I Introduction
Future wireless communication systems need to accommodate the explosive growth in data traffic demand through efficient utilization of limited frequency spectrum. Recent advances in Full Duplex (FD) communication technology demonstrate the potential of a substantial spectral efficiency improvement over the conventional frequency and timedivision duplexing systems through simultaneous UpLink (UL) and DownLink (DL) communication in the same frequency and time resources [islam2019unified, Samsung, sabharwal2014band, kolodziej2019band, smida2017reflectfx, korpi2016full]
. The exploitation of wideband MultipleInput MultipleOutput (MIMO) systems provides further spectral performance boost due to enhanced spatial degrees of freedom (DoF) offered by the plurality of Transmitter (TX) and Receiver (RX) antennas and larger bandwidths
[riihonen2011mitigation, everett2016softnull, alexandropoulos2017joint, bharadia2014full, masmoudi2017channel, anttila2014modeling, bharadia2013full]. In addition, FD MIMO radios have recently been considered for certain physicallayerbased latency improvement through simultaneous communication of data and control signals [islam2020simultaneous, mirza2018performance, Islam_2020_Sim_Multi]. Thus, enabling FD in conjunction with wideband MIMO operation can meet the stringent throughput and latency requirements of beyond 5th Generation (5G) wireless communication systems with limited spectrum resources [Samsung].The simultaneous transmission and reception in wideband FD systems induce strong inband SelfInterference (SI) signals at the FD receivers due to the inevitable limited isolation between the TX and RX blocks [sabharwal2014band]. To suppress the strong SI signal, first, analog cancellation is employed at the input of the RX blocks at the FD node to ensure that none of the reception Radio Frequency (RF) components (i.e., Low Noise Amplifiers (LNAs), Inphase Quadrature (IQ) mixers) goes into saturation due to high SI power, while ensuring that the dynamic range of the AnalogtoDigitalConverters (ADCs) is large enough to capture the residual SI and the naturally weak desired signal with sufficient precision [korpi2016full]. Compared to a narrowband FD SingleInput SingleOutput (SISO) system, where a single direct SI coupling path exists between TX and RX, the wideband FD MIMO analog cancellation design is much more challenging. This happens because each RX chain suffers from the direct SI signals introduced by all the TX antennas as well as their multipath SI components created by environmental reflections [bharadia2014full, khaledian2018inherent, kolodziej2016multitap, khaledian2018robust, korpi2016full, chen2019wideband, antonio2013adaptive, lee2019analysis, le2020beam, huberman2014mimo, duarte2020full, cao2020integrated]. For an FD MIMO transceiver, the narrowband analog canceller requires taps to suppress the direct SI coupling paths, where each cancellation tap includes time delays, tunable bandpass filters, phase shifters, and attenuators. Considering a wideband communication with multipath SI components being strong enough to run the RX RF chains into saturation, the same FD MIMO system would require an appropriate wideband analog canceller with taps; such analog cancellers are hereinafter referred to as fulltap cancellers.
After the analog cancellation, digital domain SI mitigation techniques are applied at the RX baseband to suppress the residual SI signal below the noise floor, which is still large enough to overwhelm the weak desired signal [sabharwal2014band, kolodziej2019band]. Digital cancellation is accomplished by reconstructing and reciprocally combining the residual SI signal at the FD RX through extensive SI channel modeling and exploiting the fact that each FD node has knowledge of its ideal transmit signal in the digital domain [riihonen2011mitigation, bharadia2013full]. Since the residual SI signal is impacted by TX hardware impairments, an appropriate SI channel model must include the SI coupling paths and the nonlinear distortions induced by the transceiver chain’s practical RF components, specifically the image effect due to the gain and phase imbalance of the IQ mixer and Power Amplifier (PA) nonlinearities. For singleantenna FD systems, baseband modeling of these nonlinear distortions has been performed to provide appropriate digital cancellation [bharadia2013full, ahmed2015all, korpi2014widely, islam2019comprehensive]. Akin to the analog canceller, the wideband FD MIMO operation increases the computational complexity of the digital cancellation since the number of linear and nonlinear components to be estimated increases with the number of TX/RX chains and SI channel paths. Moreover, signal modeling in wideband FD MIMO systems requires multicarrier designs (i.e. Orthogonal FrequencyDivision Multiplexing (OFDM)), as the channel becomes frequency selective due to larger bandwidth.
