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Abstract
Kramers-Kronig optical single-sideband receivers remove the signal-signal beat interference (SSBI) that occurs when detecting a signal that has electrical signals mapped onto its optical field at the transmitter; such signals support electronic dispersion compensation without the need for a coherent receiver. To use the full range of the analog-to-digital converter’s (ADC) range, it is best to a.c.-couple the photocurrent, to remove its DC content; however, the DC must be restored digitally before the KK algorithm is applied. Recent publications have concentrated on perfectly determining the restored DC’s required level from the signal, with a view this is optimal for lowering error rates. In this paper, we investigate signal-signal beat interference (SSBI) cancellation in a single photodiode receiver using Kramers-Kronig receiver algorithm, with large variations in optical carrier-to-signal power ratio (CSPR) and DC offset level. Through simulations and experiments, we find a strategy to optimize the signal quality without the need of an extensive search for the DC offset value. We also find that a theoretically perfect determination of the original DC level does not provide best signal quality especially for low CSPRs; in order to achieve maximum cancellation of signal-signal beat interference, the level of the restored DC has an optimum value that depends on the optical CSPR. We define a digital CSPR, which is the value of the CSPR in the digital domain after DC restoration. Our measurements show that we simply need to bias the signal upwards and make the minimum signal above zero by 0.1% of the r.m.s. signal amplitude when the optical CSPR is low. For higher values of optical CSPR, the optimal digital CSPR is about 2-dB lower than the optical CSPR, and the optimal DC offset can be calculated from this digital CSPR. We find that the boundary between our low optical CSPR region and high optical CSPR region depends on the noise level in the system.
1. Introduction
Single-photodiode receivers have become a recent focus of research interest due to the rising demands for data in metropolitan-area and short-reach networks, which require compact low-power technologies. High-capacity systems using single carrier PAM4 (e.g. [1]), multi-carrier direct-detection OFDM and DMT [2], carrier-less single sideband techniques such as CAP (e.g. [3]), and Nyquist sub-carrier (e.g. [4]) have been demonstrated. Single-sideband systems with added optical carriers [5–7] have gained much attention recently. A single-sideband optical signal, where the electrical signal is mapped onto the optical field of the sideband, supports dispersion compensation at the receiver using digital signal processing (DSP) [8,9]. However, photodetection creates unwanted signal-signal beat interference (SSBI) in the same frequency range as the detected signal. The reduction of SSBI is necessary, unless a frequency gap is added between signal and carrier [10] to displace the wanted signal band from the lower-frequency SSBI. SSBI reduction techniques include: using high optical power carriers [11], SSBI estimation and cancellation DSP [7], and the Kramers-Kronig method [12], which was originally developed for radio transmission [13].
The Kramers-Kronig method provides effective SSBI cancellation for carrier-to-signal power ratios of 6-13 dB, and has a reasonable algorithmic complexity. Recent KK demonstrations fall into three categories: realization of KK algorithms for different signal formats, such as orthogonal frequency division multiplexing (OFDM), to achieve higher data rates and longer propagation distances [8,9,14]; modification of the KK algorithm to simplify the computation or gain better performance [15,16]; modification of transmitter and receiver hardware to reduce cost or improve signal quality [17,18]. We have also studied the beneficial effects of clipping the negative excursions of the signals [19], the effect of sampling on the minimum-phase boundary [20], and approximating the nonlinear functions in the KK processing, leading to an analog implementation of the KK algorithm [21]. Our work shows that moving away from a theoretically perfect implementation of a KK receiver (which requires continuous signals or very high oversampling) can have benefits.
Here we show that to achieve optimal SSBI reduction in a KK system, the amount of DC ‘added back’ to the signal before the KK processing is different to the DC-component of the received signal, and that this difference depends on the optical carrier-to-signal power ratio (CSPR). We provide two methods for determining near optimal amounts of DC offset: Method 1 uses the CSPR set at the transmitter and calculates or looks-up an equivalent digitalCSPR, which is multiplied by the r.m.s. value of the a.c. photocurrent; Method 2, adds a bias to the received waveform so that its minimum value is zero, then adds a further bias of 0.1% the r.m.s. value of the a.c. photocurrent.
