adoption of an adequateinformation exchange protocol between the SNs would notbe sufficient for the latter to compute this quantity. In fact,no prior information about C F j is available at the i th SN.Furthermore, the power allocation for u j, b [ t ] at the j th SNimpacts both the power profile of the SI affecting the j th SNitself and the spectral efficiency of the link towards the i thSN. In practice, these two phenomena are strongly coupled.To cope with this issue, iterative WF algorithms should beperformed at both SNs to maximize the spectral efficiencyof the link between them. However, the coherence time ofthe channels towards the ANs (upon which the CIA precoderis based) imposes strict time requirements to the SNs. Thus,iterative approaches are not feasible, especially in the absenceof a wired connection between the SNs, and alternative strate-gies need to be devised. Interestingly, these two issues couldbe addressed by resorting to two simplifications. First, weconjecture that C F j is a unitary matrix, as per the derivationperformed for the previous case. Additionally, we assume auniform power allocation for the L symbols of u j, b [ t ] , suchthat P F j, b = ( N + L ) P b L I L . An approximation of the noise-whitening matrix in this case can then be obtained ascov (cid:0) z j, b (cid:1) = N I N + L + ρα b H ∇ s AF − P F i, o FA H (cid:0) H ∇ s (cid:1) H + ρα b H △ s AF − P F i, o FA H (cid:0) H △ s (cid:1) H + ρN P th AF − P F j, o FA H + ρN ( N + L ) P b LP th Γ F j (cid:0) Γ F j (cid:1) H . (20)Now, we let C F i = Q F i and (cid:0) U F i (cid:1) H cov − (cid:0) z j, b (cid:1) be thedecoding matrix for the signaling and, similarly to the previous case, we obtain the achievable rate over the link between thetwo SNs in the forward phase as R F i, b ( ρ ) = 1 N + L L X n =1 log ρα b ( N + L ) P b (cid:2) Σ F i (cid:3) nn L ! . (21) D. The Impact of the Splitting Ratio
By looking at (17) and (21) we see that ρ is present inthe in-log term both explicitly and implicitly via (cid:2) P F i, b (cid:3) nn and (cid:2) Σ F i (cid:3) nn . A closed-form analysis of its impact on (17)and (21) is not straightforward. However, we can address thisproblem as follows. We first define ρ ∗ as the value at which therate of the incoming signaling is maximum, i.e., the optimalsplitting ratio. Subsequently, we note that the power of both thereceived signaling and interference signals increases with ρ atthe same pace, by construction, whereas N is independentof the latter parameter. Thus, we can intuitively infer thatboth the SINR and achievable rate over the link betweenthe two SNs are monotonically increasing functions of ρ (thelatter being a logarithmic function of the splitting ratio), for ≤ ρ ≤ ρ ∗ . Consequently, the optimal splitting ratio isdeterministically given by ρ ∗ = P th P or ρ ∗ = min (cid:8) , P sat P (cid:9) if residual SI is absent or present, respectively. We note thatthese deductions will be verified in Sec VI. At this stage,we can safely assume a priori that the curve representing theachievable rate as a function of the SINR may be approximatedby g ( ρ, k ) = log (cid:0) c k ρ φ k (cid:1) , where c k and φ k depend onboth the concavity of the curve and on the absence (e.g., k = 1 ) or presence (i.e., k = 2 ) of residual SI. In practice, c k and φ k can be identified by the SNs following a simpleone-shot procedure. After the first CSI has been acquiredby the SNs, an estimation of the instantaneous achievablerate is computed using (17) or (21), for two values of ρ arbitrarily chosen. For instance, reasonable choices for theinterval ≤ ρ ≤ P th P , i.e., the absence of residual SI, are ρ = P th P and ρ = P th P . Similarly, reasonable choices withinthe interval P th P < ρ ≤ min (cid:8) , P sat P (cid:9) could be ρ = P th + ǫP ,with ǫ → , and ρ = min (cid:8) , P sat P (cid:9) . The following propositionholds. Proposition 1.