Ia Related works on FD MIMO SI Cancellation
For narrowband or frequencyflat FD MIMO systems, fulltap analog cancellers connecting all TX outputs to RX inputs are usually employed, where the number of taps increases with the number of TX/RX RF chains [riihonen2011mitigation, masmoudi2017channel]. To reduce this hardware complexity, analog SI canceller designs exploiting AUXiliary (AUX) TX structures and/or joint design of TX/RX beamformers were studied in [huberman2014mimo, duarte2020full, george2018journal], where the analog cancellation signals were injected into each of the RX inputs using separate TX RF chains. Although these techniques reduce the analog canceller hardware complexity, they are unable to suppress the nonlinear SI components due to the nonideal RF frontend hardware rendering the RX chains into saturation [kolodziej2019band]. In our previous work [islam2019unified], we presented a unified low complexity Analog and Digital (A/D) cancellation for narrowband FD MIMO systems. For wideband FD MIMO systems in [antonio2013adaptive, bharadia2014full], fulltap analog cancellers with adaptive filters were utilized to provide sufficient analog SI cancellation. In [lee2019analysis], a fulltap wideband FD MIMO RF canceller was presented with a tunable time delay circuit, which employed reflected type phase shifters to emulate the true time delays of the SI channel. Recently, a fulltap beambased RF cancellation approach was introduced in [le2020beam], which employed analog Least MeanSquared (LMS) loops as the adaptive filters for SI mitigation in FD massive MIMO systems. Those analog LMS loops include time delay generators, downconverters, LowPass Filters (LPFs), and upconverters. An integrated LMS adaptive wideband FD MIMO RF canceller was proposed in [cao2020integrated], where the time delay of the cancellation was generated using an path filter. However, the hardware complexity of all the above fulltap RF cancellers scales with the number of TX and RX RF chains as well as the number of SI multipath components, rendering the practical implementation of the analog SI cancellation unit a core design bottleneck.
Alleviating the need for analog SI cancellation, spatial suppression techniques were presented in [riihonen2011mitigation, everett2016softnull] for narrowband FD MIMO systems, where the SI suppression was solely handled by the digital TX/RX beamformers. However, those spatial suppression techniques were unable to cancel the SI in high TX power and often resulted in reductions of the data rates for both the UL and DL signals of interest. This stemmed from the fact that some of the available spatial DoFs were devoted to mitigating SI [alexandropoulos2017joint]. To avoid such issues, digital cancellation techniques exploiting SI signal modeling were utilized in practice to supplement the analog canceller in suppressing the SI signal. To achieve sufficient SI suppression, existing digital cancellation approaches capitalize on models for the PA impairments [bharadia2014full] and IQ mixer image effect [korpi2014widely], or rely on cascaded SI designs taking into account both nonidealities [anttila2014modeling]. However, the number of estimation parameters of those models grows with the number of TX/RX RF chains and SI channel components. To reduce the number of parameters for the FD MIMO system, a digital canceller based on Principle Component Analysis (PCA) was provided in [korpi2017nonlinear]. Furthermore, in [ng2012dynamic, taghizadeh2018hardware, radhakrishnan2021hardware], the authors considered FD MIMO OFDM signal modeling to design rate maximizing TX/RX beamformers. However, these techniques assumed fulltap RF cancellers to achieve certain SI suppression levels.
IB Contributions
In this paper, we present a joint A/D SI cancellation with TX/RX beamforming approach for wideband FD MIMO systems considering the effect of nonlinear hardware distortions and multipath SI components. The main contributions of this paper are summarized as follows:

We propose a novel joint wideband analog SI cancellation and TX/RX beamforming approach for multiuser FD MIMO systems in the presence of TX RF chain impairments, where the multipath SI components are suppressed using reduced analog cancellations taps compared to existing FD MIMO solutions.

