On Self-Interference Cancellation and Non-Idealities Suppression in Full-Duplex Radio Transceivers

One of the major impediments in the design and operation of a full-duplex radio transceiver is the presence of self-interference (SI), that is, the transceiver’s transmitted signal, 60–100 dB stronger than the desired signal of interest. To reduce the SI signal below the receiver’s sensitivity before coupling it to the receiver, radio frequency (RF)/analog domain cancellation is carried out. Even after SI cancellation to the required level in the analog domain, the residual SI signal still exits and lowers the transceiver’s performance. For residual SI cancellation, a digital domain cancellation is carried out. RF impairments are the major obstacle in the residual SI cancellation path in the digital domain. Linearization of RF impairments such as IQ mixer imbalance in the transmitter and receiver chain, non-linear PA with memory, and non-linear LNA are also carried out. Performance evaluation of the proposed techniques is carried out based on SINR, the power of different SI signal components, PSD, output to input relationship, SNR vs. BER, spectrum analyzer, constellation diagram, and link budget analysis. The proposed techniques provide attractive RF/analog SI cancellation of up to 80–90 dB, digital residual SI cancellation of up to 35 to 40 dB, total SI cancellation of up to 110 to 130 dB, and an SINR improvement of up to 50 dB.


Contributions
In order to resolve the problem of SI in the FD communication system, several SIG techniques have been proposed in the past. To reduce the transceiver's SI effects on the receiver's performance, both in the analog and digital domain, i.e., RF/analog and digital SI cancellation, much research have been conducted in the radio layer. Any one of the cancellation techniques alone does not have the capability to provide SI cancellation to the desired level. Moreover, existence of non-idealities in the transceiver's components act as a huge hurdle in mitigation of SI.
In this work, in order to gain high efficiency and SIC to the desired level, different SIC techniques, both in the analog and digital domains, along with linearization techniques of the transceiver's component is presented. This manuscript is focused on the functioning of the FD communication system and mitigation of its impediments at the radio layer. The FD widely linear direct conversion transceiver model is utilized to validate the performance of the proposed methods. The main contribution of this work is setting out the following: • Three RF/analog SIC techniques, i.e., CSI, GALL filter and Kautz filter for SI cancellation to level below the receiver's thermal noise floor or sensitivity level. Metrics used for FD system evaluation are link-budget analysis, power calculation of different SI signal components, and signal to interference plus noise ratio (SINR). Performance analysis of the FD transceiver's transmitter chain components is carried out separately, and results are shown through constellation diagrams and spectrum analyzers. Tx's IQ mixer performance is analyzed through constellation diagrams. PA linearity analysis is performed by examining PA power spectral density (PSD) and its input to output relationship. Unlike the Tx chain, Rx chain performance analysis is carried out by using the metric bit error rate (BER) vs. signal to noise ratio (SNR). LNA linearity analysis is performed separately and examined through PSD and the output vs. input relationship of LNA with and without linearization techniques.

Organization of the Paper
The rest of the paper is organized as follows: the proposed system model and techniques for SIC and the linearization of non-idealities of the transceiver's components are described in Section 2. The simulations, results and discussion for the proposed system and techniques along with comparison of the proposed work with the previous works are described in Section 3. Finally, Section 4 presents the conclusion of our work.

Proposed System Model and Techniques
The architecture considered for performance analysis and result verification is a wellknown transceiver model, i.e., DC transceiver, and is shown in Figure 1 along with SI cancellation techniques and transceiver component's non-idealities. The RF impairments, which the DC transceiver faces during practical realization and implementation, are IQ imbalance, phase noise, and frequency offset due to local oscillator (LO), non-linear behavior of PA and LNA, and DAC and ADC dynamic range.

Transceiver's Components Linearization
The signal that is transmitted after digital to analog conversion (represented by ( ) ( )) and amplification (represented by ( ) ) is the baseband signal (represented by its two components , and , ). DAC and low pass filter (LPF) are considered ideal, and variable gain amplifier (VGA) is omitted from calculations for the sake of simplicity. However, IQ, PA, and LNA nonlinearity is considered. The IQ imbalance problem arises because of a gain and amplitude mismatch between the inphase and quadrature branches of the IQ mixer. Phase noise and frequency offset also result from IQ mixer's LO IQ mixer mismatch in Tx, and the Rx chain is considered with the help of the image rejection ratio (IRR), as defined by the 3rd generation partnership project (3GPP) long-term evolution-advance (LTE-A) specification, i.e., 25 dB for both Tx and Rx chain. 3GPP LTE-A system specification limits the minimum image attenuation to 25 dB or 28 dB [47]. 25 dB attenuation is sufficient for the Tx chain.
However, the Rx chain IQ imbalance adds additional image attenuation. IQ imbalance in the Rx chain affects the residual SI signal cancellation. As a widely linear model is used for analysis, so both direct and image components are taken. Image rejection ratio (IRR) determines the quality of mismatch between I and Q branches. PA nonlinearity is modeled as the Hammerstein memory model. Signal after amplification from PA gets coupled to its receiver, and this coupled signal is called an SI signal. The channel between Tx and Rx is considered a multi-path coupling channel. SI signal is 60-100 dB stronger than our desired signal (the signal which we wish to receive), so it superimposes the desired signal of interest and results in degradation and distortion of the desired signal. SI also causes the saturation of the receiver chain. Hence, it is crucial to mitigate the SI below the sensitivity of the receiver chain before it enters the LNA. Now, what we obtain at the receiver antenna is the signal ℎ ( ), as shown in Equation (1).