The achievable rate over the link between the i th and the j th SN in the forward phase can be approximatedas R F i, b ( ρ ) ≈ log (cid:0) c ρ φ (cid:1) , when ρ ≤ P th P , log (cid:0) c ρ φ (cid:1) , when P th P < ρ ≤ min (cid:8) , P sat P (cid:9) , (22) where c = (cid:16) R F i, b ( P th P ) − (cid:17) (cid:18) PP th (cid:19) φ ,φ = log R F i, b ( P th P ) − R F i, b ( P th2 P ) − ! ,c = (cid:16) R F i, b ( P th+ ǫP ) − (cid:17) (cid:18) PP th + ǫ (cid:19) φ , φ = log P th+ ǫ min { P,P sat } R F i, b ( P th+ ǫP ) − R F i, b (min { , P sat P } ) − ! . Proof:
The result in (22), can be obtained by enforcing aperfect match between the aforementioned rate estimation and g ( ρ, k ) . Thus, in the case of the absence of residual SI in theRX chain, i.e., ρ ≤ P th P , we have R F i, b (cid:0) P th P (cid:1) = log (cid:16) c (cid:0) P th P (cid:1) φ (cid:17) ,R F i, b (cid:0) P th P (cid:1) = log (cid:16) c (cid:0) P th P (cid:1) φ (cid:17) . Then, the two parameters c and φ can be straightforwardlyobtained as c = (cid:16) R F i, b ( P th P ) − (cid:17) (cid:18) PP th (cid:19) φ ,φ = log R F i, b ( P th P ) − R F i, b ( P th2 P ) − ! . The two parameters c and φ can be computed analogously,in the case of the presence of residual SI in the RX chain, i.e.,when P th P < ρ ≤ min (cid:8) , P sat P (cid:9) .Proposition 1 serves a two-fold purpose. On one hand, itprovides an approximation of the achievable rate over the linkbetween the two SNs in the forward phase. On the otherhand, it allows to identify the values of ρ for which theproposed FD architecture can outperform the state-of-the-artsolutions in terms of throughput. In this sense, we recall thatthe performance of the state of the art can be easily obtained bysetting ρ = 1 , i.e., virtually deactivating the 3-port compositeelement described in Sec. II. Now, consider the relevant case P th < P ≤ P sat , i.e., the FD device implemented accordingto state-of-the-art solutions suffers from residual SI but nosaturation occurs in its RX chain. The following corollarydirectly descends from Proposition 1. Corollary 1. If P th < P ≤ P sat in the forwardphase, then the proposed FD architecture delivers a higherthroughput than the state-of-the-art FD solutions, for ρ ∈ (cid:20) φ r R F i, b(1) − c , P th P (cid:21) .Proof: When P th P < ρ ≤ , the approximated achievablerate is a monotonically increasing function. The maximum rateis then achieved for ρ = 1 , i.e., the state-of-the-art solution isalways better in this region. Alternatively, when ρ ≤ P th P , welet log (cid:0) c ρ φ (cid:1) ≥ R F i, b (1) , as per Proposition 1, and theresult of Corollary 1 can be directly obtained. E. Recycled Energy
A seen in Sec. II, the 3-port composite element present inthe proposed FD architecture includes an EH that can recyclea portion of the leaked RF energy in the form of DC energy.Using the digital representation of the signals for the sake ofcompactness of the notation, let E F i ( ρ ) be the recycled energyper symbol by the 3-port element at the i th SN, in the forwardphase. By some straightforward derivations, this quantity can be computed as E F i ( ρ ) = β (1 − ρ ) N + L × Tr (cid:16) α b (cid:16)(cid:0) H ∇ s (cid:1) H H ∇ s + (cid:0) H △ s (cid:1) H H △ s (cid:17) × (cid:16) AF − P F j, o FA H + Γ F j C F j P F j, b (cid:0) C F j (cid:1) H (cid:0) Γ F j (cid:1) H (cid:17) + (cid:16) α m (cid:0) H m △ i (cid:1) H H m △ i + (cid:0) √ α c I N + L + √ α m H m ∇ i (cid:1) H × (cid:0) √ α c I N + L + √ α m H m ∇ i (cid:1) (cid:17) × (cid:16) AF − P F i, o FA H + Γ F i C F i P F i, b (cid:0) C F i (cid:1) H (cid:0) Γ F i (cid:1) H (cid:17) (cid:17) , (23)with β as the RF-to-DC energy conversion efficiency [30].At this stage, it is worth introducing a useful approximationof (23), and increasing its qualitative descriptiveness. Thus,we neglect the received signaling and multi-path SI signalterms from (23), given their different order of magnitude ascompared of the intensity of the SI [6], and approximatethe recycled energy per symbol at the i th SN as E F i ( ρ ) ≈ βα c (1 − ρ ) P . V. T HE B ACKWARD P HASE
We switch now our focus to the backward phase.