These methods work best over different ranges of CSPRs, so an optimal solution is to calculate the signal quality using both methods then choose between them. Both methods avoid the need to sweep the bias over a range of values in order to determine the optimal value, and thus save set-up time for the system. We show by simulations and experiments that these methods give signal qualities almost identical to a fine-stepped sweep of bias level.
2. Theory
Figure 1 shows the system schematic, which is very similar to a typical KK system [12]. The transmitter modulates signal onto an optical single side band. A single a.c.-coupled photodiode mixes the sideband and carrier upon photodetection to produce the wanted signal, and unwanted signal-signal beat interference (SSBI). The photocurrent is digitized using an analog-to-digital converter (ADC); the sampling rates will be discussed in the relevant sections. The DC content of the signal is then restored by adding a DC-offset, which is the focus of this paper, to minimize negative signal samples which are removed by zero-level clipping. The signal is then processed using a conventional Kramers-Kronig algorithms where the phase of the signal is calculated in the correction path, then applied to the square-root of the intensity that passes through the main path. Fiber dispersion was not considered in this work; neither was nonlinearity because the application is for short-haul links only.
Similar to the optical CSPR that is the ratio of carrier to signal powers in the optical domain, we define a DigitalCSPR as the ratio of the DC over AC electrical powers after the ADC. Using photodetection theory, which excludes the photocurrent terms at very high frequencies, we can derive the following relationship:
In Method 2, a block of signal samples is processed in the following manner. The signal sample with the lowest photocurrent is found, which will be negative as the photocurrent is a.c. coupled. Then this value is subtracted from all samples so that the lowest sample is zero. Because the KK algorithm’s logarithm produces large negative-going peaks when the signal is close to zero, a small extra bias is then added. This extra bias equals $\rho $ (additional offset factor) multiplied by ${P_{elec\_AC}}$. In Section 3.2, we find that the optimal $\rho $ is 0.1%, and is similar to that used in our papers that clip the negative-going peaks of the waveform to a given minimum value [19]. Method 2 can be written:
In Method 3, Method 1 and Method 2 DC offsets were both processed then compared to choose the best. Two times KK processing and standard DSPs needed for this method.
3. Simulations
Here we investigate the optimal value of digital CSPR using simulations of signal quality using Method 1, and show that the optimal digital CSPR is 2-dB to 3-dB lower than optical CSPR for large optical CSPRs which is consistent with the result given by Eq. (1). We also investigate Method 2, and show that the optimal additional bias is close to 0.1% of the r.m.s. value of the a.c. photocurrent.
MATLAB was used to simulate the transmitter and Kramers-Kronig (KK) receiver. ${2^{19}}$ pseudo-random binary numbers were first generated and shaped to a 10-GBaud, 16-QAM Nyquist signal with a 10% roll-off factor. White Gaussian noise was added in Nyquist-shaped signals in the optical domain, to represent the amplified spontaneous emission due to optical amplification and set the optical signal-to noise ratio (OSNR). After 6-times digital up-sampling to 60 GSa/s, a DC offset was added back to the signal. Then standard KK schemes were applied (see Fig. 1). The receiver DSP processing includes: 10% roll-off RRC filtering, frequency offset compensation using spectral peak searching, equalization using a two-step 41-tap equalizer based on the constant-modulus algorithm with tap-weights initialized by a training-aided least-mean-squared equalizer, and phase compensation using maximum-likelihood estimation. After standard DSP, the signal quality was calculated based on signal noise ratio (SNR).
