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this report presents a new design using a beamforming technique to achieve active self-interference cancellation for full duplex MIMO system. This new design, unlike prior work, has no limitation on antenna placement and field-pattern. In theory, it can be employed in concatenation with any passive suppression method. We present the two-level design of active self-interference cancellation: (1) analog cancellation using a beamforming technique at the transmit level, (2) digital cancellation at the receive level. Simulation results demonstrate that if the self-interference channel is perfectly known, more than 80 dB of active suppression can be achieved with transmit beamforming with no reduction in SNR of the desired signal from the far field.
In practice however, the self-interference channel cannot be perfectly estimated due to phase noise in the transmit and receive chains of the full duplex node. Thus, we develop a new approach to apply transmit beamforming in the presence of an arbitrary unknown bounded channel distortion. Computer simulations show that this new design has robust performance on active self-interference cancellation over imperfect channel estimation.
Index Terms-full duplex MIMO, self-interference cancellation, beamforming, channel distortion
In full duplex communication systems, a node which has at least one Tx and one Rx antenna, can simultaneously transmit and receive signals in the same frequency band. The key challenge in operating a full duplex network is reducing self-interference generated by the Tx antenna at the Rx antenna on the same node. A general approach to achieve self-interference suppression is a combination of passive and active methods . The main purpose of passive methods is to increase path loss for self-interference signal. One passive suppression mechanism can be implemented by using antenna designs. For example, in , a nulling antenna  is used to cancel 25 dB- 30 dB of interference. In , the author shows that the directional antenna provides more interference isolation than the omnidirectional antenna. Recent research has also proposed two more passive suppression mechanisms, absorptive shielding and cross-polarization . In fact, a large portion of the total self-interference suppression in full-duplex design can be achieved by these three key passive suppression mechanisms . However, one would further require active cancellation so that full-duplex communication is feasible due to the proximity of the transmit and the receive antennas at a node.
Active methods, which include analog cancellation and digital cancellation, employ the knowledge of self-interference to actively cancel the self-interference signal from the received signal. Recently, active methods have been extensively studied. The objective of active methods is to produce a negative copy of the self-interference signal being received at the receive antennas. Then this estimated signal can be employed to create a null for self-interference signal. For example,  generates a 180 degree phase shift self-interference signal by asymmetric placement of transmit antennas. While,  uses a fixed 180 degree phase shifter to create the cancelling signal for self-interference signal at the receivers. These two methods are simple to implement, however, they work only if the self-interference channel is symmetric and stable overtime. Other proposed schemes of active cancellation require an estimation of self-interference channel between the transmit and receive antennas on the same node. For instance, in , the cancelling signal is generated in the digital baseband and unconverted via a parallel radio chain, while  uses the QHx220 chip to track and emulate the self interference channels so that a replica of self-interference signal through the air, via the receive antenna, can be created. All these schemes belong to analog cancellation, which cancels the self-interference signal before it is digitalized. If the cancellation occurs after the signal is digitalized, it should belong to digital cancellation methods. For example, in   , the digital samples of the transmitted signal are subtracted from the received signals in digital domain, removing up to 25 dB of self-interference. Analog cancellation and digital cancellation are usually used in tandem to achieve the desired level of self-interference cancellation. However, the total amount of active self-interference suppression is limited due to the phase noise as reported in .
The feasibility of full-duplex systems largely depends on the amount of the residual self-interference at receive antennas, while the self-interference is trivial for the conventional half-duplex system due to frequency-division. In fact, the key challenge in simultaneous bidirectional communication systems on the same band lies in that neither time-division nor frequency-division can be employed to avoid the high-powered self-interference generated from transmit antennas at receive antennas. With next generation wireless devices equipped with multiple antennas, however, the available spatial degrees of freedom can be employed to implement space-division so that the receivers can avoid the self-interference signal from the location of transmit antenna while receive the signal from the desired antennas.
We might envision an extension of analog cancellation scheme    for MIMO system. Unfortunately, the author report  that these cancellation methods are hardly practical to be implemented in full-duplex MIMO systems. In , a design of full-duplex MIMO has been proposed by placing the antenna symmetric and employing the 180 degree phase shifter to null the self interference signal. This scheme is able to provide 45 dB of SI cancellation in an open-space indoor environment. But when measured in an indoor multi-path rich environment, the level of self-interference suppression is only about 15 dB, much less that the suppression required. The other shortcoming of this approach is that it is hard to implement combined with passive suppression.  has demonstrated that passive suppression accounts for a large portion of the total self-interference suppression in existing full-duplex design. Hence, the required self-interference level is hard to achieve without passive suppression.
