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Abstract
In this paper, we propose a dual-polarization Mach-Zehnder modulator-based photonic nonlinear analog self-interference cancellation (SIC) technique for in-band full duplex (IBFD) systems. By using the proposed technique, an arbitrary 4th order nonlinear transfer function can be generated, meaning the performance limitation caused by the nonlinearity of the analog SIC circuit can be overcome by imitating the nonlinear transfer function of the analog SIC circuit before cancellation. This paper also presents a performance analysis through simulations and the results of a proof-of-concept demonstration. In the experiment, the proposed nonlinear SIC technique could achieve 29 dB cancellation over 500 MHz bandwidth centered at 1.25 GHz frequency along various degree of distortion caused by nonlinearity. In addition, the performance enhancements achieved by the proposed technique are evaluated in terms of error vector magnitudes (EVMs) and constellations of the signal-of-interest (SOI) in the simulation which is based on the experimental SIC results. More than 3 dB of SOI power gain could be obtained in evaluated EVM performances.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In-band full duplex (IBFD) communication systems are one of the most important techniques for the evolution to the 6G mobile communication era. This is because they can increase RF spectral efficiency and reduce communication latency by allowing the simultaneous transmission and reception of data on the same frequency band [1–3]. However, the downlink signal transmitted from the radio unit (RU) in IBFD systems is received by the uplink antenna as interference, which is termed self-interference (SI). Moreover, because the SI is much stronger than the signal-of-interest (SOI), it cannot be recovered unless most of the SI is removed. Accordingly, SI cancellation (SIC) techniques are required, which can be divided into three domains: antenna, analog, and digital. In the antenna domain, the SI components are reduced by the electrical isolation between the TX and RX RF chains including path losses. In the separated antenna scheme, techniques such as using cross-polarization [4,5], and directional antennas [6] have been studied. In addition, techniques using a circulator have been developed in the shared antenna scheme [7], which have achieved isolation of more than 50 dB. However, considering that the SIC requirement is generally in excess of 100 dB, cancellation of the residual SI remains a major issue for the realization of IBFD systems. The required SIC can be calculated as follow:
In Eq. (1), ${P_t}$ is the transmission power of the node, and $NL$ is the noise floor according to the bandwidth. For instance, a macro e-Node B (eNB) of 20 MHz bandwidth having noise floor of -96 dBm and transmission power of 46 dBm needs 142 dB SIC. Therefore, the residual SI should be removed in both the analog and digital domains. The digital domain SIC techniques are performed after analog-to-digital conversion and cancel the residual SI after the analog SIC. In digital SIC techniques, all the system characteristics, which include nonlinearity of the power amplifier (PA), wireless channel characteristics, and antenna domain and analog domain cancellation characteristics, should be precisely modeled to cancel the residual SI [8–10]. However, as the residual SI power becomes stronger after the analog SIC, the RF amplifier after the analog SIC becomes highly saturated by the residual SI, meaning the SOI will be degraded by the quantization noise of the analog-to-digital converter (ADC). Consequently, the digital SIC performance will also be degraded because it becomes difficult to model the system characteristics precisely as the quantization noise increases. Therefore, it should be possible to resolve the residual SI after analog SIC. Assuming that 12-bit ADC is used in the receiver RF chain, the residual SI should be resolved in dynamic range of 60 dB, which is the 10-bit dynamic range considering 2-bit of peak-to-average power ratio (PAPR) margin. In this case, the forementioned macro eNB requires 82 dB cancellation before digital SIC. Thus, even if 50 dB cancelation is achieved in antenna domain, more than 32 dB additional cancellation is required in analog SIC considering non-ideal digital SIC performance. Therefore, analog SIC is a very important step in SIC processes. To achieve higher cancellation performance, many researchers have studied photonic-assisted analog SIC techniques, taking advantage of the wide bandwidth and high tuning precision offered by modern photonics [11]. However, in practice, the SI component passes a time-varying wireless multipath channel and should be actively compensated. Accordingly, an