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Abstract
The carrier contribution factor (CCF) is used in the DC-value method to assure the minimum phase condition (MPC) at the transmitter. We analyze the impact of the CCF estimation accuracy at the receiver on the DC-value system performance. We found that the CCF estimation accuracy at the receiver should be within ±5% to exploit all advantages of the DC-value method. We propose an accurate method to estimate the CCF at the receiver that works effectively with both direct current (DC) and alternating current (AC) coupled photodetectors. We experimentally validate the proposed method employing a 24 GBaud 16QAM signal. The experimental results show that the minimum BER value can be approached within ±5% offset of the estimated CCF value almost independently of the carrier to signal power ratio (CSPR) value. The proposed method can also be used to reconstruct the DC component in other minimum phase signal self-coherent techniques, such as the Kramers-Kronig receiver.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, several techniques have been proposed to extract amplitude and phase information from the optical field using direct detection (DD) schemes [1–3]. Among them, the DC-Value method offers upsampling free amplitude and phase recovery and works with low carrier to signal power ratio (CSPR) [3–5]. Nevertheless, the DC-Value performance is highly dependent on the receiver estimation of the carrier power that reaches the photodetector. Therefore, an accurate carrier contribution factor (CCF) estimation method is mandatory in order to exploit all advantages of the DC-Value method.
Single sideband (SSB) methods have been explored to retrieve full complex electric field of the optical signal using direct detection (DD) [1,6]. However, the performance of SSB transmission methods tend to be limited by the inherent signal-to-signal beating noise (SSBN) generated upon square-law detection [6–8]. To address this problem, more recently, a Kramers-Kronig (KK) receiver has been proposed that significantly alleviates the impact of SSBN and improves the system performance [1,9,10]. However, the nonlinear operations in the KK method demands higher sampling rates to accommodate spectral broadening [11–13]. To address the high sampling rate problem, an upsampling free KK method was proposed that uses a Taylor expansion approximation of the associated nonlinear operations [2]. Nevertheless, the proposed approximations tend to require a higher CSPR, which results in a sensitivity penalty [2]. To address the aforementioned high sampling rate and high CSPR requirements of the conventional KK and upsampling free KK methods, respectively, a DC-Value method was recently proposed [3]. The DC-Value method provides an upsampling free phase reconstruction process at low CSPR [3]. The underlying idea of the DC-Value method is to iteratively impose the minimum phase condition (MPC) [3]. The signal reconstruction accuracy of the DC-Value method depends on the accurate estimation of CCF used in the MPC. The estimation of the CCF is simpler in the case of direct current coupled (DC-coupled) photodetectors. However, the DC component at the output of the photodetector causes unwanted bias offset to the subsequent circuits [14]. Also, the DC component degrades the signal quality when automatic gain control (AGC) is employed at the receiver [15,16]. The alternating current coupled (AC-coupled) photodetector alleviates the aforementioned problems and also allows to use full scale of the analog to digital convertor (ADC) that translates into improved signal to quantization noise ratio (SQNR). Therefore, the AC-coupled photodetector tends to become the preferred choice in the high-speed optical communication systems [16]. However, the AC-coupled photodetector makes the CCF estimation more challenging. Recently, a zero padded preamble based DC component recovery technique was proposed, which allows to recover CCF in AC-coupled photodetectors [16,17]. Nevertheless, it requires a preamble to estimate the CCF value [16,17]. Another alternative is to guess a constant bias and add it to the received signal. The guessed value can then be checked with the known CSPR value and be improved by successive iterations [18]. It means, however, that this method requires multiple iterations to estimate the DC component [18].
Here, we propose an accurate CCF estimation technique that does not require any iterative process and it is compatible with both DC and AC-coupled photodetectors. We also present an experimental validation of the proposed technique. We show that by applying this method the CCF estimation error falls below 1% for CSPR as low as 9 dB. Also, the experimental results confirm that the optimum BER value can be found within $\pm 5 \%$ offset of the estimated CCF value.