Transceiver's Components Linearization
The signal that is transmitted after digital to analog conversion (represented by x I (t) and x Q (t)) and amplification (represented by x PA (t)) is the baseband signal x n (represented by its two components x n,I and x n,Q ). DAC and low pass filter (LPF) are considered ideal, and variable gain amplifier (VGA) is omitted from calculations for the sake of simplicity. However, IQ, PA, and LNA nonlinearity is considered. The IQ imbalance problem arises because of a gain and amplitude mismatch between the in-phase and quadrature branches of the IQ mixer. Phase noise and frequency offset also result from IQ mixer's LO IQ mixer mismatch in Tx, and the Rx chain is considered with the help of the image rejection ratio (IRR), as defined by the 3rd generation partnership project (3GPP) long-term evolution-advance (LTE-A) specification, i.e., 25 dB for both Tx and Rx chain. 3GPP LTE-A system specification limits the minimum image attenuation to 25 dB or 28 dB [47]. 25 dB attenuation is sufficient for the Tx chain.
However, the Rx chain IQ imbalance adds additional image attenuation. IQ imbalance in the Rx chain affects the residual SI signal cancellation. As a widely linear model is used for analysis, so both direct and image components are taken. Image rejection ratio (IRR) determines the quality of mismatch between I and Q branches. PA nonlinearity is modeled as the Hammerstein memory model. Signal after amplification from PA gets coupled to its receiver, and this coupled signal is called an SI signal. The channel between Tx and Rx is considered a multi-path coupling channel. SI signal is 60-100 dB stronger than our desired signal (the signal which we wish to receive), so it superimposes the desired signal of interest and results in degradation and distortion of the desired signal. SI also causes the saturation of the receiver chain. Hence, it is crucial to mitigate the SI below the sensitivity of the receiver chain before it enters the LNA. Now, what we obtain at the receiver antenna is the signal y ch (t), as shown in Equation (1).
Here x PA (t) denotes PA output, x d (t) denotes the desired signal, h d (t) denotes desired signal channel, h ch (t) denotes multi-path coupling channel and n th (t) denotes thermal noise.  [48][49][50] is implemented for the compensation and estimation of gain and phase mismatch for transmitter chain IQ imbalance. For Rx chain IQ imbalance, a blind adaptive algorithm (BA) [51,52] is implemented. BAFF algorithm is simple and has low complexity. Both BAFF and BA are blind and do not require any pilot symbols or training elements for estimation and compensation. Because of their blind nature, they can be easily incorporated into any transceiver and any communication standard. It does not involve training of the desired signal; instead, only statistical properties are needed. Both schemes do not have any stability or convergence issues. Figure 2 shows the complete block diagram of the IQ mixer with gain and phase mismatch and IQ mixer linearization technique (BAFF and BA algorithm). A blind adaptive feedforward algorithm (BAFF) [48][49][50] is implemented for the compensation and estimation of gain and phase mismatch for transmitter chain IQ imbalance. For Rx chain IQ imbalance, a blind adaptive algorithm (BA) [51,52] is implemented. BAFF algorithm is simple and has low complexity. Both BAFF and BA are blind and do not require any pilot symbols or training elements for estimation and compensation. Because of their blind nature, they can be easily incorporated into any transceiver and any communication standard. It does not involve training of the desired signal; instead, only statistical properties are needed. Both schemes do not have any stability or convergence issues. Figure 2 shows the complete block diagram of the IQ mixer with gain and phase mismatch and IQ mixer linearization technique (BAFF and BA algorithm). For its implementation, three estimators are required to estimate necessary parameters, which involve 1-bit quantization. In the BAFF algorithm, first, we estimate the three parameters 1 , 2 , 3 . These three parameters are further used for the calculation of coefficients 1 , 2 for the compensation of IQ imbalance, and for calculating new gain and phase � �. Output after applying BAFF algorithm can be expressed as: where, , ( ) = For its implementation, three estimators are required to estimate necessary parameters, which involve 1-bit quantization. In the BAFF algorithm, first, we estimate the three parameters ϑ 1 , ϑ 2 , ϑ 3 . These three parameters are further used for the calculation of coefficients γ 1 , γ 2 for the compensation of IQ imbalance, and for calculating new gain and phaseĝ andφ.
Output after applying BAFF algorithm can be expressed as: where, V I,TX (t) = g TX γ 2,TX * U I,TX (t) V Q,TX (t) = γ 1,TX * U I,TX (t) + U Q,TX (t) IQ Mixer coefficients: As with the Tx chain IQ mixer, asymmetric IQ imbalance is also modeled in the Rx chain. The output of Rx IQ mixer after passing through LPF can be expressed as desired/wanted signal and image/unwanted signal, as shown in Equation (3): where, g RX and ϕ RX represents amplitude mismatch and phase mismatch respectively, y IQ,RX (t) and G 1,RX (t) are the desired/wanted signal components whereas y IQ,RX * (t) and G 2,RX (t) are the image/unwanted signal components.
The above Equation (3) can be expressed in terms of orthogonal frequency division multiplexing (OFDM) symbols as non-negative "O m (n)" and mirror "O −m (n)" OFDM sub-carriers with IQ imbalance after ADC as: : is an interference matrix "G" can be break into leakage and scaling matrix as: Here: The first one is the leakage matrix, and the second one is the scaling matrix. The leakage matrix is subjected to IQ mixer imbalance compensation and estimation, whereas the scaling matrix can be cancelled through channel estimation, andĜ q = OFDM symbols can be reconstructed by using the matrix inversion property, as in realistic case, "G" can never be singular for imbalance OFDM symbols. Here the principal or critical point is gaining knowledge about mixing matrix "G".
For the estimation of the "G" matrix and compensation, two probability statistics are used for parameter calculation: (i) cross-correlation b/w OFDM symbols "E{O m (n)O −m (n)}", and (ii) power of sum of OFDM symbols "E |O m (n) + O * −m (n)| 2 ". Here an assumption is made.
Assumption: E ŷ IQ,RX (n)ŷ −IQ,RX (n) = 0, i.e., uncorrelated OFDM symbols with zero Further: Here an assumption is made. Assumption: , ( ) , ( ) = 0, i.e., uncorrelated OFDM symbols with zero Further: , + * , = , + * , = 1 IQ imbalance parameter estimation through BA algorithm can be expressed as: And for calculation: where, The imbalance between the I and Q branches of Tx and Rx chain mixer increases with an increase in carrier frequency. A higher constellation is required for achieving high data rates and high operating SNR. To achieve all the above, a high complex analog domain circuit is needed. Hence, to eliminate the IQ imbalance in wireless transceivers with low complexity, digital compensation techniques are developed. (4) And forĜ q calculation: For the estimation of the "G" matrix and compensation, two probability statistics are used for parameter calculation: (i) cross-correlation b/w OFDM symbols " { ( ) ( )}", and (ii) power of sum of OFDM symbols " {| ( ) + * ( )| }".
Here an assumption is made. Assumption: , ( ) , ( ) = 0, i.e., uncorrelated OFDM symbols with zero Further: , + * , = , + * , = 1 IQ imbalance parameter estimation through BA algorithm can be expressed as: And for calculation: The imbalance between the I and Q branches of Tx and Rx chain mixer increases with an increase in carrier frequency. A higher constellation is required for achieving high data rates and high operating SNR. To achieve all the above, a high complex analog domain circuit is needed. Hence, to eliminate the IQ imbalance in wireless transceivers with low complexity, digital compensation techniques are developed. (5) where, M: Non-negative Subcarrier indices ï: sample time indices New estimated gainĝ RX and phaseφ TX can be defined as: Now the compensation matrix can be written as: The imbalance between the I and Q branches of Tx and Rx chain mixer increases with an increase in carrier frequency. A higher constellation is required for achieving high data rates and high operating SNR. To achieve all the above, a high complex analog domain circuit is needed. Hence, to eliminate the IQ imbalance in wireless transceivers with low complexity, digital compensation techniques are developed.