A. Signal Model
Similar to the forward phase, the ANs communicate with theSNs by means of an OFDMA transmission. The informationtransmitted by the i th AN in the t th OFDM block is given by (cid:2) v i, o [ t ] (cid:3) n = ((cid:2) v i, o [ t ] (cid:3) n ∈ C , n ∈ N i , , otherwise , (24)with covariance matrix P B i, o ∈ C N × N . The i th transmittedsignal in the backward phase reads s i, o [ t ] = AF − v i, o [ t ] ,with Tr (cid:0) E [ s i, o [ t ] (cid:0) s i, o [ t ] (cid:1) H ] (cid:1) = ( N + L ) P A , where P A is theTX power budget at the AN. We recall that the ANs are notthe only devices active in this phase of communication. Infact, the FD SNs are equally active and exchange signalingwhile receiving the backward signals from the ANs. We denotethe signaling transmitted by the i th SN in the t th block as s i, b [ t ] ∈ C N + L , with Tr (cid:0) E [ s i, b [ t ] (cid:0) s i, b [ t ] (cid:1) H ] (cid:1) = ( N + L ) P .At this stage, we recall that the SNs do not transmitinformation to the ANs during the backward phase. Thus afterthe operations performed by the 3-port composite element, theoverall received signal at the i th SN, including both the SI andthe signals from the other devices, is y i [ t ] = √ ρ (cid:16) √ α ii H ∇ ii s i, o [ t ] + √ α ii H △ ii s i, o [ t − √ α ij H ∇ ij s j, o [ t ] + √ α ij H △ ij s j, o [ t −
1] + √ α c s i, b [ t ]+ √ α b H ∇ s s j, b [ t ] + √ α b H △ s s j, b [ t − √ α m H m ∇ i s i, b [ t ] + √ α m H m △ i s i, b [ t − (cid:17) + w i [ t ] . (25)Subsequently, after cancelling the SI, the information signalto be decoded by the i th SN reads y SIC i [ t ] = √ ρ (cid:16) √ α ii H ∇ ii s i, o [ t ] + √ α ii H △ ii s i, o [ t − √ α ij H ∇ ij s j, o [ t ] + √ α ij H △ ij s j, o [ t −
1] + √ α eq s i, b [ t ]+ √ α b H ∇ s s j, b [ t ] + √ α b H △ s s j, b [ t − (cid:17) + w i [ t ] , (26) where w i ∈ CN ( , N I N + L ) is an AWGN. Now, first thebackward signal is decoded by the i th SN by means of legacyOFDM demodulation. Similar to the previous case, a block-wise decoding of the backward signal is performed at each SN.A successive interference cancellation is then performed bysubtracting the decoded backward signal from (26) and finallythe decoding of the signaling takes place. In this regard, wenote that a perfect cancellation is assumed. B. Performance of the Transmission in the Backward Phase
As for (12), residual SI can be absent or present in theRX chain, depending on the value of ρ . Thus, as before, thefollowing analysis encompasses the two cases, for the sake ofcompleteness.
1) Absence of Residual Self-interference:
The only poten-tial source of interference during the OFDM demodulation isgiven by s j, b [ t ] . In this context, the i th SN designs its signalingin order not to induce a rate loss on the link between the i th SN/AN pair in the backward phase. We know from thediscussion in Sec. IV, that if s i, b [ t ] satisfies D j FBH ∇ s s i, b [ t ] = , i = j, (27)then its impact on the SINR of the backward signal at the j th SN is zeroed, and the decoding of the latter can proceedaccording to the conventional OFDMA scheme. Now, byplugging α eq = 0 into (26), we can write the demodulatedsignal in this case as y B i, o [ t ] = √ ρα ii D i diag (cid:0) ˜h ii (cid:1) v i, o [ t ] + D i FBw i [ t ] . (28)The achievable rate over the link between the i th SN/AN pairin this case is then R B i, o ( ρ ) = 1 N + L X n ∈ N i log ρα ii [ P B i, o ] nn | [ ˜h ii ] n | N ! , (29)where (cid:2) P B i, o (cid:3) nn = (cid:16) κ ,i − N ρα ii | [ ˜h ii ] n | (cid:17) + , ∀ n ∈ N i is theWF solution, with κ ,i chosen such that Tr (cid:0) P B i, o (cid:1) = N P A .Furthermore, it is straightforward to see that ρ ∗ = P th P in thiscase.
2) Presence of Residual Self-interference:
Provided that(27) is satisfied, and plugging α eq = N P th into (26), we canwrite y B i, o [ t ] = √ ρα ii D i diag (cid:0) ˜h ii (cid:1) v i, o [ t ]+ D i FB w i [ t ] + r ρN P th s i, b [ t ] ! . (30)The equivalent noise in (30) is colored due to the presence ofresidual SI. However, a noise-whitening cannot be performedstraightforwardly in this case. In fact, each N -sized vector y B i, o [ t ] includes | N j | zero elements by construction. This issuecan be addressed without loss of generality and correctnessby removing the zero elements from y B i, o [ t ] , to obtain a | N i | -sized vector offering higher analytical tractability. Thus, wefirst let o B i = [ o B i (1) , . . . , o B i ( | N i | )] be an ordered vectorof indexes, such that o B i ( k ) ∈ N i , ∀ natural k ∈ [1 , | N i | ] . Subsequently, we define a modified ( | N i | × N ) -sized sub-carrier selection matrix ˜D i = [ e o B i (1) · · · e o B i ( | N i | ) ] H , where e o B i ( j ) is the o B i ( j ) th standard unit vector. Finally, we canrewrite (30) as ˜y B i, o [ t ] = √ ρα ii ˜D i diag (cid:0) ˜h ii (cid:1)(cid:0) ˜D i (cid:1) H ˜v i, o [ t ] + z i [ t ] , (31)where we let ˜v i, o [ t ] = ˜D i v i, o [ t ] , for the sake of consistency ofthe notation, and define z i [ t ] = ˜D i FB (cid:16) w i [ t ] + q ρN P th s i, b [ t ] (cid:17) as the equivalent noise. Now we can proceed as done for theforward phase and first write the noise-whitening matrix ascov ( z i ) = N I | N i | + ρN ( N + L ) P ( N + L − | N j | ) P th × ˜D i FBΓ B i (cid:0) Γ B i (cid:1) H B H F − (cid:0) ˜D i (cid:1) H , (32)and then compute cov − ( z i ) ˜D i diag (cid:0) ˜h ii (cid:1)(cid:0) ˜D i (cid:1) H = ˜U i ˜Λ i (cid:0) ˜U i (cid:1) H , where ˜U i ∈ C | N i |×| N i | is a unitarymatrix and ˜Λ i = [ diag (˜ λ , z , . . . , ˜ λ | N i | , z )] , where ˜ λ i, z is the i th eigenvalue of cov − ( z i ) ˜D i diag (cid:0) ˜h ii (cid:1)(cid:0) ˜D i (cid:1) H .Analogously, we can decompose the link betweenthe i th SN/AN pair into | N i | parallel Gaussianchannels by letting P B i, o = (cid:0) ˜D i (cid:1) H ˜U i ˜P B i, o (cid:0) ˜U i (cid:1) H ˜D i , and (cid:2) ˜P B i, o (cid:3) nn = (cid:16) κ ,i − ρα ii | [ ˜Λ i ] nn | (cid:17) + , ∀ n ∈ { , · · · , | N i |} ,with κ ,i chosen such that Tr (cid:0) ˜P B i, o (cid:1) = N P A . Thus, theachievable rate over this link is R B i, o ( ρ ) = 1 N + L | N i | X n =1 log (cid:16) ρα ii (cid:2) ˜P B i, o (cid:3) nn | [ ˜Λ i ] nn | (cid:17) . (33)Intuitively, the values of ρ ∗ for the backward phase are thesame as for the forward phase. Moreover, as before, assessingthe impact that ρ may have on (29) and (33), when a WFstrategy is adopted, is not straightforward. Thus, we canproceed as done for Proposition 1 and state the following. Proposition 2.
The achievable rate over the link between the i th SN/AN pair in the backward phase can be approximatedas R B i, o ( ρ ) ≈ log (cid:0) c ρ φ (cid:1) , when ρ ≤ P th P , log (cid:0) c ρ φ (cid:1) , when P th P < ρ ≤ min (cid:8) , P sat P (cid:9) , (34) where c = (cid:16) R B i, o ( P th P ) − (cid:17) (cid:18) PP th (cid:19) φ ,φ = log R B i, o ( P th P ) − R B i, o ( P th2 P ) − ! ,c = (cid:16) R B i, o ( P th+ ǫP ) − (cid:17) (cid:18) PP th + ǫ (cid:19) φ ,φ = log P th+ ǫ min { P,P sat } R B i, o ( P th+ ǫP ) − R B i, o (min { , P sat P } ) − ! . Proof:
The result can be obtained as done for Proposi-tion 1. We now focus as before on the relevant case P th < P ≤ P sat , i.e., the FD device implemented according to state-of-the-art solutions suffers from residual SI but no saturation occursin its RX chain. In this case, the following result directlydescends from Proposition 2. Corollary 2.
When P th < P ≤ P sat , the interval of ρ for which the achievable rate over the link between the i thSN/AN pair is higher if the proposed approach is adoptedover the state-of-the-art solutions can be approximated as (cid:20) φ r R B i, o(1) − c , P th P (cid:21) .Proof: The result can be obtained as done for Corollary 1.
C. Performance of the Signaling Between SNs
We start by noting that (27) implies thatdim (cid:0) null (cid:0) D j FBH ∇ s (cid:1)(cid:1) = N + L − | N j | . Hence, the i th SN can transmit up to N + L − | N j | independentinformation streams for signaling in the backward phase.We let v i, b [ t ] ∈ C N + L −| N j | , with covariance matrix P B i, b ,be the information symbol vector transmitted by the i thto the j th SN in this phase. Then, if we adopt the samerepresentation as in (8), we can express the i th signalingas s i, b = Γ B i C B i v i, b , with Γ B i ∈ C ( N + L ) × ( N + L −| N j | ) anarbitrary semi-unitary matrix such that D j FBH ∇ s Γ B i = .Now, assume a block-wise decoding of the signaling at the j th SN, as done as for the forward phase. Thus, the receivedsignal at the j th SN after the SIC, i.e., y SIC j [ t ] , can beobtained from (26) as y SIC j [ t ] = √ ρ (cid:16) √ α b H ∇ s s i, b [ t ] + √ α eq s j, b [ t ]+ √ α ji H ∇ ji s i, o [ t ] + √ α ji H △ ji s i, o [ t − (cid:17) + w j [ t ] . (35)Following the same approach adopted so far, we divide therest of the analysis into two parts: first, we will focus on thecase ρ ≤ P th P , in which no residual SI affects the decodingof the signaling; then, we will study the case P th P < ρ ≤ min (cid:8) , P sat P (cid:9) , in which residual SI remains after the SIC.