In this paper, we develop a new powerful approach to actively cancel the self-interference in a full-duplex MIMO system. Our approach is based on the beamforming technique. We develop two levels of active cancellation, at transmit antennas and receive antennas respectively, to achieve the required amount of active self-interference cancellation at a node. Usually perfect knowledge of channels is required to implement the beamforming techniques. However, it is impractical to track each self-interference channel at all times. Note that self-interference channels are stable over time but are contaminated by phase noise that is bounded, we extend the robust adaptive beamformer proposed in  to the full-duplex MIMO systems due to the good performance of this beamforming technique in the presence of an arbitrary unknown channel distortion. And we use a convex second-order cone (SOC) program to solve this problem efficiently (in polynomial time) via the well-established intertior point method (see -). Computer simulations show that the active self-interference suppression level of this beamforming scheme is over some other active cancellation methods.
Our paper is organized as follows. The model of full duplex MIMO system is presented in Section â…¡ï¼Œwhere both the far-field and self-interference channel model we used are introduced. In Section â…¢, we describe the two level design of active self-interference cancellation. In Section â…£, we introduce a new robust method to solve the transmit beamforming problem described in Section â…¢. This method is based on the optimization of worst-case performance. Then, we introduce how to convert it to a convex SOC problem that can be efficiently solved by the interior point algorithms. Sectionâ…¤presents our simulation results about active self-interference suppression of the transmit and receive antennas. Section â…¥ contains our concluding remarks.
We simplify the full duplex MIMO system into a two-node full duplex communication system. Each node has the same number of transmit antennas and receive antennas. We will use M and N to denote the number of transmit and receive antennas. The channels from transmitters on node2 to receivers on node1 can be described by a far-field model due to the long distance between the two nodes, while a near-field model is required to describe self-interference channel due to the proximity of the transmit and receive antennas at the same node. We will next introduce these two channel models that we used.
Fig.1 Two node full duplex communication model. The solid line denotes desired propagation and the dashed line denotes self-interference.
The propagation between transmitters on node2 and receivers on node1 can be characterized by a far-field MIMO model. Most studies use the idealistic uncorrelated high-rank (UHR) channel model which assumes the elements of the channel matrixto be i.i.d , we use a more practical model, uncorrelated low-rank (ULR) model . The channel matrix is given by
where and are independent receive and transmit fading vectors, with i.i.d complex-valued components ,. Every realization of has rank 1, and although diversity of the channel is present, capacity will be less than the UHR model since there is no multiplexing gain.
The propagation channel between each transmit and receive antenna pair on a same node can be simulated by the IEEE 802.15.4a channel model , which is valid spanning the frequency range from 2 to 10 GHz and a distance range from 2 to 8 m. The coherence time of the IEEE 802.15.4a channel model is 60ms, and every iteration step will produce a channel matrix over one coherence time period.
where is the complex channel vector that consists of the impulse response of channel from every transmit antenna to the receive antenna on the same node.
Two Level Beamforming Design To Achieve Self-Interference Cancellation
The nodes of a full-duplex MIMO system employs a beamforming technique to suppress self-interference while improving the SNR of desired signal. We first describe the objective and rational constraints of our cancellation design at two levels, transmit level and receive level. Then, with the objective and constraints, we show how to formulate and solve the beamforming design problem of transmit antennas. Then, we show how to extend the cancellation design to the receive antennas, thereby completing the two stages of active self-interference cancellation.
Note that the full-duplex system consists of nodes that have multiple transmit and receive antennas. This implies that each node has the available spatial degrees of freedom to implement beamforming or a spatial filter. Therefore, for each MIMO capable FD node, we can apply the beamforming technique (or spatial filter) for both transmitters and receivers based on its information of SI channels and receive/transmit fading vectors .
Formulation of Transmit Beamforming Problem
In a node, the self-interference signal generated by transmit antennas at receive antennas is given by
where is the time, is the transmitted signal, is the self-interference channel matrix ( is the number of transmit antennas and is the number of receive antennas), is the complex vector of transmit beamforming weights, and stand for the transpose and Hermitian transpose, respectively. Here, we assume that all of the transmit antennas are synchronized and send the same signal. Then, the transmit signal at far-field location is given by
where is the transmit fading vector of uncorrelated low-rank model . The transmit beamforming weight vector can be found by maintaining a distortionless response at the far-field location while minimizing the power of transmit self-interference signal at receive antennas
Here, we suppose that signal has average power 1. Then, the transmit beamforming weights is equivalent to the solution of the optimal problem
where is a constant that constrains the total transmit power. According to (2), we can rewrite the problem (6) as
where is the amplitude of transmit self-interference signal at the receive antenna. Note that problem (7) is convex, therefore the solution of can be easily found with computational complexity.
