A Comprehensive Comparison of Simplified Volterra Equalization and Kramers–Kronig Schemes in 200 Gb/s/λ PON Downlink Transmission

Abstract

The emerging high-bandwidth services of 6G, such as high-definition video transmission and real-time interaction, have promoted the progress of the fiber optic access network industry, driving its development towards the next-generation PON with higher speed and larger system capacity. In response to the future requirements of 200 Gb/s/λ PON for both 20 km and 30 km downlink transmission scenarios, this paper proposes a Simplified Volterra Equalization (SVLE) scheme based on Nyquist PAM4 single-sideband modulation direct detection (SSBM-DD) scheme. In order to verify its advantages, the IQ-modulated Kramers–Kronig reception (KK) scheme is introduced for comparison. Simulation validation platforms for two schemes are conducted, and the performance comparison of the SVLE and KK schemes is carried out. In both, the impact of the carrier signal power ratio (CSPR) on receiver sensitivity, the influence of input optical power on power budget and receiver sensitivity, and the tolerance of receiver sensitivity to the linewidth of the DFB laser are investigated in the simulation. Finally, a comprehensive comparison of the two schemes is presented in terms of system performance, cost, and DSP complexity. In the 20 km downlink transmission scenario, the SVLE scheme outperforms the KK scheme by 4.9 dB in terms of power budget. The total number of multiplications of the SVLE scheme is 37, while that of the KK scheme is 4358. Therefore, the DSP complexity of the SVLE scheme is much lower than that of the KK scheme. The results of the comparison demonstrate that, in downlink transmission scenarios, the SVLE scheme is more suitable than the KK scheme as it exhibits a higher power budget, lower DSP complexity, and lower cost. Consequently, the proposed SVLE scheme could be a highly promising solution for future ultra-high-speed PON downlink transmission.

1. Introduction

The rise of 6G networks and the Internet of Things (IoT) [1] has spurred the emergence of new big data applications such as high-definition video transmission, real-time interaction, virtual scene enhancement, and augmented reality. These application scenarios demand higher data rates and greater bandwidth capacity from optical communication links. In optical fiber communications, Passive Optical Network (PON) systems are favored for their inherent high bandwidth and capability for full-service access [2]. PON is the access technology for fiber-to-the-X (FTTX). Especially in fiber-to-the-home (FTTH) [3], it is necessary to directly extend the optical fiber to the user’s home, providing high-speed and large-capacity data transmission capabilities. The construction of FTTH involves a large amount of infrastructure investment, including the laying of optical fibers, the installation of optical splitters, and the deployment of the Optical Network Units (ONUs). The economic affordability of ordinary single users is limited, so PON systems are very cost-sensitive. Optical fiber communication systems are generally divided into two categories: coherent detection systems and direct detection systems. Coherent detection systems offer excellent transmission performance but come with higher complexity and cost, making them unsuitable for meeting the low-power and low-cost requirements of ultra-high-speed optical communication. In contrast, direct detection systems do not require a local oscillator light source, saving costs [4] and avoiding complex wavelength calibration work. Therefore, considering the sensitivity to receiver-side costs, the downlink transmission scheme based on Intensity Modulation Direct Detection (IMDD) has been a research hotspot. Moreover, to improve spectral efficiency, high-speed IMDD transmission schemes typically employ multi-level modulation techniques, such as four-level pulse amplitude modulation [5] (PAM4), discrete multitone [6] (DMT), and carrierless amplitude phase modulation [7] (CAP). Among these modulation formats, PAM4 has attracted significant attention due to its simpler configuration, lower cost, and power consumption.