Besides the introduction, this paper comprises four more sections. In Section 2, we analyze the impact of CCF in the DC-Value method. In Section 3, we present a novel procedure to estimate the CCF. In Sections 4, we discuss experimental results obtained by the proposed method. In the last section, we present the major concluding remarks.
2. Impact of the carrier contribution factor
Consider the complex envelope of the incoming MPC optical signal as [3],
where, $E_{o} \; e^{(i2\pi f_{o}t+\phi )}$ represents a carrier tone with an amplitude $E_{o}$, frequency $f_{o}$, and phase $\phi$ that coincides with the right edge of the information spectrum, and $E_{s}(t)$ is the complex information signal. When the field is detected using a DC-coupled photodetector, the following photocurrent will be generated,2.1 Impact of the CCF
The normalized mean square error (NMSE) is an important parameter to characterize signal reconstruction accuracy in the DC-Value method. It is mathematically shown in [3] that the NMSE function between the known magnitude, $|E(t)|$, and the estimated magnitude, $|E_{n}(t)|$ (see Fig. 1),
Fig. 1. Schematic diagram of the NMSE calculation between available MPS magnitude, $|E(t)|$, and the DC-Value method estimated MPS magnitude, $|E_{n}(t)|$.
Next, we quantify the impact of CCF estimation error $\Delta E_{o}$ in the signal reconstruction process using the DC-Value method. For the purpose of assessment of the CCF impact, let’s consider a 24 GBaud 16QAM DC-Value system with a CSPR of 12 dB. The electric field is detected using a single photodetector and full signal reconstructed using a DC-Value method algorithm. Results shown in Fig. 2 presents the NMSE between the known MPS magnitude, $|E(t)|$, and the DC-Value method estimated MPS magnitude, $|E_{n}(t)|$, as a function of $\Delta E_{o}$ for 20 iterations. In the signal reconstruction process, we varied the CCF error, $\Delta E_{o}$, from −25 to +25% and analyze its impact on the NMSE.
Fig. 2. Impact of CCF estimation error $\Delta E_{o}$ in signal reconstruction using the DC-Value method. Here, CSPR of 12 dB was considered in the numerical analysis.
The analysis shows that for $\Delta E_{o} = 0 \%$, the NMSE converges to zero, as it was expected. We can also see that if $\Delta E_{o}$ increases then the NMSE becomes nearly independent of the number of iterations, preventing a proper convergence of the DC-Value method, thus resulting in a limited reconstruction accuracy. Results shown in Fig. 3 confirm that the NMSE is monotonically decreasing for the $\Delta E_{o} = 0\%$. For an estimation error greater than $\pm 2.5 \%$, a 17 dB NMSE penalty is observed in the numerical analysis for 5 iterations. Results presented in Fig. 4 show that for an SNR of 30 dB, a BER threshold of $10^{-3}$ can be assured after 10 iterations if CCF estimation error $\Delta E_{o}$ falls within approximately $\pm 5 \%$ range. Further, we have performed extensive simulation for other similar systems and the obtained results are analogous, i.e. the CCF must be estimated within a $\pm 5\%$ error to fully exploit the advantages of the DC-Value method. In the subsequent section, we show an accurate method to estimate CCF for the DC-Value method compatible with both DC and AC-coupled photodetectors.
Fig. 3. Impact of CCF estimation error $\Delta E_{o}$ on the NMSE as a function of iteration number.
Fig. 4. Impact of CCF estimation error $\Delta E_{o}$ on the BER as a function of SNR. For higher negative value of $\Delta E_{o}$ (usually less than −5%), the CCF value reduces significantly and the signal no longer follows the minimum phase condition in the phase reconstruction process which results in highly degraded system performance. Therefore, we have considered only positive $\Delta E_{o}$ values in the analysis to assess the performance.
