Nonlinear selfinterference cancellation in MIMO fullduplex transceivers under crosstalk
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Abstract
This paper presents a novel digital selfinterference canceller for an inband multipleinputmultipleoutput (MIMO) fullduplex radio. The signal model utilized by the canceller is capable of modeling the inphase quadrature (IQ) imbalance, the nonlinearity of the transmitter power amplifier, and the crosstalk between the transmitters, thereby being the most comprehensive signal model presented thus far within the fullduplex literature. Furthermore, it is also shown to be valid for various different radio frequency (RF) cancellation solutions. In addition to this, a novel complexity reduction scheme for the digital canceller is also presented. It is based on the widely known principal component analysis, which is used to generate a transformation matrix for controlling the number of parameters in the canceller. Extensive waveform simulations are then carried out, and the obtained results confirm the high performance of the proposed digital canceller under various circuit imperfections. The complexity reduction scheme is also shown to be capable of removing up to 65% of the parameters in the digital canceller, thereby significantly reducing its computational requirements.
Keywords
Fullduplex MIMO Selfinterference RF impairments Crosstalk1 Introduction
Inband fullduplex communications is a promising candidate technology for further improving the spectral efficiency of the next generation wireless systems, such as the 5G networks [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The basic idea behind it is to transmit and receive at the same time at the same centerfrequency, thereby in principle doubling the spectral efficiency. The drawback of such inband fullduplex operation is the own transmit signal, which is coupling to the receiver and becomes an extremely powerful source of selfinterference (SI). The most significant challenge in implementing inband fullduplex radios in practice is thereby the development of SI cancellation solutions, which are capable of removing the SI in the receiver. There are already reports of various demonstrator implementations, which achieve relatively high SI cancellation performance, thereby allowing for true inband fullduplex operation [1, 2, 3, 6, 7, 11, 12, 13, 14].
Moreover, in order to meet the high throughput requirements of the future wireless networks, it is inevitable that the inband fullduplex concept must be combined with MIMO capabilities in the transceivers [7, 12, 13, 14, 15, 16, 17, 18, 19]. This obviously results in a higher physical layer capacity, but it also requires more elaborate SI cancellation solutions. In particular, in a MIMO transceiver, the observed SI signal in each receiver consists of a combination of all the transmit signals, which means that also the SI cancellers must have all of the transmit signals available. Furthermore, in order to perform SI cancellation, the coupling channels between all the transmitters and receivers must be estimated, which results in a somewhat more demanding SI cancellation procedure. Nevertheless, this increased complexity is justified by the higher physical layer throughputs.
Especially the complexity of the RF canceller is heavily affected by the number of transmitters and receivers [7, 15]. For an N _{ T }×N _{ R } MIMO transceiver, the RF canceller requires at least N _{ T } N _{ R } cancellation paths, or even more if using a multitap solution [7, 20]. This number can be somewhat decreased by using auxiliary transmitters to upconvert digitally generated cancellation signals, since then only N _{ R } cancellation paths are required. However, the drawback of this solution is obviously the need for additional RF transmitters, as well as the fact that the digitally generated cancellation signals do not include any of the transmitterinduced impairments, which thereby remain unaffected by this type of an RF cancellation solution [1]. Another possible solution for decreasing the complexity of RF cancellation in the context of very large transmit antenna arrays is to use beamforming to form nulls in the receive antennas [4, 21], which might even allow for completely omitting RF cancellation. In typical MIMO devices, however, the increase in the RF cancellation complexity is more or less inevitable.
In this article, we present a general signal model for the observed SI in the digital domain under a scenario where there is crosstalk between the transmit chains before and after nonlinear PAs. Moreover, it is shown that the signal model can be applied to various different RF cancellation solutions. The presented comprehensive signal model, which shows the effect of the crosstalk in terms of the original transmit signals, is then used as a basis for a highperformance digital SI canceller. The IQ imbalance occurring both in the transmitters and in the receivers is also included in the signal model, since it is typically one of the dominant sources of distortion in a practical transceiver, alongside with the PAinduced nonlinearities [30].