PA and LNA Linearization
PAs are the critical component at the transmitter end to manage these entire signals and their specific necessities. PA used must be capable of maintaining high power output, high linearity, and high efficiency, and all of them are inter-related quantities. The primary and foremost requirement at the Tx end is high linearity and efficiency.
Several techniques exist in the literature to satisfy the efficiency and linearity tradeoff. Most of these techniques are characterized based on action to reduce or avoid distortion in a non-linear PA. In this research study, the Doherty technique [53][54][55] combined with one of the linearization techniques is used for efficiency and linearity improvement. The Doherty technique is considered a mature technique and is used for efficiency enhancement.
Doherty technique operation depends on two different transfer function regimes containing two parallel PA, i.e., carrier PA (CPA) and peaking PA (PPA). One is called a low-power and the other as a high-power regime. Low-power is also defined as a linear regime, and high power is a non-linear regime. By combining the two PAs, the linear range increases where the Doherty amplifier operates. Doherty amplifier has a highly complex structure in practice, but it can be endured over its flexible architecture and highefficiency performance, and this amplifier's topology is the fundamental reason behind high efficiency. CPA and PPA are ideal current generators with zero reactive freeloading. Load impedance of CCA is increased while PPA load impedance is decreased as compared to CCA. No extra or additional circuit is required for any kind of controlling operation. Sufficient bandwidth (BW) is provided, i.e., up to 20 MHz, but is limited to only a single frequency or single band of operation.
For linearity improvement, two different linearization techniques are used combined with the Doherty technique, i.e., the pre-distortion technique [56,57] that avoids distortion and the feedforward technique that reduces the distortion being produced by non-linear PA.
Similarly, for the linearization of LNA, two techniques are implemented, i.e., negative feedback and feedforward [58]. In both LNA and PA, the feedforward linearization technique first compares the attenuated amplifier output with amplifier input, and generates the distortion signal. The distorted signal is then combined destructively at the amplifier output with the help of an auxiliary amplifier, and the resultant signal is a distortion-free signal. However, it can remove or reduce one type of harmonics at a time, i.e., to remove second and third order harmonics. The feedforward technique required an additional degree of freedom, and hence required an additional auxiliary amplifier path. The auxiliary amplifier path power efficiency is decreased, but still provides a reasonable amount of linearization at the BW range of 3-50 MHz. The negative feedback linearization technique improves A I IP2 and A I IP3 (second and third intercept points) and hence increases the linear operation range of the amplifier. Figure 3 shows the complete block diagram of PA linearization with the efficiency improvement technique. The LNA with negative feedback and feedforward linearization techniques is depicted as well.

SI Cancellation Techniques:
2.2.1. Analog/RF SI Cancellation Techniques: Several analog/RF cancellation techniques exist in the literature, as described in the previous section, to suppress the SI. After applying RF/analog cancellation techniques, the signal at the input of LNA can be described as: where " ( )" is the estimated signal attained from RF/analog cancellation techniques and can be expressed as: As stated above, the significant bottleneck in the efficiency and performance of FD transceivers is the self-interfering signal, i.e., coupling of transmit signal to its Rx. The

Analog/RF SI Cancellation Techniques
Several analog/RF cancellation techniques exist in the literature, as described in the previous section, to suppress the SI. After applying RF/analog cancellation techniques, the signal at the input of LNA can be described as: where "y est (t)" is the estimated signal attained from RF/analog cancellation techniques and can be expressed as: As stated above, the significant bottleneck in the efficiency and performance of FD transceivers is the self-interfering signal, i.e., coupling of transmit signal to its Rx. The critical challenge is to cancel the SI signal with one part per billion accuracies, and to cancel the effect of the SI signal over the desired signal up to 100% approximately. To achieve this accuracy to be the replica of the transmitted signal, SI signal is needed. It is difficult to replicate the SI signal because of multi-path channel effects or the signal from the nearby environment. Two automated operations are required to eliminate the SI in the analog domain efficiently: (1) sensing the self-interfering signal at the Rx, and (2) using adaptive integrated procedure or filter to cancel SI signal to a higher degree of accuracy.
Reliable FD communication systems need to develop an algorithm that can control SI, which is altered by the environment, i.e., reflections from nearby buildings and multi-path channel effects.

