Kramers-Kronig Receiver Research Articles

Simplified coherent chaotic optical secure communication scheme based on the Kramers-Kronig receiver.

Enhancing physical layer encryption in fiber-optic networks remains a challenging yet vital task. In this Letter, we propose a simplified coherent chaotic secure optical communication scheme based on the Kramers-Kronig (KK) receiver. This scheme incorporates a semiconductor laser with a phase-conjugated optical feedback serving as a common chaotic source, and its chaotic output is directly injected into the two slave lasers arranged separately at the transmitter and receiver end to achieve high-quality synchronization of chaotic signals, with a corresponding chaotic bandwidth of 30.6 GHz. By virtue of the common-signal-induced broad chaotic synchronization, a proof-of-principle demonstration is successfully conducted. It involves the secure transmission of a 20 Gbaud 16-level quadrature amplitude modulation (16QAM) signal over a 50 km standard single-mode fiber (SSMF) link. At the receiver end, we deploy a KK receiver to reconstruct the field of the optical signal and hence enable signal compensation and recovery with offline digital signal processing (DSP). This method simplifies device requirements in the current chaotic coherent optical secure communication, offering a cost-effective mode and promising path for advancing physical layer encryption in inter-data center communications.

Medium-reach six-mode multiplexed transmission with Kramers–Kronig receiver enabled by mode permutation technique

We investigate the transmission distance of a 20-Gbaud 6-mode dual-polarization QPSK transmission system using the Kramers–Kronig(KK) receiver based on the mode permutation (MP) scheme. Compared to the system without the use of MP scheme, the maximum transmission distance is increased from 460 km to 910 km and to over 1000 km via the cyclic mode permutation and the mirror-flipped mode permutation, respectively, at the KP4 forward error correction (FEC) threshold of 2.2×10−4. The required carrier signal power budget of KK receiver is also reduced. The KK receiver can detect the signal of each mode after the transmission of 1000 km with potential advantages of low cost and low power. It is shown as a competitive detection technique in medium-reach metropolitan networks.

Opto-electronic machine learning network for Kramers-Kronig receiver linearization.

We numerically demonstrate the use of an opto-electronic network comprising a photonic reservoir and an electronic feedforward equalizer (FFE) to linearize a Kramers-Kronig (KK) receiver. The KK receiver is operated under stringent conditions, with restricted sampling rates and low carrier powers, resulting in a nonlinear behavior. We propose two different network configurations, varying in the placement of the FFE component, and evaluate their ability to linearize the KK receiver. By training these networks on back-to-back systems, we arrive at a generic solution that significantly enhances the receiver performance, independent of specific link characteristics. The trained networks are tested in a plug-and-play manner across diverse short-reach links that employ standard digital signal processing blocks for the qualization. Our results show significant improvement in receiver linearity, resulting in a reduction in bit error rate of up to a factor of four.

Non-sweep DC component estimation method for a virtual-carrier assisted Kramers-Kronig receiver

The Kramers-Kronig (KK) receiver has attracted much attention in short-range optical interconnection because of its ability to recover the phase of the signal from the intensity information through KK algorithm. In high-speed KK systems, such as virtual-carrier (VC) assisted ones, an alternating current (AC) coupled photo-detector (PD) is preferred due to relaxing the requirements of analog-to-digital converter (ADC) and electronic amplifier by filtering direct current (DC) component. However, the loss of the DC component will cause the KK algorithm to break down, so it is necessary to accurately recover DC value in the digital domain with multiple-sweep. In this paper, we propose what we believe is a novel non-sweep DC component estimation scheme based on optimized digital carrier-to-signal power ratio (OD-CSPR) method, which can accurately estimate the DC component with only 3-4 iterations in the scenario of VC-assisted KK receiver optical transmission. The scheme utilizes the one-dimensional search optimization algorithm based on golden section search and parabolic interpolation without sweeping. The simulation and experimental results of the proposed non-sweep OD-CSPR method show that the DC component can be estimated accurately in a large CSPR range, and the system performance is close to that of the conventional DC-sweep method. Compared with the typical defined digital CSPR (DD-CSPR) based optimization method, the proposed one can realize optical signal-to-noise ratio (OSNR) gains of 0.9 dB in the back-to-back (B2B) and 0.7 dB under 80 km fiber transmission scenarios respectively with a total bit rate of 160Gb/s.

