Characterization of Nonlinear Distortion in Intensity Modulation and Direct Detection Systems

Intensity modulation and direct detection (IM-DD) systems are widely used in short-reach optical networks. Thus, developing a practical and accurate method to characterize nonlinear distortion and to estimate system performance is essential, as this is the foundation of IM-DD system design. The amount of nonlinear distortion can be characterized using the noise-to-power ratio (NPR). Although measuring the NPR using a notch requires only a spectrum analyzer, this method cannot be applied to a nonlinear system with non-Gaussian stimuli in general. We employ the method of measuring the NPR using a notch to characterize the nonlinear distortion in directly modulated distributed feedback (DM-DFB) laser-based IM-DD systems. Nonlinear distortion in these IM-DD systems can be attributed to two specific mechanisms: the imperfect power–current characteristic of the DM-DFB laser and the interaction among the modulation, chromatic dispersion (CD), and detection. Experimental results demonstrated that the nonlinear distortion can be accurately characterized with a mean absolute error of 0.47 dB, even when the stimuli are pulse amplitude modulation. The system performance subjected to nonlinearity could be estimated by the equivalent additive noise model. Besides, the method of measuring the NPR using a notch is also effective when the Mach–Zehnder modulator-based IM-DD system has nonnegligible CD. However, it failed in the case of negligible CD. Considering that this method is not always effective, we analyzed the attributes of various nonlinear systems and found that the above method is applicable only to systems dominated by second-order nonlinearity.


I. INTRODUCTION
I NTENSITY modulation and direct detection (IM-DD) systems with multilevel pulse amplitude modulation (PAM) are widely used in short-reach optical networks, such as data center interconnects (DCI), 5G fronthaul, and metro access due to Manuscript  its low-complexity and cost-effective features [1]. In IM-DD systems, the fiber nonlinear Kerr effect can be ignored because of the short transmission distance. Nonlinear distortions from devices, such as the nonlinear power-current characteristic of the directly modulated distributed feedback (DM-DFB) laser [2] and the sinusoidal function of Mach-Zehnder modulator (MZM) [3], degrade the system performance. Additionally, the interaction among direct modulation of laser, fiber chromatic dispersion (CD), and the square-law detection of photo diode (PD) also causes nonlinear distortion [4]. Eye skew is another nonlinear distortion that occurs in IM-DD systems [5].
Practically and accurately estimating the nonlinear distortion is a basic requirement in nonlinearity research. Separating various impairments, such as additive noise, linear distortion, and nonlinear distortion, is helpful to identify the dominant impairment in a specific system. The system error floor is usually determined by the nonlinear distortion. In the open network scenario, the transponders of different vendors should interoperate. Thus, the transmitted signal quality, including the nonlinear characteristics, should be specified.
In IM-DD systems, the level separation mismatch ratio (R LM ) has been widely used for characterizing the vertical linearity of the signal [6]. However, there is no clear relationship between R LM and the system performance Q factor. The transmitter and dispersion eye closure quaternary specifies the overall signal quality [6], [7] but cannot indicate how many distortions are caused by the nonlinear effect.
Orthogonal component analysis, which uses a finite impulse response (FIR) filter to separate the best linear approximation (i.e., correlated component) and residual nonlinear distortion (i.e., orthogonal component) from the nonlinear device output, has been applied to accurately estimate the nonlinear distortion of devices in optical communication systems [8]. However, this approach has many issues [9]. For example, it requires an accurate measurement of input and output waveforms, which requires expensive instruments (e.g., digital storage oscilloscope (DSO)). Furthermore, the orthogonal component is the small difference between two large signals (correlated component and nonlinear device output); thus, the required measurement accuracy is high. Supposing the nonlinear distortion is 20 dB lower than the signal, the nonlinear distortion measurement error is as large as the distortion if the waveform measurement error is 1%. In addition, if the FIR tap length is insufficient, the correlated and orthogonal components cannot be fully separated, and the nonlinear distortion will be overestimated [10].
Noise-to-power ratio (NPR) measurement using a notch is another method to estimate nonlinear distortion [11]. This method requires only a spectrum analyzer; hence, it is a cost-effective candidate compared to orthogonal component analysis. However, NPR measurement by notch is not a general method and only works correctly for Gaussian inputs or some special nonlinear systems with non-Gaussian inputs [12]. In some cases, the measurement error is as large as 8 dB [13]. Therefore, it is necessary to evaluate the NPR measurement by notch and to find its applicable range.
We previously analyzed a nonlinear mechanism of a DM-DFB laser-based IM-DD system (i.e., the interaction among modulation, CD, and detection) and experimentally demonstrated that the NPR measurement by notch method can characterize this nonlinearity and that the equivalent additive noise model can estimate the system performance [14]. In this study, we extended the previous theoretical analysis and further considered another nonlinear mechanism for DM-DFB laser-based IM-DD systems: the imperfect power-current characteristic of the DM-DFB laser.
We performed experiments to determine whether the NPR measurement by notch and equivalent additive noise model are still applicable. We also performed simulations to evaluate the NPR measurement using notch for MZM-based IM-DD system in the cases of negligible and nonnegligible CD. In addition, we analyzed various systems with different nonlinear attributes and compared the NPR results to determine the applicable range of the NPR measurement by notch.
The rest of the article is organized as follows. Section II presents theoretical analyses of the two kinds of nonlinear mechanisms in DM-DFB laser-based IM-DD system. Section III introduces the concepts of orthogonal component analysis, NPR measurement by notch, and equivalent additive noise model. Section IV gives the experimental results. Section V shows the simulation evaluation of MZM-based IM-DD system and discusses the applicable range of NPR measurement by notch. Section VI concludes the article.