Channel State Information (CSI)
Tx does not have any knowledge about the signal being transmitted over the channel. The signal that a Tx knows is a baseband signal (the signal that is sent after digital to analog conversion and amplification) and the signal present at the output of the PA (the signal before transmission over the channel). Due to the multi-path coupling channel between Tx and Rx, the Rx receives multiple copies of the transmitted signal, i.e., delayed, attenuated, and phase-shifted copies of the transmitted signal. Here, multi-path coupling channel characteristic and statistical information is unknown to the Rx [59]. This statistical information of the multi-path coupling channel is crucial for the Rx to boost the received signal or to remove the SI signal. The knowledge about channel statistical information, i.e., fading distribution type, average channel gain, line of sight (LOS) component, spatial correlation, and impulse response, is termed as channel state information (CSI) [60]. CSI depicts the flow of the signal and how it propagates from the Tx end to the Rx end.
CSI illustrates the properties of the communication link between Tx and Rx. CSI also includes the information about the statistics of received interferences and thermal noise. Using CSI in wireless communication systems aims to improve the channel's capacity, maximize the SNR, and minimize the BER at the detection stage. Primarily, CSI is categorized as: (1) CSI is known to both Tx and Rx (i.e., channel state information at the Tx (CSIT) and channel state information at Rx (CSIR)); (2) CSI is known to Rx only and Tx is unaware of it (i.e., (CSIR)); and (3) both Tx and Rx are unaware of CSI. CSIR is obtained on the Rx side and is used at the Tx side through either the reciprocity principle or feedback method. CSI helps the Tx or Rx adapt to a current channel state for reliable communication with a high data rate. As the channel is primarily a time-varying entity, its statistical information varies with time, so CSI must be predicted on a short-term basis, i.e., must be predicted after a regular interval or period. Usually, a training-based approach is used for estimating the current state of the channel. This approach is considered a popular approach for channel estimation, but the significant overhead of this approach is reduced transmission efficiency [61]. In this approach, CSI is performed using the training period in which a known training sequence or pilot is transmitted over the channel. The required channel matrix is estimated by knowing the transmitted signal and the received antenna's signal. Tx does not directly access CSI; hence, mainly, CSI is utilized at the Rx side, quantized, and fed back to the Tx. Perfect CSI holds when the channel remains static, or the variations are minimal compared to the transmission rate.
Separate transmit and receive antennae are used, and the desired signal of infinitesimal strength compared to self-interfering signal is considered for analysis, as shown in Figure 4. The signal at the Rx antenna "Y" can be expressed as: channel. The required channel matrix is estimated by knowing the transmitted signal and the received antenna's signal. Tx does not directly access CSI; hence, mainly, CSI is utilized at the Rx side, quantized, and fed back to the Tx. Perfect CSI holds when the channel remains static, or the variations are minimal compared to the transmission rate.
Separate transmit and receive antennae are used, and the desired signal of infinitesimal strength compared to self-interfering signal is considered for analysis, as shown in Figure 4. The signal at the Rx antenna " " can be expressed as: Her,e is the SI signal, is the desired signal, and W is the AWGN noise with mean zero and variance 2 .
Her,e Y SI is the SI signal, Y D is the desired signal, and W is the AWGN noise with mean zero and variance σ 2 W .
Channel State Information at Rx (CSIR) During the phase of obtaining CSI at the Rx side, the training sequence is transmitted over the channel and channel H SI is estimated from the knowledge of the received signal "Y" and the training sequence "P". Let the training sequence be p 1 , p 2 , . . . . . . . . . , p T , where P is transmitted over the channel. The statistics of the received signal can be described as: Here, Y = [y 1 , y 2 , . . . . . . . . . , y T ], P = [p 1 , p 2 , . . . . . . . . . , p T ] and W = [w 1 , w 2 , . . . . . . . . . , w T ], "T" is the length of the training sequence, and "D" is the length of the data sequence. The total training interval is defined as L = T + D and total time spent during training is defined as duty cycle factor = η = 1 − T L . Now, if each sub-carrier is orthogonal to the other, then the training symbols for N sub-carriers can be represented in matrix form as: Here, E{p[i]} = 0 is the mean of ith sub-carrier and The estimators used for channel estimation to obtain CSI at the Rx side are least square (LS) and minimum mean square error (MMSE) channel estimators. Estimated mean square error (MSE) is proportional to the trace of P H P and the MSE is minimized when P H P = TI, i.e., tr (P H P) is scaled identity matrix and MSE can also be described as MSE = the CSI obtained from the training phase is further utilized for the cancellation of SI. As the channel is assumed to be static, it remains constant for transmission duration. Here in our transceiver model, CSI gained at the receiver side can be used on the Tx side without any overhead of extra circuitry. Once CSI is obtained, SI can be cancelled by a satisfactory amount given by equations below: where Y est = H est X PA . Hence after estimation of SI signal using CSI and cancellation of estimated SI signal from the total received signal, the signal left at Rx side can be expressed as: Blind Estimation Blind estimation deals with the issues of estimating an unknown signal or input at the receiver side. Input that suffered from phase and gained attenuation due to multi-path fading channel. For the proper equalization and estimation of the channel, the receiver does not require any training sequences from the transmitter. Blind estimation primarily relies on higher-order statistics (HOS) and temporal or spatial oversampling. Many HOS-based blind approaches for blind estimation exist, i.e., Laguerre filter, Kautz filter, and gradient adaptive Laguerre lattice (GALL) filter, etc. [62].
As stated above, the Rx has no prior knowledge about the signal being transmitted over the channel. Due to the multi-path channel and reflection from the nearby environment, the Rx receives multiple copies of the transmitted signal, i.e., delayed, attenuated, and phase-shifted copies of the transmitted signal. To estimate and segregate the transmitted signal from the Rx side, two filters are used for blind estimation: (i) Kautz filter and (ii) GALL filter. After the estimation, segregation, and recovering of the SI signal at the Rx side, it is subtracted from the total received signal. This can be expressed as Here, Y SI denotes SI signal, Y D denotes the desired signal, W indicates additive white Gaussian noise (AWGN) with mean zero and variance σ 2 W and Y GALL , and Y Kautz denotes the estimated, segregated, and recovered SI signal.

