# Widely-Linear Digital Self-Interference Cancellation in Direct-Conversion Full-Duplex Transceiver

## Abstract

This article addresses the modeling and cancellation of self-interference in full-duplex direct-conversion radio transceivers, operating under practical imperfect radio frequency (RF) components. Firstly, detailed self-interference signal modeling is carried out, taking into account the most important RF imperfections, namely transmitter power amplifier nonlinear distortion as well as transmitter and receiver IQ mixer amplitude and phase imbalances. The analysis shows that after realistic antenna isolation and RF cancellation, the dominant self-interference waveform at receiver digital baseband can be modeled through a widely-linear transformation of the original transmit data, opposed to classical purely linear models. Such widely-linear self-interference waveform is physically stemming from the transmitter and receiver IQ imaging, and cannot be efficiently suppressed by classical linear digital cancellation. Motivated by this, novel widely-linear digital self-interference cancellation processing is then proposed and formulated, combined with efficient parameter estimation methods. Extensive simulation results demonstrate that the proposed widely-linear cancellation processing clearly outperforms the existing linear solutions, hence enabling the use of practical low-cost RF front-ends utilizing IQ mixing in full-duplex transceivers.

Direct-conversion radio, full-duplex radio, self-interference, image frequency, IQ imbalance, widely-linear filtering

## 1 Introduction

\IEEEPARstartFull-duplex radio communications with simultaneous transmission and reception at the same radio frequency (RF) carrier is one of the emerging novel paradigms to improve the efficiency and flexibility of RF spectrum use. Some of the recent seminal works in this field are for example [1, 2, 3, 4, 5, 6, 7, 8], to name a few. Practical realization and implementation of small and low-cost full-duplex transceivers, e.g., for mobile cellular radio or local area connectivity devices, are, however, still subject to many challenges. One of the biggest problems is the so called self-interference (SI), which is stemming from the simultaneous transmission and reception at single frequency, thus causing the strong transmit signal to couple directly to the receiver path.

In general, the transmitter and receiver may use either the same [9, 10, 11] or separate but closely-spaced antennas [2, 3, 4, 5]. In this work, we focus on the case of separate antennas where depending on, e.g., the deployed physical antenna separation and transmit power, the level of the coupling SI signal can be in the order of 60–100 dB stronger than the actual received signal of interest at receiver input, especially when operating close to the sensitivity level of the receiver chain. To suppress such SI inside the transceiver, various antenna-based solutions [2, 12, 13, 4], active analog/RF cancellation methods [2, 3, 14, 4, 15, 16, 8] and digital baseband cancellation techniques [17, 14, 18, 3] have been proposed in the literature. In addition, a general analysis about the overall performance of different linear SI cancellation methods is performed in [7], while in [19], SI cancellation based on spatial-domain suppression is compared to subtractive time-domain cancellation, and rate regions are calculated for the two methods.

However, the performance of the SI cancellation mechanisms based on linear processing is usually limited by the analog/RF circuit non-idealities occurring within the full-duplex transceiver. For this reason, some of the most prominent types of such analog/RF circuit non-idealities have been analyzed in several recent studies. The phase noise of the transmitter and receiver oscillators has been analyzed in [20, 21, 22]. It was observed that the phase noise can significantly limit the amount of achievable SI suppression, especially when using two separate oscillators for transmitter and receiver. A signal model including the effect of phase noise is also investigated in [23], where the feasibility of asynchronous full-duplex communications is studied. In [24], the effects of the receiver chain noise figure and the quantization noise are also taken into account, in addition to phase noise. The authors then provide an approximation for the rate gain region of a full-duplex transceiver under the analyzed impairments. The effect of quantization noise is also analyzed in [25], where the relation between analog and digital cancellation under limited dynamic range for the analog-to-digital converter is studied. The existence of several non-idealities in the transmit chain, including power amplifier (PA) nonlinearity, is acknowledged in [26]. As a solution, the authors propose taking the cancellation signal from the output of the transmitter chain to exclude these non-idealities from the signal path. In [27], several non-idealities in the transmitter and receiver chains are also considered, including nonlinearities and IQ mismatch. Their effect is then studied in terms of the achievable SI suppression with linear processing and received spatial covariance eigenvalue distribution. In [28], the transmitter non-idealities are modelled as white noise, the power of which is dependent on the transmit power, and their effect is analyzed in the context of cognitive radios. In [29], comprehensive distortion calculations, taking into account several sources of nonlinear distortion, among other things, are reported. The findings indicate that, especially with high transmit powers, the nonlinear distortion of the transmitter PA and receiver front-end components can significantly contribute to the SI waveform and thus hinder the efficiency of purely linear digital cancellation. Stemming from the findings of these studies, novel nonlinear cancellation processing solutions have been recently proposed in [17, 18, 14] to suppress such nonlinear SI, in addition to plain linear SI, at receiver digital baseband.

