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Abstract
We propose and evaluate a method to estimate the DC bias required for AC-coupled Kramers-Kronig receivers. The proposed method is based on a spectral analysis of the reconstructed signal without requiring an evaluation of the signal quality. The proposed method is described analytically and demonstrated experimentally using 12.5 GBaud 16-ary quadrature-amplitude modulated signals in back-to-back and after 100 km transmission.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The Kramers-Kronig (KK) coherent receiver has been proposed as a means to enable low-cost high spectral efficiency transmission [1]. In particular, high symbol-rate short-reach systems using direct detection can benefit from the advantages of coherent receivers operating at 1550 nm such as digital compensation of group velocity dispersion [2,3]. KK receivers rely on the use of minimum phase (MP) signals, which are characterized by a simple relation between their amplitude and phase based on the Hilbert transform (HT) [1]. This property enables the full reconstruction of the complex envelope of a MP signal simply through the detection of its intensity (or phase). After the signal reconstruction, digital signal processing (DSP) techniques can be used for various purposes, just as in conventional digital coherent receiver, including dispersion compensation [2,3], polarization demultiplexing [4], or even mode demultiplexing [5,6].
The MP condition that is required for the KK processing may be fulfilled using single-sideband (SSB) signals with a sufficiently large carrier-to-signal power ratio (CSPR) [1] and various practical implementations have been experimentally demonstrated in works such as [2–8]. In these implementations, one may generally assume that the signal at the receiver input takes the form of an SSB signal, which is directly detected by a single-ended DC-coupled photodetector (PD), as shown in Fig. 1. The resulting electrical signal is digitized by an analog-to-digital converter (ADC) followed by DSP electronics to recover the optical field amplitude and phase. After the KK optical field recovery, additional DSP modules may then be applied to the recovered signal to improve performance. The sensitivity of the receiver may also be improved through the use of a transimpedance amplifier (TIA) at the PD output. However, as the optical signal takes the form of an SSB signal, the dynamic range of the TIA and ADC must accommodate the entire range of the electrical current at the PD output, which includes the information-carrying signal but also the strong carrier. To avoid this, it is possible to replace the DC-coupled PD with an AC-coupled PD, eliminating the average component of the detected current, as shown in Fig. 1. This substantially reduces the required dynamic range of the TIA and ADC, thereby simplifying their implementation. However, since the KK algorithm relies on the knowledge of the incoming signal intensity, the average signal component must be re-introduced digitally, prior to field recovery. In addition, we note that even with the use of DC-coupled PDs, the receiver electronics may affect the average component of the detected signal, which would also require digital correction. As such, it is crucial for the front-end DSP of KK receivers to estimate the required bias correction to be introduced prior to signal reconstruction. This bias correction will be referred to here as DC bias.
Perhaps the most effective estimate of the DC bias may be performed using quality indicators after the demodulation of the information signal, such as pre-forward error correction bit-error rate or error vector magnitude estimates. These can be used as a metric to drive a search sub-system, which then determines the optimum DC bias. However, the use of post-demodulation quality indicators may be impractical, due to high complexity, latency or instability. This work proposes a novel method to estimate the DC bias, based on the spectral analysis of the reconstructed signal. In particular, the method introduces a DC metric based on the power level of the reconstructed signal within the small guard band which is usually introduced between the CW component and the information-carrying part of the SSB signal. It is experimentally shown that this metric, which is designed to vanish in the absence of noise and distortions, is nearly minimal when the DC-bias is correctly estimated, and may be used to drive an arbitrary DC bias estimation sub-system.
This work is structured as follows. Section 2 describes the theoretical principle behind the use of the SSB guard-band power as a metric for the DC metric. Section 3 describes an experimental demonstration, validating the proposed principle. Final conclusions are outlined in Section 4.