As the rate of single-wavelength transmission increases, ultra-high-speed PON transmission systems encounter a multitude of transmission losses, including elevated noise levels, dispersion effects, and nonlinear distortion. At higher symbol rates, linear distortion caused by dispersion is exacerbated. With increasing transmission distance, the dispersion effect is further aggravated, resulting in a reduction in channel bandwidth, which ultimately leads to power fading and decreased receiver sensitivity. Additionally, as the transmission capacity and distance increase, the input optical power needs to increase accordingly, causing nonlinear damage to become more serious. To address this issue, the academic community has proposed various solutions. Among them, the traditional Volterra scheme is cost-effective and capable of effectively compensating for both linear and nonlinear distortions, albeit with high algorithmic complexity [8]. On the other hand, the Kramers–Kronig reception (KK) scheme stands out for its advantages of simple structure [9], low costs, and superior system performance despite its high algorithmic complexity. Therefore, after in-depth research and analysis, this paper designs a Simplified Volterra Equalization (SVLE) scheme based on Nyquist PAM4 single-sideband modulation direct detection (SSBM-DD) scheme for the future single-wavelength rate 200 Gb/s PON downlink transmission. To validate its advantages, the paper constructs a corresponding downlink simulation system and conducts a comprehensive comparative analysis with the KK scheme based on IQ modulation across three aspects: system performance, cost, and digital signal processing (DSP) complexity.

This paper conducts a detailed study of the SVLE scheme by comparing it with the KK scheme. Section 1 provides a brief overview of the research background, significance, and main content of the paper. Section 2 describes the structures and principles of the two schemes and derives the general formulas for calculating the complexity of the algorithms. Section 3 constructs corresponding simulation systems for two scenarios, 20 km and 30 km downlink transmissions; completes corresponding simulations; and conducts a comprehensive comparative analysis of the simulation results for both schemes across three aspects: system performance, cost, and DSP complexity. This analysis determines the advantages and disadvantages of the two schemes in different scenarios. Section 4 concludes the entire paper.

2. Structures and Principles of the Schemes

2.1. Overall Scheme Design

As shown in Figure 1, the system comprises three main parts: the Optical Line Terminal (OLT), the ONU, and the optical fiber transmission line. On the OLT side, components include DSP algorithms, digital-to-analog converters (DACs), distributed feedback (DFB) lasers, and IQ modulators (IQMs). The ONU side, on the other hand, consists of PIN photodetectors, analog-to-digital converters (ADCs), and DSP algorithms. In order to obtain a sufficient power budget and to reduce the cost of the devices, the overall scheme is based on the PAM4-SSBM-DD scheme. And a root-raised cosine filter with a roll-off factor of 0.1 is used for the shaping filtering to further compress the spectral width of the signal, thus reducing the bandwidth requirement of the devices while minimizing the impact on the chromatic dispersion (CD).

In the transmitter link, the input bitstream is converted into real signal named I and imaginary signal named Q after a series of processes in the transmitter DSP. Actually, Q is the signal of I after Hilbert transform. The two signals are, respectively, sent to the IQM after passing through the DACs. They are then modulated with the laser emitted by the DFB laser as an optical single sideband (SSB) signal, which is then input into the optical fiber transmission line. In the receiver link, PIN photodetector is used to convert the optical signal into an electrical signal, followed by ADCs. The digitized signal is then processed in the receiver DSP to obtain the output bitstream. Subsequently, the bit error rate (BER) is statistically calculated to serve as a metric for evaluating system performance.

2.2. Analysis of Nonlinear Damage in the SSBM-DD Systems

In the SSBM-DD systems, the transmitted signal can be expressed as

$$x\left(t\right)=s\left(t\right)+A$$

(1)

where A is the DC component, and s(t) can be expressed as

$$s\left(t\right)=m\left(t\right)+jH\left(m\right(t\left)\right)$$

(2)

where m(t) is the baseband signal and H(·) represents the Hilbert transform.

Disregarding third-order dispersion, the equivalent low-pass frequency domain transfer function expression for CD in the channel is as follows:

$$H(j\omega )=\mathit{exp}\{-\frac{j}{2}\beta L{\omega }^2\}$$

(3)

where L represents the length of the optical fiber transmission, and $\beta$ is the group velocity dispersion coefficient.

After transmission through the channel, the received signal can be expressed as

$$r\left(t\right)=s\left(t\right)\otimes h\left(t\right)+A$$

(4)

where $h\left(t\right)$ is the time-domain expression of $H\left(j\omega \right)$ after the inverse Fourier transform.

If the signal obtained from square-law detection of r(t) by a PIN photodetector is R(t), R(t) can be expressed as

$$R\left(t\right)={\left|r\left(t\right)\right|}^2=A^2+2ARe\left\{s\right(t)\otimes h(t\left)\right\}+{\left|s\left(t\right)\otimes h\left(t\right)\right|}^2$$

(5)

where $2ARe\left\{s\right(t)\otimes h(t\left)\right\}$ is the expected signal and ${\left|s\left(t\right)\otimes h\left(t\right)\right|}^2$ is the nonlinear signal-to-signal beat interference (SSBI) term.