RF SIC circuit comprising a bundle of variable attenuators, variable delay lines, and phase shifters is used before cancellation. However, according to [12], RF SIC circuits have nonlinearities that limit the analog SIC performance, and analog SIC performance decreased from 30 dB to 20 dB as the received power before analog SIC increased from -60 dBm to -45 dBm. Unfortunately, the received power before analog SIC can be much higher. For instance, if a macro eNB having transmission power of 46 dBm have 60 dB antenna domain cancellation, the received power before analog cancellation will be -14 dBm, which is much larger power than that of in [12]. Therefore, the performance limitation caused by the nonlinearity of the analog SIC circuit can be much severe so that high order nonlinear terms should be considered. To address this problem, some photonic-assisted analog SIC techniques have been developed that consider a multipath SI channel with multiple optical paths having different optical wavelengths with variable optical attenuators instead of an RF SIC circuit [13–15]. In [13], multipath optical tunable delay lines (MOTDLs) for 3 × 3 MIMO radio over fiber transmission system were used instead of analog SIC circuits and 30.18 dB, 24.9 dB, 22.7 dB, and 20.88 dB at 16.6 GHz were achieved for the bandwidths of 100 MHz, 500 MHz, 1 GHz, and 2 GHz, respectively. In [14], optical phase modulation based MOTDLs were used instead of analog SIC circuit, where 26 dB and 28 dB cancellations were achieved over 100 MHz bandwidth at central frequency of 6 GHz and 10 GHz, respectively. In [15], fiber Bragg grating delay lines were used instead of analog SIC circuit and achieved 24.3 dB and 20 dB cancellation at 2.5 GHz center frequency over 100 MHz and 1 GHz bandwidths, respectively. However, these approaches significantly increase the hardware complexity of the analog SIC system. Some photonic-assisted analog SIC techniques do not employ fiber-based delay lines and the cancellation schemes are simple [16–19]. Among them, [18] performed 30 dB cancellation over 2.7 GHz bandwidth. These techniques should employ an RF-based analog SIC circuit to manage the multipath channel of SI. However, they did not consider the nonlinearity caused by the analog SIC circuit, rendering practical implementation difficult.
In this paper, we propose a dual-polarization Mach-Zehnder modulator (DP-MZM)-based photonic nonlinear analog SIC technique in IBFD. Similar SIC structures such as dual-polarization-based parallel MZMs [20,21] and dual-wavelength-based dual-parallel-MZMs were used in [22]. However, none of these studies have considered the nonlinearity caused by the analog SIC circuit. In our scheme, leveraging the nature of the nonlinear transfer curve of the MZM, an arbitrary 4th order nonlinear transfer function can be generated using the proposed scheme. Therefore, the nonlinearity created by the RF SIC circuit can be imitated, meaning the performance limitation caused by this nonlinearity can be effectively mitigated.
2. Operation principle
2.1 Transfer function of MZM
The operation principle of the proposed technique leverages the nature of the nonlinear transfer function of the MZM. Therefore, it is important to analyze the transfer function of the MZM mathematically. Figure 1 displays the MZM structure used in this work. As depicted in Fig. 1, the MZM used in this paper has push-pull operation, where the optical input is divided into two paths and each path is optically phase modulated with the same amplitude and reversed RF phase. Then, the field transfer function can be expressed as follows:
In Eq. (3), we can assume that $u(t )$ has two parts, ${V_{bias}}$ and ${V_{AC}}(t )$, where ${V_{bias}}$ is the DC bias voltage and ${V_{AC}}(t )$ is the AC signal voltage. Then, Eq. (3) can be expressed in terms of ${V_{bias}}$ and ${V_{AC}}(t )$ as follows:
As demonstrated in Eq. (4), the ratio between the cosine and sine components can be controlled by the bias voltage (${V_{bias}}$). For example, the transfer function of the MZM is purely a cosine and sine function when ${V_{bias}}$ is zero and $- \frac{{{V_\pi }}}{2}$, respectively. Assuming that the ratio between the cosine and sine components is ${c_1}:{s_1}$ and $v(t )$ is the normalized RF voltage (which can be expressed as $\frac{{{V_{AC}}(t )}}{{{V_\pi }}}\pi $), (4) can be expressed as follows using Taylor’s expansion:
According to Eq. (5), the transfer function of the MZM has high-order nonlinear terms, which are generally not considered because the intensity modulation is usually biased at the quadrature point and the linear region of the MZM transfer curve is used. However, as the input RF voltage becomes stronger, the nonlinear terms become larger compared to the linear term.