1) Absence of Residual Self-interference:
From (13) wehave that α eq = 0 in this case, then (35) can be rewrittenas y SIC j [ t ] = √ ρα b H ∇ s Γ B i C B i v i, b [ t ] + z j [ t ] , (36)with z j [ t ] = w j [ t ] + √ ρα ji (cid:0) H ∇ ji AF − v i, o [ t ] + H △ ji AF − v i, o [ t − (cid:1) as equivalent noise. Now, letcov ( z j ) = N I N + L + ρα ji H ∇ ji AF − P B i, o FA H (cid:0) H ∇ ji (cid:1) H + ρα ji H △ ji AF − P B i, o FA H (cid:0) H △ ji (cid:1) H (37)be the corresponding noise-whitening matrix. Subsequently,we resort to the same approach adopted for theforward phase and write cov − ( z j ) H ∇ s Γ B i = U B i Σ B i (cid:0) Q B i (cid:1) H , with U B i ∈ C ( N + L ) × ( N + L ) and Q B i ∈ C ( N + L −| N j | ) × ( N + L −| N j | ) unitary matrices, and Σ B i = [ diag ( σ B1 , z , . . . , σ B N + L −| N j | , z ) ( N + L −| N j | ) ×| N j | ] T where σ B i, z is the i th singular value of cov − ( z j ) H ∇ s Γ B i .Accordingly, when no residual SI is present in the RX chain, the achievable rate over the link between the i th and the j thSNs in the backward phase is R B i, b ( ρ ) = 1 N + L × N + L −| N j | X n =1 log (cid:16) ρα b (cid:2) P B i, b (cid:3) nn (cid:2) Σ B i (cid:3) nn (cid:17) , (38)with (cid:2) P B i, b (cid:3) nn = (cid:0) κ ,i − (cid:0) ρα b (cid:2) Σ B i (cid:3) nn (cid:1) − (cid:1) + , ∀ n ∈{ , · · · , N + L − | N j |} WF solution. κ ,i is chosen such thatTr (cid:0) P B i, b (cid:1) = ( N + L ) P . As before, since no residual SI ispresent in this case, ρ ∗ = P th P .
2) Presence of Residual Self-interference:
We first plug α eq = N P th into (35) and rewrite it as y SIC j [ t ] = √ ρα b H ∇ s Γ B i C B i v i, b [ t ] + z j [ t ] , (39)where z j [ t ] = w j [ t ] + √ ρ (cid:16) √ α ji H △ ji AF − v i, o [ t − √ α ji H ∇ ji AF − v i, o [ t ] + r N P th Γ B j C B j v j, b [ t ] (cid:19) is the equivalent noise. Subsequently, we letcov ( z j ) = N I N + L + N ρ ( N + L ) PP th ( N + L − | N i | ) Γ B j (cid:0) Γ B j (cid:1) H + ρα ji (cid:16) H ∇ ji AF − P B i, o FA H (cid:0) H ∇ ji (cid:1) H + H △ ji AF − P B i, o FA H (cid:0) H △ ji (cid:1) H (cid:17) be its covariance matrix, with P B j, b = ( N + L ) PN + L −| N i | I N + L −| N i | .Now, following the same procedures as in Sec. IV-C2, we writecov − ( z j ) H ∇ s Γ B i = U B i Σ B i (cid:0) Q B i (cid:1) H , with U B i , Σ B i , and Q B i defined analogously to the previous case. Finally, as before, wedefine C B i = Q B i and let (cid:0) U B i (cid:1) H cov − ( z j ) be the decodingmatrix for the i th signaling at the j th SN. Thus, when residualSI is present in the RX chain, the achievable rate over the linkbetween the i th and the j th SN in the backward phase is R B i, b ( ρ ) = 1 N + L × N + L −| N j | X n =1 log ρα b ( N + L ) P (cid:2) Σ B i (cid:3) nn N + L − | N j | ! . (40)As previously inferred, the values of ρ ∗ for the backwardphase are the same as for the forward phase Additionally,assessing the impact that ρ may have on (38) and (40), whena WF strategy is adopted, is not straightforward. Hence, wecan proceed as done for Propositions 1 and 2 and obtain. Proposition 3.
The achievable rate for the transmission be-tween the i th and the j th SN in the backward phase can beapproximated as R B i, b ( ρ ) ≈ log (cid:0) c ρ φ (cid:1) , when ρ ≤ P th P , log (cid:0) c ρ φ (cid:1) , when P th P < ρ ≤ min (cid:8) , P sat P (cid:9) , (41) where c = (cid:16) R B i, b ( P th P ) − (cid:17) (cid:18) PP th (cid:19) φ ,φ = log R B i, b ( P th P ) − R B i, b ( P th2 P ) − ! ,c = (cid:16) R B i, b ( P th+ ǫP ) − (cid:17) (cid:18) PP th + ǫ (cid:19) φ ,φ = log P th+ ǫ min { P,P sat } R B i, b ( P th+ ǫP ) − R B i, b (min { , P sat P } ) − ! . Proof:
The result can be obtained as done for Proposi-tion 1.Now, applying Proposition 3 and following the same ap-proach adopted for Corollary 1, we can state the followingresult.
Corollary 3.