Based on the conclusions above, for SSBM-DD systems, after the signal is transmitted through optical fiber, due to the square-law detection characteristic of the PD receiver, it contains nonlinear SSBI terms, which causes waveform distortion.

2.3. Principle of the SVLE Scheme

The Volterra algorithm is a method for modeling and identifying nonlinear systems based on the Volterra series expansion. Volterra series expansion is an order-by-order expansion of the nonlinear response of the system, expressing the output of the system as the weighted sum of various power combinations of inputs.

From Equation (5), the SSBI terms of SSBM-DD systems only contain the second-order terms. The second-order nonlinear distortion becomes the main factor affecting the transmission, so the second-order Volterra algorithm is enough to compensate [10], which can be written as

$$y(n)={\sum }_{l_1=0}^N{h}_{l_1}\left(n\right)x(n-l_1)+{\sum }_{l_1=0}^N{\sum }_{l_2=0}^N{h}_{l_1,l_2}\left(n\right)x(n-l_1)x(n-l_2)$$

(6)

where $h_{l_1}\left(n\right)$ and $h_{l_1,l_2}\left(n\right)$ are the kernel coefficients of the nonlinear filter model of the second-order Volterra series expansion, and $y\left(n\right)$ is prediction output signal at time instant n.

However, the complexity of overall Volterra algorithm is still at a high level. Research has shown that for the quadratic terms of the Volterra algorithm, the compensation for nonlinear impairments mainly comes from the square terms, while the influence of cross terms is small. Therefore, by retaining only the square terms in the quadratic terms, significant simplification of calculation and efficiency improvement can be achieved while ensuring the accuracy of the algorithm. Equation (6) can be rewritten as

$$y(n)={\sum }_{l_1=0}^{N_1-1}{h}_1\left(l_1\right)x(n-l_1)+{\sum }_{l_2=0}^{N_2-1}{h}_2\left(l_2\right)x^2(n-l_2)$$

(7)

where $h_1\left(l_1\right)$ and $h_2\left(l_2\right)$ are linear and quadratic coefficients, respectively. $N_1$ and $N_2$ are the memory length of first-order term and second-order term in Volterra model, respectively. The linear term is mainly used to compensate for linear damage such as dispersion, and the square term is mainly used to remove the SSBI term. It also provides some compensation for nonlinear damage. The values of $N_1$ and $N_2$ are related to the specific degree of damage. Generally speaking, the more severe the linear damage, the larger the value of $N_1$, and the more severe the nonlinear damage, the larger the value of $N_2$. For example, when dispersion intensifies, $N_1$ always needs to be increased. Due to the fact that the SSBI term is also related to dispersion, $N_2$ also needs to be increased.

Under the same dispersion condition, the difference in DSP complexity between the SVLE scheme and the KK scheme mainly lies in the SVLE algorithm and the KK algorithm. Therefore, this paper defines DSP complexity as the total number of real multiplications and the total number of real additions required by the SVLE algorithm and the KK algorithm.

From Equation (7), a total of $N_1+N_2$ taps are required to compute the complexity of the SVLE algorithm, and a total of $N_1+{2N}_2$ real multiplications, as well as $N_1+N_2-1$ real additions, are required for each symbol.

Based on this, this paper proposes a scheme to use a low-complexity simplified Volterra equalizer for damage equalization in the receiver DSP and builds its corresponding simulation system.

2.4. Principle of the KK Scheme

At the same time, due to nonlinear SSBI terms, the signal will lose the phase information of the vector field, and the lack of phase causes the receiver to be unable to reconstruct the vector field signal [11]. The KK algorithm uses the Kramers–Kronig relation to perform cancellation of SSBI terms to recover the original signal [12]. It has now become an effective receiving scheme in DD systems. Therefore, this paper introduces the IQ-modulated KK scheme, which requires CD equalization through a fixed dispersion equalization algorithm based on overlap frequency domain equalization (OFDE) [13,14] and compares it with the SVLE scheme so as to verify the superiority of the SVLE scheme.