2.2 Proposed SIC scheme
Figure 2 depicts the proposed nonlinear analog SIC scheme, where the downlink signal after the power amplifier (PA) is referenced for the analog SIC so the analog SIC performance is not degraded by the nonlinearity of the PA. In the analog SIC circuit, the multipath channel of the SI component is realized with a bundle of variable attenuators, phase shifters and delay lines to maximize the analog SIC performance. The analog SIC circuit actively imitates the multipath channel of SI, based on the RF channel information. The optimizations of the delay, phase and attenuations of the analog SIC circuit have been widely studied in [23–25]. However, these active RF components have innate nonlinearities that limit the analog SIC performance, although the PA nonlinearity can be ignored. After the analog SIC circuit, the reference signal with nonlinearity of the analog SIC circuit is applied to the upper MZM with bias voltage $\frac{{{V_\pi }}}{2}$ for linear modulation at the negative slope of the transfer curve. To mitigate the performance limitation of the analog SIC caused by the nonlinearity of the analog SIC circuit, a DP-MZM that modulates two orthogonal polarizations (x and y polarizations in parallel) is used to generate the same nonlinear transfer function with the analog SIC circuit. The DP-MZM consists of two MZMs in parallel, MZM for x polarization (MZX) and MZM for y polarization (MZY), where the optical input is divided by the polarization beam splitter (PBS). Hence, assuming that the power division ratio by the PBS is ${R_x}:{R_y}$, the power transfer function can be expressed as follows:
In Eq. (6), ${c_x}$: ${s_x}$ and ${c_y}:\; {s_y}$ are the ratios between cosine and sine components for MZX and MZY, respectively, while ${v_x}(t )$ and ${v_y}(t )$ are the normalized input RF voltages applied in MZX and MXY, respectively. Using Taylor’s expansion, Eq. (6) can be expressed as follows:
For MZX, the applied RF voltage has a wide dynamic range, where the 4th order nonlinear term should be considered and the 5th and higher order nonlinear terms are not considered. In contrast, the applied RF voltage for MXY has a small dynamic range, where the 2nd order nonlinear term should be considered and 3rd and higher order nonlinear terms are not considered. Moreover, because the applied RF voltages for MZX and MXY are the same (except for the amplitude), ${v_y}(t )$ can be expressed as $R{v_x}(t )$. Then, Eq. (7) can be simplified as follows:
As depicted in Eq. (8), the coefficients of the 3rd and 4th order nonlinear terms are in terms of ${R_x}$, ${s_x}$, and ${c_x}$. Therefore, the ratio between the coefficients of the 3rd and 4th nonlinear terms can be controlled by the DC bias voltage and the dynamic range of the input RF voltage for MZX. After the 3rd and 4th nonlinear coefficients are controlled for a desired ratio by controlling ${s_x}$ and ${c_x}$ in this stage, the coefficients of the linear and 2nd order nonlinear terms are adjusted. This can be achieved by controlling the DC bias voltage, the dynamic range of the input RF voltage for MZY, and the polarization division ratio at the PBS, which can be controlled by the polarization controller (PC). In some systems, it may be difficult to control the dynamic range of the input RF voltages for MZX and MZY, because of the additional nonlinearity issue caused by the variable RF amplifiers. However, an arbitrary 4th order nonlinear function can be emulated although the dynamic