In addition to phase noise and nonlinearities, also other RF imperfections can impact the SI waveform and its cancellation. One particularly important imperfection in IQ processing based architectures is the so-called IQ imbalance and the corresponding inband IQ image or mirror component [30]. On the transmitter side, in general, such IQ imbalance is contributing to the transmitter error vector magnitude, and possibly also to adjacent channel leakage and spurious emissions, depending on the transmit architecture. As a practical example, 3GPP Long Term Evolution (LTE)/LTE-Advanced radio system specifications limit the minimum attenuation for the inband image component in mobile user equipment transmitters to 25 dB or 28 dB, depending on the specification release [31]. Such image attenuation is sufficient in the transmission path, but when considering the full-duplex device self-interference problem, the IQ image of the SI signal is additional interference leaking to the receiver path. Furthermore, additional IQ imaging of the SI signal takes place in the receiver path. In [32], the effect of IQ imbalance on a full-duplex transceiver was noticed in the measurements, as it was causing clear residual SI after all the cancellation stages. However, there is no previous work on compensating the SI mirror component caused by the IQ imbalance of the transmitter and receiver chains. Overall, there is relatively little discussion in the existing literature about the effect of IQ imbalance on the performance of a full-duplex transceiver. In part, this is due to the high-cost high-quality equipment that has been used to demonstrate the full-duplex transceiver principle in the existing implementations. For example, the WARP platform, which has been used at least in [2, 3, 5, 14, 7], provides an attenuation in the excess of 40–50 dB for the inband image component [33]. As our analysis in this article indicates, this is adequate to decrease the mirror component sufficiently low for it to have no significant effect on the performance of the full-duplex transceiver. Furthermore, if properly calibrated high-end laboratory equipment is used, the image attenuation can easily be even in the order of 60–80 dB, meaning that the effect of IQ imbalance is totally negligible. However, for a typical mobile transceiver with low-cost mass-market analog/RF components, the image attenuation is generally significantly less, as already mentioned [31]. This means that also the effect of IQ imbalance in a full-duplex transceiver must be analyzed and taken into account.

In this article, we firstly carry out detailed modeling for the SI waveform in different stages of the transmitter-receiver coupling path, taking into account the effects of transmitter and receiver IQ image components, as well as transmitter PA nonlinearities. Incorporating then also the effects of realistic multipath antenna coupling, linear analog/RF cancellation and linear digital baseband cancellation, the powers of the remaining SI components at the output of the full coupling and processing chain are analyzed. This analysis shows that, with realistic component values and linear cancellation processing, the IQ image of the classical linear SI is heavily limiting the receiver path signal-to-noise-plus-interference ratio (SINR). Such observation has not been reported earlier in the literature. Motived by these findings, a novel widely-linear (WL) digital SI canceller is then developed, where not only the original transmit data, but also its complex conjugate, modeling the IQ imaging, are processed to form an estimate of the SI signal. Efficient parameter estimation methods are also developed to estimate the cancellation parameters of the proposed WL structure through WL least-squares model fitting. The proposed WL SI canceller is shown by analysis, and through extensive simulations, to substantially improve the SI cancellation performance in the presence of practical IQ imaging levels, compared to classical purely linear processing, and it can hence enable full-duplex transceiver operation with realistically IQ balanced low-cost user equipment RF components.

The rest of this article is organized as follows. In Section 2, the structure of the considered full-duplex transceiver and its baseband-equivalent model are presented, alongside with the overall self-interference signal model and its simplified version. Also principal system calculations, in terms of the powers of the different self-interference terms, are carried out. The proposed method for widely-linear digital cancellation and the estimation procedure for the coefficients are then presented in Section 3. In Section 4, the performance of widely-linear digital cancellation under different scenarios is analyzed with full waveform simulations. Finally, the conclusions are drawn in Section 5.