The complexity calculation of the KK scheme is relatively complicated. Digital upsampling, digital downsampling, Hilbert transform, and dispersion equalization are included in the receiver DSP. The hardware schematic diagram is shown in Figure 2. The digital upsampling and downsampling are both implemented by finite impulse response (FIR) filters. Assuming the number of taps is $N_s$, then outputting a symbol through this module requires ${ N}_s$ real multiplications and $N_s-1$ real additions. With an oversampling ratio of R, the required number of operations needs to be multiplied by R. Regarding some special operations, such as Hilbert transform, as referenced in the conclusions of researchers such as Christoph Füllner at OFC in 2018 [15], it can be approximately implemented by an FIR filter with a limited number of taps of $N_h$, which requires $N_h/2$ real additions and $N_h/2$ real multiplications. N represents the number of FFT points for the fixed dispersion equalization algorithm based on OFDE.

Due to the conclusions above, the digital signal undergoes interpolation followed by FIR filtering, then passes through an FIR filter based on the Hilbert transform. Subsequently, it is multiplied by sine and cosine functions separately, resulting in two separate signal paths. Each signal path then undergoes FIR filtering and decimation before being subjected to dispersion equalization collectively, yielding the corresponding output result.

Therefore, the total number of multiplications required by the KK receiver is

$$M_r=\left(3N_s+\frac{N_h}{2}+2\right)R+4N{\mathit{log}}_{2}N+5N+2$$

(8)

The total number of additions is

$$A_r=\left(3\left(N_s-1\right)+\frac{N_h}{2}\right)R+6Nlo{g}_{2}N+3N$$

(9)

According to the above algorithms, this paper builds its corresponding simulation system.

3. Scheme Simulation and Comparative Analysis

3.1. Simulation Design of the SVLE Scheme

The simulation system is built in the simulation software, as shown in Figure 3.

The input bitstream consists of a random sequence of 0 s and 1 s, which, after processing by the DSP algorithm in the transmitter, is divided into two paths, namely the I and Q signals for transmission. Among them, Q is the signal of I after Hilbert transform. In the transmitter DSP, PAM4 symbol mapping and double upsampling are employed, followed by Nyquist shaping filtering using a root-raised cosine filter with a roll-off factor of 0.1, and finally, Hilbert transform is applied. In the receiver DSP, the signal undergoes matched filtering, double downsampling, and SVLE processing, followed by PAM4 symbol demapping to obtain the output bitstream. By comparing the output bitstream with the input bitstream, the BER is obtained. At the same time, shot noise and thermal noise are added to the PIN photodetector, and a low-pass filter (LPF) is added to simulate the bandwidth.

The simulation parameters are as follows. The baud rate is 100 Gbaud. And the sampling rate of the DAC/ADC is 200 GSample/s, with a quantization bit depth of 8 bits. The operating wavelength of the DFB laser is 1315 nm. The extinction ratio of IQM is 25 dB. The fiber length is 20 or 30 km, and the type is G.652D, with an attenuation of 0.3 dB/km, CD of 2 ps/nm/km, and polarization mode dispersion (PMD) coefficient of 0.1 ps/(km)^1/2. The responsivity of the PIN photodetector is 0.8 A/W, with a single-sided spectral density of the mean square thermal noise current of 1.44 × 10⁻²² A²/Hz. The cutoff frequency of the LPF is 55 GHz, and the filter type is approximately an ideal rectangular shape. Furthermore, the equalizer adopts a Volterra–LMS [16] type. Variable parameters include the CSPR, input optical power, and the DFB laser linewidth.

3.2. Simulation Design of the KK Scheme

The simulation system is built in the simulation software, as shown in Figure 4.

In the transmitter DSP, PAM4 symbol mapping and double upsampling are employed, followed by Nyquist shaping filtering using a root-raised cosine filter with a roll-off factor of 0.1, and finally, Hilbert transform is applied. Q is the signal of I after Hilbert transform. In the receiver DSP, the signal first undergoes matched filtering and quadruple downsampling, followed by phase information recovery using a KK algorithm to reconstruct the complex vector signal. Subsequently, CD equalization processing and PAM4 symbol demapping are applied to obtain the output bitstream, thus calculating the BER.