range of the input voltages for MZX and MZY are fixed. Therefore, the polarization controller and the DC bias controller are enough to emulate 4th order nonlinear transfer function as shown in Fig. 2. Also, real-time adjustment of the nonlinear transfer function may be required because of the time-varying wireless channel. In order to cope with this issue, an output power after photodetection is fed into analog SIC circuit to match the power. Also, the nonlinear transfer function of the analog SIC circuit is measured by referencing the input and the output of the analog SIC circuit. Based on the input and output samples of the analog SIC circuit, the nonlinear coefficients can be obtained by the polynomial curve fitting. With the nonlinear coefficient information, the bias controller controls the polarization controller and the DC biases of MZX and MZY. In this process, an arbitrary DC bias point control technique is required. Fortunately, such techniques have been widely studied [26–28]. Hence, by precisely matching the nonlinear transfer function between the analog SIC circuit and the DP-MZM, the performance limitation caused by the nonlinearity of the analog SIC circuit can be effectively mitigated. In this process, the SOI, which is received by Rx antenna can be distorted by the emulated nonlinear transfer function. The reason why analog SIC circuit is not linearized is that, in order to linearize the nonlinearity generated by analog SIC circuit, the inverse function should be emulated, which requires 5th and higher order polynomial function. In that case, additional MZMs are required which severely increase the hardware complexity. However, the power of the SOI is much smaller than that of the SI before analog cancellation so that the nonlinear distortion in SOI is very small. Therefore, it does not worth the additional MZMs for linearization because the nonlinear distortion caused by SOI can be neglected. This issue is discussed in Section 4.
3. Simulations
As introduced in Section 2, an arbitrary 4th order nonlinear transfer function can be generated, assuming that the 5th and higher order nonlinear terms generated by MZX and 3rd and higher order nonlinear terms generated by MZY are not considered. However, these higher order nonlinear terms are generated in realty and can degrade the performance of the proposed analog SIC. Therefore, the analog SIC performance proposed in this paper is analyzed through MATLAB simulations in the presence of these undesirable high order nonlinear terms. In order to focus on the evaluation of the performance limitation by higher order nonlinear terms, the delay, phase and amplitude mismatches are assumed to be zero. The amplitude matching is performed based on 4th order polynomial. Before examining the limitations caused by these high order nonlinear terms, there must be an understanding of the relationship between the PAPR of the SI and the SIC performance. Because the coefficients of the higher order nonlinear terms are very small, as highlighted in (5), the performance degradations caused by these terms mainly occur when the input RF voltage is high. Hence, assuming that the peak-to-peak input RF voltages are the same, these performance degradations become severe when the PAPR of the SI is small. Accordingly, the PAPR of the SI is also considered in the simulation. The SI component used in the simulation is the orthogonal frequency division multiplexing (OFDM) signal with the parameters specified in Table 1.