3. Carrier contribution factor estimation method
In a DC-coupled photodetector, the photocurrent generated upon direct-detection can be written from Eq. (2) as (considering responsivity $R=1$),
The DC-coupled photocurrent contains nonzero DC component contributed from the carrier-carrier beating, $|E_{o}|^{2}$, and the signal-signal beating, $|E_{s}(t)|^{2}$. The carrier-signal beating does not contain any non-zero DC component as it has zero mean value. Thus, the DC component in Eq. (6) can be written as,
therefore, where $P_{c}=|E_{o}|^{2}$, $P_{s}(t)=|E_{s}(t)|^{2}$ and $\langle \cdot \rangle$ indicates an average operator. To make the notations simpler in the CCF estimation process, we can re-write Eq. (6) as,Notice that the photodetector sensitivity is assumed as $R=1$ for simplicity. In practice, the value of responsivity may differ from 1, nevertheless, the proposed CCF estimation method is insensitive to responsivity value.
In the case of AC-coupled photodetector, the DC component given by Eq. (8) gets filtered out by the DC block. Therefore, for the AC-coupled photodetector the current is given by,
From Eq. (8) and (9), we can write $I_{AC}(t)$ as,
In case of sufficiently large CSPR (usually higher than 8 dB), we can assume that (refer to Fig. 5 for the approximation error),
Now considering $\langle P_{s}(t) \rangle = P_{c}/\textrm {CSPR}$ in the Eq. (12),
Therefore,
In a similar way, considering $P_{c} = \textrm {CSPR} \; \langle P_{s}(t) \rangle$ in Eq. (12), we obtain,
Therefore, from Eq. (11), (14) and (15), the photocurrent with its recovered DC component can be written as,
The CCF estimation error in percentage can be written as,
where $\hat {E}_{o}$ represents the estimated CCF from Eq. (14), and it can be written as,Finally, the Eq. (18) can be used to formulate the MPC in the DC-Value method.
The results in Fig. 6 show the numerical analysis of CCF estimation accuracy of the purposed method as function of the CSPR. The analysis shows that the CSPR is not a relevant factor for the CCF estimation as the $\Delta E_{o}$ lies below 3.5% for the given CSPR range. The results in the Fig. 6 shows that less than 1% error can be ensured for the CSPR value as low as 8.5 dB for the given QPSK, 16QAM, and 64QAM modulation formats. The algorithmic flow shown in Fig. 7 presents the proposed method of DC component recovery and estimation of CCF. In practice, the known value of CSPR, and the photodetected current signal are supplied to the CCF estimation algorithm. First, the signal is passed through the DC remover and followed by a DC component recovery estimation (as per Eq. (16)) to calculate the photocurrent with its recovered DC component. Similarly, by using the known value of CSPR and output of the DC remover, the estimation of CCF can be carried out (as per Eq. (18)) before employing the DC-Value method.
Fig. 6. Numerical analysis of CCF estimation error $\Delta E_{o}$ in % using Eq. (17) with respect to CSPR for different modulation formats. The analysis shows that the CSPR is not a relevant factor for the CCF estimation as the $\Delta E_{o}$ lies below 3.5% for the given CSPR range.
Fig. 7. The algorithmic flow of DC component recovery and estimation of CCF compatible with both DC and AC-coupled photodetectors, i.e. $I_{AC}(t)$ or $I_{DC}(t)$ as an input.
4. Experimental results
In this section, we experimentally validate the proposed CCF estimation method and its impact on the overall system performance. We start this section by describing the used experimental setup in detail.