Formulation of Receive cancellation Problem
The complex array observation vector of receive antennas is given by
where , , and are the desired signal, self-interference signal and noise components, respectively. Here, is the far-field signal from the intended transmit antennas and is the receive fading vectors . The receive spatial filter weight vector should maximize the signal-to-interference-plus-noise ratio (SINR)  while maintaining the distortionless response to the far-field desired signal .
is the interference-plus-noise covariance matrix, and is the signal power. The maximum problem of (9) is equivalent to (11) from 
The solution of this optimal problem is equivalent to the complex spatial filter weight vector of receive antennas. In fact, (11) has a well-known solution so that the filter weight vector has the closed-form expression:
where is the normalization constant that does not affect the optimal SINR of
The exact interference-plus-noise covariance matrix of any node remains unknown in practical full duplex MIMO systems. Therefore, we use the sample covariance matrix of the received signal instead of
The receive weight vector is finally given by
The receive weight vector can be employed as the FIR digital filter at the receivers, which can provide SINR performance of receive signal rapid convergence to the optimal value (13).
An Approach To Robust Beamforming Filter
In this section, we develop a new adaptive beamformer at the transmit antennas that is robust against an arbitrary channel distortion vector. This approach is based on the worst-case performance optimization. We begin with the formulation of the robust adaptive beamforming problem and then convert it to a convex form which can be solved by using SOC programming.
In a full duplex MIMO system, the self-interference channel at a node should be stable overtime due to the fixed location and the proximity of the transmit and receive antennas. But the self-interference channel can be affected by the phase noise in the local oscillators in the transmit and receive chain of the full-duplex node .
The unpredictable phase noise is the root cause of channel estimation error which causes performance bottleneck of active self-interference cancellation.
The amplitude of phase noise is limited by the signal power. Therefore, it is reasonable for us to assume that the norm of the channel distortion vector can be bounded by some known constant:
Then, the actual self-interference channel vector belongs to the set
Indeed, if , then, . Since can be any vector in (17), we impose a constraint that for all channel vectors that belong to, the absolute value of the self-interference at receive antenna should not larger than , i.e.,
Using (19), the robust formulation of transmit beamforming problem can be written as the following constrained minimization problem:
Note that (20) represent a modified version of (7). The main modification of (7) is that instead of requiring fixed amplitude of self-interference signal at receive antenna, in (20), such constraints is maintained by means of inequality constraints for a continuum of all possible channel vectors given by the set . Hence, the constraints in (20) guarantee that the SI signal amplitude will be limited in the worst case, i.e., for the particular channel vector that corresponds to the largest value of . Therefore, such a design should improve the beamformer robustness against channel vector distortions that satisfy (17).
For each channel vectors of, the condition represents a nonlinear and nonconvex constraint on . Since there is an infinite number of channel vectors in, there is an infinite number of such constraints. Hence, (20) is a semi-infinite nonconvex quadratic program. It is well known that the general nonconvex quadratically constrained quadratic programming problem is NP-hard and, thus, intractable . However, due to the structure of the constraints, (20) can be converted into a convex problem which can be solved by SOC program.
Let us first convert the semi-infinite nonconvex constraint to a single constraint that corresponds to the worst-case constraint from (19). In particular, (20) can be equivalently described as
According to (18), we can rewrite the constraint of (21) as
where the set is defined as
Applying the triangle and Cauchy-Schwarz inequalities along with the inequality, we have that
Moreover, it is easy to verify that
Then combining (22) and (23) ,we conclude that
Therefore, the semi-infinite nonconvex quadratically constrained problem (21) can be written as the following quadratic minimization problem with a nonlinear constraint:
Note that the problem (29) has much simpler formulation than (21) and is convex. This problem can be solved by interior point method.
In our simulations, we use a uniform linear array with M=3 omnidirectional antennas spaced half a wavelength apart as our transmit antennas on a node. The central frequency of the narrowband signal is f=2.4 GHz. And the distance between transmit and receive antennas is 2m. In all simulations, the channels are produced according to the channel model in section â…¡. The SeDuMi convex optimization MATLAB toolbox has been used as SOC program to compute the robust transmit beamforming weight vector .