Furthermore, to address the increase in the computational complexity due to the MIMO operation and crosstalk modeling, a novel principal component analysis (PCA)based solution is proposed, which can be used to control the complexity of the signal model. In particular, PCA processing is used to identify the insignificant terms in the observed SI signal, which are then omitted in the further cancellation processing. This results in a significant reduction of the unknown parameters that must be estimated, which obviously decreases the computational requirements of the digital SI canceller. Moreover, since the most dominant SI terms are retained by such processing, there is no essential degradation in the cancellation performance. To the best of our knowledge, such complexity reduction schemes have not been previously proposed in the context of SI cancellation solutions.

We derive the most comprehensive MIMO signal model for the observed SI presented so far in the literature. It covers various RF cancellation scenarios, while also modeling the crosstalk between the transmitters under lowcost nonlinear PAs and IQ imbalance.

We propose a novel nonlinear digital SI canceller, which utilizes the aforementioned advanced signal model.

We propose a novel complexity reduction scheme based on PCA, which can be used to control the computational complexity of the digital canceller, while minimizing the decrease in the cancellation performance.

We present numerical results, which illustrate various aspects of the proposed digital SI cancellation solution with realistic waveform simulations.
The rest of this article is organized as follows. In Section 2, the MIMO signal model is derived. Then, in Section 3, the actual nonlinear digital SI canceller is presented, alongside with the parameter estimation procedure and the PCAbased complexity reduction scheme. After this, in Section 4, the proposed digital SI cancellation solution is evaluated with realistic waveform simulations. Finally, the conclusions are drawn in Section 5.
2 Baseband equivalent signal modeling
In this section, we build a complete SI channel model for a MIMO fullduplex device, including the effects of transmitter impairments (PA nonlinearity, IQ imbalance, and transmitter crosstalk), the linear MIMO SI channel, and RF cancellation. In the forthcoming analysis, the nonlinearities produced by the digitaltoanalog and analogtodigital converters (DACs and ADCs) [31], alongside with phase noise, are omitted from the signal model for simplicity, although phase noise is still included in the reported simulation results.
2.1 Power amplifier and IQ modulator models with crosstalk
with K _{ 1 , j }=1 / _{2}(1+g _{ j } exp(j φ _{ j })),K _{ 2 , j }=1 / _{2}(1−g _{ j } exp(j φ _{ j })), where g _{ j },φ _{ j } are the gain and phase imbalance parameters of transmitter j. Notice that under typical circumstances K _{ 1 , j }≫K _{ 2 , j }. The magnitude of the IQ image component, represented by the conjugated signal term in (1), can be characterized with the image rejection ratio (IRR) as 10 log10(K _{ 1 , j }^{2}/K _{ 2 , j }^{2}).
and h _{ p , j }(n) denote the impulse responses of the PH branches for transmitter j, while M and P denote the memory depth and nonlinearity order of the PH model, respectively [34, 35, 36]. The PH nonlinearity is a widely used nonlinear model for direct as well as inverse modeling of PAs [34, 35, 36, 37].
where α _{ 1 , i j }=α _{ i j } K _{ 1 , i } and α _{ 2 , i j }=α _{ i j } K _{ 2 , i }.
where \(h_{p,j,q_{0},\dotso,r_{N_{T}1}}(m)\) are the coefficients for the basis function of the form \(\prod _{i=1}^{N_{T}}x_{i}(n)^{a_{i}}x_{i}^{*}(n)^{b_{i}}\) such that \(\sum _{i=1}^{N_{T}}\left (a_{i}+b_{i} \right) = p\). This signal model is of similar form as the one presented in [26], with the exception that the model in (7) also incorporates the effect of IQ imbalance and is thus more complete.
