Hardware Impairments Aware Transceiver Design for Bidirectional Full-Duplex MIMO OFDM Systems
Omid Taghizadeh, Vimal Radhakrishnan, Ali Cagatay Cirik, Rudolf Mathar, Lutz Lampe
11 Hardware Impairments Aware Transceiver Design forBidirectional Full-Duplex MIMO OFDM Systems
Omid Taghizadeh, Vimal Radhakrishnan, Ali Cagatay Cirik,
Member, IEEE , RudolfMathar,
Senior Member, IEEE , Lutz Lampe
Senior Member, IEEE
Abstract —In this paper we address the linear precoding and de-coding design problem for a bidirectional orthogonal frequency-division multiplexing (OFDM) communication system, betweentwo multiple-input multiple-output (MIMO) full-duplex (FD)nodes. The effects of hardware distortion as well as the channelstate information error are taken into account. In the first step,we transform the available time-domain characterization of thehardware distortions for FD MIMO transceivers to the frequencydomain, via a linear Fourier transformation. As a result, theexplicit impact of hardware inaccuracies on the residual self-interference (RSI) and inter-carrier leakage (ICL) is formulatedin relation to the intended transmit/received signals. Afterwards,linear precoding and decoding designs are proposed to enhancethe system performance following the minimum-mean-squared-error (MMSE) and sum rate maximization strategies, assumingthe availability of perfect or erroneous CSI. The proposed designsare based on the application of alternating optimization overthe system parameters, leading to a necessary convergence.Numerical results indicate that the application of a distortion-aware design is essential for a system with a high hardwaredistortion, or for a system with a low thermal noise variance.
Keywords — Full-duplex, MIMO, OFDM, hardware impairments,MMSE.
I. I
NTRODUCTION F ULL-Duplex (FD) transceivers are known for their ca-pability to transmit and receive at the same time andfrequency, and hence have the potential to enhance the spec-tral efficiency [2]. Nevertheless, such systems suffer fromthe inherent self-interference (SI) from their own transmitter.Recently, specialized self-interference cancellation (SIC) tech-niques, e.g., [3]–[6], have demonstrated an adequate level ofisolation between transmit (Tx) and receive (Rx) directions tofacilitate an FD communication and motivated a wide rangeof related studies, see, e.g., [7]–[10]. A common idea of suchSIC techniques is to subtract the dominant part of the SI signal,e.g., a line-of-sight (LOS) SI path or near-end reflections, inthe radio frequency (RF) analog domain so that the remainingsignal can be further processed in the baseband, i.e., digitaldomain. Nevertheless, such methods are still far from perfectin a realistic environment mainly due to i) aging and inherentinaccuracy of the hardware (analog) elements, as well as ii) O. Taghizadeh, V. Radhakrishnan, and R. Mathar are with the Institutefor Theoretical Information Technology, RWTH Aachen University, Aachen,52074, Germany (email: { taghizadeh, radhakrishnan, mathar } @ti.rwth-aachen.de).A. C. Cirik and L. Lampe are with the Department of Electrical andComputer Engineering, University of British Columbia, Vancouver, BC V6T1Z4, Canada (email: { cirik, lampe } @ece.ubc.ca).Part of this paper has been presented at the 28th IEEE Annual Interna-tional Symposium on Personal, Indoor, and Mobile Radio Communications(PIMRC’17) [1]. inaccurate channel state information (CSI) in the SI path,due to noise and limited channel coherence time. In thisregard, inaccuracy of the analog hardware elements used insubtracting the dominant SI path in RF domain may result insevere degradation of SIC quality. This issue becomes morerelevant in a realistic scenario, where unlike the demonstratedsetups in the lab environment, analog components are proneto aging, temperature fluctuations, and occasional physicaldamage. Moreover, an FD link is vulnerable to CSI inaccuracyat the SI path in environments with a small channel coherencetime, see [11, Subsection 3.4.1]. A good example of suchchallenge is a high-speed vehicle that passes close to an FDdevice, and results in additional reflective SI paths .In order to combat the aforementioned issues, an FDtransceiver may adapt its transmit/receive strategy to the ex-pected nature of CSI inaccuracy, e.g., by directing the transmitbeams away from the moving objects or operating in thedirections with smaller impact of CSI error. Moreover, theaccuracy of the transmit/receiver chain elements can be con-sidered, e.g., by dedicating less power, or ignoring the chainswith noisier elements in the signal processing. In this regard,a widely used model for the operation of a multiple-antennaFD transceiver is proposed in [12], assuming a single carriercommunication system, where CSI inaccuracy as well as theimpact of hardware impairments are taken into account. Agradient-projection-based method is then proposed in the samework for maximizing the sum rate in an FD bidirectional setup.Building upon the proposed benchmark, a convex optimizationdesign framework is introduced in [13], [14] by defining aprice/threshold for the SI power, assuming the availabilityof perfect CSI and accurate transceiver operation. Whilethis approach reduces the design computational complexity,it does not provide a reliable performance for a scenariowith erroneous CSI, particularly regarding the SI path [15].Consequently, the consideration of CSI and transceiver errorin an FD bidirectional system is further studied in [16], [17]by maximizing the system sum rate, in [18] by minimizingthe sum mean-squared-error (MSE), and in [19], [20] forminimizing the system power consumption under a requiredquality of service.The aforementioned works focus on modeling and designmethodologies for single-carrier FD bidirectional systems,under frequency-flat channel assumptions. In this regard, theimportance of extending the previous designs for a multi-carrier (MC) system with a frequency selective channel isthreefold. Firstly, due to the increasing rate demand of the Since the object is moving rapidly, the reflective paths are more difficultto be accurately estimated. a r X i v : . [ c s . I T ] F e b wireless services, and following the same rationale for thepromotion of FD systems, the usage of larger bandwidths be-comes necessary. This, in turn, invalidates the usual frequency-flat assumption and calls for updated design methodologies.Secondly, unlike the half-duplex (HD) systems where theoperation of different subcarriers can be safely separated inthe digital domain, an FD system is highly prone to theinter-carrier leakage (ICL) due to the impact of hardwaredistortions on the strong SI channel . This, in particular,calls for a proper modeling of the ICL as a result of non-linear hardware distortions for FD transceivers. And finally,the channel frequency selectivity shall be opportunisticallyexploited, by means of a joint design of the linear transmitand receive strategies at all subcarriers, in order to enhancethe system performance. A. Related works on FD MC systems
In the early work by Riihonen et al. [21], the performanceof a combined analog/digital SIC scheme is evaluated foran FD orthogonal-frequency-division-multiplexing (OFDM)transceiver, taking into account the impact of hardware distor-tions, e.g., limited analog-to-digital convertor (ADC) accuracy.The problem of resource allocation and performance analysisfor FD MC communication systems is then addressed in[22]–[28], however, assuming a single antenna transceiver.Specifically, an FD MC system is studied in [22]–[24] inthe context of FD relaying, in [26], [27] and [25] in thecontext of FD cellular systems with non-orthogonal multipleaccess (NOMA) capability, and in [28] for rate region analysisof a hybrid HD/FD link. Moreover, an MC relaying systemwith hybrid decode/amplify-and-forward operation is studiedin [29], with the goal of maximizing the system sum ratevia scheduling and resource allocation. However, in all of theaforementioned designs, the behavior of the residual SI signalis modeled as a purely linear system. As a result, the impactsof the hardware distortions leading to ICL, as observed in [21],are neglected.
B. Contribution and paper organization
In this paper we study a bidirectional FD MIMO OFDMsystem , where the impacts of hardware distortions leading toimperfect SIC and ICL are taken into account. Our main con-tributions, together with the paper organization are summarizedas follows: leftmargin=* • In the seminal work by Day et al. [12], an FD MIMOtransceiver is modeled considering the impacts of hard-ware distortions in transmit/receiver chains, which isthen extensively used for the purpose of FD systemdesign and performance analysis, e.g., [16], [19], [30]–[34]. In the first step, we extend the available time-domain characterization of hardware distortions into anFD MIMO OFDM setup via a linear discrete Fourier For instance, a high-power transmission in one of the subcarriers will resultin a higher residual self-interference (RSI) in all of the sub-channels due to,e.g., a higher quantization and power amplifier noise levels. The modeling and the obtained design frameworks can be applied also forany multi-carrier system with orthogonal waveforms, i.e., with zero intrinsicinterference. transformation. The obtained frequency-domain charac-terization reveals the statistics of the RSI and ICL, inrelation to the intended transmit/receive signals at eachsubcarrier. Please note that this is in contrast to theavailable prior works on FD MC systems [22]–[29],where ICL is neglected and RSI signal is modeled viaa purely linear system. • Building on the obtained characterization, linear trans-mit/receive strategies are proposed in order to enhancethe system performance. In Section III, an alternatingquadratic convex program (QCP), denoted as AltQCP, isproposed in order to obtain a minimum weighted MSEtransceiver design. The known weighted-minimum-MSE(WMMSE) method [35] is then utilized to extend theAltQCP framework for maximizing the system sum rate.For both algorithms, a monotonic performance improve-ment is observed at each step, leading to a necessaryconvergence. • In Section IV, we extend the studied system to anasymmetric OFDM FD bidirectional setup, where anFD transceiver with a large antenna array simultane-ously communicates with multiple single-antenna FDtransceivers. The extended scenario is particularly rel-evant, both due to the recent advances in building FDmassive MIMO transceivers [36] as well as the signifiedimpact of hardware distortions due to the lower per-element cost (e.g., low resolution quantization [37]). Analgorithm for joint power and subcarrier allocation isthen proposed, following the successive inner approxi-mation (SIA) framework [38], with a guaranteed con-vergence to a solution satisfying Karush–Kuhn–Tucker(KKT) conditions. • In Section V the proposed design in Section III isextended by also taking into account the impact ofCSI error. In particular, a worst-case MMSE design isproposed as an alternating semi definite program (SDP),denoted as AltSDP. Similar to the previous methods,a monotonic performance improvement is observed ateach step, leading to a necessary convergence. Moreover,a methodology to obtain the most destructive CSI errormatrices is proposed. This is done by converting theresulting non-convex quadratic problem into a convexprogram, in order to facilitate worst-case performanceanalysis under CSI error.Numerical simulations show that the application of adistortion-aware design is essential, as transceiver accuracydegrades, and ICL becomes a dominant factor.