## 2 Full-Duplex Transceiver and Self-Interference

The structure of the analyzed full-duplex transceiver is presented in Fig. 1. It can be observed that the transceiver follows a typical direct-conversion architecture [34, 35, 36], which is well-known in previous literature, and thus it is not discussed here in detail. This architecture is chosen due to its simple structure and wide applications in modern wireless transceivers. The actual IQ imaging problem is caused by the IQ mixers at both the transmitter and receiver chains. Due to the inherent mismatches between the amplitudes and phases of the I- and Q-branches, the mirror image of the original signal is added on top of it, with certain image attenuation [30]. In this paper, we assume that the level of this image attenuation is similar to what is specified in 3GPP LTE specifications [31], namely 25 dB.

The actual analysis of the full-duplex transceiver and SI waveform at different stages of the transceiver is done next by using baseband-equivalent models. The block diagram of the overall baseband-equivalent model is shown in Fig. 2, alongside with the principal mathematical or behavioral model for each component, propagation of the transmitted signal, and the corresponding variable names. In the following subsection, a complete characterization for the effective SI waveform in different stages of the transceiver is provided, using the same notations as in Fig. 2, taking into account transmitter IQ imaging and PA distortion, realistic multipath coupling channel, realistic linear analog/RF cancellation, receiver IQ imaging, and receiver linear cancellation. Stemming from this, the powers of the different SI terms at receiver digital baseband are then analysed in Subsection 2.2.

### 2.1 Self-Interference Signal Model with Practical RF Components

The complex baseband transmitted signal is denoted by , or by after digital-to-analog conversion. It is assumed that the power of is such that the desired transmit power is reached after the amplification by IQ mixer and PA, i.e., the transmitter VGA is omitted from the baseband-equivalent model. This is done to make the notation simpler and thus more illustrative. In addition, the signal is perfectly known, as it is generated within the transceiver. In this analysis, the digital-to-analog converters (DACs) and low-pass filters (LPFs) are assumed to be ideal, but the IQ mixer is assumed to have some imbalance between the I and Q branches. The signal at the output of the transmit IQ mixer can be now written as

(1) |

where is the response for the direct signal component, and is the response for the image component [30]. Here indicates the complex conjugate and denotes the convolution operation. Above-kind of transformations, where both direct and complex-conjugated signals are filtered and finally summed together, are typically called widely-linear in the literature, see e.g. [37, 38].

The quality of the IQ mixer can be quantified by the image rejection ratio (IRR). With the variables used in this analysis, it can be defined for the transmitter as

(2) |

where and are the frequency-domain representations of and , respectively [30]. A similar characterization can obviously be established also for the receiver IQ mixer, referred to as .

Before transmission, the signal is amplified with a nonlinear PA. In this analysis, we model the PA response with a Hammerstein nonlinearity [39, 40] given as

(3) |

where is the linear gain, is the gain of the third order component, and is the memory model of the PA. For simplicity, we write , and use this to refer to the third order nonlinear component. Thus, we can rewrite (3) as

(4) |

It is obvious that true PAs contain also distortion components beyond third-order, but in this analysis we make a simplification and focus only on the third-order distortion, as that is in practice always the strongest nonlinearity at PA output.

The transmitted signal is next coupled back to the receive antenna, thus producing SI. In this analysis, to simplify the notations, it is assumed that there is no actual received signal of interest. This will decrease the complexity of the equations while having no significant effect on the results, as the purpose is to characterize the final SI waveform at receiver baseband. Thus, the signal at the input of the receiver chain is of the following form:

(5) |

where is the multipath coupling channel between transmit and receive antennas, and denotes thermal noise. To suppress the SI before it enters the LNA, RF cancellation is performed. The signal after RF cancellation can be expressed as

(6) |

where is typically an estimate for the main path of the coupling channel [7, 2, 3]. In other words, is a one tap filter, depicting the delay, phase, and attenuation of the main coupling propagation. Thus, in this analysis it is assumed that RF cancellation attenuates only the direct coupling component. Furthermore, in the numerical experiments and results, is chosen so that it provides the desired amount of SI attenuation at RF, e.g., 30 dB, as has been reported in practical experiments [5, 7, 15]. This is done by tuning the error of its attenuation and delay, modeling realistic RF cancellation.