When P th < P ≤ P sat , the interval of ρ for which the achievable rate of the link between thetwo SNs is higher, if the proposed approach is adoptedover the state-of-the-art solutions, can be approximated as (cid:20) φ r R B i, b(1) − c , P th P (cid:21) .Proof: The result can be obtained as done for Corollary 1.
D. Recycled Energy
Using the digital representation of the signals for the sakeof compactness of the notation, and by some straightforwardcalculations, we can express the recycled energy per symbolat the i th SN as E B i ( ρ ) = β (1 − ρ ) N + L × h α ii Tr (cid:16)(cid:16)(cid:0) H ∇ ii (cid:1) H H ∇ ii + (cid:0) H △ ii (cid:1) H H △ ii (cid:17) AF − P B i, o FA H (cid:17) + α ij Tr (cid:16)(cid:16)(cid:0) H ∇ ij (cid:1) H H ∇ ij + (cid:0) H △ ij (cid:1) H H △ ij (cid:17) AF − P B j, o FA H (cid:17) + α b Tr (cid:16) (cid:16)(cid:0) H ∇ s (cid:1) H H ∇ s + (cid:0) H △ s (cid:1) H H △ s (cid:17) Γ B j C B j P B j, b × (cid:0) C B j (cid:1) H (cid:0) Γ B j (cid:1) H (cid:17) + Tr (cid:16)(cid:16) α m (cid:0) H m △ i (cid:1) H H m △ i + (cid:0) √ α c I N + L + √ α m H m ∇ i (cid:1) H (cid:0) √ α c I N + L + √ α m H m ∇ i (cid:1) (cid:17) × Γ B i C B i P B i, b (cid:0) C B i (cid:1) H (cid:0) Γ B i (cid:1) H (cid:17)i . (42)At this stage, we know from Sec. IV-E that a safe approx-imation of E B i ( ρ ) can be found by neglecting the receivedsignaling and multi-path SI signals terms from (42). Thus, wecan approximate the recycled energy per symbol at the i th SNas E B i ( ρ ) ≈ βα c (1 − ρ ) P .VI. N UMERICAL R ESULTS
In this section, the performance of the proposed FD archi-tecture is studied in the considered four-node setting. In partic-ular, we consider a realistic indoor scenario in which we let thedistance between the SNs and the ANs be d sa = 10 m, and thedistance between SNs be d ss ∈ { , } m. The transmittedsignals are modulated at carrier frequency f c = 1800 MHz, and experience a distance dependent path loss modeled ac-cording to [39]. Furthermore, for the sake of consistency withthe state of the art, we let α c = − dB, α m = − dB, β = 0 . , P sat = 28 dBm, and P th = 20 dBm [5], [6], [28].As previously discussed, we are particularly interested in char-acterizing the behavior of the FD radio in critical conditions,i.e., when the state-of-the-art solutions fail to cancel all theSI in the RX chain. Thus, we let the TX power of the FDradio be P ∈ [ P th , P sat ] , with P o = P b = P . Concerningthe OFDMA transmission, we set N = 64 and L = 16 according to [40]. We assume channels characterized by theexponentially decreasing PDP, such that ξ n = exp (cid:26) − nT s τ h (cid:27) with τ h the root mean square delay spread of the channel and T s the sampling period at each device. In particular, we restrictour study to channels with l = L taps and the slowly decayingPDP, i.e., T s τ h ∈ [2 , , for matters of space economy.Finally, we define the two metrics that may offer a straight-forward way to identify the range of values of ρ for whichthe proposed FD architecture outperforms the state-of-the-art solution [5]. In particular, we let η R = R · i, · ( ρ ) R · i, · (1) , and η E = E · i ( ρ ) E tx , with E tx energy per transmitted symbol bythe FD device. These metrics can be directly interpreted asfollows. If the proposed FD architecture achieves a largerspectral efficiency than the state of the art, then η R > .Similarly, if a portion of the energy of the transmitted signal isactually recycled by the proposed architecture, then η E > . A. Forward Phase
We first focus on the forward phase. We recall that notransmission is performed by the ANs in this phase. For thefirst study, we consider only two values for P within theaforementioned range, i.e. P = 24 dBm and P = 28 dBm,for ease of representation. Subsequently, we let ρ ∈ [0 , and compute η R , both analytically, e.g., by means of thederived approximations, and numerically, in Fig. 3. We start ρ η R Numerical, d ss = 15 m, 28 dBmNumerical, d ss = 15 m, 24 dBmNumerical, d ss = 20 m, 28 dBmNumerical, d ss = 20 m, 24 dBmApproximated, d ss = 15 m, 28 dBmApproximated, d ss = 15 m, 24 dBmApproximated, d ss = 20 m, 28 dBmApproximated, d ss = 20 m, 24 dBm State-of-the-art{0.21, 0.21}{0.1, 0.1}{0.1, 0.1} {0.2, 0.2}
Figure 3. [Forward] Numerical and approximated η R for the signaling as ρ varies. by observing that a range of values of ρ for which η R ≥ can be found in each of the tested configurations. In thisregard, we note that the couples of numbers (in the form {· , ·} ) P [dBm]