The simulation parameters are as follows. The baud rate is 100 Gbaud. And the sampling rate of the DAC/ADC is 400 GSample/s, with a quantization bit depth of 8 bits. The operating wavelength of the DFB laser is 1315 nm. The extinction ratio of IQM is 25 dB. The fiber length is 20 or 30 km, and the type is G.652D, with an attenuation of 0.3 dB/km, CD of 2 ps/nm/km, and PMD coefficient of 0.1 ps/(km)^1/2. The responsivity of the PIN photodetector is 0.8 A/W, with a single-sided spectral density of the mean square thermal noise current of 1.44 × 10⁻²² A²/Hz. The LPF has a 3 dB bandwidth of 75 GHz and is a fourth-order Bessel filter. Variable parameters include the CSPR, input optical power, and the DFB laser linewidth.

3.3. The Results and Analysis of Simulation

3.3.1. Feasibility Verification of the Schemes

• Optical Power Spectrum analysis

When the input optical power is 0 dBm, the Optical Power Spectrums obtained by simulation are shown in Figure 5. It can be clearly seen that the schemes use optical SSBM, and the signal power is significantly reduced after optical fiber transmission. Due to the low dispersion of the schemes and the small nonlinear effect of the optical fiber at this time, it is difficult to observe the influence of dispersion and nonlinear effects on the signal from Figure 5.

• The Simplified Volterra Equalization scheme

When the CSPR is 10 dB, the eye diagrams and scatter diagrams of the received signal before and after being processed by the SVLE algorithm are shown in Figure 6 and Figure 7. It can be easily seen that the scatter diagram after being processed is more obviously concentrated than before being processed, and the opening degree of the eye diagram after being processed is obviously larger. This shows that after being processed by the SVLE algorithm, the performance of the system has been significantly improved, and the BER has been significantly reduced. Therefore, the SVLE scheme is indeed a feasible scheme.

• The IQ-modulated Kramers–Kronig reception scheme

When the CSPR is 7 dB, the eye diagrams and scatter diagrams of the received signal before and after being processed by the KK algorithm are shown in Figure 8 and Figure 9. Similarly, the KK scheme is also an effective scheme.

3.3.2. Identify the Optimal CSPR

It is necessary to identify the optimal CSPR for each of the two schemes in the back-to-back (BtB) 20 km and 30 km downlink transmission scenarios. According to the simulation conditions, the input optical power is set to 0 dBm, and the DFB laser linewidth is 1 MHz to explore the relationship between CSPR and receiver sensitivity.

In the SVLE scheme, CSPR is set in the range of 7 to 15 dB with a step size of 1 dB for simulation, and the relationship is depicted in Figure 10a. As the CSPR increases, the receiver sensitivity gradually improves. When the CSPR is 10 dB, the receiver sensitivity reaches the optimum values. Specifically, they are −12.2 dBm for the BtB, −12.0 dBm for 20 km, and −11.9 dBm for 30 km. With a further increase in CSPR, the receiver sensitivity gradually deteriorates. Therefore, the optimal CSPR values are determined to be 10 dB. In the KK scheme, CSPR is set in the range of 4 to 12 dB [17] with a step size of 1 dB for simulation, and the relationship is illustrated in Figure 10b. The receiver sensitivity gradually improves as the CSPR increases. When the CSPR is 7 dB, the receiver sensitivity reaches the optimal values. They are −13.0 dBm for the BtB, −12.8 dBm for 20 km, and −12.6 dBm for 30 km. Subsequently, with a further increase in CSPR, the receiver sensitivity gradually deteriorates. Therefore, the optimal CSPR values are determined to be 7 dB.

In both schemes, the influence of CSPR on receiver sensitivity is essentially identical. It has been analyzed that when the CSPR is less than a certain value, the signal suffers severe nonlinear signal–signal beat interference (SSBI) degradation due to the relatively low carrier power [18], and the receiver DSP algorithm is unable to provide an effective compensation. In this case, even if the received optical power is 0 dBm, regardless of the adjustments made to the relevant algorithm parameters, the BER cannot reach 0.01. As the CSPR increases, the damage to the signal caused by nonlinear SSBI decreases. This allows the receiver to effectively compensate, resulting in an improved receiver sensitivity that reaches an optimal value. As the CSPR continues to increase, the proportion of useful signals declines, resulting in a decrease in the signal-to-noise ratio (SNR). Therefore, the receiver sensitivity gradually deteriorates.