GALL Filter
A GALL filter is a filter in which a gradient-based approach is used for pole adaption [63]. Pole adaption using a gradient-based method is a straightforward and less complicated technique. Hence, this makes GALL filter implementation easy and straightforward. A GALL filter is used because it wraps time in filter coefficients, i.e., requires less time to calculate and obtain filter coefficients. It also has a lower delay, reduced-order modeling, and requires a smaller number of coefficients and fewer computational operations [64]. It has a fast convergence response and can attain stability in less time. The GALL filter contains an additional free parameter, i.e., a free variable for the independence of memory and the number of adaptive coefficients. Thus, the GALL filter can easily model an extended response filter by using relatively few parameters. Both perform well in steady-state and under tracking conditions [65]. Here, in this paper, for pole adaption, a sign-based gradient algorithm is used.
The transfer function of nth order Laguerre filter: Laguerre filters have an all-pass filter A m (z, u) with a low pass pre-processing filter stage L(z, u). w n,i shows adaptive coefficients of the filter. The Laguerre filter's correlation matrix can be expressed as: Here, x m (t, u) represents a Toeplitz matrix. For a lattice description of the Laguerre filter, consider a lattice predictor that transforms the filtered reference signal "v(t, u)" into orthogonal filter backward prediction errors "b m (t, u)", 0 < m < M − 1, where M is the order of lattice predictor andĥ(t, u) will be the estimated secondary path. The filtered reference signal "v(t, u)" can be expressed as: Lattice backward and forward prediction error orthogonalization can be obtained as per Figure 5 and can be defined as: Mathematics 2021, 9, 1434 16 of 34 Figure 5. Analog/RF SI cancellation using GALL filter.
The assumption for orthogonality: The adjustment of ( ) is carried out as follows by the recursive formula: The assumption for orthogonality: The adjustment of κ m (u) is carried out as follows by the recursive formula: where µ is a step size parameter and δ is a regularization constant, and the power of both forward and backward prediction error of the mth stage can be defined as: where β is the smoothing parameter.
In the form of correlation matrix lattice, backward and forward prediction error can be expressed as: where, With filtered prediction error: and: y(t) is the desired output, and x m (t, u) signal is considered linearly independent. Now the prediction error can be defined as: and in the normalized form: In the GALL filter, the expectation operator is defined with the help of exponential forgetting window 0 < γ ≤ 1, as: Here, all the terms involved in expectation are dependent upon time and result in a time-variant system. In the GALL filter, correct pole adaption "u," i.e., change in pole direction, the time-variant approach is used in which all the parameters are estimated on a sample-by-sample basis, and for the sign verification of c m (u) and c m+1 (u), the adaption rule is: Additionally, α(t) should be very small, i.e., order of 10 −6 . Any change in "u" will result in a transient state in the GALL filter algorithm, increasing error. GALL filter operates only in the stationary case, so rapid adaption of poles is also worthless, and secondly as |u| → 0 , the value of α(t) must decrease. Coefficients of the filter are adapted to diminish the power of the estimated error. and: Now the updated coefficient equation is defined as: and hence the residual error can be calculated as follows: In the next stage, the estimated transmitted signal is used to cancel SI from the received signal to obtain the desired one. Let the estimated SI signal from the GALL filter be y GALL (t) and error be ε GALL (t) now the signal at receiver after analog/RF cancellation by GALL filter can be expressed as: As can be seen from Figure 6, GALL filter error "ε GALL (t)" decays at a very rapid pace and approaches to zero within a time of about 50 µs.
and hence the residual error can be calculated as follows: In the next stage, the estimated transmitted signal is used to cancel SI from the received signal to obtain the desired one. Let the estimated SI signal from the GALL filter be ( ) and error be ( ) now the signal at receiver after analog/RF cancellation by GALL filter can be expressed as: As can be seen from Figure 6, GALL filter error " ( )" decays at a very rapid pace and approaches to zero within a time of about 50 µs.

Kautz Filter
The Kautz filter is an infinite impulse response (IIR) based tapped delay line transversal filter. It can be well-defined by a set of stable poles and a corresponding set of tap-output weights. It offers valuable means for optimization and control. It provides linear in-parameter models and adaptive filtering for both non-fixed and fixed structures and various systems identification and approximation schemes [66]. Kautz filter designing can be seen as two different aspects of the optimizing system model and its equalization: (i) choosing an optimal set of poles of Kautz filter � �, and (ii) evaluating a corresponding optimal set of Kautz filter tap coefficients or weights { }. These are the two parameters (poles and tap coefficients or weights) that determine the filter structures. In signal processing terms, a generic form of the Kautz filter function as per Figure 7 can be described by

Kautz Filter
The Kautz filter is an infinite impulse response (IIR) based tapped delay line transversal filter. It can be well-defined by a set of stable poles and a corresponding set of tapoutput weights. It offers valuable means for optimization and control. It provides linear in-parameter models and adaptive filtering for both non-fixed and fixed structures and various systems identification and approximation schemes [66]. Kautz filter designing can be seen as two different aspects of the optimizing system model and its equalization: (i) choosing an optimal set of poles of Kautz filter z j , and (ii) evaluating a corresponding optimal set of Kautz filter tap coefficients or weights {w i }. These are the two parameters (poles and tap coefficients or weights) that determine the filter structures. In signal processing terms, a generic form of the Kautz filter function as per Figure 7 can be described by where, Here, i = {0, 1, . . . , N} and G i (z) is determined by the chosen optimal set of poles z j N j = 0 such that z j < 1. The following assumptions are made for modeling a real response: (i) poles should be real or (ii) poles occur in complex conjugate pairs.  Normalization terms of the Kautz filter are determined by the corresponding optimal pole pair � , * � and are described as: where = −2 { } = | | 2 Here, pole adaption is carried out using the BU method. This method is used because of its capability of producing stable and accurate poles, and this method operates in the Normalization terms of the Kautz filter are determined by the corresponding optimal pole pair z j , z j * and are described as: where r i = −2Re{z i } and p i = |z i | 2 Here, pole adaption is carried out using the BU method. This method is used because of its capability of producing stable and accurate poles, and this method operates in the time domain for signal processing.

Digital SI Cancellation Techniques
In a wireless communication system, accurate estimation of desired signal strength is critical [67]. Estimating the desired signal is defined as the process of characterizing the effects of the physical channel, i.e., the transceiver's components and the wireless channel between Tx and Rx on the desired input sequence [68]. Perfect suppression of SI is impossible in practice; even after applying antenna and analog cancellation techniques, residual SI still exists. The statistical properties of SI are different from those of thermal noise, but it still acts in a similar way to noise and is also treated as noise. Digital cancellation techniques are applied at the baseband side of the transceiver and are used to suppress residual SI, i.e., low power multi-path components of SI [6,7,24,25]. They are also applied to compensate for non-linear components of the Tx and Rx front end [30]. The performance of the FD transceiver is highly degraded due to non-idealities in different stages of transceiver's architecture, i.e., Tx and Rx front-end components (low cost, low-quality components) and due to the circuitry used for active and passive cancellation.
In this research study, after applying linearization techniques, digital SI cancellation is performed, to mitigate residual SI. Linearization techniques are used to strengthen the residual SI cancellation. Non-linear transmitter and receiver chain components degrade the transceiver's performance and affect the cancellation of residual SI or make residual SI mitigation difficult. In this section, the technique for residual SI cancellation in the digital domain will be addressed. The method used for residual SI cancellation in the digital domain is the Kalman filter. In our previous work [69], we investigated the performance and digital SI cancellation using the extended Kalman filter (EKF) and achieved SINR improvement of about 90 dBm.