2. Principle
We define the complex envelope of the MP signal at the photodetector input as:
where $P_c$ and $s \!\left (t\right )$ are the power of the signal carrier and the frequency-shifted information-carrying signal, respectively. Figure 2 shows a representation of the signal complex envelope in the frequency domain, which assumes a guard band $f_{\mathrm {GB}}$ and a bandwidth $2B$. Assuming AC-coupled square-law detection followed by the inclusion of a DC bias, $P_{\mathrm {DC}}$, the detected signal is given by:
Fig. 2. Possible implementation of the DC bias estimation (a) and simulated dependence of the DC metric on the DC bias for CSPR values of 6 dB, 10 dB and 16 dB.
where the MP signal to be reconstructed is multiplied by a component resulting from the DC bias error. Taking the first term of the Taylor series expansion of (4) on $\xi$ yields the approximation:
Figure 2 shows a possible implementation of the proposed technique in the time domain. The value of $P_{\mathrm {DC}}$ is estimated by taking the recovered signal and sampling frequency components using two band-pass filters. The powers of the signals at the filters output are then used to compute $\mathrm {DC}_{\mathrm {metric}}$, which can then be used by a search algorithm to estimate $P_{\mathrm {DC}}$. Figure 3(a) shows the dependency of the DC metric on the DC bias error computed by numerical simulation for CSPR values of 6 dB, 10 dB, and 16 dB. The simulated signal was a 16-QAM signal with raised cosine pulse shape and 0.01 roll-off. The guard band was set at 10% of the symbol rate. As shown, the proposed metric varies strongly with $\xi$ and is minimum for $\xi \approx 0$, i. e. when the included DC bias matches the value removed by the AC coupler. Figure 3(b) shows examples of the signal spectrum for normalized DC bias error of 0, 0.25, and 0.5. Note that for a CSPR of 6 dB, the reduction of the guard-band power when the error approaches zero is not as evident as the cases for higher CSPR. This results from the validity limits of the assumed approximations. The following section validates the proposed approach experimentally.
Fig. 3. Simulation of the proposed DC metric computation. a) DC metric dependence on the normalized DC bias error and CSPR. b) Example spectrums of the simulated signal.
3. Experimental demonstration
3.1 Experimental setup
This section presents an experimental validation of the DC metric for optimization of the DC bias on AC-coupled KK receivers proposed in the previous section. Figure 4 shows the experimental setup. An SSB signal was produced by combining the lightwaves from two external cavity lasers (ECLs) centered around 1550 nm. One of the lightwaves was modulated by a single-polarization dual-parallel Mach-Zehnder modulator (DP-MZM) driven by an arbitrary waveform generator (AWG) to produce 12.5 GBaud 16QAM signals with a root-raised cosine pulse shape and a roll-off of 0.01. The AWG had a 3-dB cut-off frequency of 12 GHz and operated at 50 GS/s. Its outputs were electrically delayed to decorrelate the in-phase and quadrature components of the signal. The frequency offset between lasers was set to an average of 13.1 GHz, corresponding to a 600 MHz guard band, and the carrier frequencies wandered within a range of approximately 20 MHz. To ensure that the two lightwaves were co-polarized, their polarizations were adjusted using polarization controllers (PCs) before a 10 dB coupler followed by a polarizer (POL) to reduce non-co-polarized components. The relative powers of the two lightwaves were adjusted using variable optical attenuators (VOAs) in order to set the desired CSPR at the POL output, measured using a power meter. An erbium doped fiber amplifier (EDFA) followed by a VOA were used to adjust the average signal power to 3 dBm prior to the transmission fiber. The latter was a 100.1 km standard single-mode fiber (SSMF) with a total span loss of approximately 19 dB. For back-to-back measurements, the fiber was replaced by an attenuator with similar loss. After transmission, the signal at the fiber output was pre-amplified using an EDFA followed by a VOA. The pre-amplified signal was combined with amplified spontaneous emission (ASE) noise at a 3-dB coupler for noise loading tests. In this work, the OSNR was estimated considering the total power of the signal and carrier. The coupler was followed by a 0.2 nm tunable band-pass filter (TBPF) to limit the noise bandwidth. The filtered signal was detected by a photodetector (PD) with a 36 GHz 3-dB cut-off frequency, which was AC-coupled to a transimpedance amplifier (TIA). The inset A of Fig. 4 shows a spectrum of the signal at the PD input. The TIA output was digitized by a 40 GS/s real-time oscilloscope and the resulting traces were stored for offline processing.