II. NONLINEARITY IN DM-DFB LASER-BASED IM-DD SYSTEMS
A DM-DFB laser-based IM-DD system operating with a single-mode fiber generally comprises a digital-to-analog converter (DAC), electrical driver, DM-DFB laser, optical fiber, PD, transimpedance amplifier (TIA), and analog-to-digital converter (ADC), all of which cause nonlinearities. Here we focus on two specific nonlinearities: the imperfect power-current characteristic of the DM-DFB laser and the interaction among the modulation, CD, and detection. Fig. 1 shows a simple schematic of a DM-DFB laser-based IM-DD system. According to Neto et al. [15], the optical field of the DM-DFB laser output can be represented as

A. Schematic of a DM-DFB Laser-Based IM-DD System
where t is the time, f 0 is the center frequency, P (t) is the optical power, and ϕ(t) is the phase term. The optical power is expressed  by where η denotes the power-current characteristic of the DM-DFB laser, b is the bias current, and s(t) is the transmitted electrical signal. The instantaneous frequency is described by where Δf (t) = 1/2π · dϕ(t)/dt is the frequency chirp, α is the linewidth enhancement factor, and κ is the adiabatic chirp constant. According to (2) and (3), the amplitude modulation, phase modulation, and frequency modulation coexist on the transmitter side, and all of them have a nonlinear relationship with the input signal s(t). In the transmission link, the CD response is described by where β 2 is the group velocity dispersion parameter, z is the transmission distance, and ω is the angular frequency. At the receiver side, the PD has a square operation for the received optical field, and the received electrical signal is expressed by where R is the PD responsivity, | | 2 is the square modulus operation, and ℱ{ } and ℱ −1 { } are the Fourier transform and inverse Fourier transform, respectively.

B. Nonlinearity Caused By an Imperfect Power-Current Characteristic
The transmission distance for intra-DCI is limited to 2 km [16], in which case the CD effect is negligible. According to (1) and (5), the square root of the optical power of the DM-DFB laser and the square-law detection of the PD are a pair of inverse nonlinear operations. As shown in Fig. 2, the nonlinear  operations of modulation and detection are concatenated so that the total response is linear. Thus, by substituting (1) and (2) into (5), the received electrical signal can be described by Although the nonlinear modulation and nonlinear detection are eliminated, the system still has nonlinearity as long as the power-current characteristic of DM-DFB laser is nonlinear (i.e., η changes with s(t)).

C. Nonlinearity Caused By Interaction Among Modulation, CD, and Detection
The CD effect is nonnegligible at greater transmission distances, such as 10 km for inter-DCI [16], 10-20 km for 5G fronthaul [1], and 40-80 km for metro access [1]. In such cases, even if the power-current characteristic of the DM-DFB laser is linear, the system has a nonlinear-linear-nonlinear structure, as shown in Fig. 3. Such a distributed nonlinear structure is widely known as the Hammerstein-Wiener model [17], [18]. An analysis of (2) and (3) clearly shows that different PAM levels have different optical powers and optical frequencies. Different optical frequencies correspond to different propagation velocities in a dispersive fiber. After transmission with nonnegligible CD, different PAM levels have different propagation delays as shown in Fig. 4. As a result, eye skew occurs, which is a typical nonlinear effect in IM-DD systems [5]. This cannot be effectively mitigated by conventional linear equalization, which only uses one set of coefficients to equalize all PAM levels [19].