Table 1. Parameters of simulated OFDM
As shown in Table 1, the PAPRs used in the simulation were obtained by signal clipping, which is performed as follows:
Figure 3 displays the simulated SIC performance of MZX without band-pass filtering. Because MZX was used to adjust the 3rd and 4th nonlinear terms, the RF input for MZX was operated by a large signal to consider the 3rd and 4th nonlinear terms. Figure 4 displays the simulated SIC performance of MZY without band-pass filtering, which was operated by a small signal to adjust only the linear and 2nd order nonlinear terms. As demonstrated in Figs. 3 and 4, the cancellation depth performances degraded as the normalized RF swing range increased. This was because as the RF swing range increased, the undesired high order nonlinear terms became larger. Furthermore, as the PAPR reduced, the cancellation depth performances degraded because the average power of the SI increased when the PAPR reduced with a fixed peak-to-peak voltage. Therefore, the SI was more affected by the undesired high order nonlinear terms. The cancellation depth and the normalized RF swing range used in Figures can be calculated as follows:
${P_{in}}$ and ${P_{out}}$ in (10) are the AC powers of SI before and after SIC, respectively. Furthermore, the cancellation performances of the cosine component in Figs. 3 and 4 were evaluated when ${V_{bias}}$ was zero, which is the maximum transmission point of MZM. In contrast, the cancellation performance of the sine component were evaluated when ${V_{bias}}$ was $- \frac{{{V_\pi }}}{2}$, which is the quadrature point of MZM. As depicted in Figs. 3 and 4, the cancellation performances of the cosine components were superior to those of the sine components, because the undesired nonlinear terms were 5th and higher order terms. By using Taylor’s expansion, the cosine component only had even order terms, meaning there was no 5th order nonlinear term. Therefore, for the cosine components, the 6th and higher order terms were the undesired high order nonlinear terms that degraded the cancellation performance, while the 5th and higher order terms degraded the cancellation performance for the sine component. It should be noted that Figs. 3 and 4 only exhibit the theoretical cancellation performance of the entire frequency component. However, a band-pass filter would be applied in reality to maximize the SIC performance. Accordingly, the SIC performance with a band-pass filter should be analyzed. To estimate the in-band SIC performance, the spectrums of the SI after the cosine and sine transfer function should be studied. Figure 5 displays the spectrum of OFDM after the cosine and sine transfer functions, where the PAPR of OFDM was 14 dB and the normalized RF swing range (${V_\pi }$) was 3. As depicted in Fig. 5(a), there was only a small in-band frequency component because the cosine transfer function only had even-order nonlinear terms and no linear term. As the carrier frequency was significantly higher than the bandwidth of the OFDM, these even order nonlinear terms only caused small nonlinear distortions in the signal frequency band. In comparison, the spectrum after the sine transfer function in Fig. 5(b) displayed odd order nonlinear terms, causing nonlinear distortion in the signal frequency band.
Fig. 3. Large signal SIC performances of (a) cosine and (b) sine transfer function for MZX without band-pass filtering.
Fig. 4. Small signal SIC performances of (a) cosine and (b) sine transfer function for MZY without band-pass filtering.
Therefore, to evaluate the in-band SIC performance, cancellation depth after the sine transfer function with band-pass filtering was performed, as displayed in Fig. 6. Figure 6(a) displays the band-pass filtered SIC performance of the sine transfer function for MZX, which was for large signal operation. Figure 6(b) displays the band-pass filtered SIC performance of the sine transfer function for MZY, which was for small signal operation. Similar to Figs. 3 and 4, the cancellation performance degraded as PAPR reduced and the peak-to-peak RF voltage increased. According to (8), any normalized RF swing range $({{V_\pi }} )$ could be used because the ratio of the nonlinear coefficients could be adjusted by ${c_x}$, ${s_x}$, ${c_y}$, ${s_y}$, ${R_x}$, and ${R_y}$. Therefore, it can be selected according to the nonlinear transfer function of the analog SIC circuit and noise floor level. According to Fig. 6, the smaller the normalized RF swing range $({{V_\pi }} )$, the better the SIC performance. However, in this case, the SOI power would also be smaller after SIC, meaning the error vector magnitude (EVM) performance of the SOI could be worse due to noise, although the cancellation depth performance would be improved. The optimization for the values of normalized RF swing range $({{V_\pi }} )$ were not examined in this study because this would require addressing the entire SIC system, including the antenna and digital domain SICs.
Fig. 6. SIC performances of sine transfer function for (a) MZX and (b) MZY with band-pass filtering.