Kalman Filter
The Kalman filter is one of the most significant discoveries of the 20th century. One of its most promising and great immediate applications is controlling the complex dynamic systems. It propagates the current state of a dynamic system that is extremely important for the statistical analysis of a system. The propagated information includes the statistical properties of random dynamic perturbations and the influence of all the past measurements [70]. The Kalman filter utilizes the recursive structure, and hence helps in real-time execution. It is a statistical technique. It effectively describes the random forms of experimental measurements. The Kalman filter has relatively low complexity and is a discrete-time linear, finite dimension system [71]. The Kalman filter is usually applied to non-linear systems, and it does not require that the system, i.e., dynamic or random systems under analysis, to have static properties. It involves estimating the system's initial state, and the initial state is obtained by detecting it. The Kalman filter is used most solely for two purposes: (i) estimation of desired signal and (ii) estimator's performance analysis [70]. The Kalman filter is habitually applied to real-time applications because of its robustness and ability to deal with unknown signals. In the Kalman filter, mostly dynamic disturbances are modeled as additional unknown parameters or signals [72]. The Kalman filter has fewer computational requirements, it smoothes the noisy data, and estimates the parameters of interest [73].
Our purpose for applying the Kalman filter as a digital cancellation technique at the digital side of the transceiver is to construct an estimator that can accurately estimate the SI signal. This mitigates the residual SI signal before the digital signal processing of the desired signal, as shown in Figure 1. The Kalman filter provides the best estimate of x n,I (n) and x n,Q (n) at the Rx side through estimator of the form: y n,I (n) = c k,I (n)x n,I (n) + c k,I (n − 1)x n,I (n − 1) + . . . + c k,I (1)x n,I (1), (42) y n,Q (n) = c k,Q (n)x n,Q (n) + c k,Q (n − 1)x n,Q (n − 1) + . . . + c k,Q (1)x n,Q (1), (43) whereŷ n,I (n) andŷ n,Q (n) are the estimated symbols and c k,I (l) and c k,Q (l) , l = n, n − 1, . . . , 1 are the coefficients to be adapted to the knowledge of the model, "n" represents the total number of symbols. The Kalman filter involves two successive stages: (i) Prediction stage: (ii) Updating stage: y n,I (n|n) =ŷ n,I (n|n − 1) + k I (n) x n,I (n) − H I (n)ŷ n,I (n|n − 1) , y n,I (n) andŷ n,Q (n) are updated when x n,I (n) and x n,Q (n) become available; the amounts to modify theŷ n,I (n|n − 1) andŷ n,Q (n|n − 1), i.e., former estimates by a quantity proportional to the innovative part of the new observation.
The updating stage of the Kalman filter includes an innovative process which is one of the essential concepts in the Kalman filter technique. The innovative approach can be described as:ŷ The estimation error can be explained by the equation below, when x n,I (n) and x n,Q (n) is estimated byŷ n,I (n) andŷ n,Q (n). e I (n) =ŷ n,I (n) − x n,I (n), (50) e Q (n) =ŷ n,Q (n) − x n,Q (n),

Results and Discussion
The performance analysis of the assumed FD DC transceiver model with its nonidealities and their linearization techniques, analog/RF SI cancellation, and digital residual SI cancellation technique is simulated and verified explicitly with the help of well-known simulator "MATLAB/Simulink". OFDM is used as a baseband test signal for simulating a realistic signal. Non-idealities simulated with their linearization techniques are IQ mixer imbalance and PA nonlinearity in the Tx chain and LNA linearity and IQ mixer imbalance in the Rx chain. The mismatch between the I and Q branch of the IQ mixer is adjusted such that the required IRR can be achieved. PA nonlinearity is modeled with the help of a non-linear memory polynomial, and LNA nonlinearity with a quasi-memoryless model. Other components such as VGA, DAC, and ADC are considered linear. VGA is considered linear with complex gain and ADC and DAC as uniform quantization processes.
Two separate antennae are used for transmission and reception, and the antenna separation for isolation is considered and modeled as path loss of: (i) 30 dB and (ii) 40 dB. The multi-path channel between transmit and receive antenna is static for SI signal and is modeled, such that the line of sight (LOS) path and two non-line of sight (NLOS) paths of the multi-path channel have an average power ratio of 35 dB. However, the rest of the channel is dynamic in nature, i.e., due to motion of surrounding things such as humans, vehicles or other voices, and is modeled as noise. Noise is modeled as AWGN with the power of −174 dBm/Hz, mean zero and variance σ 2 W . Parameters of test signal and system are given in Tables 1 and 2. Table 3 shows the non-linear and linear parameters of components of the Tx and Rx chain.  The efficiency and performance of the FD transceiver are calculated based on link budget analysis, thereby computing powers of different components of SI signal and SINR.

Link Budget Analysis
Link budget analysis describes the details of signal components at every stage of Tx and Rx. These details include noise figure, gain, SNR, and IIP3, as shown in Figure 8.