C. Mathematical Notation
Throughout this paper, column vectors and matrices aredenoted as lower-case and upper-case bold letters, respec-tively. Mathematical expectation, trace, inverse, determinant,transpose, conjugate and Hermitian transpose are denoted by E {·} , tr ( · ) , ( · ) − | · | , ( · ) T , ( · ) ∗ and ( · ) H , respectively. TheKronecker product is denoted by ⊗ . The identity matrix withdimension K is denoted as I K and vec ( · ) operator stacks theelements of a matrix into a vector. m × n represents an all-zero matrix with size m × n . (cid:107) · (cid:107) and (cid:107) · (cid:107) F respectively Table I. U
SED SYMBOLS k, i, l index of a subcarrier, communication direction,and a transmit/receive chain I , V , U set of comm. directions, precoder and decoder matrices N i , M i , d i number of transmit and receive antennas and data streams s ki (˜ s ki ) transmitted (estimated) data symbol U ik ( V ik ) linear decoder (precoder) matrix y ki (˜ y ki ) received signal before (after) SI cancellation H kij , ˜ H kij the exact, and estimated CSI matrix ∆ kij , D kij CSI error, and the set of feasible CSI errors ζ kij , D kij radius and shaping matrix for the feasible CSI error region e k r ,i ( e k t ,i ) receiver (transmitter) distortion over the subcarrier k Θ rx ,i ( Θ trx ,i ) diagonal matrix of receive (transmit) distortion coefficients ν ki , Σ ki collective residual SI plus noise signal, and its covariance n ki , σ i,k additive thermal noise and its variance u ki ( v ki ) undistorted received (transmitted) signal x ki , P i transmit signal, and the maximum transmit power represent the Euclidean and Frobenius norms. diag ( · ) returnsa diagonal matrix by putting the off-diagonal elements tozero. (cid:98) A i (cid:99) i =1 ,...,K denotes a tall matrix, obtained by stackingthe matrices A i , i = 1 , . . . , K . R{ A } represents the range(column space) of the matrix A . The set F K is definedas { , . . . , K } . The set of real, positive real, and complexnumbers are respectively denoted as R , R + , C .II. S YSTEM M ODEL
A bidirectional OFDM communication between two MIMOFD transceivers is considered. Each communication directionis associated with N i transmit and M i receive antennas, where i ∈ I , and I := { , } represents the set of the communicationdirections. The desired channel in the communication direction i and subcarrier k ∈ F K is denoted as H kii ∈ C M i × N i where K is the number of subcarriers. The interference channel from i to j -th communication direction is denoted as H kji ∈ C M j × N i .All channels are quasi-static , and frequency-flat in eachsubcarrier.The transmitted signal in the direction i , subcarrier k isformulated as x ki = V ki s ki (cid:124) (cid:123)(cid:122) (cid:125) =: v ki + e k t ,i , (cid:88) k ∈ F K E (cid:8) (cid:107) x ki (cid:107) (cid:9) ≤ P i , (1)where s ki ∈ C d i and V ki ∈ C N i × d i respectively representthe vector of the data symbols and the transmit precodingmatrix, and P i ∈ R + imposes the maximum affordabletransmit power constraint. The number of the data streamsin each subcarrier and in direction i is denoted as d i , and E { s ki s ki H } = I d i . Moreover, v ki ∈ C N i represents the desiredsignal to be transmitted, where e k t ,i models the inaccuratebehavior of the transmit chain elements, i.e, transmit distortion,see Subsection II-A for more details.The received signal at the destination can be consequentlywritten as y ki = H kii x ki + H kij x kj + n ki (cid:124) (cid:123)(cid:122) (cid:125) =: u ki + e k r ,i , (2) It indicates that the channel is constant in each communication frame, butmay vary from one frame to another frame. where n ki ∼ CN (cid:16) M i , σ i,k I M i (cid:17) is the additive thermal noise.Similar to the transmit signal model, e k r ,i represents the receiverdistortion and models the inaccuracies of the receive chainelements. The known , i.e., distortion-free, part of the SI isthen subtracted from the received signal, employing an SICscheme. This is formulated as ˜ y ki : = y ki − H kij V kj s kj = H kii V ki s ki + ν ki , (3)where ˜ y ki is the received signal in direction i and subcarrier k , after SIC. Moreover, the aggregate interference-plus-noiseterm is denoted as ν ki ∈ C M i , where ν ki = H kij e k t ,j + H kii e k t ,i + e k r ,i + n ki , j (cid:54) = i. (4)Finally, the estimated data vector is obtained at the receiver as ˜ s ki = (cid:16) U ik (cid:17) H ˜ y ki , (5)where U ki ∈ C M i × d i is the linear receive filter. A. Limited dynamic range in an FD OFDM system
In the seminal work by Day et al. [12], a model for theoperation of an FD MIMO transceiver is given, relying onthe experimental results on the impact of hardware distortions[39]–[42]. In this regard, the inaccuracy of the transmit chainelements, e.g., DAC error, PA and oscillator phase noise, arejointly modeled for each antenna as an additive distortion, andwritten as x l ( t ) = v l ( t ) + e t ,l ( t ) , see Fig. 1, such that e t ,l ( t ) ∼ CN (cid:16) , κ l E (cid:8) | v l ( t ) | (cid:9)(cid:17) , (6) e t ,l ( t ) ⊥ v l ( t ) , e t ,l ( t ) ⊥ e t ,l (cid:48) ( t ) , e t ,l ( t ) ⊥ e t ,l ( t (cid:48) ) ,l (cid:54) = l (cid:48) ∈ L T , t (cid:54) = t (cid:48) , (7)please see [12, Section II. B,C], [30, Section II. C,D], [16],[19], [31]–[33] for a similar distortion characterization for FDtransceivers . In the above arguments, t denotes the instanceof time, and v l , x l , and e t ,l ∈ C are respectively the basebandtime-domain representation of the intended transmit signal, theactual transmit signal, and the additive transmit distortion atthe l -th transmit chain. The set L T represents the set of alltransmit chains. Moreover, κ l ∈ R + represents the distortioncoefficient for the l -th transmit chain, relating the collectivepower of the distortion signal, over the active spectrum, to theintended transmit power.In the receiver side, the combined effects of the inaccuratehardware elements, i.e., ADC error, AGC and oscillator phasenoise, are presented as additive distortion terms and written as It is worth mentioning that the accuracy of the above-mentioned modelingvaries for different implementations of FD transceivers, depending on the com-plexity and the used SIC method. In this regard, the statistical independenceof distortion elements defined in (iii) and (iv) also hold for an advancedimplementation of an FD transceiver, assuming a high signal processingcapability. This is since any correlation structure in the distortion signal canbe exploited and removed in order to reduce the RSI via advanced signalprocessing, see [4, Subsection 3.2]. However, the linear dependence of theremaining distortion signal variance to the desired signal strength varies fordifferent SIC implementations, and should be estimated separately for eachtransceiver. y l ( t ) = u l ( t ) + e r ,l ( t ) such that e r ,l ( t ) ∼ CN (cid:16) , β l E (cid:8) | u l ( t ) | (cid:9)(cid:17) , (8) e r ,l ( t ) ⊥ u l ( t ) , e r ,l ( t ) ⊥ e r ,l (cid:48) ( t ) , e r ,l ( t ) ⊥ e r ,l ( t (cid:48) ) ,l (cid:54) = l (cid:48) ∈ L R , t (cid:54) = t (cid:48) , (9)where u l , e r ,l , and y l ∈ C are respectively the basebandrepresentation of the intended (distortion-free) received signal,additive receive distortion, and the received signal from the l -th receive antenna. The set L R represents the set of all receivechains. Similar to the transmit chain characterization, β l ∈ R + is the distortion coefficient for the l -th receive chain, seeFig. 1. For each communication block, the frequency domainrepresentation of the sampled time domain signal is obtainedas x kl = 1 √ K K − (cid:88) m =0 x l ( mT s ) e − j πmkK =1 √ K K − (cid:88) m =0 v l ( mT s ) e − j πmkK (cid:124) (cid:123)(cid:122) (cid:125) =: v kl + 1 √ K K − (cid:88) m =0 e t ,l ( mT s ) e − j πmkK , (cid:124) (cid:123)(cid:122) (cid:125) =: e k t ,l (10)and y kl = 1 √ K K − (cid:88) m =0 y l ( mT s ) e − j πmkK =1 √ K K − (cid:88) m =0 u l ( mT s ) e − j πmkK (cid:124) (cid:123)(cid:122) (cid:125) =: u kl + 1 √ K K − (cid:88) m =0 e r ,l ( mT s ) e − j πmkK , (cid:124) (cid:123)(cid:122) (cid:125) =: e k r ,l (11)where T s is the sampling time, and KT s is the OFDM blockduration prior to the cyclic extension, see [43] for a detaileddiscussion on OFDM technology. Lemma II.1.