The channel distortion is defined as
where is the deviation of the amplitude of the channel vector, andis the mean amplitude of channel vector . In fact, represents the path loss between the transmitters and receivers, which can be leveraged to achieve passive suppression. And the amount of passive suppression largely relies on the path loss . depends on some unpredictable factors in environment, for example, the phase noise of the system. From the simulation of IEEE 802.15.4a, we find that environment reflection is another factor which influences the standard deviation.
A. In this example, we simulate the scenario that the number of transmit antennas is 3 and the number of receive antennas is 1. We employ the robust beamforming method at transmit antennas and measure the amount of active analog cancellation at the receive antenna when the distortion of channel vectors are different.
Fig.2. The active SI cancellation versus channel distortion
The mean of cancellation at one channel distortion value is the average result of 200 simulations. The lower bound of cancellation is derived from the worst-case cancellation of all 200 simulations.
As we can see in Fig2, the robust beamforming enjoys the good performance on active self-interference cancellation. If the self-interference channel distortion is less than 0.1, the active cancellation can be more than 45dB by using the robust beamforming technique. However, designs  report anywhere between 20-45 dB of active analog cancellation.
In this example, the number of transmit antennas is 3 and the number of receive antenna is 1. We compare the active cancellation performance of the robust TX beamforming in Section â…£ with the performance of traditional TX beamforming in Section â…¢.
At one value of channel distortion, we use the traditional and robust beamforming respectively to cancel the self-interference signal at the receive antenna and measured the active analog self-interference cancellation in dB. The mean of active cancellation is derived by averaging the results of 200 simulations. And the lower bound is the worst cancellation performance in all 200 simulations.
As we can see in Fig3, the robust beamforming technique enjoys a better performance than the traditional beamforming. Even the lower bound of the robust beamforming is better than the mean performance of the traditional beamforming when the channel distortion is larger than 0.25. This can be explained by the fact that the robust beamforming belongs to the class of diagonal loading techniques , which had been proved to be a better beamformer in noisy environment than the traditional beamformer.
Fig.3. The mean of active SI cancellation by using different TX beamforming.
In this example, the number of transmit antenna is 3, and the number of receive antennas is 1, 2, 3 respectively. We compare the mean of active cancellation at one channel distortion value when the number of receive antennas differs. In the scenario that the number of receive antenna is one, the active cancellation is directly measured at this antenna. In the scenario that more than one receive antennas are involved, we measured the self-interference cancellation at each receive antenna. Then, the mean active cancellation can be derived by calculating the mean of cancellation at all the receive antennas.
Fig.4. The lower bound of cancellation by using different TX beamforming
As we can see in Fig4, if the channel distortion is less than 0.6, the active cancellation improves dramatically with the decreasing number of the receive antennas. However, when the channel distortion is larger than 1.2, the cancellation performances are almost same while the number of the receive antennas are different.
The active analog cancellation of the robust beamforming will be impaired when the channel distortion increases. Usually the channel distortion will increase when the passive suppression in full duplex MIMO node increases. Because the phase noise will not linearly decrease with the decrease of the channel amplitude. This implies that highly passive suppression may impair the active self-interference cancellation of the robust beamforming.
The cancellation performance varies when the ratio of the transmit antennas number and the receive antennas number changes which raises the question, "How many receive antennas should we choose to achieve best active cancellation if the number of transmit antennas is fixed." Since the total active cancellation includes analog cancellation and digital cancellation, we need to take both of these two cancellations into account.
In Fig4, it demonstrates that, when the channel distortion is small, the less receive antennas we use, the more analog cancellation we will achieve with the robust beamforming methods. However, the digital cancellation will increase with the increase of the order of the receive filters. This implies that the increase of the receive antennas will improve the performance of the digital cancellation. From , we know that the digital cancellation can achieve up to 20 dB cancellation, much less than the analog cancellation. Therefore, we can conclude that, if the channel distortion is small, the less receive antennas will provide us better performance on the total amount of active cancellation. While, if the channel distortion is large, the analog cancellation we can achieve will not rely on the number of the receive antennas. Therefore, the increase of the total amount of active cancellation will purely depend on the increase of digital cancellation. Therefore, we should improve the digital cancellation by increasing the number of the receive antennas.
We have highlighted the various self-interference methods for full duplex system. Given that MIMO is crucial for next generation multiple antenna devices, we propose a new method to achieve self-interference cancellation for the full-duplex MIMO system based on the beamforming technique. The simulation results show that our approach performs well in active cancellation. However, we need more theoretical work to explore the limitation of active cancellation in full duplex MIMO system.