Modified Kramers-Kronig receiver based on memory polynomial compensation for photonics-assisted millimeter-wave communications.

Photonics-assisted millimeter-wave (MMW) wireless communications are advancing rapidly driven by the escalating congestion in the lower-band spectrum and the growing demand for higher data rates. Concurrently, Kramers-Kronig (KK) receivers provide an economical solution ideally suited for cost-sensitive deployment and application. However, the conventional KK receiver is subject to performance degradation due to the nonlinearity and memory effects introduced by practical electronic devices. In this work, the performance degradation of the conventional KK receiver is investigated and quantitatively simulated, showing that the KK receiver exhibits greater sensitivity to nonlinearity and memory effects compared to the conventional coherent receiver. To enhance the performance of KK receivers deployed in MMW communication systems, we propose a modified KK receiver employing memory polynomial compensation, namely MP-KK receiver, capable of effectively compensating memory effects whilst simultaneously addressing nonlinearity. Crucially, the memory polynomial model is employed prior to the KK algorithm to prevent further signal degradation caused by the nonlinear operator in the KK algorithm in the scenario of photonics-assisted MMW wireless communication based on the KK receiver. For verification, we present a 95 GHz W-band MMW wireless transmission demonstration with 20 Gb/s QPSK and 40 Gb/s 16-QAM signals. The experimental results indicate that the MP-KK receiver can achieve more than 3.5 dB improvement in EVM and 71.25% reduction in BER compared to the conventional approaches.

Performance of Kramers–Kronig and coherent receivers in Medium-Reach optical communication

As the capacity and the transmission distance of medium-reach optical fiber communication systems continue to increase, it is a major trend in demanding lower optical link budgets while maintaining the quality of signal transmission. We experimentally compared the transmission performance of the Kramers–Kronig (KK) and the coherent receivers in a 20-Gbaud quadrature phase shift keying (QPSK) system over a ∼ thousand-kilometer standard single-mode fiber (SMF) link. Experimental results show that the KK receiver requires lower cost and simpler structure, and behaves similarly with the coherent receiver in the transmission under 1260 km, despite higher transmitted power and computational complexity.

2.4-Tb/s/λ*km fundamental mode transmission over OM2 fiber employing Kramers-Kronig receiver.

We propose a digital-carrier Kramers-Kronig (DC-KK) scheme based high-speed multimode fiber short-reach optical interconnect system with fundamental mode transmission. After optimization of the parameters, including the roll-off factor of the root-raised-cosine (RRC) filter, and the guard interval (GI) between signal and carrier tone, as well as the carrier signal power ratio (CSPR), 200-Gb/s 32-quadrature amplitude modulation (32QAM) signal transmission over 12-km OM2 fiber has been experimentally demonstrated with a bit error ratio (BER) below the soft-decision forward error correction (SD-FEC) threshold of 4 × 10-2. To the best of our knowledge, this is the highest experimental record of single lambda bitrate-distance-product (SLBDP) achieved by direct-detection (DD)-based transmission over a standard multimode fiber (MMF). The proposed scheme has potential to improve the system performance without replacing massive deployed legacy MMFs for future large-capacity data center interconnects (DCIs).

Impact of resampling interpolation FIR filter in the practical Kramers-Kronig receiver.

The practical Kramers-Kronig (KK) receiver has been a competitive receiving technique in the data-center, medium reach, and even long-haul metropolitan networks. Nevertheless, an extra digital resampling operation is required at both ends of the KK field reconstruction algorithm due to the spectrum broadening caused by adopting the nonlinear function. Generally, the digital resampling function can be implemented by using linear interpolation (LI-ITP), the Lagrange cubic interpolation (LC-ITP), the spline cubic interpolation (SC-ITP), time-domain anti-aliasing finite impulse response (FIR) filter method (TD-FRM) scheme, and fast Fourier transform (FFT)-based scheme. However, the performance and the computational complexity analysis of different resampling interpolation schemes in the KK receiver have not been thoroughly investigated yet. Different from the interpolation schemes of conventional coherent detection, the interpolation function of the KK system is followed by the nonlinear operation, which will broaden the spectrum significantly. Due to the frequency-domain transfer function of different interpolation schemes, the broadened spectrum will have a potential spectrum aliasing, which will cause serious inter-symbol interference (ISI) and further impair the KK phase retrieval performance. We experimentally investigate the performance of different interpolation schemes under different digital up-sampling rates (i.e. the computational complexity) as well as the cut-off frequency, the tap number of the anti-aliasing filter, and the shape factor of the TD-FRM scheme in a 112-Gbit/s SSB DD 16-QAM system over 1920-km Raman amplification (RFA)-based standard single-mode fiber (SSMF). The experimental results involve that the TD-FRM scheme outperforms other interpolation schemes and the complexity is reduced by at least 49.6%. In fiber transmission results, take 20% soft decision-forward error correction (SD-FEC) of 2×10-2 as the threshold, the LI-ITP and LC-ITP schemes only reach 720-km while others can reach up to 1440-km.