The new memory length of the received signal model is also increased from M to M+L. The input signal of the ith receiver (z _{ i }(n)) is then further processed by the RF canceller and the actual receiver chain. Note that the above signal model in (11) also applies to circulator and electrical balance duplexerbased implementations, where each transmitter and receiver pair share the same antenna [32], and hence it is generic in that respect.
2.2 RF cancellation
To ensure an extensive analysis and derivation for the proposed digital cancellation algorithm, we consider three different RF cancellation solutions. The first technique is similar to what has been used, e.g, in [5, 6], and it involves directly tapping the transmitter outputs to obtain the reference signals for RF cancellation. This method is based on purely analog processing, as the whole cancellation procedure is performed in the RF domain. The two other considered methods are based on auxiliary TX chains, which are used to produce the RF cancellation signal from digital baseband samples [1, 38, 39]. We call this latter approach hybrid RF cancellation to distinguish it from purely analog cancellation. Furthermore, we consider both linear and nonlinear preprocessing to be used with this auxiliary transmitter based RF cancellation.
2.2.1 RF cancellation with transmitter output signals
Hence, the structure of the RF canceller output signal model is still of the same form as in (11), but with modified coefficients expressed as \(\breve {h}_{i,p,\mathbf {s}^{k}}(m) = \widetilde {h}_{i,p,\mathbf {s}^{k}}(m) \check {h}^{RF}_{i,p,\mathbf {s}^{k}}(m)\).
This type of purely analog RF cancellation calls for N _{ T }×N _{ R } canceller circuits to be implemented in the device, one canceller from each transmitter to each receiver. The complexity may become prohibitive when the number of antennas is significantly increased and, thereby, when implementing a high order fullduplex MIMO device, alternative methods for RF cancellation might have to be considered.
2.2.2 Hybrid RF cancellation using auxiliary transmitters with linear preprocessing
Also this model is essentially of the same form as (11), with the coefficients of the linear SI terms being affected by the hybrid RF cancellation procedure, while the other terms remain unchanged. This means that the observed SI signal in the receiver digital domain can still be modeled with the same signal model as in the case of pure analog RF cancellation (or no RF cancellation at all). Thus, from the perspective of the digital cancellation algorithm, it makes no difference whether RF cancellation is performed by tapping the transmitter output or by using auxiliary TX chains with linear preprocessing, although the RF cancellation performance itself might obviously be different for the considered methods.
2.2.3 Hybrid RF cancellation using auxiliary transmitters with nonlinear preprocessing
where P ^{′} is the nonlinearity order of the RF cancellation signals. Note that this signal model neglects IQ imbalance and crosstalk, since the RF canceller must only attenuate the SI such that the receiver is not saturated. Also this RF cancellation signal can be easily represented with a signal model of the same form as in (11). The coefficients \(\check {h}^{\mathit {RF}}_{i,p,\mathbf {s}^{k}}(m)\) of the signal model now consist of \( h^{\mathit {RF}}_{ij,p}(l)\) with the parameters p and s ^{ k } that correspond to the basis functions \(x_{j}(nl)^{\frac {p+1}{2}} x^{*}_{j}(nl)^{\frac {p1}{2}}\), and other coefficients are set to zero. Similar to the other RF cancellation schemes, after subtracting the cancellation signal from the received signal, as in (13), the signal model remains the same and its coefficients are \(\breve {h}_{i,p,\mathbf {s}^{k}}(m) = \widetilde {h}_{i,p,\mathbf {s}^{k}}(m)  \check {h}^{\mathit {RF}}_{i,p,\mathbf {s}^{k}}(m)\). Now, also some of the nonlinear SI terms are attenuated by RF cancellation, as they are modeled in the preprocessing stage.
Overall, it can be concluded that the essential structure of the observed SI signal in the digital domain is independent of the chosen method for RF cancellation. This means that, in the forthcoming analysis, the same digital cancellation algorithm can be applied in all the situations since the only difference between the three alternative RF cancellation schemes are the relative power levels of the various SI terms. However, as already mentioned, the RF cancellation performance is likely to differ between these techniques, and also the hardware and computational requirements are different for each RF canceller structure.