3.1 Joint optimization of digital CSPR and optical CSPR
Figure 3 plots the signal quality vs. optical CSPR for various digital CSPRs. The OSNR was 30 dB (unpolarized noise into 0.1 nm). The signal quality changes dramatically with the amount of DC added back. At optical CSPRs less than 6 dB, the signal quality increases when the digital CSPR is decreased. This would suggest that the lower extremes of the signal now become clipped, as only positive-values can be processed by the logarithm function of the KK algorithm; this will be discussed in Section 3.2. When optical CSPR is larger than 8 dB, the signal quality is able to reach its peak value (about 24 dB), which is limited by the OSNR. There are two main causes of this decreasing signal quality for optical CSPRs less than 8 dB. Firstly, the minimum phase condition will be violated more frequently for low optical CSPRs. Secondly, even though the minimum phase condition is satisfied at the transmitter, the received signal can easily approach zero due to the noise in the system, causing large short negative peaks after the logarithm in the KK algorithm. This causes large inter-modulation distortion (IMD) when digital sampling rate is not sufficient [20]; according to our simulations, this the main cause of decreased signal quality for 6-8 dB optical CSPRs. Both causes can be solved through providing sufficient large DC offset, but this also causes signal distortion due to the square-root function being applied to a signal with the wrong DC level, which reduces signal quality at larger digital CSPRs.
Figure 4 plots the relationship between the optimal digital CSPR and the optical CSPR; the optimal digital CSPR is 2-3 dB below the optical CSPR for optical CSPR’s higher than 8 dB. For lower optical CSPRs the two values converge, but these low optical CSPR levels would not support low error rates. The relationship between the optimal digital CSPR and optical CSPR is not linear for low CSPRs (in dB scales), because increasing the digital CSPR avoids strong negative peaks after the logarithm, when the signal approaches zero. For higher CSPRs (larger than 8 dB) the difference would be 2 to 3 dB as expected according to Eq. (1). However, for extremely large optical CSPRs, a few dB variation in digital CSPR does not affect the signal quality significantly; thus −2 dB is a reasonable choice. The blue line shows the theoretical relationship between optimal digital CSPR and the optical CSPR, from Eq. (1). The simulated optimal digitalCSPR agrees well with the theoretical value, noting that the simulated value was derived from the discrete values of digitalCSPR in Fig. 3.
Fig. 4. Simulated results showing the optimal digital CSPR versus optical CSPR (red line: simulated; blue line: theoretical, Eq. (1)).
3.2 DC offset for low optical CSPRs
For optical CSPRs less than 8 dB, the signal amplitude will often approach zero, causing, large negative spikes to occur at the output of the logarithm [19]. These large spikes will cause inter-modulation distortion (IMD) in the signal spectrum [20]; so, cause large amplitude/phase errors. Figure 5 plots parts of the waveforms (250 samples) through the processing chain when the optical CSPR equals 6 dB, the digital CSPR equals 4 dB (left side); and the optical CSPR equals 6 dB and digital CSPR equals to 6 dB (right side). As shown on the left, there is an extremely large signal spike near sample 120 after the square root and log. When the signal is moved upwards (right side), there are no large spikes. References [19,20] provide a detailed analysis. The signal quality improves by 6 dB by simply adding more DC offset (from 14.5 dB to 20.8 dB). So, optimizing the DC offset level can be an alternate way of optimizing the clipping level in low optical CSPRs [19]. Note that when the digital CSPR = 4 dB, the DC level added back is lower than the original DC level; when the digital CSPR = 6 dB, the DC level added back is higher than the original DC.
Fig. 5. Simulated waveforms (250 samples) throughout the processing chain, with low (left) and higher (right) digital CSPRs.
We now re-bias the photocurrent to optimize the additional offset factor ($\rho $) according to Eq. (3). Figure 6 shows the signal qualities for various positive additional offset factors when the optical CSPR is 6 dB. With $\rho = {10^{ - 3}}$, the signal quality reaches an optimum, improving by about 1.5 dB compared with using the photodetected DC level (not shown in the figure). With extremely small $\rho $ (smaller than ${10^{ - 50}}$), the signal quality drops quickly. With a large $\rho $ (larger than ${10^{ - 2}}$), the signal becomes distorted by the KK processing as it has an incorrect level when entering the square-root and other nonlinear functions. These two extreme regions have low signal qualities as expected. For $\rho $ between ${10^{ - 50}}$ and ${10^{ - 5}}$, large negative-going spikes exist after the logarithm, but scheme still able to provide reasonable signal qualities (show in Fig. 6 left part). With the same data and different noise loading (same noise level), the signal quality varies considerably for $\rho $ between ${10^{ - 50}}$ and ${10^{ - 5}}$. Sometimes the signal quality reduced to less than 14 dB and other times the signal could not be processed by the algorithms in the standard DSP.