Fig. 4. Experimental setup. The insets A, B and C show the optical spectrum of the SSB signal, the DSP chain and an example constellation of the detected signal in back-to-back, respectively.
The inset B of Fig. 4 shows a diagram of the DSP. It consisted of a resampling stage to 4 samples per symbol followed by a normalization stage. This stage eliminated residual DC components introduced by the TIA or the oscilloscope and scaled the signal to unitary power. Afterwards, a DC bias was applied. Since it was applied after the normalization stage, we will refer to it here as normalized DC bias. Two replicas of the signal were created to compute the square root of the signal and its logarithm. A frequency domain HT was applied to the latter with a resolution of 512 samples using an overlap and save approach [9]. The result was used to set the phase of the recovered signal who’s amplitude was set by the square root component. A copy of the recovered signal was used to compute the DC bias estimate. The signal demodulation was performed by first resampling to 2 samples per symbol and using a 17-tap adaptive equalizer. The equalizer taps were updated using a blind least-mean squares algorithm and carrier recovery was embedded in the equalizer loop. The inset C of Fig. 4 shows an example constellation of the recovered signal. Q-factor estimates were based on error counting over 7 traces, each with 2.5$\times$105 symbols. The DC bias estimate was produced by filtering the recovered signal between 100 MHz and 300 MHz to estimate the guard-band power and between 9.7 GHz and 11.7 GHz to estimate the signal power. The ratio of the measured powers normalized to the respective bandwidths of the band-pass filters was used to compute the DC metric.
Figure 5 shows the spectrum of the reconstructed signal using 4 different values for the normalized DC bias. For this example, we have chosen a back-to-back configuration and 600 MHz guard band. The ASE source was disabled to maximize the OSNR of the detected signal. From Fig. 5, it is clear that the magnitude of the frequency components within the guardband varies significantly with the chosen value of DC bias correction, as predicted by Eq. (7) and shown through numerical simulation. In contrast, the magnitude of remainder of the signal spectrum does change very significantly.
Fig. 5. Detail of the spectrum of the reconstructed signal with normalized DC bias between 1.8 and 3.5. The inset shows the RF spectrum of the recovered field for a wider frequency range
3.2 Estimation error induced Q-factor penalty
Figure 6 shows the dependence of the Q-factor and the proposed DC metric on the normalized DC bias for a back-to-back signal with a guard band of 600 MHz. The considered CSPR and OSNR values were 10 dB and 26 dB, respectively. It is shown that the normalized DC offset for minimum DC metric fairly approximates the corresponding value for maximum Q-factor. The differences between the two values can be attributed to the approximations taken for the proposed approach as well as the impact of ASE and transceiver noise contributions. Nevertheless, Fig. 6 shows that the difference between the maximum Q-factor and the Q-factor obtained for minimum DC metric is relatively small. This penalty will be used in the remainder of this work to quantify the effectiveness of the proposed metric.
Fig. 6. Dependence of the Q-factor and DC metric on the normalized DC bias. CSPR and OSNR values of 10 dB and 26 dB, respectively, were used on a back-to-back configuration. The vertical dashed lines show the normalized DC bias corresponding to maximum Q-factor (black) and minimum DC metric (red).