The impact of hardware distortions in thefrequency domain is characterized as e k t ,l ∼ CN (cid:32) , κ l K (cid:88) m ∈ F K E (cid:110) | v ml | (cid:111)(cid:33) , e k t ,l ⊥ v kl , e k t ,l ⊥ e k t ,l (cid:48) , (12) e k r ,l ∼ CN (cid:32) , β l K (cid:88) m ∈ F K E (cid:110) | u ml | (cid:111)(cid:33) , e k r ,l ⊥ u kl , e k r ,l ⊥ e k r ,l (cid:48) , (13) transforming the statistical independence, as well as the pro-portional variance properties from the time domain.Proof: See the Appendix.The above lemma indicates that the distortion signal vari-ance at each subcarrier, relates to the total distortion powerat the corresponding chain, indicating the impact of ICL.This can be interpreted as a variance-dependent thermal noise,where the temporal independence of signal samples results ina flat power spectral density over the active communicationbandwidth. In this part we consider a general framework
Tx chain Rx chain
Figure 1. Limited dynamic range is modeled by injecting additive distortionterms at each transmit or receive chain. e t ,l and e r ,l denote the distortionterms, and n l represents the additive thermal noise. where the transmit (receive) distortion coefficients are notnecessarily identical for all transmit (receive) chains belongingto the same transceiver, i.e., different chains may hold differentaccuracy due to occasional damage and aging. This assumptionis relevant in practice since it enables the design algorithms toreduce communication task, e.g., transmit power, on the chainswith noisier elements. Following Lemma II.1, the statistics ofthe distortion terms, introduced in (1), (2) can be inferred as e k t ,i ∼ CN (cid:32) N i , Θ tx ,i (cid:88) k ∈ F K diag (cid:16) E (cid:110) v ki v ki H (cid:111)(cid:17)(cid:33) , (14) e k r ,i ∼ CN (cid:32) M i , Θ rx ,i (cid:88) k ∈ F K diag (cid:16) E (cid:110) u ki u ki H (cid:111)(cid:17)(cid:33) , (15)where Θ tx ,i ∈ R N i × N i ( Θ rx ,i ∈ R M i × M i ) is a diagonalmatrix including distortion coefficients κ l /K ( β l /K ) for thecorresponding chains .Via the application of (14)-(15) on (4), the covariance of thereceived collective interference-plus-noise signal is obtained as Σ ki := E (cid:110) ν ki ν ki H (cid:111) ≈ (cid:88) j ∈ I H kij Θ tx ,j diag (cid:88) l ∈ F K V lj V ljH H kijH + σ i,k I M i + Θ rx ,i diag (cid:18) (cid:88) l ∈ F K (cid:18) σ i,l I M i + (cid:88) j ∈ I H lij V lj V ljH H lijH (cid:19)(cid:19) , (16) where Σ ki ∈ C M i × M i is obtained considering ≤ β l (cid:28) , ≤ κ l (cid:28) , and hence ignoring the terms containing higherorders of the distortion coefficients in (16). B. Remarks • In this section, we have assumed the availability ofperfect CSI and focused on the impact of non-lineartransceiver distortions. This assumption is relevant forthe scenarios with stationary channel, e.g., a backhauldirective link with zero mobility [44], where an ade-quately long training sequence can be applied, see [12,Subsection III.A]. The impact of the CSI inaccuracy islater addressed in Section V. • As expected, the role of the distortion signals on theRSI, including the resulting ICL, is evident from (16).It is the main goal of the remaining parts of this chapterto incorporate and evaluate this impact on the design ofthe defined MC system. A simpler mathematical presentation can be obtained by assuming thesame transceiver accuracy over all antennas, similar to [12], [30]. In such acase, the defined diagonal matrices can be replaced by a scalar.
III. L
INEAR T RANSCEIVER D ESIGN FOR M ULTI -C ARRIER C OMMUNICATIONS
Via the application of V ki and U ki , as the linear transmitprecoder and receive filters, the mean-squared-error (MSE)matrix of the defined system is calculated as E ki : = E (cid:110)(cid:0) ˜ s ki − s ki (cid:1) (cid:0) ˜ s ki − s ki (cid:1) H (cid:111) = (cid:16) U ki H H kii V ki − I d i (cid:17) (cid:16) U ki H H kii V ki − I d i (cid:17) H + U ki H Σ ki U ki , (17)where Σ ki is given in (16). In the following we proposetwo design strategies for the defined system, proposing analternating QCP framework. A. Weighted MSE minimization via Alternating QCP (AltQCP)
An optimization problem for minimizing the weighted sumMSE is written asmin V , U (cid:88) i ∈ I (cid:88) k ∈ F K tr (cid:0) S ki E ki (cid:1) (18a)s.t. tr (cid:18) ( I N i + K Θ tx ,i ) (cid:88) l ∈ F K V li V liH (cid:19) ≤ P i , ∀ i ∈ I , (18b)where X := { X ki , ∀ i ∈ I , ∀ k ∈ F K } , with X ∈ { U , V } ,and (18b) represents the transmit power constraint. It is worthmentioning that the application of S ki (cid:31) , as a weight matrixassociated with E ki is two-folded. Firstly, it may appear asa diagonal matrix, emphasizing the importance of differentdata streams and different users. Secondly, it can be applied asan auxiliary variable which later relates the defined weightedMSE minimization to a sum-rate maximization problem, seeSubsection III-B.It is observed that (18) is not a jointly convex problem.Nevertheless, it holds a QCP structure separately over thesets V and U , in each case when other variables are fixed.In this regard, the objective (18a) can be decomposed over U for different communication directions, and for differentsubcarriers. The optimal minimum MSE (MMSE) receive filtercan be hence calculated in closed form as U ki, mmse = (cid:16) Σ ki + H kii V ki V ki H H kiiH (cid:17) − H kii V ki . (19)Nevertheless, the defined problem is coupled over V ki , dueto the impact of inter-carrier leakage, as well as the powerconstraint (18b). The Lagrangian function, corresponding tothe optimization (18) over V is expressed as L ( V , ι ) := (cid:88) i ∈ I (cid:18) ι i P i ( V ) + (cid:88) k ∈ F K tr (cid:0) S ki E ki (cid:1) (cid:19) , (20) P i ( V ) := − P i + tr (cid:18) ( I N i + K Θ tx ,i ) (cid:88) l ∈ F K V li V liH (cid:19) , (21)where ι := { ι i , i ∈ I } is the set of dual variables. The dualfunction, corresponding to the above Lagrangian is defined as F ( ι ) : = min V L ( V , ι ) (22) where the optimal V ki is obtained as V ki (cid:63) = (cid:16) J ki + ι i ( I N i + K Θ tx ,i ) + H kiiH U ki S ki U ki H H kii (cid:17) − × H kiiH U ki S ki , (23)and J ki : = (cid:88) l ∈ F K (cid:88) j ∈ I (cid:18) H kjiH diag (cid:16) U lj S lj U ljH Θ rx ,j (cid:17) H kji + diag (cid:16) H ljiH U lj S lj U ljH H lji Θ tx ,i (cid:17) (cid:19) . (24)Due to the convexity of the original problem (18) over V , thedefined dual problem is a concave function over ι , with P i ( V ) as a subgradient, see [45, Eq. (6.1)]. As a result, the optimal ι is obtained from the maximization ι (cid:63) = argmax ι ≥ F ( ι ) , (25)following a standard subgradient update, [45, Subsec-tion 6.3.1].Utilizing the proposed optimization framework, the alternat-ing optimization over V and U is continued until a stable pointis obtained. Note that due to the monotonic decrease of theobjective in each step, and the fact that (18a) is non-negativeand hence bounded from below, the defined procedure leadsto a necessary convergence. Algorithm 1 defines the necessaryoptimization steps. Algorithm 1
Alternating QCP (AltQCP) for weighted MSE mini-mization (cid:96) ← (set iteration number to zero) V ← right singular matrix initialization, see [46, Appendix A] U ← solve (19) repeat (cid:96) ← (cid:96) + 1 V ← solve (23) or QCP (18), with fixed U U ← solve (19) or QCP (18) with fixed V until a stable point, or maximum number of (cid:96) reached return { U , V } B. Weighted MMSE (WMMSE) design for sum rate maximiza-tion
Via the utilization of V ki as the transmit precoders, theresulting communication rate for the k -th subcarrier and forthe i -th communication direction is written as I ki = log (cid:12)(cid:12)(cid:12) I d i + V ki H H kiiH (cid:0) Σ ki (cid:1) − H kii V ki (cid:12)(cid:12)(cid:12) , (26)where Σ ki is defined in (16). The sum rate maximizationproblem can be hence presented asmax V (cid:88) i ∈ I (cid:88) k ∈ F K ω i I ki , s.t. (18b) . (27)where ω i ∈ R + is the weight associated with the communica-tion direction i . The optimization problem (27) is intractablein the current form. In the following we propose an iterativeoptimization solution, following the WMMSE method [35].Via the application of the MMSE receive linear filters from (19), the resulting MSE matrix is obtained as E ki, mmse = (cid:16) I d i + V ki H H kiiH (cid:0) Σ ki (cid:1) − H kii V ki (cid:17) − . (28)By recalling (26), and upon utilization of U ki, mmse , we observethe following useful connection to the rate function I ki = − log (cid:12)(cid:12) E ki, mmse (cid:12)(cid:12) , (29)which facilitates the decomposition of rate function via thefollowing lemma, see also [35, Eq. (9)]. Lemma III.1.