Deep Learning-Based Phase Retrieval Scheme for Minimum-Phase Signal Recovery

We propose a deep learning-based phase retrieval method to accurately reconstruct the optical field of a single-sideband minimum-phase signal from the directly detected intensity waveform. Our method relies on a fully convolutional Neural Network (NN) model to realize non-iterative and robust phase retrieval. The NN is trained so that it performs full-field reconstruction and jointly compensates for transmission impairments. Compared to the recently proposed Kramers-Kronig (KK) receiver, our method avoids the distortions introduced by the nonlinear operations involved in the KK phase-retrieval algorithm and hence does not require digital upsampling. We validate the proposed phase-retrieval method by means of extensive numerical simulations in relevant system settings, and we compare the performance of the proposed scheme with the conventional KK receiver operated with a 4-fold digital upsampling. The results show that the 7% hard-decision forward error correction (HD-FEC) threshold at BER 3.8e-3 can be achieved with up to 2.8 dB lower carrier-to-signal power ratio (CSPR) value and 1.8 dB better receiver sensitivity compared to the conventional 4-fold upsampled KK receiver. We also present a comparative analysis of the complexity of the proposed scheme with that of the KK receiver, showing that the proposed scheme can achieve the 7% HD-FEC threshold with 1.6 dB lower CSPR, 0.4 dB better receiver sensitivity, and 36% lower complexity.

Optical Direct-Detection System Based on Chaotic Encryption and Kramers-Kronig Receiver

Optical Direct-Detection System Based on Chaotic Encryption and Kramers-Kronig Receiver

Symbol-interleaved precoding technique for DDO-OFDM systems with the KK receiver.

Receiver sensitivity of direct-detection optical orthogonal frequency-division multiplexing (DDO-OFDM) systems can be significantly improved by using the Kramers-Kronig (KK) receiver. However, the KK algorithm is sensitive to the high signal-to-average power ratio (PAPR) of the OFDM signal, which may violate the minimum phase condition (MPC) and lead to severe signal distortions during the optical field reconstruction. Channel-independent discrete Fourier transform (DFT) precoding technique can efficiently reduce the PAPR and realize subcarrier signal-to-noise ratio (SNR) equalization, but the PAPR of the precoded signal will increase due to chromatic dispersion. Besides, the relevant researches exhibit that it is hard to accurately reconstruct the complex field from the sampled signals based on KK relation at the low carrier-to-signal power ratio (CSPR) even if the MPC is satisfied. To combat the signal distortions induced by the KK algorithm and improve the bit error rate (BER) performance, we propose and verify DFT/Walsh-Hadamard transform (WHT)-based precoding with a simple symbol-interleaving technique for DDO-OFDM transmission systems. In addition to PAPR reduction, the proposed technique can equalize both subcarrier and inter-symbol SNRs efficiently. The simulated results show that the BER performance can be improved by up to one order of magnitude at a CSPR of 6 dB after up to 1000km single-mode fiber transmission, compared to the conventional DFT precoding technique. It is expected that the proposed technique can be employed to improve the receiver sensitivity of medium-reach DDO-OFDM transmission systems with the KK receiver.

Photonic reservoir computing for nonlinear equalization of 64-QAM signals with a Kramers–Kronig receiver

Abstract Photonic reservoirs are machine learning based systems that boast energy efficiency and speediness. Thus they can be deployed as optical processors in fiber communication systems to aid or replace digital signal equalization. In this paper, we simulate the use of a passive photonic reservoir to target nonlinearity-induced errors originating from self-phase modulation in the fiber and from the nonlinear response of the modulator. A 64-level quadrature-amplitude modulated signal is directly detected using the recently proposed Kramers–Kronig (KK) receiver. We train the readout weights by backpropagating through the receiver pipeline, thereby providing extra nonlinearity. Statistically computed bit error rates for fiber lengths of up to 100 km fall below 1 × 10−3 bit error rate, outperforming an optical feed-forward equalizer as a linear benchmark. This can find applications in inter-datacenter communications that benefit from the hardware simplicity of a KK receiver and the low power and low latency processing of a photonic reservoir.