4. Experiments
A proof-of-concept experiment was performed to demonstrate the performance enhancement achieved by the proposed analog nonlinear SIC technique. Figure 7 displays the experimental scheme for the proposed technique, in which we used an optical I/Q modulator that consisted of two optical couplers and two MZMs instead of the DP-MZM. This was due to the absence of a commercial DP-MZM in our experimental environment. Therefore, in this scheme, it was difficult to adjust the optical power ratio ${R_x}$ and ${R_y}.$ Also, because we used I/Q modulator instead of DP-MZM, the power transfer function is different from the DP-MZM. The electric field transfer function can be expressed as follow:
In Eq. (13), ${c_I}$ and ${s_I}$ is the cosine and sine components of in-phase component while ${c_Q}$ and ${s_Q}$ are the cosine and sine components of quadrature component. Note that Eq. (13) is similar to Eq. (6) except for ${R_x}$ and ${R_y}$. Therefore, as long as the I/Q relationship is perfectly maintained, similar results can be obtained compared to the DP-MZM. Therefore, by Eq. (13), an arbitrary 4th order nonlinear transfer function could still be generated by the experimental scheme, although the flexibility of the normalized RF swing range $({{V_\pi }} )$ was limited. The I/Q modulator, which is displayed with two optical couplers and two MZMs in Fig. 7, imitated an arbitrary 4th order nonlinear transfer function generated in the analog SIC circuit. The 4th order nonlinear transfer function of the analog SIC circuit was implemented by the nonlinear digital filter, as depicted in the right side of Fig. 7. The cancellation was performed by off-line processing in the digital domain to minimize any performance degradation caused by the RF phase and power mismatches.
Received signals from the DP-MZM side were upsampled by a factor of 10. After upsampling, a precise delay tuning is performed by shifting the samples and downsampling. Therefore, the delay tuning resolution in our experiment is 4 picoseconds, which is ten times smaller than the sampling interval. For the amplitude matching, the signal powers from DP-MZM and single MZM sides matched based on the RMS value of received signals. After that, the amplitude of one of these signals is adjusted with a step size of 0.001 dB in order to maximize the cancellation performance. The parameters of the SI signal, which was an OFDM signal, are displayed in Table 2. The OFDM parameters used in the experiment were basically the same as those used in the simulations, except for the RF center frequency of the OFDM and the evaluated PAPRs. The reason for using a lower RF center frequency in the experiment was to minimize any performance degradation caused by the RF phase mismatch. The nonlinear transfer function implemented in the nonlinear digital filter was fixed, which meant that the peak-to-peak RF voltage applied in the I/Q modulator was fixed. Because the 4th order nonlinear transfer function was implemented by the nonlinear digital filter, the MZM in the right side of Fig. 7 was operated in quadrature bias and the linear transfer region. The bias voltages of the I/Q modulator were controlled to imitate the nonlinear transfer function generated by the nonlinear digital filter. To eliminate any out-of-band noise including optical beating interference caused by the mismatch between optical in-phase and quadrature relationship and obtain realistic in-band SIC performance, a sharp digital band-pass filter with a pass band of 0.8 to 1.7 GHz was applied. Figure 8 displays the nonlinear and linear transfer curves generated by the I/Q modulator with the digital band-pass filter applied. Figure 9 displays the cancellation depth performance enhancement achieved by the proposed nonlinear SIC scheme.
Table 2. Parameters of experimental OFDM used for SI
The cancellation depth curve without the proposed technique was plotted when the transfer function of I/Q modulator was linear, as displayed in Fig. 8(b). The cancellation depth curve with the proposed technique was plotted when the transfer function of I/Q modulator was nonlinear, as displayed in Fig. 8(a).