Let E ∈ C d × d be a positive definite matrix.The maximization of the term − log | E | is equivalent to themaximization max E , S − tr ( SE ) + log | S | + d, (30) where S ∈ C d × d is a positive definite matrix, and we have S = E − , (31) at the optimality.Proof: See [47, Lemma 2].By recalling (29), and utilizing Lemma III.1, the originaloptimization problem over V can be equivalently formulatedasmax V , U , S (cid:88) i ∈ I ω i (cid:88) k ∈ F K (cid:18) log (cid:12)(cid:12) S ki (cid:12)(cid:12) + d i − tr (cid:0) S ki E ki (cid:1) (cid:19) s.t. (18b) , (32)where S := { S ki (cid:31) , ∀ i ∈ I , ∀ k ∈ F K } . The obtainedoptimization problem (32) is not a jointly convex problem.Nevertheless, it is a QCP over V when other variables arefixed, and can be obtained with a similar structure as for(18). Moreover, the optimization over U and S is respectivelyobtained from (19), and (31) as S ki = E ki − . This facilitates analternating optimization where in each step the correspondingproblem is solved to optimality, see Algorithm 2. The definedalternating optimization steps results in a necessary conver-gence due to the monotonic increase of the objective in eachstep, and the fact that the eventual system sum rate is boundedfrom above. Algorithm 2
AltQCP-WMMSE design for sum rate maximization Algorithm 1, Steps 1-2 (initialization) repeat Algorithm 1, Steps 5-7 S ← S ki = (cid:0) E ki (cid:1) − until a stable point, or maximum number of (cid:96) reached return { V } IV. B
IDIRECTIONAL
FD M
ASSIVE
MIMO S
YSTEMS :J OINT P OWER AND S UBCARRIER A LLOCATION
In this part, we extend the studied system into an asymmetricsetup, where an FD transceiver equipped with a large antennaarray (e.g., a basestation) performs a bidirectional commu-nication with multiple FD single-antenna nodes (e.g., users).Thanks to the FD capability and multi-user beamforming, thecommunication at different directions can flexibly coexist onshared subcarriers, improving the spectral efficiency, or canbe accommodated on different subcarriers in order to control the interference. Please note that the impact of hardwareimpairments is known to be significant for a system with alarge antenna array, due to the lower per-element cost, e.g.,low resolution ADC and DAC [37]. This signifies the role ofthe characterization in Lemma II.1 regarding the impact ofhardware impairments for an FD MIMO OFDM system. Inorder to extend the defined setup to an asymmetric one, wedenote the set of communication directions from (to) the usersto (from) the massive MIMO transceiver as I UL ( I DL ), suchthat I = I UL ∪ I DL . Moreover, the lower-case notations ( ˜ f ki ) f ki and u ki are used to represent the (normalized) transmit andreceive linear filters . Moreover, we have f ki = ˜ f ki √ p i,k where p i,k denotes the transmit power. In this part, we perform a jointsubcarrier and power allocation with the goal of maximizingthe system sum rate. An upper bound on the achievableinformation rate is obtained as R UB i,k = γ log (cid:18) (cid:12)(cid:12)(cid:12)(cid:0) u ki (cid:1) H h kii ˜ f mi (cid:12)(cid:12)(cid:12) p i,k σ i,k + (cid:80) j ∈ I (cid:80) m ∈ F K γ kmij p j,m (cid:19) (33)where < γ < indicates the portion of the frame durationdedicated to data communication, and γ kmij : = δ klij (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) u ki (cid:17) H H mij ˜ f mj (cid:12)(cid:12)(cid:12)(cid:12) (cid:124) (cid:123)(cid:122) (cid:125) co-channel interference ++ (cid:16) u ki (cid:17) H H kij Θ tx ,j diag (cid:18) ˜ f mj (cid:16) ˜ f mj (cid:17) H (cid:19) (cid:16) H kij (cid:17) H u ki (cid:124) (cid:123)(cid:122) (cid:125) transmitter distortion + (cid:16) u ki (cid:17) H Θ rx ,i diag (cid:18) H mij ˜ f mj (cid:16) ˜ f mj (cid:17) H (cid:0) H mij (cid:1) H (cid:19) u ki (cid:124) (cid:123)(cid:122) (cid:125) receiver distortion , (34) where δ kmij = 0 if k (cid:54) = m or j ∈ I DL , i ∈ I UL and otherwise δ kmij = 1 . Please note that the given upper bound in (33)is obtained similar to [49] assuming an accurate CSI, pleasesee Section V for the consideration of CSI error. It is worthmentioning the impact of hardware distortions on the RSI,as well as the ICL is evident from (34) . The optimizationproblem for maximizing the system sum rate is formulated as The channel dimensions are accordingly obtained as H kij ∈ C when i ∈ I UL , j ∈ I DL , h kii ∈ C ˜ M ( C × ˜ N ) when i ∈ I UL ( I DL ) , and H kji ∈ C ˜ M × ˜ N when i ∈ I DL , j ∈ I UL . ˜ N ( ˜ M ) represent the number of transmit(receive) antennas at the massive MIMO transceiver. Due to the properties of the large antenna arrays, the transmit precoder andreceive filters are usually chosen via a maximum ratio transmission/combining(MRT/MRC) strategy [37], a projection to the null-space of the SI channel [36]or via a joint user and SI spatial zero-forcing [48], [49], resulting in a differentperformance-complexity tradeoff. In particular to a massive MIMO transceiver, where low-resolution quanti-zation is used, the distortion coefficient κ l in Θ tx ,i (and similarly β l in Θ rx ,i )is obtained as κ l [dB] = − . b l , where b l is the number of quantizationbits at the chain l . max p i,k ≥ (cid:88) i ∈ I ω i (cid:88) k ∈ F K R UB i,k , (35a)s.t. (cid:88) k ∈ F K p i,k ≤ P i , i ∈ I UL , (cid:88) i ∈ I DL (cid:88) k ∈ F K p i,k ≤ P i , i ∈ I DL . (35b)It can be observed that (35) is not a jointly convex optimizationproblem. However, it falls into the class of smooth difference-of-convex (DC) optimization problems. In this regard, wepropose an iterative optimization, following the successiveinner approximation (SIA) framework [38] which is provento converge to a point satisfying KKT optimality conditions.Let p i,k, be a feasible transmit power value. Then, employingthe first order Taylor’s approximation on the concave terms,the value of R UB i,k is lower-bounded as R UB i,k ≥ γ log (cid:18) (cid:13)(cid:13)(cid:13) h kii (cid:13)(cid:13)(cid:13) p i,k + (cid:88) j ∈ I (cid:88) m ∈ F K γ kmij p j,m + σ i,k (cid:19) − γ log (cid:18) (cid:88) j ∈ I (cid:88) m ∈ F K γ kmij p j,m, + σ i,k (cid:19) − γ (cid:80) j ∈ I (cid:80) m ∈ F K γ kmij ( p j,m − p j,m, ) log (2) (cid:80) j ∈ I (cid:80) m ∈ F K γ kmij p j,m, + σ i,k =: R UB i,k , (36) where R UB i,k is a jointly concave function over p i,k , facilitat-ing an iterative update where in each iteration the convexproblem max p i,k ≥ (cid:80) i ∈ I (cid:80) k ∈ F K R UB i,k s.t. (35b) is solved to theoptimality. The proposed iterative update is continued untila stable solution is obtained. It can be observed that R UB i,k represents a tight and global lower bound to R UB i,k , with a sharedslope at the point of approximation p j,m, . As a result, theproposed iterative update follows the requirements set in [38,Theorem 1], with a proven convergence to a solution satisfyingthe KKT conditions.V. R OBUST D ESIGN WITH I MPERFECT
CSIIn many realistic scenarios the CSI matrices can not beestimated or communicated accurately due to the limitedchannel coherence time as a result of, e.g., reflections froma moving object, or due to dedicating limited resource on thetraining/feedback process. This issue becomes more significantin an FD system, due to the strong SI channel which callsfor dedicated silent times for tuning and training process,see [12, Subsection III.A]. In particular, the impact of CSIerror on the defined MC FD system is three-fold. Firstly,similar to the usual HD scenarios, it results in the erroneousequalization in the receiver, as the communication channelsare not accurately known. Secondly, it results in an inaccurateestimation of the received signal from the SI path, and therebydegrades the SIC quality. Finally, due to the CSI error, theimpact of the distortion signals may not be accurately known,as the statistics of the distortion signals directly depend on thechannel situation. In this part we extend the proposed designsin Section III where the aforementioned uncertainties, resultingfrom CSI error, are also taken into account. This is directly concluded for a first-order Taylor’s approximation on anysmooth convex function [50].