3.3 AC and DC coupled Kramers-Kronig receiver comparisons
Sections 3.1 and 3.2, identified a strategy to optimize DC offset: (a) for optical CSPRs <8 dB, bias the signal upwards and make the minimum signal above zero by 10−3 of r.m.s. signal amplitude; (b) for higher optical CSPRs, the we only add a proportion of the optical CSPR (−2 dB) back to the AC signal. Figure 7 compares our strategies (Method 1, Method 2, Method 3 being the best of Method 1 and Method 2) with: adding back original DC level, identifying the optimum by sweeping (1 dB steps) for the optimal digital CSPR, extensively searching for the optimal digital CSPR (0.01 dB steps) and the theoretical method of Eq. (1), for a range of optical CSPRs. Method 3 is able to provide better signal quality for all optical CSPRs when compared with adding back the original DC level (i.e., a DC-coupled photodiode). For optical CSPRs larger than 8 dB, all four methods provide the same signal quality. For lower optical CSPRs, we need to add a greater proportion of the DC back to get the optimal signal quality. The sweep method’s signal quality depends on the coarseness of the sweep step. When the sweep step is small enough (extensive search), it should provide the best signal performance but will require the most computations. In our case, the sweep step was 1 dB of the digital CSPR. The extensive search can provide best signal quality all the time, but it is computationally expensive. The theoretical method (Eq. (1)) is able to provide similar signal quality to Method 3; however, did not provide sufficient DC levels to prevent large negative-going spikes after logarithm for our experimental waveforms.
4. Experimental investigations
4.1 Experimental setups
Figure 8 shows the experimental set-up. Light from an external cavity laser (ECL) is modulated using a 35-GHz bandwidth InP complex Mach-Zehnder modulator (MZM). The modulation is a 10-Gbd 16-QAM signal, shaped by a 10% roll-off digital root-raised cosine (RRC) filter, generated by a 33-GHz bandwidth arbitrary waveform generator (AWG) running at 90 GSa/s. Optical noise loading is performed using two erbium-doped fiber amplifiers (EDFAs) with a wavelength selective switch (WSS) to flatten the noise floor. The OSNR is monitored by and optical spectrum analyzer (OSA). To isolate the impact of optical CSPR from the optical-signal-to-noise ratio (OSNR) degradation associated with a strong transmitted carrier, we introduce the carrier as a local oscillator at the receiver side. A second ECL, boosted by an EDFA, is used as this carrier. The carrier and signal with noise are injected into the photodiode together via a polarization-maintaining 3-dB coupler. Polarization controllers (PC) and a polarization beam splitter (PBS) ensure that the LO and signal with noise are co-polarized. The photodiode is a.c. coupled. The output of the photodiode is digitized by a 160-GSa/s 62.5-GHz bandwidth real-time oscilloscope. The resulting waveforms were then processed off-line using MATLAB. The offline DSP processing is the same with simulation (see Fig. 2).
Fig. 8. Experimental schematic. VOA: variable optical attenuator, PD: photodiode, DSO: digital oscilloscope.
4.2 Experimental results
Figure 9 plots the experimental signal quality ${\textrm{Q}^2}$ vs. optical CSPR for different values of digital CSPR. The OSNR was again 30 dB. The trends are similar to the simulation results of Fig. 3; however, the maximum signal quality is more than 2-dB lower than our simulation results, due to nonideal components, limited bandwidths and limited ADC and DAC conversion resolutions. These effects also broaden the peak of each trace, though these peaks lie at similar optical CSPRs to those in the simulation results.