Power Level of Different Components of SI. Signal After RF/Analog SI Cancellation
The power of different components of the SI signal at Rx input without any SI cancellation is shown in Figure 9a. and , are much stronger signal components and have much higher powers compared to the desired signal, which has a power of approximately −55 dBm at an Rx sensitivity level of −88.9 dBm. At Tx power of 5 dBm, is ~−15 dBm and , I s~−40 dBm, hence is ~40 dBm and , is ~15 dBm stronger than the desired signal, and is ~73.9 dBm and , is ~48.9 dBm stronger than the Rx sensitivity. However, and , are not very strong but are still higher than the sensitivity level of Rx at higher transmit powers, i.e., above 25 dBm. Figure 9b,c shows the results of the proposed SI cancellation technique, i.e., CSI with LS estimator. With this technique, along with 30 dB antenna isolation as shown in Figure 9b, the power of all the SI signal components drops below the desired signal and Rx sensitivity level, except the image component , .
Hence, now the only SI component with a power level above the Rx sensitivity is , . It can be observed in Figure 9c that by using the CSI technique along with 40 dB antenna isolation, , power also falls below Rx sensitivity. For antenna isolation of 30 dB with CSI, , remains below the desired signal power level for transmit power levels higher than 20 dBm. Similarly, for antenna isolation of 40 dB with CSI, , remains below the desired signal power level for transmit power levels higher than 30 dBm. The receiver can also be protected from high power SI signal by using only a single stage of a Kautz and GALL filter. Figure 9d,e shows that after passing through the Kautz and GALL filter, SI signal power is reduced to a much lower level that is below sensitivity level, and the power received at the input of the Rx is approximately equal to the desired signal, which shows that SI signal is almost cancelled, and the remaining error can be treated as low power noise. From Figure 9, SI signal power is significantly dominant over the desired signal before applying the proposed SI cancellation techniques. As a result, the total power at the Rx input is approximately equal to the SI signal power. However, after applying SI cancellation techniques, the SI power becomes negligibly small, and the total power is predominantly similar to the signal power.

Power Level of Different Components of SI. Signal After RF/Analog SI Cancellation
The power of different components of the SI signal at Rx input without any SI cancellation is shown in Figure 9a. P SI and P SI,im are much stronger signal components and have much higher powers compared to the desired signal, which has a power of approximately −55 dBm at an Rx sensitivity level of −88.9 dBm. At Tx power of 5 dBm, P SI is~−15 dBm and P SI,im I s~−40 dBm, hence P SI is~40 dBm and P SI,im is~15 dBm stronger than the desired signal, and P SI is~73.9 dBm and P SI,im is~48.9 dBm stronger than the Rx sensitivity. However, P I MD and P I MD,im are not very strong but are still higher than the sensitivity level of Rx at higher transmit powers, i.e., above 25 dBm. Figure 9b,c shows the results of the proposed SI cancellation technique, i.e., CSI with LS estimator. With this technique, along with 30 dB antenna isolation as shown in Figure 9b, the power of all the SI signal components drops below the desired signal and Rx sensitivity level, except the image component P SI,im .
Hence, now the only SI component with a power level above the Rx sensitivity is P SI,im . It can be observed in Figure 9c that by using the CSI technique along with 40 dB antenna isolation, P SI,im power also falls below Rx sensitivity. For antenna isolation of 30 dB with CSI, P SI,im remains below the desired signal power level for transmit power levels higher than 20 dBm. Similarly, for antenna isolation of 40 dB with CSI, P SI,im remains below the desired signal power level for transmit power levels higher than 30 dBm. The receiver can also be protected from high power SI signal by using only a single stage of a Kautz and GALL filter. Figure 9d,e shows that after passing through the Kautz and GALL filter, SI signal power is reduced to a much lower level that is below sensitivity level, and the power received at the input of the Rx is approximately equal to the desired signal, which shows that SI signal is almost cancelled, and the remaining error can be treated as low power noise. From Figure 9, SI signal power is significantly dominant over the desired signal before applying the proposed SI cancellation techniques. As a result, the total power at the Rx input is approximately equal to the SI signal power. However, after applying SI cancellation techniques, the SI power becomes negligibly small, and the total power is predominantly similar to the signal power.  Figure 10a shows SINR after the implementation of analog SI cancellation techniques. Kautz and GALL filters are better than CSI techniques, because for Kautz and GALL filters, although separate antennae are used, no antenna cancellation techniques are required. Figure 10b shows the SINR before digital signal processing and after implementing the digital SI cancellation technique, i.e., the Kalman filter. By applying the Kalman filter on the digital side, the system SINR improved to a significant value, i.e., approximately equal to SI NR Ideal , which falls because of the presence of the residual SI signal.  Figure 10a shows SINR after the implementation of analog SI cancellation techniques. Kautz and GALL filters are better than CSI techniques, because for Kautz and GALL filters, although separate antennae are used, no antenna cancellation techniques are required. Figure 10b shows the SINR before digital signal processing and after implementing the digital SI cancellation technique, i.e., the Kalman filter. By applying the Kalman filter on the digital side, the system SINR improved to a significant value, i.e., approximately equal to , which falls because of the presence of the residual SI signal.

Linearization
A performance analysis of the FD transceiver's Tx chain components is carried out separately, and results are shown through constellation diagrams and spectrum analyzers. Figure 11a-c shows the signal trajectory of the ideal, imbalance, and compensated IQ mixer of the Tx chain, respectively. Figure 11c shows that the compensated IQ mixer follows nearly the same signal trajectory as that of an ideal IQ mixer (Figure 11a). Figure  11d-f shows the constellation diagram of the ideal, imbalance, and compensated IQ mixer of the TX chain, respectively. It is clear from the constellation diagram Figure 11f that the compensated results are approximately nearer to those of the ideal IQ mixer (Figure 11d). The IQ mixer in the Tx chain is tested separately with its linearization techniques before putting it into the transceiver design, and then the combined results are evaluated. The signal used to test the IQ mixer model is a simple 16-quadrature amplitude modulation (QAM) signal.
PA linearity analysis is carried out by examining PA PSD and its input to output relationship. Figure 12 shows the output vs. input relationship of PA with and without linearization techniques. The non-linear amplifier shows great diversity due to intermodulation products of the third order. However, after applying linearization techniques, the output is the linear product of input. Figure 13a,b shows the results of PA before and after applying linearization techniques. It is clear from the figures that inter-modulation products have approximately equal power spectral density as linear gain and, hence, result in large signal distortion. In contrast, linearization techniques help with insufficient reduction of spectral regrowth