A. Norm-bounded CSI error
In this part we update the defined system model in Section IIto the scenario where the CSI is known erroneously. In thisrespect we follow the so-called deterministic model [51],where the error matrices are not known but located, with asufficiently high probability, within a known feasible errorregion . This is expressed as H kij = ˜ H kij + ∆ kij , ∆ kij ∈ D kij , i, j ∈ I , (37)and D kij := (cid:8) ∆ kij (cid:12)(cid:12) (cid:107) D kij ∆ kij (cid:107) F ≤ ζ kij (cid:9) , ∀ i, j ∈ I , k ∈ F K , (38)where ˜ H kij is the estimated channel matrix and ∆ kij representsthe channel estimation error. Moreover, D kij (cid:23) and ζ kij ≥ jointly define a feasible ellipsoid region for ∆ kij which gener-ally depends on the noise and interference statistics, and theused channel estimation method. For further elaboration on theused error model see [51] and the references therein.The aggregate interference-plus-noise signal at the receiver ishence updated as ν ki = H kij e k t ,j + H kii e k t ,i + e k r ,i + ∆ kij V kj s kj + n ki , j (cid:54) = i ∈ I , (39)where Σ ki , representing the covariance of ν ki , is expressed in(40). B. Alternating SDP (AltSDP) for worst-case MSE minimiza-tion
An optimization problem for minimizing the worst-caseMSE under the defined norm-bounded CSI error is writtenas min V , U max C (cid:88) i ∈ I (cid:88) k ∈ F K tr (cid:0) S ki E ki (cid:1) , s.t. (18b) , ∆ kij ∈ D kij , ∀ i, j ∈ I , k ∈ F K , (41)where C := { ∆ kij , ∀ i, j ∈ I , ∀ k ∈ F K } , and E ki isobtained from (17) and (40). Note that the above problem isintractable, due to the inner maximization of quadratic convexobjective over C , which also invalidates the observed convexQCP structure in (18). In order to formulate the objective into The feasible error region can be obtained from the statistical distributionof the true CSI values, as a minimum radius ball or ellipsoid containing thetrue CSI values with a desired confidence probability, or via the knowledge ofthe CSI quantization strategy, in case the CSI error is dominated by feedbackquantization. Σ ki = ∆ kij V kj V kj H ∆ kijH + (cid:88) j ∈ I H kij Θ tx ,j diag (cid:88) l ∈ F K V lj V ljH H kijH + Θ rx ,i diag (cid:18) (cid:88) l ∈ F K (cid:18) σ i,l I M i + (cid:88) j ∈ I H lij V lj V ljH H lijH (cid:19)(cid:19) + σ i,k I M i . (40) a tractable form, we calculate (cid:88) k ∈ F K tr (cid:16) S ki E ki (cid:17) = (cid:88) k ∈ F K (cid:32) (cid:13)(cid:13)(cid:13) W ki H (cid:16) U ki H H kii V ki − I d i (cid:17)(cid:13)(cid:13)(cid:13) F + (cid:13)(cid:13)(cid:13) W ki H U ki H ∆ ki − i V k − i (cid:13)(cid:13)(cid:13) F + σ i,k (cid:13)(cid:13)(cid:13) W ki H U ki H (cid:13)(cid:13)(cid:13) F + (cid:88) j ∈ I (cid:88) l ∈ F Nj (cid:88) m ∈ F K (cid:13)(cid:13)(cid:13) W ki H U ki H H kij ( Θ tx ,j ) Γ lN j V mj (cid:13)(cid:13)(cid:13) F + (cid:88) j ∈ I (cid:88) l ∈ F Mi (cid:88) m ∈ F K (cid:13)(cid:13)(cid:13) W ki H U ki H ( Θ rx ,i ) Γ lM i H mij V mj (cid:13)(cid:13)(cid:13) F + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ki H U ki H Θ rx ,i (cid:88) q ∈ F K σ i,q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F (cid:33) (42) = (cid:88) j ∈ I (cid:88) k ∈ F K (cid:13)(cid:13)(cid:13) c kij + C kij vec (cid:16) ∆ kij (cid:17) (cid:13)(cid:13)(cid:13) , (43) where Γ lM is an M × M zero matrix except for the l -thdiagonal element equal to . In the above expressions W ki = (cid:0) S ki (cid:1) , and c kij := δ ij vec (cid:16) W ki H (cid:16) U ki H ˜ H kij V kj − I d j δ ij (cid:17)(cid:17)(cid:106) vec (cid:16) W ki H U ki H ˜ H ij ( Θ tx ,j ) Γ lN j V mj (cid:17)(cid:107) l ∈ F Nj ,m ∈ F K (cid:106) vec (cid:16) W mi H U mi H ( Θ rx ,i ) Γ lM i ˜ H kij V kj (cid:17)(cid:107) l ∈ F Mi ,m ∈ F K δ ij vec (cid:18) W ki H U ki H (cid:16) σ i,k I M i + Θ rx ,i (cid:80) m ∈ F K σ i,m (cid:17) (cid:19) , (44) C kij := V kj T ⊗ (cid:16) W ki H U ki H (cid:17)(cid:22)(cid:16) ( Θ tx ,j ) Γ lN j V mj (cid:17) T ⊗ (cid:16) W ki H U ki H (cid:17)(cid:23) l ∈ F Nj ,m ∈ F K (cid:106) V kj T ⊗ (cid:16) W mi H U mi H ( Θ rx ,i ) Γ lM i (cid:17)(cid:107) l ∈ F Mi ,m ∈ F K M i d i × M i N i , (45) where δ ij is the Kronecker delta where δ ij = 1 for i = j and zero otherwise. Moreover we have c kij ∈ C ˜ d ij × , C kij ∈ C ˜ d ij × M i N j such that ˜ d ij := d i d j (1 + K ( N j + M i )) + d i M i . (46)Please note that (42) is obtained by recalling (17) and (40) andthe known matrix equality [52, Eq. (516)], and (44)-(45) arecalculated via the application of [52, Eq. (496), (497)].By applying the Schur’s complement lemma on the epigraphform of the quadratic norm (43), i.e., (cid:13)(cid:13) c kij + C kij vec (cid:0) ∆ kij (cid:1) (cid:13)(cid:13) ≤ τ kij , the optimization problem (41) is equivalently written asmin V , U , T max C (cid:88) i ∈ I (cid:88) k ∈ F K τ kij , s.t. (18b) , (cid:107) b kij (cid:107) F ≤ ζ kij , (47a) (cid:34) b kijH ˜ D kijH C kijH C kij ˜ D kij b kij ˜ d ij × ˜ d ij (cid:35) + (cid:20) τ kij c kijH c kij I ˜ d ij (cid:21) (cid:23) , (47b)where T := { τ kij , ∀ i, j ∈ I , ∀ k ∈ F K } and ˜ D kij := I N j ⊗ (cid:0) D kij (cid:1) − , (48) ˜ ∆ kij := D kij ∆ kij , b kij := vec (cid:16) ˜ ∆ kij (cid:17) , (49)are defined for notational simplicity. The problem (47) isstill intractable, due to the inner maximization. The followinglemma converts this structure into a tractable form. Lemma V.1.