3.3.3. Explore the Influence of the Input Optical Power

Both schemes take their own optimal CSPR to explore the effect of the input optical power. Among them, the linewidth of the DFB laser is set to 1 MHz.

• The 20 km downlink transmission scenario

In the SVLE scheme, the optimal value of CSPR is set to 10 dB. Simulations are conducted for input optical power ranging from −6 to 18 dBm, with a step size of 1 dBm. For input optical power ranging from 18 dBm to 19 dBm, simulations are conducted with a step size of 0.25 dBm. Using power budget as the metric, the optimal power budget is found by adjusting equalizer parameters, and the relationship is depicted in Figure 11a,c. When the input optical power is less than 18 dBm, the receiver sensitivity remains approximately constant as the input optical power increases, with a mean value of −12.2 dBm. The power budget gradually increases, reaching its maximum value of 29.9 dB when the input optical power is 18 dBm. With a further increase in the input optical power, the receiver sensitivity sharply deteriorates, making the power budget rapidly decrease. Therefore, the input optical power value is determined to be 18 dBm. At this juncture, the equalizer exhibits a first-order memory length ($N_1$) of 15 and a second-order memory length ($N_2$) of 11.

In the KK scheme, the CSPR is set to the optimal value of 7 dB. When the input optical power ranges from −7 to 14 dBm, simulations are conducted with a step size of 1 dBm. Similarly, for input optical power within the range of 14 dBm to 15 dBm, simulations are conducted with a step size of 0.25 dBm, resulting in the relationship graphs as shown in Figure 11b,d. When the input optical power is less than 14 dBm, the receiver sensitivity remains approximately constant as the input optical power increases, with a mean value of −12.6 dBm. The power budget, on the other hand, gradually increases, reaching its maximum value of 25.0 dB when the input optical power is 14 dBm. As the input optical power continues to increase, there is a noticeable decline in receiver sensitivity, leading to a swift reduction in the power budget. Therefore, the optimal input optical power value is determined to be 14 dBm. At this point, the FFT computation point number (N) is 128.

The trend of the input optical power exerts a comparable influence on the two schemes. Analysis suggests that when the input optical power is less than a certain value, the low power of the optical signal results in small nonlinear effects within the fiber, insufficient to cause significant nonlinear distortion [19]; thus, the receiver sensitivity remains almost unchanged. As the input optical power increases, the power budget gradually grows, reaching its maximum when the input optical power reaches that value. Subsequently, with a further increase in the input optical power, the nonlinear effects within the fiber escalate sharply, causing a rapid deterioration in receiver sensitivity and consequently leading to a swift decrease in the power budget.

The performance advantages and disadvantages of the two schemes are discussed below in terms of optimal receiver sensitivity and power budget in total, respectively. In terms of the optimal receiver sensitivity, when the input optical power is low, the optimal receiver sensitivity for the Volterra equalization scheme is approximately −12.2 dBm, while for the KK reception scheme, it is also approximately −12.6 dBm. The difference between the two schemes is 0.4 dB, indicating that the KK scheme slightly outperforms the SVLE scheme. Regarding the power budget, for the SVLE scheme, when the CSPR is 10 dB and the input optical power is 18 dBm, the power budget reaches its maximum of 29.9 dB. On the other hand, for the KK scheme, when the CSPR is 7 dB and the input optical power is 14 dBm, the power budget also reaches its maximum of 25.0 dB. The difference between the two schemes is 4.9 dB, suggesting that the SVLE scheme is superior to the KK scheme.

In 2023, Ricardo Rosales [20] et al. demonstrated 2 × 100 Gb/s 20 km fiber transmission, achieving −25 dBm sensitivity around 1340 nm, resulting in a 33 dB link budget. The performance of the SVLE scheme is slightly worse than theirs. The gap is analyzed as possibly being due to two EML-SOAs, the pre-SOA, and the double-sideband modulation method. At the same time, the receiver algorithm in their scheme uses feed-forward equalization (FFE) and maximum likelihood sequence estimation (MLSE), which is more complex than the SVLE scheme.