A comprehensive OFDM signal modeling of the proposed FD MIMO system is derived, including baseband equivalent models of the TX RF chain impairments, corresponding wideband channels, and A/D SI cancellers.

We present a novel adaptive digital canceller based on the Truncated Singular Value Decomposition (TSVD) that reduces the computational complexity of conventional digital SI cancellation while successfully suppressing the residual SI signal after analog cancellation below the RX noise floor.

A joint optimization framework for A/D cancellation and TX/RX beamforming is presented to maximize the achievable sumrate performance of the considered threenode wideband FD MIMO OFDM system.

Finally, we perform extensive waveform simulations to illustrate the proposed A/D SI cancellation performance and provide comparisons with the relevant stateoftheart methods. It is demonstrated that our proposed wideband canceller in conjunction with TX/RX beamforming exhibits superior SI mitigation capability with reduced complexity (less than analog taps) compared to the existing fulltap cancellers in the presence of TX hardware impairments.
IC Organization and Notations
The rest of the paper is organized as follows. In Section II, the baseband signal model of the considered wideband FD MIMO OFDM system is presented. Then, in Sections III and IV, we present the proposed A/D SI canceller alongside the joint optimization framework. Section V includes the performance evaluations of the proposed SI cancellation approach via extensive waveform simulations. Finally, the conclusions are drawn in Section VI.
Notations:Vectors and matrices are denoted by boldface lowercase and boldface capital letters, respectively. The transpose, Hermitian transpose, and conjugate of are denoted by , , and , respectively, and is ’s determinant, while () is the identity matrix. stands for the Euclidean norm of , denotes the Hadamard power operation to the factor , operand represents the Hadamard entrywise product, is a column vector resulting after vertically concatenating vectors , and denotes a square diagonal matrix with ’s elements in its main diagonal. , , and represent ’s th element, th row, and th column, respectively, while denotes the th element of . represents the complex number set, is the expectation operator, and denotes the amplitude of a complex number. The rest of the notations used throughout this paper are listed in Table I.
Variable  Definition 

Number of TX antennas at node  
Number of RX antennas at node  
Number of RX antennas at node  
Number of TX antennas at node  
Number of subcarriers  
Subcarrier index  
Number of data streams at node  
Number of data streams at node  
th subcarrier symbol vector at node  
th subcarrier TX Beamformer at node  
th subcarrier symbol vector at node  
th subcarrier TX Beamformer at node  
Node symbol vector at time  
Node symbol vector at time  
Linear power allocation matrix at node  
Linear power allocation matrix at node  
Nonlinear TX signal at node  
Nonlinear TX signal at node  
Node TX output at time instant  
Node TX output at time instant  
Wideband DL channel between node and  
DL channel delay taps  
Wideband UL channel between node and  
UL channel delay taps  
Wideband SI channel  
SI channel delay taps  
Coefficients of the analog SI canceller  
Delay taps of analog SI canceller  
Baseband received signal at node 
Variable  Definition 

Baseband received signal at node  
AWGN vector at node  
AWGN vector at node  
Digital SI cancellation signal at node  
th subcarrier FFT output at node  
Estimated symbol vector of  
Estimated symbol vector of  
th subcarrier RX beamformer at node  
th subcarrier RX beamformer at node  
Frequency representation of DL channel  
Frequency representation of UL channel  
Frequency representation of SI channel  
Frequency representation of analog SI canceller  
Frequency representation of  
Frequency representation of  
Frequency representation of  
Frequency representation of  
Frequency representation of  
Analog SI canceller MUX configurations  
Analog SI canceller coefficients of th tap  
Analog SI canceller DEMUX configurations  
Achievable UL rate of th subcarrier  
Achievable DL rate of th subcarrier  
Estimated IpN covariance matrix at node  
Estimated IpN covariance matrix at node  
Maximum transmit power at node  
Maximum transmit power at node  
RF chain saturation threshold at node 
Ii System and Signal Models
We consider the threenode FD wireless communication system of Fig 1 comprising of an FD MIMO Base Station (BS) node communicating concurrently with two Half Duplex (HD) multiantenna nodes: node in the downlink and node in the uplink direction. The FD MIMO node is assumed to be equipped with TX and RX antenna elements. Each TX antenna is attached to a dedicated RF chain that consists of a Digital to Analog Converter (DAC), IQ mixer, and PA; similarly holds for the RX antennas and their attached RF chains, each containing LNA, IQ mixer, and ADC. The HD multiantenna nodes and are assumed to have and antennas, respectively, with each of their antennas connected to a dedicated RF chain. All three nodes are considered capable of performing digital beamforming and OFDM operations with subcarriers.