Generalized Petersen’s sign-definiteness lemma:Let Y = Y H , and X , P , Q are arbitrary matrices withcomplex valued elements. Then we have Y (cid:23) P H XQ + Q H X H P , ∀ X : (cid:107) X (cid:107) F ≤ ζ, (50) if and only if ∃ λ ≥ , (cid:20) Y − λ Q H Q − ζ P H − ζ P λ I (cid:21) (cid:23) . (51) Proof:
See [53, Proposition 2], [54].By choosing the matrices in Lemma V.1 such that X = b kij , Q = (cid:104) − , × ˜ d ij (cid:105) and Y = (cid:20) τ kij c kijH c kij I ˜ d ij (cid:21) , P = (cid:2) M i N j × , ˜ D kijH C kijH (cid:3) , the optimization problem in (47) is equivalently written asmin V , U , T , M (cid:88) i,j ∈ I (cid:88) k ∈ F K τ kij (52a)s.t. F ki,j (cid:23) , G i (cid:23) , ∀ i, j ∈ I , k ∈ F K , (52b)where M := { λ kij , ∀ i, j ∈ I , k ∈ F K } , and G i : = (cid:20) P i ˜ v Hi ˜ v i I (cid:21) , ˜ v i := (cid:106) vec (cid:16) ( I + K Θ tx ,i ) V ki (cid:17)(cid:107) k ∈ F K , F ki,j : = τ kij − λ kij c kijH × M i N j c kij I ˜ d ij − ζ kij C ijk ˜ D kij M i N j × − ζ kij ˜ D kijH C kijH λ kij I M i N j . Similar to (32), the obtained problem in (52) is not a jointly, buta separately convex problem over V and U , in each case whenthe other variables are fixed. In particular, the optimization over V , T , M is cast as an SDP, assuming a fixed U . Afterwards,the optimization over U , T , M is solved as an SDP, assuminga fixed V . The described alternating steps are continued untila stable point is achieved, see Algorithm 3 for the detailedprocedure. Algorithm 3
Alternating SDP (AltSDP) for worst-case MMSEdesign under CSI error. (cid:96) ← (set iteration number to zero) V , U ← similar initialization as Algorithm repeat (cid:96) ← (cid:96) + 1 V , T , M ← solve SDP (52), with fixed U U , T , M ← solve SDP (52), with fixed V until a stable point, or maximum number of (cid:96) reached return { U , V } C. WMMSE for sum rate maximization
Under the impact of CSI error, the worst-case rate maxi-mization problem is written asmax V min C (cid:88) i ∈ I (cid:88) k ∈ F K I ki (53a)s.t. (18b) , ∆ kij ∈ D kij , ∀ i, j ∈ I , k ∈ F K . (53b)Via the application of Lemma III.1, and (29) the rate maxi-mization problem is equivalently written asmax V min C max U , W (cid:88) i ∈ I (cid:88) k ∈ F K (cid:18) log (cid:12)(cid:12)(cid:12) W ki W ki H (cid:12)(cid:12)(cid:12) + d i − tr (cid:16) W ki H E ki W ki (cid:17) (cid:19) (54a)s.t. (53b) , (54b)where W := { W ki , ∀ i ∈ I , ∀ k ∈ F K } . The above problemis not tractable in the current form, due to the inner min-maxstructure. Following the max-min exchange introduced in [47,Section III], and undertaking similar steps as in (43)-(52a) theproblem (54) is turned intomax V , U , W , T , M (cid:88) i ∈ I (cid:88) k ∈ F K (cid:18) log (cid:12)(cid:12) W ki (cid:12)(cid:12) + d i − (cid:88) j ∈ I τ kij (cid:19) (55a)s.t. F ki,j (cid:23) , G i (cid:23) , ∀ i, j ∈ I , k ∈ F K , (55b)where F ki,j , G i are defined in (52). It is observable that thetransformed problem holds a separately, but not a jointly, con-vex structure over the optimization variable sets. In particular,the optimization over V , T , M and U , T , M are cast as SDPin each case when other variables are fixed. Moreover, theoptimization over W can be efficiently implemented usingthe MAX-DET algorithm [55], see Algorithm 4. Similar toAlgorithm 3, due to the monotonic increase of the objectivein each optimization iteration the algorithm convergences to astationary point. See [47, Section III] for arguments regardingconvergence and optimization steps for a problem with asimilar variable separation. Algorithm 4
AltSDP-WMMSE algorithm for worst-case rate max-imization under CSI error Algorithm 1, Steps 1-3 (initialization) repeat W , T , M ← solve MAT-DET (52), with fixed V , U Algorithm 3, Steps 4-6 until a stable point, or maximum number of (cid:96) reached return { U , V } D. Worst case CSI error
It is beneficial to obtain the least favorable CSI errormatrices, as they provide guidelines for the future channelestimation strategies. For instance, this helps us to choosea channel training sequence that reduces the radius of theCSI error feasible regions in the most destructive directions.Moreover, such knowledge is a necessary step for cutting-set-based methods [56] which aim to reduce the design complexityby iteratively identifying the most destructive error matricesand explicitly incorporating them into the future design steps.In the current setup, the worst-case channel error matrices areidentified by maximizing the weighted MSE objective in (41)within their defined feasible region. This is expressed asmax C (cid:88) i ∈ I (cid:88) k ∈ F K tr (cid:0) W ki H E ki W ki (cid:1) , (56a)s.t. (cid:13)(cid:13) D kij ∆ kij (cid:13)(cid:13) F ≤ ζ kij , ∀ i, j ∈ I , k ∈ F K . (56b)Due to the uncoupled nature of the error feasible set, and thevalue of the objective function over ∆ kij , following (43), theabove problem is decomposed asmin b kij − (cid:13)(cid:13)(cid:13) C kij ˜ D kij b kij (cid:13)(cid:13)(cid:13) − Re (cid:110) b kijH ˜ D kijH C kijH c kij (cid:111) − c kijH c kij (57a)s.t. b kijH b kij ≤ ζ kij , (57b)where Re {·} represents the real part of a complex value. Notethat the objective in (57a) is a non convex function and can notbe minimized using the usual numerical solvers in the currentform. Following the zero duality gap results for the non-convexquadratic problems [57], we focus on the dual function of (57).The corresponding Lagrangian function to (57) is constructedas L (cid:0) b kij , ρ kij (cid:1) = b kijH A kij b kij − Re (cid:110) b kijH ˜ D kijH C kijH c kij (cid:111) − c kijH c kij − ρ kij ζ kij , (58)where ρ kij is the dual variable and A kij := ρ kij I N j M i − ˜ D kijH C kijH C ijk ˜ D kij . (59)Consequently, the value of the dual function is obtained as g (cid:0) ρ kij (cid:1) = − c kijH C kij ˜ D kij (cid:0) A kij (cid:1) − ˜ D kijH C kijH c kij − c kijH c kij − ρ kij ζ kij , if A kij (cid:23) , and ˜ D kijH C kijH c kij ∈ R{ A kij } , and otherwise isunbounded from below . By applying the Schur complementlemma, the maximization of the dual function is written using If one of the aforementioned conditions is not satisfied, an infinitely largevalue of b ij can be chosen in the negative direction of A ij , if A kij is notpositive semi-definite, or in the direction ˜ D kijH C kijH c kij within the null-spaceof A kij . the epigraph form asmax ρ kij ≥ , φ kij − φ kij (60a) s . t . (cid:20) φ kij − c kijH c kij − ρ kij ζ kij c kijH C kij ˜ D kij ˜ D kijH C kijH c kij A kij (cid:21) (cid:23) , (60b)where φ kij ∈ R is an auxiliary variable . By plugging theobtained dual variable ρ kij into (58), and considering the factthat − ˜ D kijH C kijH C kij ˜ D kij + ρ kij(cid:63) I N j M i (cid:23) as a result of (60),the optimal value of b kij is obtained from (58) as b kij(cid:63) = (cid:16) − ˜ D kijH C kijH C kij ˜ D kij + ρ kij(cid:63) I N j M i (cid:17) − ˜ D kijH C kijH c kij , where ( · ) (cid:63) represents the optimality and the worst case ∆ kij isconsequently calculated via vec ( ∆ kij ) = ˜ D kij b kij(cid:63) . E. Computational complexity
The proposed designs in Section III and V are based on thealternative design of the optimization variables. Furthermore,it is observed that the consideration of non-linear hardwaredistortions, leading to inter-carrier leakage, as well as theimpact of CSI error, result in a higher problem dimension andthereby complicate the structure of the resulting optimizationproblem. In this part, we analyze the arithmetic complexityassociated with the Algorithm V. Note that Algorithm V isconsidered as a general framework, containing Algorithm IIIas a special case, since it takes into account the impacts ofhardware distortion jointly with CSI error.The optimization over V , U are separately cast as SDP. Ageneral SDP problem is defined asmin z p T z , s.t. z ∈ R n , Y + n (cid:88) i =1 z i Y i (cid:23) , (cid:107) z (cid:107) ≤ q, where the fixed matrices Y i are symmetric block-diagonal,with M diagonal blocks of the sizes l m × l m , m ∈ F M , and de-fine the specific problem structure, see [58, Subsection 4.6.3].The arithmetic complexity of obtaining an (cid:15) -solution to thedefined problem, i.e., the convergence to the (cid:15) -distance vicinityof the optimum is upper-bounded by O (1) (cid:32) M (cid:88) m =1 l m (cid:33) (cid:32) n + n M (cid:88) m =1 l m + n M (cid:88) m =1 l m (cid:33) digit ( (cid:15) ) , where O (1) is a positive constant and invariant to the problemdimensions [58], and digit ( (cid:15) ) is obtained from [58, Subsec-tion 4.1.2] and affected by the required solution precision. Therequired computation of each step is hence determined by sizeof the variable space and the corresponding block diagonalmatrix structure, which is obtained in the following:
1) Optimization over V , T , M : The size of the variablespace is given as n = 2 K (cid:0) (cid:80) i ∈ I d i N i (cid:1) . Moreover, theblock sizes are calculated as l m = 2 + 2 Kd i N i , ∀ i ∈ I ,corresponding to the semi-definite constraint on G i , and as l m = 2 + 2 ˜ d ij + 2 M i N j , ∀ i, j ∈ I , k ∈ F K , corresponding Note that the semi-definite presentation in (60b) automatically satisfies A kij (cid:23) , and ˜ D kijH C kijH c kij ∈ R{ A kij } . Number of Interations O b j ec t i v e [ d B ] MinAvgRSM
Optimality gap κ = − κ = − κ = − κ = − κ = − κ = − AltQCPAltSDP
Figure 2. Average convergence behavior for AltQCP and AltSDP algorithms.AltQCP converges with fewer steps, and leads to a smaller optimality gapcompared to AltSDP. Both algorithms converge in - iterations. M l o g (cid:2) CT (cid:3) AltSDPAltQCP P th − High P th − Low (a) CPU Time vs M K l o g (cid:2) CT (cid:3) AltSDPAltQCP P th − High P th − Low (b) CPU Time vs K Figure 3. Comparison of the algorithm computational complexities, in termsof the required CPU time (CT), for different system dimensions, i.e., different K and M . κ represents the hardware inaccuracy, i.e., Θ rx ,i = κ I M i , Θ tx ,i = κ I N i . to the semidefinite constraint on F ki,j from (52). The overallnumber of the blocks is calculated as M = 2 + 4 K .