Note that this signal model implicitly incorporates also the IQ imbalance occurring in the receiver, even though it is omitted in the derivations for brevity [15].
2.3 Total number of basis functions in the overall model
Luckily, many of the terms arising from the cascade of the impairments are so insignificant that they can be neglected with very little effect on the overall modeling accuracy. This will reduce the computational cost of such modeling and the corresponding cancellation procedure. In this work, we propose a specific preprocessing stage which can be used to decrease the dimensionality of the full signal model in (16). This is elaborated in more details in Section 3.2.
2.4 Nonlinear signal model without crosstalk
where \(\breve {h}_{i,j,p,q}(m)\) represents now the coupling channel corresponding to the considered SI signal terms propagating from the jth transmitter to the ith receiver. This signal model is also derived in [15], where it is briefly discussed and analyzed. For this reason, the detailed derivation process of (18) is omitted in this article.
When investigating Fig. 3, it can be seen that this signal model results in a significant reduction of basis functions, when compared to the full signal model with crosstalk. With moderate crosstalk levels, it is therefore likely that using this signal model will provide a very favorable tradeoff between cancellation performance and computational complexity. However, as already discussed, in highly integrated transceivers explicit modeling of the crosstalk between the transmitters is likely required in order to ensure sufficient cancellation performance [28].
3 Selfinterference parameter estimation and digital cancellation
In this section, building on the previous modeling in, e.g., [15, 29], we will describe the proposed digital cancellation algorithm that models both IQ imbalance and PA nonlinearity in a MIMO fullduplex transceiver with crosstalk between the transmitters. In general, there are two possible approaches for nonlinear digital SI cancellation: (i) construct a linearinparameters model of the observed SI signal in the digital domain, including the different impairments, the MIMO propagation channel, and RF cancellation, estimate the unknown parameters of the model, and finally recreate and cancel the SI from the received signals; (ii) have separate models for the MIMO propagation channel and the transmitter impairments, estimate the unknown model parameters sequentially, and recreate and cancel the SI from the received signals. Typically the latter approach is computationally less demanding, but it requires a more elaborate estimation procedure. In this article, we consider the former approach, while the latter is left for future work.
3.1 Linearinparameters model
Here, \(\bar {P}\) is the nonlinearity order of the digital canceller, M _{1} is the number of precursor taps, M _{2} is the number of postcursor taps, and \(\hat {\breve {h}}_{i,p,\mathbf {s}^{k}}(m)\) contains the estimated parameters of the signal model. The precursor taps are introduced to model all the memory effects produced by the transmitter and RF cancellation circuitry.
3.1.1 Leastsquaresbased estimator
In this work, the actual parameter learning is performed with the widely used least squares (LS) estimation. For brevity, the parameter learning and digital cancellation procedure is here outlined only for the ith receiver, since the procedure is identical for all the receivers.
with \(p=1,3,\ldots,\bar {P}\), and s ^{ k } is each combination for which ∥s ^{ k }∥_{1}=p, similar to the sum limits shown in (16). Overall, the number of concatenated matrices is given by the total number of basis functions in (17), since this is the amount of different combinations of s ^{ k } for all the nonlinearity orders.
assuming full column rank in Ψ.
3.2 Computationally efficient estimation with principal component analysis
Another approach to simplify the estimation procedure is to retain the crossterms, and instead determine which of them are actually significant in terms of the cancellation performance. In this analysis, principal component analysis (PCA) [41] is used to decrease the number of parameters to be estimated. The idea behind the PCA is to determine which of the terms have the highest variance, providing valuable information regarding the significance of the different basis functions. In practice, PCA results in a transformation matrix, with which the original data matrix is multiplied. The size of the transformation matrix can be chosen to provide the desired number of parameters for the final estimation procedure.