Iia Downlink TX and RX Signal Modeling
During DL transmission, the FD MIMO node sends data streams multiplexed at each subcarrier to the HD node . The unit power symbol vector of th subacarrier is denoted as , which, in practice, is selected from a discrete modulation set. The symbol vector is linearly precoded by the TX beamformer . Without loss of generality, we assume that
have unit norm columns. The precoded symbols are converted to timedomain samples using the Inverse Fast Fourier Transform (IFFT) operation. To prevent InterSymbol Interference (ISI), a cyclic prefix is appended in front of each IFFT block; in the subsequent analysis, we ignore the cyclic prefix for simplicity. The output of the node
TX baseband block after the IFFT operation at a discrete time instant is represented as(1) 
Baseband Modeling of TX RF Chain Impairments: As shown in Fig 1, the node baseband samples in are fed to the TX RF chains for upconversion and amplification. We introduce a baseband equivalent model for each of these RF chains incorporating IQ imbalances and PA nonlinearities [korpi2014widely], assuming that the RF chains are identical. Upon entering the TX RF chain of node , each baseband sample goes through the IQ mixer for upconversion to the carrier frequency. In practical IQ mixers, a mirror image of the original signal with certain image attenuation is induced by the IQ phase and gain imbalances. Denoting the input at the th () TX RF chain of node at time instant as , the IQ mixer output can be written as [korpi2014widely, eq. (8)]
(2) 
where and with and are representing the gain and phase imbalances, respectively. It is noted that the Image Rejection Ratio, defined as , represents the strength of the IQ induced conjugate term[korpi2014full].
Before transmission, the upconverted signal is fed into the PA for amplification while satisfying the TX power constraint. Note that practical PAs exhibit varying degrees of nonlinearity. However, we consider a quasi memoryless PA model of thirdorder nonlinearity, as it is the most dominant distortion in practice, and all the evenpower harmonics lie out of the band and will be cut off by the RF low pass filter at the RXs [gu2005rf, korpi2014widely, zhou2005baseband]. For this PA model, the baseband equivalent of each th PA output at time instant is given using (2) as
(3) 
where represents the nonlinearity order^{1}^{1}1Note that (3) is general enough to model various degrees of nonlinearities in the TX RF chains; this can be accomplished by setting to the desired nonlinearity order. and the six gain components with are derived as
(4) 
where denotes the PA linear gain and is the gain of the thirdorder nonlinear distortions with representing the thirdorder Inputreferred Intercept Point of the PA[gu2005rf].
Based on (3) and after some algebraic manipulations, the baseband representation of the impaired transmitted signal from the TX antennas of the FD MIMO node in the DL direction can be expressed as
(5) 
where is the power allocation matrix of the linear components of the TX signal and denotes its nonlinear part, which is given by
(6) 
In the latter expression, for representing the coefficient matrices for the nonlinear components of is defined in the similar way to . In (5), we also introduce the notation for the augmented power allocation matrix, and the vertically arranged signal vector including the image and nonlinear components. The matrices are given by
(7) 
We finally make the practical assumption that the DL signal transmission is power limited to such that .