2) Optimization over U , T , M : The size of the variable spaceis given as n = 2 K (cid:0) (cid:80) i ∈ I d i M i (cid:1) . The block sizes arecalculated as l m = 2 + 2 ˜ d ij + 2 M i N j , ∀ i, j ∈ I , k ∈ F K ,corresponding to the semidefinite constraint on F ki,j from (52).The overall number of the blocks is calculated as M = 4 K .
3) Remarks:
The above analysis intends to show how thebounds on computational complexity are related to differentdimensions in the problem structure. Nevertheless, the actualcomputational load may vary in practice, due to the structuresimplifications and depending on the used numerical solver.Furthermore, the overall algorithm complexity also dependson the number of optimization iterations required for conver-gence. See Subsection VI-A for a study on the convergencebehavior, as well as a numerical evaluation of the algorithmcomputational complexity.VI. S
IMULATION R ESULTS
In this section we evaluate the behavior of the studiedFD MC system via numerical simulations. In particular, weevaluate the proposed designs in Sections III and V forvarious system situations, and under the impact of transceiver −60 −40 −20 0 205678910 κ [dB] W C − M S E [ d B ] AltSDPAltQCPHD κ = 0SC P th − ∞ P th − High P th − Low
Assuming no distortionFD−HD gainMC−SC gain (a) MSE vs. hardware inaccuracy −40 −30 −20 −10 0234567891011 ζ [dB] W C − M S E [ d B ] AltSDPAltQCPHD κ = 0SC P th − ∞ P th − High P th − Low
Distortion−awareness gainAssuming perfect CSIFD−HD gain (b) MSE vs. CSI error −50 −40 −30 −20 −10 0456789101112 σ [dB] W C − M S E [ d B ] AltSDPAltQCPHD κ = 0SC P th − ∞ P th − High P th − Low
Perfect CSI and hardware assumptionMC−SC gain (c) MSE vs. noise variance −80 −60 −40 −20 0 20 40−15−10−5051015 ρ [dB] W C − M S E [ d B ] AltSDPAltQCPHD κ = 0SC P th − ∞ P th − High P th − Low
Robustness gainFD−HD gain (d) MSE vs. comm. channel strength K W C − M S E [ d B ] AltSDPAltQCPHD κ = 0SC P th − ∞ P th − High P th − Low
MC−SC gainFD−HD gain (e) SMSE vs. number of subcarriers M W C − M S E [ d B ] AltSDPAltQCPHD κ = 0SC P th − ∞ P th − High P th − Low
MC−SC gainFD−HD gain (f) MSE vs. number of subcarriers −60 −50 −40 −30 −20 −10 0−20246810 κ l , ∀ l [dB] W C − M S E [ d B ] AltSDP K sum = − { , , , , , } [dB] β l = − { , , , , , , , } (g) MSE vs. distribution of quantization M i , ∀ i W C − M S E [ d B ] AltSDP M sum = { , , } N i = { , , , , , , } (h) MSE vs. distribution of antenna resources −60 −50 −40 −30 −20 −10 0 10024681012 κ [dB] Su m r a t e AltQCPHD κ = 0SC P th − High P th − Low
Robustness gainMC−SC gainFD−HD gain (i) Sum rate vs. hardware inaccuracy −60 −50 −40 −30 −20 −10 00510152025303540 σ [dB] Su m r a t e AltQCPHD κ = 0SC P th − High P th − Low
MC−SC gain FD−HD gain (j) Sum rate vs. noise variance −10 0 10 20 30 400510152025303540 P max [dB] Su m r a t e AltQCPHD κ = 0SC P th − High P th − Low
FD−HD gain MC−SC gain (k) Sum rate vs. maximum transmit power −80 −60 −40 −20 00510152025303540 Su m r a t e κ [dB] HD FD M = ˜ N = 2562 ˜ M = ˜ N = 32 (l) Sum rate vs. hardware inaccuracyFigure 4. System performance under various specifications. The application of a distortion-aware design is essential for a system with erroneous hardware orwith a high signal-to-noise ratio (SNR). The default parameter set is κ = − dB, σ n := σ i,k = − dB for (i)-(l). inaccuracy and CSI error. Communication channels H kii followan uncorrelated Rayleigh flat fading model with variance ρ .For the SI channel we follow the characterization reported in[42], indicating a Rician distribution for the SI channel. In thisrespect we have H ij ∼ CN (cid:16)(cid:113) ρ si K R K R H , ρ si K R I M i ⊗ I N j (cid:17) where ρ si represents the SI channel strength, H is a deter-ministic term, and K R is the Rician coefficient. For eachchannel realization, the resulting performance is evaluated byemploying different design strategies and for various systemparameters. The overall system performance is then averagedover channel realizations. Unless otherwise is stated, thefollowing values are used to define our default setup: K = 4 , K R = 10 , M := M i = N j = 2 , ρ = − dB, ρ si = 1 , σ n := σ i,k = − dB, P max := P i = 1 , d i = 1 , κ = − dBwhere Θ rx ,i = κ I M i and Θ tx ,i = κ I N i , and ζ kij = − dB, ω i = 1 , ∀ i, j ∈ I , k ∈ F K . A. Algorithm analysis
Due to the alternating structure, the convergence behavior ofthe proposed algorithms is of interest, both as a verification foralgorithm operation as well as an indication of the algorithmefficiency in terms of the required computational effort. Inthis part, the performance of AltQCP and AltSDP algorithmsare studied in terms of the average convergence behavior andcomputational complexity. Moreover, the impact of the choiceof the algorithm initialization is evaluated.In Fig. 2 the average convergence behavior is depicted for dif-ferent values of κ [dB]. In particular, ”Min” and ”Avg” curvesrespectively represent the minimum, and the average value ofthe algorithm objective at the corresponding optimization stepover the choice of random initializations. Moreover, ”RSM”represents the right-singular matrix initialization proposed in[46, Appendix A]. It is observed that the algorithms converge,within − optimization iterations, specially as κ is small.Although the global optimality of the final solution can not beverified due to the possibility of local solutions, the numer-ical experiments suggest that the applied RSM initializationshows a better convergence behavior compared to a randominitialization. Moreover, it is observed that a higher transceiverinaccuracy results in a slower convergence and a gap withoptimality. This is expected, as larger κ leads to a morecomplex problem structure. Note that the algorithm AltQCPshows a smaller value of objective compared to that of AltSDPfor any value of κ , since the impact of CSI error is notconsidered in the algorithm objective.In addition to the algorithm convergence behavior, therequired computational complexity is affected by the problemdimension, and the required per-iteration complexity, see Sub-section V-E. In Fig. 3, the required computation time (CT) isdepicted for different number of antennas, as well as differentnumber of subcarriers . It is observed that the AltSDP results For simplicity, we choose H as a matrix of all- elements. The reported CT is obtained using an Intel Core i − M processorwith the clock rate of . GHz and GB of random-access memory (RAM).As our software platform we have used MATLAB a, on a -bit operatingsystem. in a significantly higher CT, compared to AltQCP. This isexpected as the consideration of CSI error in AltSDP resultsin a larger problem dimension, and hence higher complexity.Moreover, the obtained closed-form solution expressions in Al-tQCP result in a more efficient implementation. Nevertheless,the required CT for AltQCP is still higher than the threshold-based low-complexity approaches, see Subsection VI-B1, dueto the expanded problem dimension associated with the impactof RSI and ICL. B. Performance comparison
In this part we evaluate the performance of the proposedAltSDP and AltQCP algorithms in terms of the resultingworst-case MSE, see Subsection V-D, under various systemconditions.
1) Comparison benchmarks:
In order to facilitate a mean-ingful comparison, we consider popular approaches for the de-sign of FD single-carrier bidirectional systems, or the availabledesigns for other MC systems with simplified assumptions, seeSubsection I-A. The following approaches are hence imple-mented as our evaluation framework: leftmargin=* • AltSDP : The AltSDP algorithm proposed in Section V.The impact of the hardware distortions leading to inter-carrier leakage, as well as CSI error are taken intoaccount. • AltQCP : The AltQCP algorithm proposed in Section III.The algorithm operates on the simplified assumption thatthe CSI error does not exist, i.e., ζ = 0 , and hencefocuses on the impact of hardware distortions. • HD : The AltSDP algorithm is used on an equivalent HDsetup, where the communication directions are separatedvia a time division duplexing (TDD) scheme. • κ = 0 : The impact of CSI error is taken into accountsimilar to, e.g., [15], [29]. Nevertheless the impact ofhardware distortion, leading to inter-carrier leakage, isignored. • SC : The optimal single carrier design applied to thedefined MC system, following a similar approach asin [12], [16]. The impact of CSI error and hardwaredistortions are taken into account.Other than the approaches that directly deal with the impactof RSI, e.g., [12], [16], a low complexity design framework isproposed in [13], [14], by introducing an interference powerthreshold, denoted as P th . In this approach, it is assumed thatthe SI signal can be perfectly subtracted, given the SI poweris kept below P th . In this regard, we evaluate the extendedversion of [14] on the defined MC setup for three values of P th : leftmargin=* • P th − {∞ , High , Low } : representing a design by re-spectively choosing P th = ∞ , P i , P i / , representinga system with perfect, high, and low dynamic rangeconditions.