There are also various alternative solutions for model complexity reduction, such as compressed sampling (CS) based techniques. Nevertheless, in this work, we choose to use the PCA since it is a straightforward method for the complexity reduction of the proposed signal model, while also providing nearly the same performance as CS when high modeling accuracy is required [42]. Experimenting with different complexity reduction methods is an important future work item for us.
The first step in obtaining the desired PCA transformation matrix is to determine the least squares channel estimate given in (28) using all the basis functions. This estimate should be calculated with the highest possible transmit power, since the nonlinear SI terms that are negligible with the highest power will also be negligible with any lower transmit power. Hence, this reveals the terms, which can be omitted under the whole considered transmit power range. If the transceiver in question has more than one receiver chain, the channel estimation can be done individually for all of them, after which the mean value of the estimates is calculated. This is done to avoid having separate transformation matrices for each receiver, resulting in a decreased amount of required data storage. The hereby obtained coefficient vector, which is denoted by \(\hat {\breve {\mathbf {h}}}_{0}\), is used as an initial channel estimate for the full set of basis functions.
where 1 is a column vector consisting of 1s, and × denotes elementwise multiplication between two matrices. The matrix Ψ _{0} now contains all the SI terms in its columns, each multiplied with the corresponding coefficient of the initial channel estimate.
The hereby obtained data matrix is then used in the least squares estimation as a replacement for the original data matrix Ψ. It should also be noted that when generating the actual digital cancellation signal, the cancellation data matrix must be transformed with the same matrix \(\widetilde {\mathbf {W}}\), as the SI channel estimate is only valid in this transformed space.
An important aspect to point out is that the transformation matrix \(\widetilde {\mathbf {W}}\) is calculated only once with the highest transmit power, after which it can be used with all transmit powers to reduce the number of basis functions. Namely, since the strengths of the nonlinearities are directly proportional to the transmit power, the SI terms that are negligibly weak with the highest transmit power are at least as weak with the lower transmit powers, which means that the same SI terms can be omitted also then. This is also proven by the waveform simulations, the results of which will be discussed in Section 4. However, should the SI channel change drastically at any point, then the matrix \(\widetilde {\mathbf {W}}\) must be recalculated to ensure that no significant memory taps are neglected.
In general, perhaps the most crucial design problem in the context of the PCA is to determine the optimal number of parameters to be included in the final model. This can be most easily determined experimentally by reducing the number of parameters until the obtained cancellation performance starts to drop. Also, the singular values in Σ can be used to calculate the percentage of the variance accounted for by the included basis functions. We will address this issue more closely with the help of waveform simulations in Section 4.
4 Performance simulations and analysis
The evaluation of the proposed scheme is now done with realistic waveform simulations, utilizing a comprehensive inband fullduplex transceiver model. It incorporates all the relevant impairments, and thereby the SI waveform represents a realworld scenario rather well. Below, we describe the waveform simulator in detail, after which the results are shown. As an important future work item, we aim to evaluate the proposed scheme also with actual RF measurements to confirm the results obtained here with the simulations.
4.1 Simulation setup and parameters
The relevant parameters of the waveform simulator
Parameter  Value 

Bandwidth  20 MHz 
Sampling frequency  122.88 MHz 
Number of TX/RX antennas  2/2 
PA gain  27 dB 
PA IIP3  13 dBm 
Level of TX crosstalk before the PAs  −10 dB/varied 
Level of TX crosstalk after the PAs  −10 dB 
Receiver noise floor  −96.9 dBm 
Phase noise characteristics  See Fig. 5 
Transmit power  25 dBm/varied 
SI channel length  20 taps 
Antenna attenuation  40 dB 
RF cancellation  30 dB 
IRR (TX/RX)  25 dB 
ADC bits  12 
Parameter estimation sample size (N)  30,000/varied 
Parameter estimation sample size for PCA  10,000 
Nonlinearity order of the canceller (P)  5 
Number of precursor taps (M _{1})  10 
Number of postcursor taps (M _{2})  20 

Digital canceller with the full signal model in (22), including PCA processing to decrease the dimensionality and computational complexity

Digital canceller with the full signal model in (22), but without any dimensionality reduction

Digital canceller utilizing the Ninput memory model from [26], which considers the nonlinearity of the PA and both linear and nonlinear crosstalk.