Next, the received signal is amplified by the low-noise amplifier (LNA). The output signal of the LNA can be written as

(7) |

where is the complex gain of the LNA, and is the increase in the noise floor caused by the LNA. The nonlinearity effects of the receiver chain amplifiers are not taken into account in this paper, as they are insignificant in comparison to the other distortion components under typical circumstances [29]. This will also significantly simplify the analysis.

Similar to the transmitter, the receiver IQ mixer has IQ imbalance, and thus it produces an image component of the total signal entering the mixer, including the SI. The signal at the output of the receiver IQ mixer can be now expressed as

(8) |

where is the response for the direct signal component, and is the response for the receiver image component.

Finally, the signal is amplified by the variable gain amplifier (VGA) to match its waveform dynamics to the voltage range of the analog-to-digital converter (ADC) and then digitized. The digitized signal can be written as

(9) |

where is the complex baseband gain of the VGA, is the sampling time, and denotes quantization noise. In the continuation, we drop the sampling interval from the equations for brevity and use only the discrete-time index .

To express the residual SI signal at the digital domain in terms of the known transmit data , a complete equation for is next derived by substituting (1) to (4), (4) to (5) and so on. After these elementary manipulations, we arrive at the following equation for , with respect to and fundamental system responses:

(10) |

where we have defined a total noise signal as , including the thermal noise at the input of the receiver chain and the additional noise produced by the LNA. The channel responses of the individual signal components can be written as follows:

(11) | ||||

(12) | ||||

(13) | ||||

(14) | ||||

(15) | ||||

(16) |

As can be seen in (10), the total SI at receiver digital baseband contains not only the linear SI but also its complex conjugate. These different components of the SI signal are hereinafter referred to as linear SI and conjugate SI, respectively. In addition to these signal components, PA-induced IMD and its complex-conjugate, which will similarly be referred to as IMD and conjugate IMD, are also present in the total SI signal.

Using the above equations, it is possible to describe the effect of conventional linear digital SI cancellation, which can attenuate only the linear SI component. Corresponding to the notation in Fig. 2, where the linear channel estimate is denoted by , the signal after linear digital cancellation can be expressed as

(17) |

From (17) it can be observed that should estimate the channel of the linear SI component, i.e., . However, even with a perfect estimate of , the signal can still be substantial interference from weak desired signal perspective. This is mainly due to the conjugate SI, IMD, and conjugate IMD. Obviously, there is also some thermal noise, but typically it is not significantly limiting the performance of a full-duplex transceiver.

Notice that nonlinear distortion has been shown earlier in the literature to limit the achievable SINR of a full-duplex radio, and there are also methods for attenuating it [17, 18, 14]. However, there is no previous work on analyzing and attenuating the conjugate SI signal, which is relative to in our notations. In the next subsection, we will analyze the relative strength of this conjugate SI through principal power calculations, and show that with typical RF component specifications, it is the dominant SINR limiting phenomenon. In Section 3, we then also provide a method for suppressing the conjugate SI in the digital domain by processing the original transmit data in a widely-linear manner with two filters and , marked also in Fig. 2. However, in the following subsection it is still first assumed that , and thus no compensation is done for the conjugate SI, in order to properly quantify and illustrate the limitations of classical linear SI cancellation.

### 2.2 Principal System Calculations for Different Distortion Terms

In order to analyze and illustrate the relative levels of the different distortion components of the overall SI signal, a somewhat simplified scenario is first considered. More specifically, the frequency-dependent characteristics of different distortion components are neglected, which allows for the equations to be presented in a more compact and illustrative form. Furthermore, as we are here primarily interested in the average powers of different distortion components, neglecting inband frequency-dependency is well justified. If we now denote the general impulse function by , the different system impulse responses of the distortion components can be expressed as follows: , , , , , , , and . By substituting these simplified terms into (11)–(16), we can rewrite them as