Nonlinear Distortion by Stimulated Brillouin Scattering in Kramers-Kronig Receiver Based Optical Transmission.

Nonlinear distortion for single-sideband (SSB) signals will significantly reduce the performance of Kramers–Kronig (KK) receiver-based optical transmission. In this work, we present a proof-of-concept study of stimulated Brillouin scattering (SBS)-induced nonlinear distortion for 10 Gbaud and 28 Gbaud SSB QAM16 transmission over 80 km standard single mode fiber (SSMF) based on a KK receiver. Significantly reduced bit error rate (BER) has been experimentally observed due to the SBS and the threshold of SBS at about 7 dBm is detected for such an 80 km SSMF link. With left sideband (LSB) modulation of SSB, together with optical filtering, reduced SBS nonlinear distortion has been achieved with ~2 dB power tolerance improvement. The results reveal an important issue of SBS-induced nonlinear distortion, which would be of great significance for KK receiver-based optical transmission applications.

Comparison of DC-Value Method and Kramers–Kronig Receiver in Optical OFDM SSB-DD Transmission

Short reach optical communications are desperate for cost-effective solutions to improve capacity and compatibility in dealing with massive access and applications. Single sideband-direct detection (SSB-DD) is highly recognized for improving capacity with single PD, but it is confronted with signal-signal beating interference (SSBI). Recently, a DC-Value method has been proposed for optical signal phase retrieval for single carrier signal transmission. To further explore the effectiveness and compatibility of DC-Value method, in this letter for the first time, the orthogonal frequency division multiplexing (OFDM)-based multi-carrier modulation formats are processed by the DC-Value method in the receiver. Simulation results demonstrate that, for both OFDM and Nyquist-shaped single carrier (SC), the DC-Value method is beneficial in applying lower sampling rate to achieve comparable transmission performance than that of Kramers-Kronig receiver. In addition, althrough OFDM holds 2 dB increment of the optimum carrier-signal power ratio (CSPR) than Nyquist-SC, it offers more robustness to fibre transmission.

Digital Signal Processing for High‐Speed THz Communications

The use of advanced digital signal processing (DSP) technology in the high-speed terahertz (THz) communication can effectively compensate the linear and nonlinear effects of the system and further improve the transmission performance of the system. This paper introduces the principle and application of advanced DSP algorithms such as probability shaping (PS) technology, Volterra series nonlinear compensation algorithm, Kramers-Kronig receiver, look-up table (LUT) pre-distortion compensation technology, pre-equalization and decision-directed least-mean-square equalization algorithm. Combined with DSP algorithms such as PS and LUT pre-distortion, using photon-assisted technology to successfully realize the wireless transmission of vector THz signals higher than 1 Tbit/s on the sub-THz band (D-band) 4 × 4 multiple input multiple output system.

Free Space Ground to Satellite Optical Communications Using Kramers-Kronig Transceiver in the Presence of Atmospheric Turbulence.

Coherent detection provides the optimum performance for free space optical (FSO) communication systems. However, such detection systems are expensive and require digital phase noise compensation. In this paper, the transmission performance of long-haul FSO system for ground-to-satellite communication based on a Kramers–Kronig (KK) transceiver is evaluated. KK transceivers utilize inexpensive direct detection receivers and the signal phase is retrieved from the received current using the well-known KK relations. KK transceivers are not sensitive to the laser phase noise and, hence, inexpensive lasers with large linewidths can be used at the transmitter. The transmission performance of coherent and KK transceivers is compared in various scenarios such as satellite-to-ground, satellite-to-satellite, and ground-to-satellite for weak, moderate, and strong turbulence. The results show that the transmission performance of a system based on the KK transceiver is comparable to that based on a coherent transceiver, but at a significantly lower system cost and complexity. It is shown that in the absence of turbulence, the coherent receiver has a ~3 dB performance advantage over the KK receiver. However, in the presence of strong turbulence, this performance advantage becomes negligible.