When the proposed nonlinear SIC technique was not applied, as presented in Fig. 9, the cancellation depth performance degraded as the PAPR of the SI reduced. This was because the peak-to-peak voltage applied in the I/Q modulator was fixed, which meant that the average signal power increased when the PAPR reduced. Consequently, when the PAPR was small, the SI was more affected by the nonlinear transfer curve because the nonlinear distortion became severe at the peak of the transfer curve. However, when the proposed technique was applied, a cancellation of approximately 29 dB could be achieved, irrespective of the PAPR of the SI. Figure 10 displays the measured electrical spectrum of the SI according to its PAPR. The linear SIC in Fig. 10 meant that the transfer curve generated by the I/Q modulator was linear, as displayed in Fig. 8(b). This meant that the nonlinearity of the analog SIC circuit was not considered. The nonlinear SIC in Fig. 10 meant that the proposed nonlinear SIC technique was applied by generating a nonlinear transfer function in the I/Q modulator, as displayed in Fig. 8(a). It is evident from Fig. 10 that the proposed analog nonlinear SIC technique surpassed the linear SIC technique by effectively cancelling the in-band nonlinear distortion. The residual in-band nonlinear distortion after nonlinear SIC evident in Fig. 10(a) was not due to the nonlinear transfer function generated by the I/Q modulators and nonlinear digital filter. Rather, it was because of signal clipping, which was performed to meet the PAPR conditions. Because the nonlinearity caused by clipping was generated before the entire SIC process, this nonlinearity did not need to be considered, according to the same principle that the nonlinearity of the PA did not need to be considered.
Fig. 10. Measured electrical spectrum of SI without SIC, with linear SIC, and with nonlinear SIC when the PAPR of SI is (a) 8 dB, (b) 10 dB, (c) 12 dB, and (d) 14 dB.
To further estimate the quantitative performance enhancement achieved by the proposed nonlinear SIC technique, the EVM performance of the SOI was evaluated. In order to measure the EVM performance under nonlinear degradation of SOI including inter-modulation between SI and SOI, SOI and SI are added and applied to the nonlinear digital filter. The output of the nonlinear digital filter and the received output from the I/Q modulator are cancelled as shown in Fig. 7. In practice, the SOI should be added in the I/Q modulator side with the SI. However, in that case, it is very difficult to precisely match the power and phase of the SI and reference signal, resulting in poor cancellation performance measurement accuracy. The SOI was also an OFDM signal with the parameters displayed in Table 3.
Table 3. Parameters of experimental OFDM used for SOI
To meet the condition that the signal power of the SOI should be significantly smaller than the SI, while simultaneously obtaining communicable EVM performance, the bandwidth of the SOI was set as ten times narrower than the SI. Figure 11 displays the EVM performance enhancements achieved by the proposed analog nonlinear SIC compared to the analog nonlinear SIC. As displayed in Fig. 11, the EVM performance enhancement became more significant as the PAPR of SI decreased. This was because when the PAPR of the SI was small, the SI components were more distributed to the peak of the nonlinear transfer curve, which meant that the distortion caused by the nonlinear transfer curve was severe. The relative SOI power displayed in Fig. 11 is the relative power of the SOI compared to the power of the SI in dB, which can be calculated as follows:
Fig. 11. EVM performance enhancements by proposed nonlinear SIC compared to linear SIC when the PAPR of SI is (a) 8 dB, (b) 10 dB, (c) 12 dB, and (d) 14 dB.