(18) | ||||

(19) | ||||

(20) | ||||

(21) | ||||

(22) | ||||

(23) |

Furthermore, as the magnitude of the term is very small in comparison to the magnitude of , even when considering a relatively low image attenuation, we can write (18) as

(24) |

Using (19)–(24), can then be expressed in a simplified form as follows:

(25) |

From (25), it is now possible to calculate the powers of the different signal components. These powers are defined as follows:

(26) | ||||

(27) | ||||

(28) | ||||

(29) | ||||

(30) | ||||

(31) | ||||

(32) |

where denotes the expected value, is the power of the signals and , as conjugation does not affect the power of the signal, is the power of the signals and , is the power of the signals and , and is the power of quantization noise. Here, is defined in terms of the total signal power at the input of the ADC, , and the SNR of the ADC, , where is the number of bits at the ADC, and is the peak-to-average-power ratio in dB [34].

For further simplicity, the power of the total noise term can be expressed in a more compact manner. As the LNA constitutes for most of the noise factor of the receiver chain, denoted by , it can be appoximated with very little error that LNA has a noise factor of , and no additional noise is produced throughout the rest of the receiver chain. Thus, if the power of the thermal noise at the input of the receiver chain is defined as , we can write , based on the definition of the noise factor [34].

In addition, to express the power levels using the defined parameters, the amount of achieved RF cancellation with respect to the error between and must be defined. This can be done by comparing the power of the total SI signal before and after RF cancellation. Based on (5), the SI signal before RF cancellation is . After RF cancellation, the SI signal is , based on (6). Taking the previously defined simplifications into account, we can write and . Thus, the amount of RF cancellation can be defined as

(33) |

Here it is assumed that the instantaneous path loss of the SI coupling channel, denoted by , and the error of the RF cancellation channel estimate, denoted by , are circular and normally distributed. The amount of antenna attenuation is now defined as . Using (33), it is then possible to define the power of the error between the SI coupling channel and the channel estimate for RF cancellation as

(34) |

The amount of achieved linear digital cancellation is defined next as the decrease in the power of the linear SI component . Before digital cancellation, the linear SI signal can be expressed as , and after linear digital cancellation as . The attenuation of the linear SI power by linear digital cancellation can then be expressed as follows:

(35) |

Now, by using (35), it is possible to express in terms of as follows:

(36) |

where .

Now, by substituting (34) to (26)–(29), (36) to (26), and to (30)–(31), we can finally express the power levels of all the different signal components as follows:

(37) | ||||

(38) | ||||

(39) | ||||

(40) | ||||

(41) | ||||

(42) | ||||

(43) |

In the above equations it is assumed that the complex gains of the different RF components are static and deterministic, whereas the error of the RF channel estimate is assumed to be a circular random variable, as explained earlier. Furthermore, the final term of the first equation for can be omitted, as due to the circularity assumption. The above set of derived formulas in (37)–(43) can now be used to evaluate the powers of different distortion terms at detector input, and in particular how they depend on the transmit power, antenna isolation, active RF cancellation and linear digital cancellation as well as on the transceiver RF imperfections.

#### System Calculations Example

To illustrate the relative strengths of the different signal components, typical component parameters are chosen, and (37)–(43) are used to determine the corresponding power levels, which will then be shown for a specified transmit power range. The used parameters are listed in Table 1, and they correspond to a typical wideband transceiver with low-cost mass-product components. The value for IRR, describing the image attenuation, is chosen based on 3GPP LTE specifications [31]. The baseband VGA of the receiver chain is assumed to match the total waveform dynamics at the ADC input to the available voltage range.

Parameter | Value |
---|---|

Bandwidth | 12.5 MHz |

Thermal noise floor at receiver input | -103.0 dBm |

Receiver noise figure | 4.1 dB |

SNR requirement at detector input | 10 dB |

Sensitivity level | -88.9 dBm |

Power of the received signal | -83.9 dBm |

Transmit power | varied |

PA gain | 27 dB |

PA IIP3 | 20 dBm |

Antenna separation | 40 dB |

RF cancellation | 30 dB |

LNA gain | 25 dB |

IQ mixer gain (RX and TX) | 6 dB |

IRR (RX and TX) | 25 dB |

VGA gain (RX) | 1–51 dB |

ADC bits | 12 |

ADC P-P voltage range | 4.5 V |

PAPR | 10 dB |

In addition to the distortion powers, the power of the received signal of interest at detector input, denoted by , is also shown in the figures as a reference, although it is not included in the signal model. This is done in order to be able to put the various distortion powers to proper context. The power of the signal of interest is chosen to be 5 dB above the sensitivity level at the input of the receiver chain. As the receiver sensitivity is defined for a 10 dB thermal noise SNR at detector input, this means that the signal of interest will be 15 dB above the thermal noise floor in the digital domain, which is then also the detector input SINR if no SI is present.