Figure 10 plots the optimal digital CSPR vs. optical CSPR when OSNR equals to 30 dB. The boundary used to choose between Method 1 and Method 2 is when the optical CSPR = 10 dB. Because of the non-ideal system (thermal noise, non-ideal components etc.), the signal voltage often becomes close to zero. For 8-dB optical CSPR, 6-dB digital CSPR is not enough to prevent large negative-going spikes after the logarithm at these times. The low optical CSPR region becomes broader than simulation. The experimental optimal digitalCSPR is always larger than the theoretical value (Eq. (1)). For optical CSPRs lower than 10 dB, the optimal digitalCSPR is above the theoretical value, which effectively lifts the minimum signal value above zero, preventing large negative-going peaks after the logarithm, that are known to cause sporadic errors [19,20]. For optical CSPRs higher than 10 dB, larger values of DC-offset are still needed due to noise and non-ideal electronic components.
Fig. 10. Experimental results showing optimal digital CSPR versus optical CSPRs (red line). Grey dashed lines indicate set ratios between digital and optical CSPR.
Figure 11 plots the signal qualities comparison of different methods when OSNR is 30 dB. The methods are: Method 3; sweeping digital CSPR method in 1-dB steps; extensively searching method; Method 1; Method 2; theoretical method. Figure 11 experimentally confirms that Method 3 is able to provide a similar signal quality compared with the computationally expensive extensive searching method. Interestingly, Method 3 provides better signal qualities compared with the theoretical method for low optical CSPRs. This is in-line with our previous results [19], which showed that clipping the negative-going peaks before the logarithm improved signal quality: adding excess bias has a similar effect in that both reduce strong negative-going peaks at the output of the logarithm.
To verify Method 3 for typical noise levels in an experimental system, we lowered the OSNR to 20 dB. This produced enough errors to be able to compare performance using bit-error counting, which should be more accurate at low CSPRs where the errors are caused by extreme events, where the signal approaches zero. We calculated an equivalent Q from the BER, where ${Q^2}_{\textrm{BER}} = 20\textrm{lo}{\textrm{g}_{10}}\left( {\sqrt {10} erf{c^{ - 1}}[{8 \cdot \textrm{BER}/3} ]} \right)$. Figure 12 plots this measure for Method 3, the sweeping method, an extensive search, and using the theoretical CSPR (Eq. (1)). The signal quality for Method 3 is only slightly lower than the extensive search method. On the other hand, using the theoretical value optimum digitalCSPR provides the poorest signal quality at lower CSPRs, because the noise drives the signal close to zero, causing bit errors due to large negative peaks after the logarithm and errors even though the Minimum Phase condition may have been met [19,20]. This poor performance of the theoretical digitalCSPR was less evident in Fig. 11, because we assumed Gaussian statistics.
Fig. 12. Experimental signal quality calculated from the bit errors vs. optical CSPR with different strategies (OSNR = 20 dB).
5. Conclusions
Using simulations and experiments, we have shown that the restored DC offset level needs to be optimized in an a.c.-coupled photodiode KK receiver. We have identified a simple and computationally cheap algorithm to predict the optimal DC restoration, which adds only a proportion of optical DC level back before the DSP. This strategy does not need to iteratively optimize the signal quality, nor does the preamble of zeros before data transmission showed in [22]. We divided the optical CSPR into two different regions: low optical CSPRs and higher optical CSPRs. At low optical CSPRs, we remove some DC to move the lowest signal point to zero, then add 0.1% of the mean signal amplitude. For higher optical CSPRs, theoretically and by simulation, the digital CSPR (dB) should be set to 3-dB below the optical CSPR (dB); however, the experimental results showed that 2 dB is a better choice. This may be because this higher-bias moves the signal further away from zero, so reduces the frequency severity of the negative-going peaks (and subsequent phase correction), which can cause errors. An experimental system will have more distortion of the electrical signals, and more sources of noise, which may also cause the signal to approach zero more frequently; thus, a larger bias (or digital CSPR), may beneficially avoid these extreme conditions.
In situations where the receiver may not be able to provide reliable CSPR information, our strategy is to calculate the signal using both methods, and then choose the best. Alternatively, Method 2 could be used on its own if the receiver cannot provide any information about the digital CSPR.
Funding
Australian Research Council (DP190101576).
Acknowledgements
This work was supported by the Australian Research Council’s Discovery Project scheme (DP190101576).
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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