A. Orthogonal Component Analysis
Fig. 5 shows a nonlinear system under test with the input and output signals x(t) and y(t), respectively. The basic concept of orthogonal component analysis is to decompose the nonlinear system output y(t) into the correlated component y c (t) and orthogonal component y d (t) [20]. The correlated component is the best linear approximation [21] of the nonlinear system output: where k is the tap index, l is the tap length, and g(k) is the tap coefficient that satisfies the minimum mean square error (MMSE) criterion. Then, the orthogonal component is the remaining part and can be considered nonlinear noise [20]: B. Noise-to-Power Ratio Measurement By Notch Fig. 6 shows a schematic of NPR measurement. Some frequency component of the input spectrum is notched by a digital or analog band stop filter, and the output spectrum is measured directly. The NPR is defined as the power ratio of the regrowth component (i.e., yellow part) to the output signal (i.e., blue part). Since any linear effect does not generate new frequency component, the regrowth component is considered as nonlinear distortions. This is the standard measurement method for the NPR, and the result is denoted as NPR Notch . Because the orthogonal component can be considered actual equivalent nonlinear noise [8], we can define the power ratio of the spectrum of the orthogonal component (i.e., red curve) at the notch region to the output spectrum as NPR Orth . Fig. 7 shows the equivalent additive noise model, which was established to estimate the system performance [12]. In this study, we established two models. The first is the same NPR model, where the equivalent linear model has an amplitudefrequency response that is the difference between the input and output spectra of the nonlinear system under test. The equivalent additive noise is generated by an additive white Gaussian noise sequence passing through the noise profile filter, whose in-band response is obtained by standard NPR measurement and whose out-of-band response is the same as the output spectrum. The second is the same orthogonal model, where the equivalent linear model is described by the tap coefficient g(k) and an orthogonal component with a time-cycle shift of 1000 symbols is used as the equivalent additive noise. This implies that the additive noise and nonlinear system input are independent [12]. To focus on the nonlinear effect, we assumed that the receiver includes an ideal linear equalizer.

IV. EXPERIMENT
A. Setup   Fig. 8 shows the experimental setup of the DM-DFB laserbased IM-DD system. For digital signal processing (DSP) on the transmitter side, the 4 Sa/sym 25 Gbaud PAM8 signal was generated first. After the pulse was shaped with a roll-off factor of 0.15, notch processing was implemented. The notch width was 400 MHz, and the notch center frequencies were 1.04, 3.13, 5.21, 7.29, 9.38, and 11.46 GHz to cover the signal band. Then, the signal amplitude was adjusted by the root mean square (RMS) value of the digital signal, whose full swing was −127 to 127 corresponding to 8-bit quantization. A two-tone signal was generated to verify the nonlinear order of the system. After passing through a DAC with a 100 GSa/s sampling rate, the electrical signal was amplified by an electrical driver operating under the linear condition. Then, the amplified electrical signal was sent to the DFB module.
In the back-to-back (B2B) case, we applied a 40 mA bias current so that the nonlinear effect of the DM-DFB laser would be significant. In the 10 km transmission case, the bias current was 60 mA so that the nonlinear effect of the DM-DFB laser could be ignored. The laser center wavelength was ∼1545. 5 nm. An optical power meter (OPM) was used to measure the power-current characteristic of the DM-DFB laser. In both the B2B and transmission cases, the received optical power was fixed to −6.5 dBm by adjusting the variable optical attenuator (VOA). Thus, the nonlinearity induced by the TIA in receiver was ignored. After the receiver and direct current (DC) block, the waveforms of PAM8 with and without a notch were measured by using a DSO at a sampling rate of 50 GSa/s, and the spectrum of the two-tone signal was measured by using an electrical spectrum analyzer (ESA). The DAC and ADC generally have much smaller nonlinearities [22]. Therefore, the main nonlinear distortions of this system were caused by the imperfect powercurrent characteristic of the DM-DFB laser and the interaction among modulation, CD, and detection.
For DSP on the receiver side, frame synchronization was applied first. Next, 4-frame averaging and 64-frame averaging were applied for the B2B and 10 km transmission cases, respectively. Nonlinear distortion is a deterministic distortion; therefore, the frame averaging cannot change it. Conversely, the additive noise is random; thus, the averaging setup changes the noise level and system Q. We selected different averaging setups for two cases so that they have similar Q values, and we could focus on the nonlinear distortions. For PAM8 with a notch, we then calculated NPR Notch in the frequency domain directly. For PAM8 without a notch, we calculated NPR Orth after orthogonal decomposition. We established equivalent additive noise models to estimate the system performance subjected to nonlinearity. A 1251-tap linear equalization based on the MMSE criterion was applied to handle linear distortion in the experiment and model. Then, the Q factors of the experiment and model were calculated.