4.1 DC-value transceiver experimental setup
4.1.1 Tx-DSP
At the transmitter side, see Fig. 8(a), a single-channel 30 Gbaud QPSK and 24 Gbaud 16QAM signals are generated using a root-raised cosine (RRC) pulse shaping filter with 0.1 roll-off factor, respectively. Following that, a complex tone $E_{o}e^{2 \pi f_{o}t}$, at frequency $f_{o}=16.5 \;\textrm{GHz}$ (considering 0.1 RRC roll-off factor, $\textrm{1}.1 \times \; 30\; \textrm{Gbaud}/2~=~16.5 \; \textrm{GHz}$), is added to the 30 Gbaud QPSK signal, $E_{s}(t)$, in the digital domain (similarly, complex tone at $f_{o}=13.2 \;\textrm{GHz}$ is added to the 24 Gbaud 16QAM signal). The digitally added complex tone coincides with the right edge of the information signal spectrum to generate an SSB information signal. The amplitude $E_{o}$ of the complex tone should be kept sufficiently high (usually higher than 9 to 10 dB) to satisfy the MPC condition upon photodetection. The desired CSPR is obtained by varying the amplitude $E_{o}$ of the complex tone.
Fig. 8. Experimental setup of a self-coherent transceiver employing DC-Value method. Here, virtual carrier added in the digital domain to reduce optical complexity. To reduce cost, only one optical amplifier employed (as a preamplifier) in the transmission link.
4.1.2 Setup
After Tx-DSP, in our case, a Keysight M8194A arbitrary waveform generator (AWG) containing 120 GSa/s digital to analog converter (DAC) is used to generate the signal. It should be noted that no pre-equalization was employed at the transmitter end. The output of the DAC is modulated at $1550 \; \textrm{nm}$ using a single polarization IQ Mach-Zehnder modulator (IQ-MZM). The output of the IQ-MZM signal is directly launched to the standard single-mode fiber (SSMF). It should be noticed that the digitally added carrier tone could make the IQ-MZM bias control circuitry unstable due to its very high amplitude and make it difficult to lock IQ-MZM bias conditions. To circumvent this problem, first the bias conditions are locked using only the information signal without digitally added carrier tone and then after locking bias conditions, we can digitally add carrier tone to generate the SSB signal. Following SSMF, the optical signal is amplified by an Erbium-doped fiber amplifier (EDFA) and detected using a low-cost $40 \; \textrm{GHz}$ single photodetector without trans-impedance amplifier (TIA). The photodetector output is then sampled by a $100 \; \textrm{GSa}/s$ Tektronix real-time oscilloscope (RTO) with $33 \; \textrm{GHz}$ bandwidth. Finally, the experimental data captured by the RTO is processed offline using Matlab software.
4.1.3 Rx-DSP
In the RX-DSP, first, pre-equalization is carried to compensate the non-flat response of the photodetector. Following that, the proposed CCF estimation technique is applied to estimate the CCF and recover the DC component of the photodetected signal. Next, the signal is passed to the DC-Value method to recover the missing phase information. Notice that the pre-equalization at the receiver end includes the inversion of $S_{21}$ response of the photodetector. Moreover, the transmitter response is compensated by an adaptive equalizer after the signal reconstruction process. Also, it should be noted that no digital upsampling or downsampling is required in the DC-Value method.