23 23.5 24 24.5 25 25.5 26 26.5 27 27.5 28 η ∗ R η ∗ E Signaling, d ss = 20 mSignaling, d ss = 15 mRecycled energy Figure 4. [Forward] η ∗ R and η ∗ E as the transmit power P varies. depicted in the figure denote the numerical and approximatedlower bounds of the aforementioned range of values of ρ , i.e., η R ≥ , ∀ ρ ∈ [ {· , ·} , ρ ∗ ] . At a first glance, we observe thatour approximations do not provide the same accuracy for eachvalue of ρ . However, they perfectly match the numerical valuesfor ρ ∈ [ {· , ·} , ρ ∗ ] , which is the most relevant region in termsof system design insights. This is a remarkable result since itallows the system designer to design methods to set suitablevalues of ρ on-the-fly, depending on the performance target,without the need for long offline simulations. Quantitatively,the TX power of the FD can be significantly increased ascompared to the state of the art. In fact, a performance increaseup to and is achieved for P = 24 dBm and P = 28 dBm, respectively. In practice, the larger P , thesmaller ρ , i.e., the lower the power of the IC of the receivedsignal and the resulting equivalent SNR. In this sense, thebenefits brought by the proposed architecture are evident, eventhough they decrease as the TX power increases. An in-depthanalysis of the theoretical limits of the performance gain is thematter of our future research. Finally, we note that the impactof P on η R is larger than the impact of d ss . Intuitively, thiscan be explained as follows. According to the adopted pathloss model [39], the received signal at the SN experiences areduction of dB per octave. Conversely, the SI suffered bythe device increases at a much higher pace if the TX poweris augmented, in turn reducing η R . Given the extent of theimpact of P on the performance of the FD radio, we nowconsider P ∈ [23 , dBm and compute the optimal ρ , i.e., ρ ∗ ,for each of the considered values. Subsequently, we computeboth η R and η E when ρ ∗ is adopted, and depict them in Fig. 4as η ∗ R and η ∗ E for consistency with the notation. We start bynoting that η ∗ R > for each of considered configurations.Moreover, as previously inferred, we see that η ∗ R increasesas P decreases. In particular, we observe that a TX powerincrease of dB induces a reduction of and for d ss = 15 m and d ss = 20 m, respectively. Similarly to what hasbeen observed in Fig. 3 for η R , η ∗ R also increases with d ss . Inpractice, the lower the power of the received signal at the SNs, the higher the performance gain the proposed FD architecturecan deliver. This interesting result could have been expected.In fact, the impact of an effective SIC on the achievable rateof the incoming signal is larger when the latter experiencesa severe path loss during its propagation (i.e., the power ofthe received signal by the FD radio is lower). We now switchour focus to η ∗ E . In Fig. 4, its value ranges between . and . for the considered values of P . In other words,up to around of the leaked energy can be recycled ascompared to the state of the art, while guaranteeing betterperformance in terms of throughput. Quantitatively, η ∗ E is notinfluenced by a change of d ss , since it uniquely depends onthe energy of the signal leakage at the circulator. Additionally,we note that the aforementioned values should actually beseen as lower bounds for η E , since they are obtained when ρ ∗ is adopted. In fact, if any other value of ρ within therange [ {· , ·} , ρ ∗ ] was adopted, then the resulting η E would belarger (in turn reducing η R ). This further confirms the potentialof the proposed architecture, whose benefits in the forwardphase can be summarized as: 1) higher achievable rate for thetransmission towards the AN (thanks to the higher TX power),2) higher achievable rate for the incoming signaling, and 3)lower power consumption. B. Backward Phase
Differently from the forward phase, both SNs and ANstransmit during the backward phase. In this regard, we notethat no OFDMA signal is transmitted by the SNs in this phase.Thus, we will lower the values of P considered in the previoussection by dB, to make sure that the same TX power isinvested for the signaling in both phases, and obtain a faircomparison. As before, we start our study by focusing onthe achievable rate for the signaling between the two SNsand compute η R as ρ varies in Fig. 5. Qualitatively, both ρ η R Numerical, d ss = 15 m, 25 dBmNumerical, d ss = 20 m, 25 dBmNumerical, d ss = 15 m, 21 dBmNumerical, d ss = 20 m, 21 dBmApproximated, d ss = 15 m, 25 dBmApproximated, d ss = 20 m, 25 dBmApproximated, d ss = 15 m, 21 dBmApproximated, d ss = 20 m, 21 dBm State-of-the-art{0.3, 0.3}{0.3, 0.3} {0.34, 0.34}{0.34, 0.34}
Figure 5. [Backward] Numerical and approximated η R for the signaling as ρ varies. the values of η R for the signaling and the accuracy of ourapproximations in the backward phase are extremely similarto what we observed for the forward phase. Accordingly, theimpact of P on η R is larger than the impact of d ss , and thelarger P , the lower the resulting η R . However, we note that the performance gain brought by the proposed architecture overthe state of the art in this case is quantitatively lower. Thisresult can be understood by comparing (15) and (37). Thereinwe see that side knowledge about the covariance matrix ofthe interference generated to the signaling by the OFDMtransmission can be safely considered available during theforward phase. The same is not true for the backward phase,where such information is hardly available, unless non-linearand/or complex iterative decoding procedures are adopted atthe SNs. Thus, the achievable rate for the signaling is loweredby the presence of the interference caused by the OFDMsignals transmitted by the ANs. We will further discuss thisaspect in the last study of the section.As a matter of fact, the signals transmitted by the ANsare not only a source of interference for the signaling, butconvey information for the SNs as well. Hence, in the nextstudy we will focus on their achievable rates and compute η R w.r.t. these transmissions, as ρ varies, and depict it in Fig. 6.Interestingly, the qualitative behavior of η R for the backward ρ η R Numerical, 21 dBmNumerical, 25 dBmApproximated, 21 dBmApproximated, 25 dBm