B. Results
First, we used an OPM to measure the output power of the DM-DFB laser when only a bias current was applied. In Fig. 9, the red solid line shows the measured static power-current characteristic of the DM-DFB laser, and the black dashed line shows the linear fitting for the measured result higher than the threshold. The threshold current was ∼10 mA, and the DM-DFB laser had a nonlinear power-current characteristic.  Then, we transmitted PAM8 signals with and without a notch to measure the nonlinear distortion of the system. In Fig. 10(a) and (b), the black lines denote the spectrum of the orthogonal component of PAM8 without a notch, and the other colors denote the spectra of PAM8 with a notch when the RMS of the digital signal was 54.5. The orthogonal component can be considered equivalent to the actual nonlinear noise [8]. The nonlinear noises measured by notch clearly agreed with the orthogonal component, and the nonlinear distortions caused by different nonlinear mechanisms had significant spectral differences. Compared to the orthogonal component, notch measurement is a more practical method that can be applied to measuring the nonlinear noise caused by different nonlinear mechanisms in this system. The raised signal spectrum at high-frequency range (from ∼8 GHz to ∼12 GHz in Fig. 10(b)) was caused by the interaction between the chirp and CD [5], and the spurs at 6.25, 12.5, and 18.75 GHz ( Fig. 10(a) and (b)) were caused by DSO imperfections.
We calculated the NPRs based on the measured spectra, and the results are shown in Fig. 11(a) and (b). The solid lines denote the NPRs measured by notch, and the dotted lines denote the NPRs measured by orthogonal decomposition. For the nonlinearity caused by the imperfect power-current characteristic of the DM-DFB laser, the higher RMS of the modulation signal naturally caused a greater nonlinear distortion, as shown in Fig. 11(a). In the 10 km case, a higher RMS of the modulation   signal caused a higher frequency chirp according to (3); it also caused a larger eye skew, as shown in Fig. 12, and a greater nonlinear distortion, as shown in Fig. 11(b). At different RMS levels, the two kinds of NPR measurements showed agreement. The mean absolute errors in the B2B and 10 km transmission cases were 0.36 and 0.58 dB (overall mean absolute error was 0.47 dB), respectively. Note that the NPRs measured at a low nonlinear region had relatively large measurement errors. The maximum error was ∼2.16 dB (RMS = 27.3 at 3.13 GHz in Fig. 11(b)). However, this large error limitedly affected performance estimation because the nonlinear noises of the low-frequency range are much lower than those of the high-frequency range.
Finally, we established equivalent additive noise models to estimate the system performance subjected to nonlinearity. Fig. 13 shows the experimental and model Q factors at different RMS levels. In the B2B case ( Fig. 13(a)), the mean absolute errors of the same orthogonal model and same NPR model compared with the experimental Q factor were 0.06 and 0.5 dB, respectively. In the 10 km transmission case (Fig. 13(b)), the mean absolute errors of the same orthogonal model and same NPR model were 0.1 and 0.26 dB, respectively. The same orthogonal model had much higher estimation accuracy because the equivalent additive noise included the full information of the actual nonlinear noise. In contrast, the same NPR model is more practical even though the estimation error was slightly larger.