Figure 7(a) shows the dependence of the Q-factor on the OSNR for CSPR values of 6 dB, 10 dB, and 16 dB when performing an extensive search and when using the proposed metric to find the optimum DC bias. We assumed a back-to-back configuration. Note that for a CSPR of 16 dB, the performance improves steadily with the OSNR, it begins to floor for a CSPR of 10 dB. This is even more evident for a CSPR of 6 dB as the performance becomes limited by the ability of the receiver front end to reconstruct the signal. This behavior was expected and has been previously reported in [1]. It is shown that using the use of the proposed DC metric follows closely the OSNR dependence observed when using extensive search. This behavior is not dependent on the OSNR within the considered range. For CSPRs of 10 dB and 16 dB, the Q-factor penalty is negligible. Nevertheless, it can be observed that a substantially larger penalty is obtained for a CSPR of 6 dB, with a maximum value of 0.14 dB. This penalty may be attributed to the approximations taken for the proposed metric, as indicated in the previous section.
Fig. 7. Dependence of the Q-Factor on the OSNR in back-to-back (a) and after 100 km transmission (b) for CSPR values of 6 dB, 10 dB, and 16 dB. The DC bias values were obtained by extensive search and using the proposed metric.
Figure 7(b) shows the corresponding analysis considering a transmission distance of 100 km. A very similar behavior to the back-to-back case can be observed although we note a 0.4 dB penalty in performance. As for the Q-factor penalty of the proposed metric, we note the same behavior as in the back-to-back case. CSPRs of 10 dB and 16 dB yielded negligible penalty and a CSPR of 6 dB had a maximum penalty of 0.19 dB.
We have shown the effectiveness of the proposed DC bias estimation metric. It enabled relatively penalties under 0.2 dB with a CSPR of 6 dB and negligible penalty with CSPRs of 10 dB and 16 dB. The proposed method was shown to provide valid estimates in back to back and after 100 km transmission with a 12.5 GBaud signal. Nevertheless, it can be foreseen that the impact of higher dispersion may distort the power spectrum of the signal to a point where it becomes ineffective. This remains an open issue. Nevertheless, the proposed method should provide sufficient accuracy for short reach applications. Another issue to be considered is the width of the guard-band. In this work, we have considered a relatively wide guard-band of 600 MHz for a 12.5 GBaud signal, which could be reduced to improve spectral efficiency. It may be expected that reducing the guard-band reduces the effectiveness of the proposed method as the frequency integration window available to evaluate the signal reconstruction error is smaller. However, this potential issue may possibly be handled using longer integration times. In general, we note that the fine adjustment of the various implementation parameters of the proposed method needs to be carried out on a case-by-case basis.
Finally, we note that the overall complexity gain of a KK receiver DSP supporting the proposed method is not obvious. Signal quality estimation after demodulation may be more effective than the proposed method. Nonetheless, it may require long feedback mechanisms, which are notoriously costly to implement in parallel DSP sub-systems, as well as an association between the quality metric (pre-FEC BER, EVM or similar) and the reconstruction error, which is not trivial. The proposed method then becomes interesting for the cases where quality estimation mechanisms are undesirable due to limited resources, or cases where it is beneficial to decouple the KK signal reconstruction front-end from the demodulation and FEC modules.
4. Conclusion
This work proposed a novel method to estimate a DC bias correction for AC-coupled Kramers-Kronig receivers based on the spectral evaluation of the digitally reconstructed optical field. The proposed method was derived analytically and demonstrated experimentally using 16-QAM signals in back-to-back and after 100 km transmission. It was shown that the Q-factor penalty due to the error of the proposed method was limited to less than 0.2 dB and is mostly degraded by decreasing the CSPR. Nevertheless, it was shown effective to predict the optimum DC bias correction with reasonable precision without the need to use higher level performance metrics, such as pre or pos-FEC BER techniques. Hence it can be applied to the front-end of the receiver, operating independently of the remaining functional blocks of the receiver.
Funding
Ministero dell’Istruzione, dell’Università e della Ricerca (PRIN 2017 FIRST); (INCIPIT).
Acknowledgments
The authors would like acknowledge the technical staff of the Photonic Network System Laboratory of the National Institute of Information and Communication Technology for the support with the experimental work.
Disclosures
The authors declare no conflicts of interest.
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