DL Received Signal Model: The transmitted DL signal is received at the HD RX node after passing through the wideband DL channel denoted by , , where represents the number of DL channel paths. The received baseband signal of node at the discrete time instant is mathematically expressed as
(8) 
where represents the Additive White Gaussian Noise (AWGN) vector at node with covariance matrix . It is to be noted that we assume no internode interference between nodes and due to appropriate node scheduling[alexandropoulos2016user, atzeni2016performance].
The baseband received signal
is transformed to frequency domain using the Fast Fourier Transformation (FFT) operation, which is followed by RX beamforming for each subcarrier to obtain the estimated symbol vectors denoted by
, . Denoting th subcarrier RX beamformer as , the linearly processed estimated symbol vector is written as(9) 
where is defined as th subcarrier frequency domain representation of the wideband DL channel. Similarly, and represent the frequency transform of the TX nonlinear components vector and the AWGN vector at node , respectively. The detailed derivation of (9) is provided in Appendix A.
IiB Uplink TX and RX Signal Modeling
Now, we model the UL signal transmitting from HD multiantenna node to the FD MIMO node . Similar to the node TX, the th subcarrier symbol vector with is precoded by the unit norm TX beamformer . The precoded symbol vectors are transformed to time domain samples using FFT operation identical to (1). The time domain samples are upconverted and amplified by the TX RF chains of node following the similar operation of node TX. Therefore, the TX output at node is expressed as
(10) 
where is the power allocation matrix of the linear components of the TX signal and denotes its nonlinear part, defined similarly as (6). The UL signal transmission is power limited to such that .
UL Received Signal Model: The transmitted UL signal is received at node after passing through the wideband UL channel denoted by , , where represents the number of UL channel paths. Due to FD operation, the transmitted signal from node TX is also received at the node RX input after passing through the wideband SI channel denoted by , , where is the number of SI channel delay taps. In addition to the UL and SI signals, the analog cancellation signal stemming from the output of the wideband analog SI canceller is fed into the RX inputs of node , as shown in Fig 1. Therefore, similar to (8), the received signal is expressed as
(11) 
where is the AWGN vector at this node with covariance matrix . In this expression, , represents the coefficients of the wideband analog SI canceller, which is modeled as an th order Finite Impulse Response (FIR) filter and will be described in the following Sec. III. Recall that the wideband analog canceller utilizes the TX RF chain output , which contains the transmitter nonlinear impairments , as defined in (6). Therefore, the analog canceller is capable of suppressing both linear and nonlinear SI components.
After the downconversion of the received signals at node , the RF chain outputs are added with the digital SI cancellation signal. The resulting signals are then transformed to frequency domain using the FFT operation, as shown in Fig 1. Assuming that the digital cancellation signal at the discrete time instant is given by and using (1), (5), as well as (11), the frequencydomain received signal vector at the th subcarrier of the FFT output can be expressed as
(12)  
where
(13) 
denote the frequency domain representations of the UL, SI channel, and the wideband analog SI canceller, respectively. Similarly, , , , and are defined as the th subcarrier frequency transform of the TX nonlinear components vector and , digital cancellation signal vector , and the AWGN vector , respectively.
At the FFT output, the th subcarrier symbol vector is linearly processed by the RX combiner to obtain the estimated symbol vector , which is derived as
(14) 
Iii Join Digital TX/RX Beamforming and Wideband Analog Cancellation
In this section, we present the joint design of digital TX/RX beamforming with wideband analog SI cancellation. We first describe the proposed analog SI canceller for wideband FD MIMO OFDM radios and then present the mathematical formulation for the codesign of the analog cancellation matrix with the digital TX/RX beamformers.