In Eq. (14), ${P_{SOI}}$ and ${P_{SI}}$ represent the power of the SOI and SI, respectively. In Fig. 11, the relative SOI power was used to control the power of the SOI. In other words, the power of SI is fixed and only the power of SOI is changed. Consequently, the relative SOI power can be regarded as the signal power of SOI. However, the reason for using relative SOI power is to consider the nonlinear distortion in SOI including inter-modulation components between SOI and SI. In this case, when the relative SOI power becomes higher, SOI is more distorted by the nonlinear transfer function. Therefore, the performance enhancement achieved by the proposed analog nonlinear SIC technique would be reduced as the relative SOI power increased. However, the relative SOI power would be significantly smaller in practical analog SIC scenarios, meaning the distortion of the SOI caused by the nonlinear transfer function would be negligible. Because power of SOI is much smaller than SI, by the binomial approximation can be applied as follow:
According to Eq. (15), the nonlinear distortion of SOI including inter-modulation with SI can be ignored when the SI is much larger than SOI. Therefore, the nonlinear distortion in SOI goes smaller as the power of SOI goes smaller, resulting in negligible degradation in EVM performances of SOI. In Fig. 11, the possible insertion loss from DP-MZM was not imitated by the nonlinear digital filter. While the insertion loss does not affect the SIC performance because the power matching is performed by the analog SIC circuit, the insertion loss can still affect on the SOI recovery performance because the insertion loss can degrade the signal-to-interference-plus-noise ratio of the SOI. When the insertion loss is applied, the relative SOI power will not be changed but the EVM performance will be degraded because the signal power of SOI becomes smaller. In this case, the EVM curves in Fig. 11 will be horizontally shifted to the right side as long as the nonlinear distortion of SOI can be neglected as Eq. (15). Therefore, when insertion loss is applied, the SOI recovery performance will be different from Fig. 11. However, the performance enhancement compared to the linear SIC, such as the power gain at a target EVM will remain the same.
Figure 12 displays the constellations of SOI to demonstrate the qualitative performance enhancement achieved by the proposed analog nonlinear SIC scheme. Figures 12(a)–(d) displays the constellations of the SOI with linear SIC, while Fig. 12(e) presents the corresponding constellation after the proposed analog nonlinear SIC, meeting the EVM requirement of 16QAM (12.5%). The EVM values of Figs. 12(a)–(d) were 28.5%, 24.2%, 20.8%, and 18.2%, respectively. In practice, the EVM performance of the SOI could be measured after the fiber transmission and digital SIC processes. Therefore, the EVM performance could be different from the practical cases. However, considering that the SIC requirements generally exceed 100 dB, the digital SIC performance was affected by the performance of the antenna and analog SIC, which were related to the quantization noise of the ADC. As the analog SIC performance improved, the effect of quantization noise would decrease, meaning more precise digital SIC would be possible. Therefore, the practical performance enhancements would be more significant compared to the results presented in this section.
Fig. 12. Constellations of SOI with linear SIC when the PAPR of SI is (a) 8 dB, (b) 10 dB, (c) 12 dB, (d) 14 dB, and (e) corresponding constellation with nonlinear SIC meeting 16QAM EVM requirement.
5. Conclusion
In this paper, we proposed a novel analog nonlinear SIC technique assisted by DP-MZM. The main contribution of this paper was consideration of the cancellation performance degradation caused by the nonlinearity of the analog SIC circuit as well as overcome of the forementioned performance degradation by the proposed technique. Using the proposed technique, an arbitrary 4th order nonlinear transfer function could be generated in the analog domain, meaning it could cope with an arbitrary analog SIC circuit. The cancellation performances were analyzed in simulations according to the PAPR of the SI and the RF swing range for the input of the DP-MZM. In the experiments, the proposed nonlinear analog SIC technique achieved 29 dB cancellation over 500 MHz bandwidth centered at 1.25 GHz for various degrees of nonlinear effects, which were controlled by the PAPRs of the SI. Compared to the linear cancellation which did not consider the nonlinearity, the proposed technique outperformed along all PAPRs and 8 dB additional cancellation could be achieved when the nonlinearity was severe having 8 dB PAPR. In addition, based on the experimental SIC results, SOI simulation was performed in order to evaluate quantitative and qualitative performance enhancements in terms of EVMs and constellations of SOI, respectively. Furthermore, more than 3 dB SOI power gain could be obtained when the target EVM is 12.5%, which became more significant when the distortion caused by the nonlinearity became severe. The proposed technique could be a key technique for realizing IBFD communication systems due to its superior performance allied to simplicity and versatility.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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