In the DC-Value method, first, a square-root operation is carried out to obtain the magnitude $|E(t)|$ of the optical field and multiplied by a phase correction factor $e^{i\delta \theta _{n-1}(t)}$ which outputs the complex signal $E'_{n}(t)$. In the first iteration ($n=1$), the phase correction vector is assumed to be zero, i.e. $\delta \theta _0(t) = 0$. Next, the MPC is imposed (see Fig. 8(c) on the Fourier transformed signal $\tilde{E'}_{n}(\omega )$ to attain $\tilde{E}_n(\omega )$ as discussed in Eq. (3). The scaling factor $p$ used in the MPC greatly speeds up the convergence process (see Fig. 8(d)). It is worth noticing that the scaling factor $p$ improves the convergence speed because the input signal, $E'_n(t)$, for the first iteration consists of a real-valued amplitude signal $|E(t)|$, i.e. $\delta \theta _0(t) = 0$ [3]. The minimum phase condition imposed on the signal forces the positive frequency components to zero and it generates complex-valued signal $E_1(t)$ in the time domain (starts acquiring the phase information). As a consequence of forcing positive frequency components to zero, the amplitude of the signal $E_1(t)$ will have its real and imaginary parts scaled by a factor 0.5. Therefore, we set a scaling factor $p=2$ to adjust the amplitude of the information signal in the first iteration, which speeds up the convergence of the method (see Fig. 8(d)) [3]. It shows that the implementation of the scaling factor $p$ requires two less iterations to achieve the same accuracy without scaling factor. After imposing MPC, the IFFT of the $\tilde{E}_n(\omega )$ is computed to obtain the first estimate of the minimum phase signal $E_{1}(t)$. The phase correction vector corresponds to $E_{1}(t)$ and can be calculated as $e^{i \delta \theta _{1}(t)} = \frac {E_{1}(t)}{|E_{1}(t)|}$. This phase estimated is then supplied as an updated phase correction vector for the subsequent iteration ($n=2$). This process continues until the desired accuracy is reached. Also, the normalized mean squared error (NMSE) between the known magnitude $|E(t)|$ and the estimated magnitude $|E_{n}(t)|$ is monotonically decreasing after each iteration (see Fig. 8(d)), therefore, the reconstruction process converges as discussed earlier in Section 2. Afterwards, the reconstructed minimum phase signal $E_{n}(t)$ is passed through the DC remover and then upconverted by multiplying it with $e^{i2\pi f_{o}t}$. Finally, the chromatic dispersion compensation and equalization can be carried out before symbol recovery.
4.2 Performance analysis
In this subsection, we present the experimental analysis of the impact of the CCF (estimated by proposed method, and referred as $\hat {E}_{o}$) on the overall system BER performance in the DC-Value method. To analyze the impact of $\hat {E}_{o}$, we first calculate the $\hat {E}_{o}$ value as discussed previously in the Section 3., and add an offset error $\delta$ in the estimated $\hat {E}_{o}$ ranging from −25 to +25% with the step size of 0.55 % to assess its impact on the overall system performance. The $\hat {E}_{o,\textrm {offset}}$ can be given as,
Results shown in Fig. 9 present the impact of $\delta$ on the overall BER performance for the different CSPR values for 24 GBaud 16QAM signal after 40 km SSMF. The dotted red line in Fig. 9 represents the estimated $\hat {E}_{o}$ value by the proposed CCF estimation method (with no added offset, i.e. $\delta =0 \%$). The surface between two dotted yellow lines represent the range between $\pm 5 \%$ offset. For CSPR values ranging from 7 to 15 dB, we change the offset value $\delta$ as discussed above and calculate the BER performance of the system. For each CSPR value, we find the lowest achievable BER in the given range of $\delta$ and it is represented as blue dots in the graph. Also, small green dots shown in the results represent valid BER values considering the soft decision forward error correction (SD-FEC) threshold of $2.4 \times 10^{-2}$. It shows that the value of BER becomes valid for the CSPR $\geq 9$ dB considering the same SD-FEC threshold. The major highlight of Fig. 9 is that the lowest BER lies between $\pm 5 \%$ of the estimated $\hat {E}_{o}$ value which shows the higher CCF estimation accuracy of the proposed method. Nevertheless, Fig. 9 shows that the estimation error $\delta$ increases for CSPR higher than $\sim$ 12 dB. This peculiar behaviour comes from the Mach-Zehnder modulator as it starts operating in a nonlinear regime, which effectively changes the CSPR at the transmitter end. The proposed method requires the knowledge of the CSPR at the receiver end to recover the DC component. Additionally, it should be noted that the actual CSPR may change due to the impairments caused by transmission and optoelectronic components. Therefore, starting from the known value of CSPR at the transmitter, the DC-Value method can be used as a tool to improve the CSPR estimation at the receiver by proper adjustments of the offset $\delta$ such that the NMSE becomes a monotonically decreasing function as we discussed earlier in Fig. 3.