2) Visualization:
In Figs. 4 (a)-(h) the average performanceof the defined benchmark algorithms in terms of the worst-case (WC) MSE are depicted. The average sum rate behaviorof the system is depicted in Fig. 4 (i)-(l). In Fig. 4 (a) the impact of transceiver inaccuracy is depictedon the resulting WC-MSE. It is observed that the estimationaccuracy is degraded as κ increases. For the low-complexityalgorithms, where the impact of hardware distortion is notconsidered, the resulting MSE goes to infinity as κ increases.Nevertheless, the resulting MSE reaches a saturation point forthe distortion-aware algorithms, i.e, AltSDP and AltQCP. Thisis since for the data streams affected with a large distortionintensity, the decoder matrices are set to zero which limits theresulting MSE to the magnitude of the data symbols. More-over, the AltSDP method outperforms the other performancebenchmarks for all values of κ . It is worth mentioning thatthe significant gain of an FD system with low κ over theHD counterpart, disappears for a larger levels of hardwaredistortion where AltSDP and HD result in a close performance.In Fig. 4 (b) the impact of the CSI error is depicted. It isobserved that the estimation MSE increases for a larger valueof ζ . For the low-complexity algorithms where the impact ofCSI error is not considered, the resulting MSE goes to infinity,as ζ increases. Nevertheless, the performance of the AltSDPmethod saturates by choosing zero decoder matrices, followinga similar concept as for Fig. 4 (a). It is observed that theperformance of the AltSDP and AltQCP methods deviate as ζ increases, however, they obtain a similar performance for asmall ζ . Similar to Fig. 4 (a), a significant gain is observedin comparison to the HD and SC cases, for a system withaccurate CSI.In Fig. 4 (c) the impact of the thermal noise variance isdepicted. It is observed that the resulting performance degradesfor the distortion-aware algorithms, as the noise varianceincreases. Nevertheless, we observe a significant performancedegradation for the threshold-based algorithms, particularly P th − Low, in the low noise regime. This is since the imposedinterference power threshold tends to reduce the transmitpower, which results in a larger decoder matrices in a low-noise regime. This, in turn, results in an increased impactof distortion. Nevertheless, as the noise variance increases,the algorithm chooses decoding matrices with a smaller normin order to reduce the impact of noise. This also reducesthe impact of hardware distortions. Similar to Fig. 4 (a), theproposed AltSDP method outperforms the other comparisonbenchmarks. It is observed that the performance degradationcaused by ignoring the CSI error in AltQCP, or by applying asimplified single carrier design, is significant particularly fora system with a small noise variance.In Fig. 4 (d) the impact of the communication channelstrength is observed on the resulting system performance. Itis observed that the MSE decreases in most parts as thecommunication channel becomes stronger. Nevertheless, thesystem performance saturates, due to the impact of hardwaredistortion which increase proportional to the transmit/receivepower at each chain. Moreover, the performance of the meth-ods with a perfect hardware/CSI assumption saturates at ahigher MSE, due to the impact of the ignored effect. Moreover,the algorithms AltQCP and AltSDP result in an approximatelysimilar performance for a system with a high channel strength.This is since for a high ρ regime, the impact of thermal noiseand CSI error become less significant. As a result the system performance is dominated by the impact of distortion whichis amplified due to the higher channel strength.In Fig. 4 (e) the impact of the number of subcarriers isobserved on the resulting MSE. It is observed that a highernumber of subcarriers result in a higher error for all benchmarkmethods. This is expected as a higher number of subcarriersenables a higher number of communication streams, resultingin a lower available per-stream power. The performance ofthe SC design reaches optimality of a single carrier system,as expected. Nevertheless the performance of the SC schemedeviates from optimality as K increases, and results in thehighest MSE in comparison to the evaluated benchmarks, for K ≥ . This is expected, as higher independent subcarriersrepresent a channel with a higher frequency selectivity whichcalls for a specialized MC design.In Fig. 4 (f) the impact of the number of antennas is ob-served. As expected, a higher number of antennas results in anincreased performance for all of the performance benchmarks.In particular, a higher number of antennas enables the systemto better overcome the CSI error, for a fixed ζ , and also todirect the transmit power in the desired channel and not in theself-interference path.In Fig. 4 (g) the impact of the accuracy of transmit andreceiver chains are studied, where κ [dB] + β [dB] = K sum , i.e.,the sum-accuracy (in dB scale) is fixed. For instance, for an FDtransceiver with massive antenna arrays where the utilizationof analog cancelers is not feasible, and also the quantizationbits are considered as costly resources, the value of K sum isrelated to the total number of quantization bits. The similarevaluation regarding the number of transmit/receive antennas isperformed in Fig. 4 (h), where M t + M r = M sum . It is observedthat different available resources, i.e., K sum , M sum , result indifferent optimal allocations. However, as a general insight, itis observed that the performance is degraded when resourcesare concentrated only on transmit or receive side. Please notethat similar approach can be used for evaluating different costmodels for accuracy and antenna elements, regarding differentsystem setups, or SIC specifications.In Fig. 4 (i)-(l) the average sum rate behavior of the systemdepicted. In Fig. 4 (i), the impact of hardware inaccuracy isdepicted. It is observed that a higher κ results in a smallersum rate. Moreover, the obtained gains via the applicationof the defined MC design in comparison to the designs withfrequency-flat assumption, and via the application of FD setupin comparison to HD setup, are evident for a system withaccurate hardware conditions. Conversely, it is observed that adesign with consideration of hardware impairments is essentialas κ increases. In Fig. 4 (j) and (k), the opposite impacts ofnoise level, and the maximum transmit power are observed onthe system sum rate. It is observed that the system sum rateincreases as noise level decreases, or as the maximum transmitpower increases . In both cases, the gain of AltQCP method,in comparison to the methods which ignore the impact of For the algorithms with a zero-distortion assumption, the maximumallowed transmit power is utilized to reduce the impact of thermal noise.However, this result in an amplified distortion effect, and a reduced perfor-mance as SNR increases, also see the MSE peaks in Fig. 4 (c)-(d) for thesame algorithms. hardware distortions are observed for a high SNR conditions,i.e., for a system with a high transmit power or a low noiselevel. In Fig 4 (l) the performance of the asymmetric setup,studied is Section IV, is depicted, assuming | I | = 5 , and d i = 1 . It is observed that the gain of FD system (over theHD counterpart) vanishes rapidly as the hardware distortionsincrease. VII. C ONCLUSION
The application of bi-directional FD communicationpresents a potential for improving the spectral efficiency.Nevertheless, such systems are limited due to the impact ofresidual self-interference. This issue becomes more crucialin a multi-carrier system, where the residual self-interferencespreads over multiple carriers, due to the impact of hardwaredistortion. In this work we have presented a modeling anddesign framework for an FD MIMO OFDM system, taking intoaccount the impact of hardware distortions leading to inter-carrier leakage, as well as the impact of CSI error.It is observed that the application of a distortion-awaredesign is essential, as transceiver accuracy degrades, andinter-carrier leakage becomes a dominant factor. Moreover,a significant gain is observed compared to the usual single-carrier approaches, for a channel with frequency selectivity.However, the aforementioned improvements are obtained atthe expense of a higher design computational complexity.A
PPENDIX
We start the proof with the characterization of the impactof distortion on the transmit chains. The proof to the receivercharacterization is obtained similarly. The statistical indepen-dence properties at the frequency domain directly followsfrom the time domain statistical independence e t ,l ( t ) ⊥ v l ( t ) ,and e t ,l ( t ) ⊥ e t ,l (cid:48) ( t ) , and the linear nature of the transformationin (10). The Gaussian and zero-mean properties similarlyfollow for e k t ,l as a linearly weighted sum of the zero-meanGaussian values e t ,l ( mT s ) . The variance of e k t ,l can be henceobtained as E (cid:110)(cid:12)(cid:12) e k t ,l (cid:12)(cid:12) (cid:111) = E (cid:40) K (cid:32) K − (cid:88) m =0 e t ,l ( mT s ) e − j πmkK (cid:33) × (cid:32) K − (cid:88) n =0 e ∗ t ,l ( nT s ) e j πnkK (cid:33) (cid:41) (61) = 1 K K − (cid:88) m =0 K − (cid:88) n =0 E (cid:8) e t ,l ( mT s ) e ∗ t ,l ( nT s ) (cid:9) e − j π ( m − n ) kK (62) = κ l E (cid:110) | v l ( t ) | (cid:111) (63) = κ l K K (cid:88) m =1 E (cid:8) | v ml | (cid:9) (64)where (61) is obtained via direct application of (10), and (63) isobtained from (6), and the statistical independence of e t ,l at thesubsequent time samples from (7). The identity (64) followsfrom the Parseval’s theorem on the energy conversation overorthonormal Fourier basis. R EFERENCES[1] O. Taghizadeh, V. Radhakrishnan, A. C. Cirik, S. Shojaee, R. Mathar,and L. Lampe, “Linear precoder and decoder design for bidirectionalFull-Duplex MIMO OFDM systems,” in
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