• The 30 km downlink transmission scenario

In the SVLE scheme, CSPR is set to the optimal value of 10 dB. When the input optical power ranges from −2 dBm to 16 dBm, simulations are conducted with a step size of 1 dBm. When the input optical power ranges from 16 dBm to 17 dBm, simulations are conducted with a step size of 0.25 dBm. The relationship graphs are shown in Figure 12a,c. The optimal input optical power value is determined to be 16 dBm. At this juncture, the first-order memory length of the equalizer ($N_1$) is 23, and the second-order memory length ($N_2$) is 15. In the KK scheme, CSPR is set to the optimal value of 7 dB. When the input optical power ranges from −3 dBm to 12 dBm, simulations are conducted with a step size of 1 dBm. When the average input optical power ranges from 12 dBm to 13.25 dBm, simulations are conducted with a step size of 0.25 dBm. The relationship graphs are shown in Figure 12b,d. The optimal average input optical power value is determined to be 12 dBm. At this point, the FFT computation point number (N) is 256.

In terms of optimal receiver sensitivity, when the input optical power is low, the optimal receiver sensitivity for the SVLE scheme is approximately −12.0 dBm, while for the KK scheme, it is approximately −12.4 dBm. The difference between the two schemes is 0.4 dB, and the KK scheme is slightly better than the SVLE scheme. In terms of power budget, for the SVLE scheme, when CSPR is 10 dB and the input optical power is 16 dBm, the power budget reaches its maximum value of 26.8 dB. For the KK scheme, when CSPR is 7 dB, and the input optical power is 12 dBm, the power budget reaches its maximum value of 23.1 dB. The difference between the two schemes is 3.7 dB, with the SVLE scheme being superior to the KK scheme.

3.3.4. Investigate the Effect of DFB Laser Linewidth

Both schemes take their own optimal CSPR to investigate the tolerance of the receiver sensitivity to the linewidth of the DFB laser. According to the simulation conditions, the input optical power is set to 0 dBm.

• The 20 km downlink transmission scenario

In the SVLE scheme, the CSPR is set to the optimal value of 10 dB, the laser linewidth is in the range of 0 to 20 MHz, and the simulation is performed with a step size of 2 MHz. The resulting relationship diagram is shown in Figure 13a. In the KK scheme, the CSPR is set to the optimal value of 7 dB, the laser linewidth is in the range of 0 to 80 MHz, and the simulation is performed with a step size of 10 MHz. The resulting relationship diagram is shown in Figure 13b.

As the linewidth increases, receiver sensitivity declines in both schemes. In the SVLE scheme, when the laser linewidths are 7.9 MHz and 14.7 MHz, respectively, the receiver sensitivity deteriorates by 0.1 dB and 0.2 dB. Similarly, in the KK scheme, when the laser linewidths are 19.1 MHz and 48.4 MHz, the receiver sensitivity deteriorates by 0.1 dB and 0.2 dB, respectively. It is evident that the SVLE scheme is more susceptible to alterations in the laser linewidth.

The Volterra receiver employs the Volterra series for the suppression of nonlinear distortion [21]. As depicted in Equation (7), this scheme primarily compensates for nonlinear distortion through the square terms. The Volterra receiver reconstructs the vector field signal by removing the SSBI term of the received signal. However, the KK receiver utilizes the Kramers–Kronig relation to demodulate the intensity signal, which is less sensitive to phase information. Based on the relationship between phase and amplitude, the KK algorithm reconstructs the original signal. The differing operational principles and signal processing methods of these two receivers result in varying sensitivities to linewidth.

• The 30 km downlink transmission scenario

In the SVLE scheme, the CSPR is set to the optimum value of 10 dB, with the laser linewidth ranging from 0 to 16 MHz. Simulations are conducted with a step size of 2 MHz, as illustrated in Figure 14a. In the KK scheme, the CSPR is set to the optimum value of 7 dB, while the laser linewidth varies from 0 to 40 MHz. Simulations are executed with a step size of 5 MHz, depicted in Figure 14b.

For the SVLE scheme, the sensitivity of the receiver deteriorates by 0.1 dB and 0.2 dB when the linewidths of the laser are 5.4 MHz and 10.4 MHz, respectively. For the KK scheme, when the linewidths of the laser are 16.3 MHz and 31.4 MHz, respectively, the sensitivity of the receiver deteriorates by 0.1 dB and 0.2 dB. The SVLE scheme is more sensitive to the laser linewidth.