Short-Reach MCF-Based Systems Employing KK Receivers and Feedforward Neural Networks for ICXT Mitigation

This paper proposes and evaluates the use of machine learning (ML) techniques for mitigating the effect of the random inter-core crosstalk (ICXT) on 256 Gb/s short-reach systems employing weakly coupled multicore fiber (MCF) and Kramers–Kronig (KK) receivers. The performance improvement provided by the k-means clustering, k nearest neighbor (KNN) and feedforward neural network (FNN) techniques are assessed and compared with the system performance obtained without employing ML. The FNN proves to significantly improve the system performance by mitigating the impact of the ICXT on the received signal. This is achieved by employing only 10 neurons in the hidden layer and four input features for the training phase. It has been shown that k-means or KNN techniques do not provide performance improvement compared to the system without using ML. These conclusions are valid for direct detection MCF-based short-reach systems with the product between the skew (relative time delay between cores) and the symbol rate much lower than one (skew×symbol rate≪1). By employing the proposed FNN, the bit error rate (BER) always stood below 10−1.8 on all the time fractions under analysis (compared with 100 out of 626 occurrences above the BER threshold when ML was not used). For the BER threshold of 10−1.8 and compared with the standard system operating without employing ML techniques, the system operating with the proposed FNN shows a received optical power improvement of almost 3 dB.

2×2 MIMO Equalizer Enabled Transmitter Side IQ Imbalance Compensation for Optical Single Sideband Direct Detection System

Recently, optical single sideband (SSB) based direct detection (DD) has been considered as a promising candidate for inter data center interconnects (inter-DCIs). Kramers-Kronig (KK) receiver is always applied to eliminate the signal-to-signal beating interference (SSBI) caused by the square law-detection of a single-ended photodiode (PD) in SSB DD system. The electro-to optical (E/O) conversion of the complex signal can be realized by IQ modulator (IQM). However, IQ imbalance exists at the transmitter since the in-phase and quadrature-phase ports of IQM are driven by the real and imaginary parts of the signal, respectively. In this paper, we show the theoretical analysis of the transmitter side (Tx-side) IQ imbalance includes time delay, amplitude mismatch and phase mismatch. Tx-side IQ imbalance will seriously deteriorate the system performance. To deal with the impairment caused by Tx-side IQ imbalance, a 22 real-value multiple-input multiple-output (MIMO) equalizer is proposed and implemented, and the system with the impact of these three factors are investigated in simulation, respectively. Besides, a transmission of 180 Gb/s/ discrete Fourier transform spread (DFT-S) orthogonal frequency division multiplexing (OFDM) signal over 80-km single mode fiber (SMF) is experimentally demonstrated with bit error rate (BER) below the soft-decision forward error correction (SD-FEC) threshold of 210-2 enabled by the proposed 22 MIMO equalizer.

Kramers-Kronig Receiver With Digitally Added Carrier Combined With Digital Resolution Enhancer

Digital phase reconstruction of directly detected minimum phase (MP) optical signals using the Kramers-Kronig (KK) algorithm has recently drawn attention because of its low hardware complexity. Only a single photodiode is required to detect the intensity of the optical signal and KK relations are used to retrieve the phase information. To make sure the signal detected by the photodiode is MP, a carrier tone needs to be inserted. In this work, we digitally add the carrier tone at the transmitter, omitting the need for additional hardware to generate the carrier tone. However, digitally adding the carrier tone increases quantization noise resulting from the digital-to-analog conversion at the transmitter. A digital resolution enhancer (DRE) can reduce effective transmitter quantization noise by moving the quantization noise outside the signal band. We investigate the applicability of DRE to compensate for the increased transmitter quantization noise caused by the digital carrier tone insertion. We experimentally validate this concept by demonstrating 64-ary quadrature amplitude modulation (QAM) transmission at <inline-formula><tex-math notation="LaTeX">$25 \,{\mathrm GBd}$</tex-math></inline-formula>, showing that DRE increases transmission performance for a 6 bits digital-to-analog converter (DAC) system. Furthermore, employing DRE allows the reduction of the resolution of the DAC system to 4 or 5 bits with a limited performance penalty.

DC Restoration by Data-Aided Sequence in Kramers-Kronig Receiver

DC Restoration by Data-Aided Sequence in Kramers-Kronig Receiver