A. Nonlinear Behavior Model Versus Equivalent Additive Noise Model
A nonlinear behavior model is another way to characterize nonlinearity. This is because it provides the amount and waveform of nonlinear distortion. However, establishing a nonlinear behavior model is difficult, as it requires the model structures and accurate model parameters. There are many research activities in the field of nonlinear system identification [23]. The model structure of the DM-DFB laser-based IM-DD system is clear, and (1)-(5) completely describe the nonlinear behavior. The power-current characteristic η seems to be easily measured by sweeping the laser bias current and measuring the output power. However, this measuring approach only acquires the static power-current characteristic with the assumption of a constant electrical signal. Practically, the electrical signal varies with time, and the dynamic power-current characteristic differs from the static power-current characteristic. In addition, measuring the linewidth enhancement factor α and adiabatic chirp constant κ requires a digital communication analyzer and a high-resolution optical spectrum analyzer, and the measurement procedure is complicated [24].
Here, we aim to obtain the amount of nonlinear distortion rather than the waveform of nonlinear distortion and subsequently estimate the nonlinear system performance. Thus, we utilized a spectrum analyzer to measure the NPR using a notch and estimate the nonlinear system performance through an equivalent additive noise model.

B. Nonlinear Distortion in MZM-Based IM-DD Systems
In addition to the DM-DFB laser-based direct modulation approach, the MZM-based external modulation scheme has been extensively used in IM-DD systems. The MZM output optical field can be expressed as follows [3]: Here, P 0 and ω 0 are the optical power and angular frequency of continuous-wave light, respectively. φ s and φ b are the phase shifts caused by the transmitted electrical signal and bias voltage, respectively. When the MZM is biased at quadrature point (i.e., ϕ b = π/2), the received electrical signal in B2B case can be described as follows: Although the cascaded response of the square root of optical power and the square-law detection of the PD is linear, the system has a sinusoidal characteristic. When the RMS value of the electrical signal is not sufficiently small (e.g., 0.5 V π ), the sinusoidal characteristic-induced nonlinearity cannot be ignored. Moreover, the dominant nonlinearity is the third-order term according to Taylor series expansion. Fig. 14(a) shows that the nonlinear noises measured using the notch are inconsistent with orthogonal component (black line) for the case of 25 Gbaud PAM8.
With an increase in transmission distance, the CD effect becomes nonnegligible, and the system maintains a nonlinearlinear-nonlinear structure. The interaction among modulation, CD, and detection causes even-order nonlinearities [25] and a power-fading effect [26]. The RMS value of the electrical signal was set to 0.1 V π so that the nonlinearity of MZM could be ignored. Fig. 14(b) shows that the nonlinear noises measured using the notch agreed with the orthogonal component in the most of frequency range when the transmission distance was 35 km. In the low-frequency range, the nonlinear noise measured by the notch is larger than the orthogonal component. According to a previous study [3], the amount of second-order nonlinearity decreases with a decrease in frequency. Thus, the sinusoidal nonlinearity of MZM becomes dominant in the low-frequency range. However, the level of the NPR in the low-frequency range is much lower than that in the high-frequency range so that the disagreement in the low-frequency range does not significantly affect the total nonlinear distortion power estimation and nonlinear system performance estimation.