In both 20 km and 30 km downlink transmission scenarios, the SVLE scheme is more sensitive to the laser linewidth. In order to further verify the impact of linewidth, this paper conducts more specific simulations on two schemes for 20 km and 30 km downlink transmission scenarios when the linewidths are 0 and 20 MHz.

As shown in Figure 15, in the SVLE scheme, when the linewidth of the simulation system is 20 MHz and the transmission distance is 20 km, the BER is always higher than that of the system with the linewidth of 0 Hz and the transmission distance of 30 km. It can be seen that for the SVLE scheme, the effect of increasing the linewidth from 0 to 20 MHz is greater than the effect of increasing the fiber length from 20 to 30 km. In the KK scheme, the situation is the opposite. Therefore, the SVLE scheme is indeed more sensitive to changes in linewidth.

3.3.5. Comparison of Algorithmic Complexity Based on DSP

In

Table 1. The general formulas and specific results of DSP complexity for each of the two schemes.

		Scenario	Real Number Multiplication		Real Number Addition
			General Formula	Calculation Result	General Formula	Calculation Result
SVLE		20 km	$N_1+{2N}_2$	37	$N_1+N_2-1$	25
		30 km		53		37
KK		20 km	$\left(3N_s+\frac{N_h}{2}+2\right)R+4N{\mathit{log}}_{2}N+5N+2$	4358	$\left(3\left(N_s-1\right)+\frac{N_h}{2}\right)R+6Nlo{g}_{2}N+3N$	6202
		30 km		9926		13,498

, N represents the FFT points for the dispersion equalization algorithm based on OFDE, with value of 128 at 20 km and 256 at 30 km. $N_1$ represents the first-order memory length for the SVLE algorithm, $N_2$ represents the second-order memory length, with values of 15 for $N_1$ and 11 for $N_2$ at 20 km, and 23 for $N_1$ and 15 for $N_2$ at 30 km. $N_s$ and $N_h$ represent the tap numbers of the FIR filters, with typical values set at 64. R stands for the oversampling ratio, with a typical value of 2.

To sum up, the DSP complexity of the KK scheme is much greater than the SVLE scheme. Therefore, KK systems require more powerful processors, larger memory capacity, higher power consumption requirements, and higher software development costs.

3.3.6. System Cost

In the SVLE scheme, the ADC/DAC sampling rate is 200 GSample/s. However, in the KK scheme, the ADC/DAC sampling rate is 400 GSample/s. It is evident that the implementation of high sampling rate ADCs and DACs will result in a notable increase in system design complexity, as well as in the associated analog front-end requirements, digital processing requirements, data transmission bandwidth, storage costs, and power consumption. Consequently, the overall cost of the system will be significantly elevated. The cost of the KK scheme will be considerably higher than that of the SVLE scheme.

4. Conclusions

In this paper, an SVLE scheme based on the Nyquist PAM4-SSBM-DD scheme is designed. In order to verify its advantages, this paper makes a comprehensive comparison between the SVLE scheme and the KK scheme in terms of system performance, cost, and DSP complexity. In the 20 km downlink transmission scenario, the maximum power budget of the SVLE scheme is 29.9 dB, while that of the KK scheme is 25.0 dB, so the SVLE scheme is 4.9 dB better. The total number of additions and multiplications of the SVLE scheme is 25 and 37, respectively, while that of the KK scheme is 6202 and 4358, respectively. In the 30 km downlink transmission scenario, the maximum power budget of the SVLE scheme is 26.8 dB, while that of the KK scheme is 23.1 dB. Therefore, the SVLE scheme is superior by 3.7 dB. The total number of additions and multiplications of the SVLE scheme are 37 and 53, respectively, while for the KK scheme, they are 13,498 and 9926, respectively. In the SVLE scheme, the ADC/DAC sampling rate is 200 GSample/s. However, the ADC/DAC sampling rate is 400 GSample/s in the KK scheme. In the downlink transmission scenarios of PON, where cost sensitivity on the ONU side is crucial, the lower cost of the SVLE scheme makes it more conducive to large-scale deployment by network operators. Therefore, the SVLE scheme with a high power budget, low complexity, and low costs is expected to become the mainstream scheme for downlink transmission in future 200 Gb/s/λ PON systems.