The power levels of the different signal components after linear digital cancellation, for transmit powers from -5 dBm to 25 dBm, are shown in Fig. 3. The amount of digital cancellation is chosen so that linear SI is attenuated below the thermal noise floor. This has been observed to be close to the true performance of digital cancellation under realistic conditions [29, 14]. In this example, this requires 27–57 dB of linear digital cancellation, depending on the transmit power.

From Fig. 3 it can be observed that the conjugate SI signal is clearly the most dominant distortion under a wide range of transmit powers. Actually, with transmit powers above 9 dBm, the power of the conjugate SI is even more powerful than the power of the signal of interest. Thus, with the chosen parameters, the conjugate SI is seriously degrading the achievable SINR of the full-duplex transceiver, which motivates the study of possible methods for attenuating it. Now, the ideal SINR of 15 dB is unreachable with the whole transmit power range from -5 dBm onwards, due to the powerful conjugate SI.

To investigate a different scenario, the amount of antenna separation is next assumed to be 30 dB, and the amount of RF cancellation 20 dB, which are achievable figures, even with very small antenna distance and low quality components for RF cancellation [2, 4]. In addition, the IRR of the IQ mixers is increased to 35 dB to model the effect of using expensive, higher quality IQ mixers. Again, the amount of digital cancellation is chosen so that it attenuates the linear SI below the thermal noise, now requiring 47–77 dB of linear digital SI attenuation.

The resulting power levels are shown in Fig. 4. Now it can be observed that the power of the conjugate SI is even higher with respect to the power of the signal of interest, due to less analog SI cancellation. Thus, with more pessimistic values for antenna separation and RF cancellation, the effect of conjugate SI is very severe, even when using higher quality IQ mixers. It should be noted, however, that with transmit powers above 15 dBm, also the IMD produced by the PA can be observed to be a significant factor. This indicates that, in order to achieve higher transmit powers with these parameters, attenuating only the linear and conjugate SI may not be sufficient, as also the nonlinear distortion will degrade the signal quality.

## 3 Proposed Widely-Linear Digital Cancellation

It was observed in the previous section that, with typical component parameters, conjugate SI is the dominating source of distortion after classical linear digital cancellation. Thus, the performance of the analyzed full-duplex transceiver can be enhanced by attenuating also the SI image component in the cancellation processing. This can be done by utilizing widely-linear digital SI cancellation, the principle of which is proposed and formulated in this section.

In order to focus on the IQ image induced problematics, we assume below that the IMD produced by the PA, alongside with the conjugate IMD, are clearly weaker than the conjugate SI. This is also clearly visible in Fig. 3. We wish to acknowledge, though, that under certain circumstances, also the PA-induced nonlinear SI component must be further suppressed [17, 18, 14], but below we focus for simplicity on the dominating conjugate SI stemming from the IQ imbalances. A combined canceller for suppressing both the IMD and the conjugate SI is left for future work.

### 3.1 Widely-Linear Cancellation Principle

Our starting point is the fundamental signal model in (10), which we can first re-write as

where

and denotes in general the sum of all other signal terms, most notably thermal noise and quantization noise, and also PA-induced nonlinear distortion if present. Notice that here the general case of frequency-dependent antenna coupling and frequency-dependent IQ imbalances is considered, without any approximations.

The cancellation of the conjugate SI can be done in a similar manner as the cancellation of the direct component in (17). In this case, however, the known transmitted signal must first be conjugated, and the conjugated samples are then filtered with the channel estimate of the effective image channel, , to produce the cancellation signal. If the estimate for the channel of the conjugate SI signal is denoted by , corresponding to the notation in Fig. 2, then the total signal after the cancellation stage can be expressed as