C. Applicable Range of NPR Measurement by Notch
The NPR is a straightforward nonlinear specification because linear effects do not generate any new frequency components. Compared to NPR measurement by orthogonal component, NPR measurement by notch only uses a spectrum analyzer, so it is more practical for characterizing nonlinear distortion. Gharaibeh [13] showed that NPR measurement by notch agrees well with NPR measurement by orthogonal component when the input signal is Gaussian, and there is 8 dB error when the input signal is not Gaussian. In that case, the dominant nonlinear distortion is third-order nonlinearity. For an electrical driver designed for optical communication application that operates in differential mode and that is dominated by third-order nonlinearity, the NPR measurement by notch works well with a Gaussian input signal and fails with a PAM8 input signal [27]. In the simulation evaluation of the MZM-based IM-DD system with negligible CD where the third-order nonlinearity is dominant, the method of measuring the NPR using the notch fails with PAM8 input. Moreover, we showed that, for a nonlinear system modeled as a third-order memoryless polynomial y(t) = C 1 · x(t) + C 2 · x 2 (t) + C 3 · x 3 (t) where C 1 , C 2 , and C 3 are 0.476, −1.63e−3, and −2.45e−3, respectively, the NPR measurement by notch fails for a PAM4 signal [28]. The results for these nonlinear systems indicate that NPR measurement by notch cannot characterize nonlinearity correctly in those cases when the input signal does not have a Gaussian distribution.
However, we found that NPR measurement by notch works well in a vertical-cavity surface-emitting laser (VCSEL)-based IM-DD system with a PAM4 signal [28]. The VCSEL has an imperfect response-voltage characteristic in that it has a faster response when the applied voltage is increased, which causes eye skew [29], [30]. This nonlinearity is dominated by second-order terms, which was verified by two-tone test [28]. For a nonlinear system modeled as a second-order memoryless polynomial y(t) = C 1 · x(t) + C 2 · x 2 (t)where the coefficients are the same as above, NPR measurement by notch works well for PAM4 signals [28].
In the present study, our results showed that NPR measurement by notch also works well in a DM-DFB laser-based IM-DD system with a PAM8 signal. In this system, the nonlinear distortions are caused by the imperfect power-current characteristic of the DM-DFB laser and the interaction among the modulation, CD, and detection. We applied a two-tone signal at 7 and 8 GHz to verify the dominant orders of these nonlinear distortions. Fig. 15 shows the results of the two-tone measurement in the B2B case. The nonlinearity caused by the imperfect power-current characteristic of the DM-DFB laser was dominant at a bias current of 40 mA. When the RMS of the two-tone signal was 27.3, the power ratio of the fundamental frequency to the nonlinear distortion was 19 dB, as shown in Fig. 15(a). Then, we increased the RMS of the two-tone signal to 54.5, which reduced the power ratio to 13 dB as shown in Fig. 15(b). Fig. 16 shows the two-tone measurement results in the 10 km transmission case. In this case, the RMS of the two-tone signal was fixed to 54.5, and the bias current was increased from 40 mA to 60 mA. Thus, the nonlinearity caused by the imperfect power-current characteristic of the DM-DFB laser was negligible. Before the 10 km transmission, the power ratio of the fundamental frequency to the nonlinear distortion was 24 dB, as shown in Fig. 16(a). The power ratio decreased to 8 dB after the 10 km transmission, as shown in Fig. 16(b) indicates that the interaction among the modulation, CD, and detection caused significant nonlinear distortion. These results also show that the second-order nonlinear distortions were 18 and 15 dB larger than the third-order nonlinear distortions in the B2B and 10 km transmission cases, respectively. The noise floors in Figs. 15 and 16 are caused by the ESA so that they cover the whole spectrum range. Similar to the VCSEL-based IM-DD system and second-order memoryless polynomial, the DM-DFB laser-based IM-DD system was dominated by second-order nonlinearities. Moreover, our simulation results in the previous section demonstrated that the method of NPR measurement by the notch is effective for the MZM-based IM-DD system with nonnegligible CD where the second-order nonlinearity is dominant. Table I summarizes the attributes of various nonlinear systems including the dominant nonlinear mechanism, dominant nonlinear order, nonlinear structure, and NPR results with non-Gaussian inputs. The nonlinear order is the most important factor, whereas the specific nonlinear mechanism and nonlinear structure do not demonstrate any correlation. As long as the nonlinear system is dominated by second-order terms, the nonlinearity can be correctly characterized by NPR measurement by notch, even with non-Gaussian input signals. As a reference, NPR measurement by orthogonal component is not affected by the nonlinear order. Although NPR measurement by notch is more practical for nonlinear characterization, its applicability is limited to systems dominated by second-order nonlinearity. Applying NPR measurement by notch to systems dominated by third-order nonlinearity with a non-Gaussian input signal is a significant challenge.

VI. CONCLUSION
We theoretically analyzed two specific nonlinear mechanisms in DM-DFB laser-based IM-DD systems: the imperfect powercurrent characteristic of the DM-DFB laser and the interaction among the modulation, CD, and detection. Then, we conducted experiments to demonstrate that the nonlinear noise can be accurately estimated by the very simple method of NPR measurement by notch, which notches some frequency components of the nonlinear input spectrum and measures the regrowth of the nonlinear output spectrum. Compared to the reference NPR, the mean absolute error was 0.47 dB. Meanwhile, equivalent additive noise models were established to estimate the system performance subject to nonlinearity. In addition, we conducted simulations to illustrate that measuring the NPR using the notch is effective when the MZM-based IM-DD system has nonnegligible CD. We analyzed various systems with different nonlinear mechanisms, nonlinear orders, and nonlinear structures, and we concluded that the NPR measurement by notch is applicable to nonlinear systems if they are dominated by second-order nonlinearities. In contrast, NPR measurement by notch needs Gaussian stimuli when the nonlinear systems are dominated by third-order nonlinearities.