State-of-the-art{0.16, 0.16} {0.32, 0.32}
Figure 6. [Backward] Numerical and approximated η R for the backwardsignals as ρ varies. signals is very similar to the behavior for the signaling inboth forward and backward phases. Remarkably, the accuracyof our approximations is higher in this case, substantiatingeven more their importance in terms of system design insights.Quantitatively, the obtained results are significantly better thanthe performance increase achieved for the signaling in thebackward phase. This could have been expected since nointerference is affecting the OFDMA decoding at the SNs,provided that the signaling satisfy the constraints in (27). Theabsence of interference also implies that the impact of a TXpower change on η R is less evident in this case than in any ofthe previous tests. In particular, we note that the performancegain over the state of the art is rather remarkable, i.e., between and for both considered values of P . Now, as forthe forward phase we conclude the analysis of the backwardphase by letting P ∈ [20 , dBm and compute ρ ∗ for eachof the considered values and transmissions (i.e., signaling andbackward signals). Analogously, η ∗ R and η ∗ E are computed anddepicted in Fig. 7. We first focus on the signaling. As for the P [dBm]
20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25 η ∗ R η ∗ E Signaling, d ss = 20 mSignaling, d ss = 15 mTransmissionRecycled energy Figure 7. [Backward] η ∗ R and η ∗ E as the transmit power P varies. forward phase, η ∗ R decreases as P increases. This is due tothe fact that ρ ∗ decreases as P increases, in turn reducing theeffective SNR of the IC of the received signal, and thus theachievable rate for the signaling. Additionally, as noted forFig. 5, η ∗ R ≈ for P = 25 dBm, due to the aforementionedimpact of the interference generated by the OFDM signalstransmitted by the ANs. We note that this issue is more evidentas the distance between the SNs increases, as intuitively itshould be. Switching our focus to the backward signals, wenote a very consistent behavior of η ∗ R as P increases, asexpected. The decreasing trend of η ∗ R is still present, but witha much lower pace. In a way, this confirms the potential ofthe proposed architecture in terms of the performance gainw.r.t. the state of the art and highlights its merit in termsof overall throughput enhancement for the considered system,regardless of the considered phase of the transmission. Finally,concerning the amount of energy recycled in this phase, wenote that values up to η ∗ E = 3 . are obtained in this phase.In practice, this implies that almost of the leaked energycan be recycled as compared to the state of the art, whileguaranteeing better performance in terms of throughput. Thesame observations made for Fig. 4 in the study of the forwardcase hold for the backward phase as well, hence they will notbe repeated. VII. C ONCLUSIONS
In this work, we have introduced a novel energy-recyclingFD radio architecture that can provide spectral efficiency andenergy consumption improvements over the state of the art.The performance gain is achieved thanks to the introductionof a 3-port element between the circulator and the RX chainincluding a power divider and an RF energy harvester. Theimpact of this element is two-fold. First, it allows for an arbi-trary attenuation of the incoming signal, in turn increasing theeffectiveness of the state-of-the-art SIC strategies subsequentlyadopted in the RX chain. Second, it recycles a non-negligibleportion of the energy leaked through the non-ideal circulator.We have characterized the performance of this architecture in a hybrid FD/HD four-node network, framed according topractically relevant considerations, in which 2 nodes operatein FD and 2 nodes in HD. More specifically, we provideanalytical approximations of: 1) the achievable rates for thetransmissions performed by the FD and HD radios as thedirection of the communication with the HD radios changes,e.g., the forward and backward phases, and 2) the amount ofenergy recycled by the FD radio in these phases. The accuracyof these derivations has been substantiated by our numericalfindings, by which the gains that the proposed architecture canyield over its state-of-the-art alternatives have been illustrated.The next steps of our research are the design of a variantof the proposed FD architecture in the presence of multipleantennas, and an in-depth analysis of the theoretical limits ofthe performance gain brought by its adoption.R EFERENCES[1] Z. Zhang, X. Chai, K. Long, A. Vasilakos, and L. Hanzo, “Full duplextechniques for 5G networks: Self-interference cancellation, protocoldesign, and relay selection,”
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