Linearization
A performance analysis of the FD transceiver's Tx chain components is carried out separately, and results are shown through constellation diagrams and spectrum analyzers. Figure 11a-c shows the signal trajectory of the ideal, imbalance, and compensated IQ mixer of the Tx chain, respectively. Figure 11c shows that the compensated IQ mixer follows nearly the same signal trajectory as that of an ideal IQ mixer (Figure 11a). Figure 11d-f shows the constellation diagram of the ideal, imbalance, and compensated IQ mixer of the TX chain, respectively. It is clear from the constellation diagram Figure 11f that the compensated results are approximately nearer to those of the ideal IQ mixer (Figure 11d). The IQ mixer in the Tx chain is tested separately with its linearization techniques before putting it into the transceiver design, and then the combined results are evaluated. The signal used to test the IQ mixer model is a simple 16-quadrature amplitude modulation (QAM) signal.
PA linearity analysis is carried out by examining PA PSD and its input to output relationship. Figure 12 shows the output vs. input relationship of PA with and without linearization techniques. The non-linear amplifier shows great diversity due to intermodulation products of the third order. However, after applying linearization techniques, the output is the linear product of input. Figure 13a,b shows the results of PA before and after applying linearization techniques. It is clear from the figures that inter-modulation products have approximately equal power spectral density as linear gain and, hence, result in large signal distortion. In contrast, linearization techniques help with insufficient reduction of spectral regrowth and reduce PA intermodulation product. However, the results obtained from PD linearization are better than those received by applying the FF linearization technique. The downside of PD linearization techniques is that they have computational complexities, but they provide better results.
Mathematics 2021, 9, 1434 26 of 34 and reduce PA intermodulation product. However, the results obtained from PD linearization are better than those received by applying the FF linearization technique. The downside of PD linearization techniques is that they have computational complexities, but they provide better results.   and reduce PA intermodulation product. However, the results obtained from PD linearization are better than those received by applying the FF linearization technique. The downside of PD linearization techniques is that they have computational complexities, but they provide better results.   Unlike the transmitter chain, receiver chain performance analysis is carried out using the metric BER vs. SNR. LNA linearity analysis is performed separately and examined through spectral power density and output to input relationship of LNA with and without linearization techniques. Figure 14 shows the relationship between LNA output and input. Non-linear LNA output shows non-linear behavior as input increases above 0 dBm and offers a sizeable non-linear behavior. However, after applying linearization techniques, the output and input relationship become linear, and output increases linearly with the increase of input. Figure 15 shows the PSD of LNA. Linearization techniques help in a reduced amount of spectral regrowth and decrease the intermodulation product power spectral density.  Unlike the transmitter chain, receiver chain performance analysis is carried out using the metric BER vs. SNR. LNA linearity analysis is performed separately and examined through spectral power density and output to input relationship of LNA with and without linearization techniques. Figure 14 shows the relationship between LNA output and input. Non-linear LNA output shows non-linear behavior as input increases above 0 dBm and offers a sizeable non-linear behavior. However, after applying linearization techniques, the output and input relationship become linear, and output increases linearly with the increase of input. Figure 15 shows the PSD of LNA. Linearization techniques help in a reduced amount of spectral regrowth and decrease the intermodulation product power spectral density. Unlike the transmitter chain, receiver chain performance analysis is carried out using the metric BER vs. SNR. LNA linearity analysis is performed separately and examined through spectral power density and output to input relationship of LNA with and without linearization techniques. Figure 14 shows the relationship between LNA output and input. Non-linear LNA output shows non-linear behavior as input increases above 0 dBm and offers a sizeable non-linear behavior. However, after applying linearization techniques, the output and input relationship become linear, and output increases linearly with the increase of input. Figure 15 shows the PSD of LNA. Linearization techniques help in a reduced amount of spectral regrowth and decrease the intermodulation product power spectral density.   After the linearization of LNA and IQ mixer, Rx performance is carried out, and BER is calculated before and after the linearization of Rx front-end components. Figure 16 shows the curve of Rx BER vs. SNR. Due to linearization techniques, improved BER is achieved. However, in the presence of Rx impairments, BER offers quite strange behavior, and up to 20 dB SNR, BER is relatively high, i.e., even below 10 −1 . Comparison between proposed techniques for self-interference cancellation with work previously carried out in the field of FD communication systems is presented in Table 4. After the linearization of LNA and IQ mixer, Rx performance is carried out, and BER is calculated before and after the linearization of Rx front-end components. Figure 16 shows the curve of Rx BER vs. SNR. Due to linearization techniques, improved BER is achieved. However, in the presence of Rx impairments, BER offers quite strange behavior, and up to 20 dB SNR, BER is relatively high, i.e., even below 10 −1 . After the linearization of LNA and IQ mixer, Rx performance is carried out, and BER is calculated before and after the linearization of Rx front-end components. Figure 16 shows the curve of Rx BER vs. SNR. Due to linearization techniques, improved BER is achieved. However, in the presence of Rx impairments, BER offers quite strange behavior, and up to 20 dB SNR, BER is relatively high, i.e., even below 10 −1 . Comparison between proposed techniques for self-interference cancellation with work previously carried out in the field of FD communication systems is presented in Table 4. Comparison between proposed techniques for self-interference cancellation with work previously carried out in the field of FD communication systems is presented in Table 4.

Conclusions
FD has gained a great deal of attention with regard to next-generation wireless radio communication systems such as 5G. Recently, the attention of researchers in FD wireless radio communication systems has led to the study of SI cancellation techniques for colocated Tx and Rx pairs. For efficient and reliable FD systems, SI needs to be mitigated to a sufficiently low level, i.e., below the Rx sensitivity and noise floor both on the analog and the digital side. Even after applying RF/analog cancellation techniques, there is still residual SI signal. Digital SI cancellation techniques are applied to mitigate the residual SI signal. In this article, two SI cancellation techniques, a digital SI cancellation technique, and different techniques for reducing the nonlinearity are presented. System performance was evaluated based on SINR, the power of different components of SI signal, and link budget analysis. Through the CSI active cancellation technique employed in the LS estimator, the SI signal cancellation up to 70 dB was achieved. SINR obtained, in this case, is almost equal to the ideal, i.e., the one calculated at the Rx input. The Kautz and GALL filters are adaptive filters used to cancel the SI signal at the Rx side, and they achieve 70-75 dB and 80-85 dB SI cancellation, respectively. The Kalman filter is applied to mitigate the residual SI signal on the digital side, and the SI cancellation of about 30 to 35 dB is achieved. SINR obtained at the Rx side after applying the Kalman filter is almost equal to the ideal, i.e., the one calculated at the Rx output, i.e., before digital signal processing. Hence, the SI signal mitigation achieved after both RF/analog and digital SI cancellation techniques is approximately up to 100 to 130 dB.