Iiia Wideband FD MIMO Analog SI Canceller
Upon signal reception at the FD MIMO node , analog SI cancellation is applied to the signals received at the RX antennas before entering the RF chains, as shown in Fig. 1. As previously described in the considered signal model, the wideband analog SI canceller intended for suppressing the multipath SI channel with is modeled as an th order FIR filter with the coefficients , . This filter takes the outputs of the TX RF chains as inputs and routes its outputs to the inputs of the RX RF chains. Note that the special case of was considered in [alexandropoulos2017joint, islam2019unified, cao2020integrated] for suppressing narrowband SI signals. As will be shown in Sec. V with the performance evaluation results, such narrowband SI cancellers are not capable of suppressing the SI power below the maximum RX input power limit, and therefore, lead to RX RF chains’ saturation. To avoid such saturation in wideband FD MIMO systems, fulltap wideband analog SI cancellation was utilized in [antonio2013adaptive, bharadia2014full] that requires . It is apparent that the hardware complexity of the fulltap cancellers scales with the number of TX/RX antenna elements as well as the multipath channel components.
The proposed wideband analog SI canceller for the FD MIMO OFDM node is depicted in Fig 2. As illustrated in the figure, it consists of analog taps, each including a delay line, a phase shifter, and an attenuator, to suppress the SI signal between the output of any th TX RF chain and the input of any th RX RF chain with and . It will be shown in the sequel that the proposed codesign of the analog SI canceller with the digital TX/RX beamformers allows choosing , thus reducing the hardware complexity of the canceller compared to the fulltap analog canceller case [antonio2013adaptive, bharadia2014full] in stateofthearts. As shown in Fig 2, outputs of the TX RF chains are routed to the analog canceller via MUltipleXers (MUXs), and the outputs of the canceller are added to selected RX RF chains input using DEMUltipleXers (DEMUXs). The analog canceller settings in Fig 2 are repeated for all the TX and RX RF chains resulting in tap wideband analog SI canceller. Therefore, for the th filter delay, the baseband representation of the tap analog canceller is modeled as
(15) 
where and represent the MUX and DEMUX configurations of the th order of the canceller, respectively, and they take the binary values or . Therefore, it must hold that and ,. Here, is a diagonal matrix whose complex entries represent the attenuation and phase shift of the th sample delayed canceller taps. For example, we consider a wideband MIMO system with available analog cancellation taps at the th filter delay. One possible MUX and DEMUX configuration of the th order of the canceller can be written as and , respectively. Evidently, the above constraints are satisfied for the MUX and DEMUX configurations.
Compared to the narrowband [alexandropoulos2017joint, islam2019unified, cao2020integrated] and fulltap wideband cancellers [antonio2013adaptive, bharadia2014full], our proposed wideband analog SI canceller reduces the complexity in two ways: firstly, the analog canceller has filter order , and secondly, the canceller is capable of selecting the minimum number of TX/RX antenna pairs, whose SI impact is to be suppressed to avoid the RX RF chains’ saturation. Therefore, the hardware complexity of the proposed wideband analog SI canceller does not scale with the number of antennas nor with the number of multipath components. Assuming as the total number of taps of the canceller , it holds that .
IiiB Digital TX/RX Beamforming and Wideband Analog SI Cancellation
Suppose that the UL, DL, and SI wireless channels in the considered system of Fig 1 are estimated using pilot signals as , , and , , respectively. Using these estimations and the representation for the analog canceller as well as digital TX/RX beamformers, estimates for the th subcarrier achievable UL and DL rates can be respectively calculated as
(16) 
where and denote the estimated InterferenceplusNoise (IpN) covariances matrices at multiantenna nodes and , respectively, which can be computed as
(17) 
where . The latter expressions have been obtained from (14) and (9) assuming estimation of the TX impairments at RXs.
Extending the design approach of [islam2019unified], we focus on the estimated achievable FD rate of th subcarrier defined as the sum of and , and formulate the following general optimization problem for the joint design of the tap wideband analog SI canceller and the digital TX/RX beamformers:
s.t  (C1)  
(C2)  
(C3)  
(C4)  
(C5) 
where (C1) represents the analog SI canceller settings as in (15), constraints relate to the average transmit power at node and , respectively, (C4) enforces the unit norm condition of the considered beamformers, and (C5) imposes the threshold residual power level, at the RX antenna input to avoid RF chain saturation at node . The threshold power is limited by the ADC dynamic range.
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