Digital canceller with the crosstalkfree signal model in (18), from [15], where both the nonlinearity of the PA and the IQ imbalance are modeled.

Digital canceller with a traditional linear signal model, where P=1.
In all the cases, the same parameter estimation sample size is used for the different cancellers with M _{1}=10 and M _{2}=20 to ensure a fair comparison. The PCA matrix is calculated using 10 000 samples in the initial channel estimation stage. Furthermore, to avoid overfitting when estimating and cancelling the SI, separate portions of the signal are used for calculating the SI channel estimate and evaluating the actual SI cancellation performance.
4.2 Results
Note that in this case the phase noise has no significant effect on the residual SI power since a common local oscillator between the transmitters and receivers is assumed. This results in a certain level of selfcancellation of the phase noise upon downconversion, considerably reducing its significance [44].
With transmit powers beyond 20 dBm, the crosstalk effects begin to decrease also the accuracy of the crosstalkfree nonlinear signal model from [15]. On the other hand, the full signal models perform relatively well even with the highest transmit powers, resulting in only a very minor increase in the noise floor. Furthermore, as observed earlier, retaining only 35% of the terms after the PCA processing does not seem to decrease the accuracy of the signal model when compared to the full signal model with all the terms included. In fact, the performance of the digital canceller with the lower transmit powers is slightly improved by the dimensionality reduction since the smaller number of parameters results in a more accurate parameter vector estimate, and hence in more efficient cancellation.
In order to minimize the computational complexity of the cancellation procedure, the number of included terms must obviously be minimized. Hence, the smallest number of terms that still provides the required performance is in this sense the optimal choice. Figure 8 indicates that, with the parameters considered in these simulations, the optimal percentage of included terms is roughly 35%, which corresponds to 840 coefficients with the considered nonlinearity order and number of memory taps.
It can also be observed from Fig. 9 that a larger number of terms is required with the very high crosstalk levels. In particular, having only 35% of the terms retained results in a somewhat higher residual SI power than retaining all of the terms. This is explained by the fact that higher crosstalk levels also result in a larger number of significantly powerful SI terms. Nevertheless, the cancellation performance differences between the full signal models, with or without PCA processing, are still relatively small with these reasonable crosstalk levels.
5 Conclusions
In this paper, a novel digital selfinterference canceller for a nonlinear MIMO inband fullduplex transceiver was presented. The canceller is based on a comprehensive signal model for the SI observed in the digital domain, which includes the effect of crosstalk occurring between the transmit chains, while also incorporating the most significant RF imperfections. Furthermore, it was also shown that the signal model is valid for various different RF cancellers. To control the complexity of the cancellation procedure, a novel principal component analysis based scheme was then proposed, which can be used to control the number of parameters in the signal model. With the help of waveform simulations, the proposed digital canceller was shown to cancel the SI nearly perfectly, even when its computational complexity was significantly reduced using principal component analysis.
6 Appendix: Power amplifier output signal under crosstalk
where \(A_{k_{1},\dotso,k_{N1}}\) is a constant.
Notes
Acknowledgements
The research work leading to these results was funded by the Academy of Finland (under the projects #259915, #301820, and #304147), the Finnish Funding Agency for Technology and Innovation (Tekes, under the TAKE5 project), Tampere University of Technology Graduate School, Nokia Foundation, Tuula and Yrjö Neuvo Research Fund, and Emil Aaltonen Foundation.
Competing interests
The authors declare that they have no competing interests.
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