Fig. 9. Performance analysis in terms of BER for the 24 GBaud 16QAM signal after 40 km SSMF as function of CSPR and estimated $\hat {E}_{o}$ offset $\delta$.
Results shown in Fig. 10(a) presents the $\log_{10}(BER)$ as a function of $\delta$ for the 24 GBaud 16 QAM signal received after 40 km SSMF. The shaded gray area in Fig. 10(a) represents the $\pm 5 \%$ range of $\delta$. It is shown that the optimum BER value can be found in the close proximity of the estimated $\hat {E}_{o}$ value by the proposed CCF estimation method. Additionally, $\sim$ 0.2 dB penalty in terms of Q factor is observed for the $\delta \geq$ 10 % for CSPR of 11 dB. Similarly, the analysis of the received signal after 70 km SSMF shown in Fig. 10(b) represents the identical result like Fig. 10(a) such that the optimum BER values for the given CSPR lie within $\pm 5\%$ range of $\delta$. Another important observation is that after 13 dB CSPR value, an optimum value of $\log_{10}(BER)$ is found a little bit far from the $\hat {E}_{o}$ value. The possible reason behind this could be that the digitally added carrier tone makes the IQ-MZM bias controller unstable when a very high power tone (usually higher than 13 dB observed in our experiment) is inserted in the signal. In this situation, the actual CSPR value of the optical signal differs from the virtual carrier added digital signal. Also, notice that slightly higher CSPR can help reduce the phase jump [20].
Fig. 10. Impact of $\hat {E}_{o}$ estimation error on the system performance after (a) 40 km, and (b) 70 km. Here, virtual carrier assisted 24 GBaud 16 QAM signal used for the performance assessment. Results shows that the lowest BER can be assured in the DC-Value method when the $\hat {E}_{o}$ estimation falls below $\pm 5\%$.
5. Conclusion
The CCF is an important parameter in the DC-Value method and it used to form the MPC for the DC-Value method algorithm. Using the knowledge of CSPR value, we proposed a novel CCF estimation method compatible with both DC and AC-coupled photodetector for the DC-Value self-coherent transceiver. We present the analysis and impact of the CCF in the DC-Value method and show that the CCF estimation error should be no more than $\pm 5\%$ to ensure better system performance. The numerical analysis shows that the CCF estimation is quite insensitive to the CSPR value. In practice, results shows that the optimum CSPR value usually lies around 12 to 13 dB and we can ensure less than 1% CCF estimation error by the proposed method. Also, we performed experimental analysis of 24 GBaud 16QAM signal transmission and employed proposed method to estimate the CCF.
Funding
Fundação para a Ciência e a Tecnologia (DSPMetroNet (POCI-01-0145-FEDER-029405), ORCIP (CENTRO-01- 0145-FEDER-022141), UIDB/50008/2020-UIDP/50008/2020 (action DigCORE)).
Acknowledgments
This work was supported in part by Fundação para a Ciência e a Tecnologia (FCT) through national funds, by the European Regional Development Fund (FEDER), through the Competitiveness and Internationalization Operational Programme (COMPETE 2020) of the Portugal 2020 framework, under the projects DSPMetroNet (POCI-01-0145-FEDER-029405), UIDB/50008/2020-UIDP/50008/2020 (action DigCORE), and ORCIP (CENTRO-01-0145-FEDER-022141). Fernando P. Guiomar acknowledges a fellowship from "la Caixa" Foundation (ID 100010434). The fellowship code is LCF/BQ/PR20/11770015. Marco Fernandes acknowledges a PhD fellowship from FCT. The fellowship code is 2020.07521.BD.
Disclosures
The authors declare no conflicts 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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