Kramers-Kronig Receiver Research Articles

Optimizing DC restoration in Kramers-Kronig optical single-sideband receivers.

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.

Fast and simple calibration of frequency response and IQ skew for a coherent optical transmitter using a low-bandwidth photodetector.

A fast, precise, and low-cost coherent optical transmitter calibration scheme is proposed that uses multi-tone signals of a novel, to the best of our knowledge, design with unequal frequency intervals. With a single measurement, the proposed scheme can simultaneously calibrate the frequency response and the IQ skew of the transmitter using only a low-bandwidth photodiode. Simulation and experimental results indicate that the measurement error in the frequency response and coherent transmitter (Tx) skew is less than 0.3 dB and 0.2 ps, respectively. The feasibility of the proposed scheme is verified by an experiment involving 25 Gbaud 16-quadrature amplitude modulation (QAM) signal transmission using a Kramers-Kronig (KK) receiver. With the help of this calibration method, the measured bit error ratio performance was increased from 1.77e-2 to 3.52e-3 when the received optical power was -8 dBm.

Digital Compensation of SOA-Induced Nonlinearities in Field-Modulated Direct-Detection Systems

In this work we experimentally demonstrate the compensation of nonlinearities introduced by a semiconductor optical amplifier (SOA) operating as a power booster in a field-modulated direct detection fiber-optic system transmitter. We show that a combination of digital pre -compensation to deal with SOA nonlinearities, and digital post -compensation after Kramers-Kronig receiver for mitigating the impacts of chromatic dispersion and signal-signal beat interference (SSBI) is an effective scheme to gain ∼3dB in ROSNR in presence of SOA nonlinearity at 3dB of gain compression.

Impact of the carrier contribution factor in the self-coherent DC-value method

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.

Kramers-Kronig Optical OFDM for Bandlimited Intensity Modulated Visible Light Communications

Visible light communication (VLC) operates on optical intensity channels that are inherently limited in bandwidth. Though Kramers-Kronig (KK) receivers have been considered as a lower complexity alternative for coherent fiber optic communication (FOC) systems, in this paper we extend this concept to bandlimited IM/DD VLC channels and propose KK optical OFDM (KKO-OFDM). In KKO-OFDM the optical power of a low-cost LED is directly modulated by the double-sideband (DSB) squared modulus of a minimum phase single-sideband (SSB) signal. This results in a real-valued, non-negative and strictly bandlimited transmit signal which is suitable for VLC channels. At the receiver, the phase of the transmitted SSB signal is reconstructed via the KK relations. The required DC bias, average electrical signal-to-noise ratio (SNR), bit error rate (BER), and capacity are analyzed with approximate closed-forms and through simulation. Numerical results show that KKO-OFDM achieves the same spectral efficiency as the existing DC biased optical OFDM (DCO-OFDM), however, realizes approximately 1 dB optical SNR gain at a BER <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$=10^{-4}$</tex-math></inline-formula> while simultaneously having a small peak-to-average power ratio.

A Study on Sampling Penalties Reduction of Kramers-Kronig Receivers

Kramers-Kronig (KK) relation-based receiver can recover the field using a single-photodiode. However, the KK receiver requires a high carrier-to-signal power ratio (CSPR), and digital signal processing performed a few times faster than the Nyquist sampling rate. We propose a method to relax the requirements for sampling rate and CSPR by using digital upsampling with harmonic filtering. In the proposed method, log calculation of the KK algorithm is preceded by digital upsampling and followed by harmonics elimination and downsampling, respectively. Our results predict that the combination of digital upsampling and harmonic elimination can comply with the requirement of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nd</sup> generation FEC limit in 16-QAM 40-km transmission with the following signal parameters, CSPR 6.7 dB, ADC sampling rate 2.5 sps, digital upsampling 3 sps. We propose performing harmonic elimination in combination with Hilbert transform using a single finite impulse response (FIR) filter to increase computation efficiency. Compared with the frequency-domain implementation, similar performances were observed for FIR filters of length 33 or higher.

Joint equalization of CD and RSOP using a time–frequency domain Kalman filter structure in Kramers–Kronig system

Kramers–Kronig (KK) system is a direct detection (DD) optical short-reach interconnection system which is suitable for high capacity data center interconnect (DCI) applications. For this kind of short-reach high capacity polarization-division multiplexing (PDM) optical transmission system, the fiber channel impairments contain chromatic dispersion (CD) and rotation of state of polarization (RSOP) will lead to a severe signal distortion and need to be equalized. This paper proposed a time–frequency domain Kalman filter scheme for joint equalization of CD and RSOP simultaneously in KK system. The proposed scheme is capable of tracking the fast RSOP in time domain and equalizing CD in frequency domain using a window-split structure. The performance of the Kalman scheme is investigated in a 28GBaud PDM 16QAM DD system with KK receivers. The simulation results prove that the proposed Kalman scheme exhibits a good performance of RSOP tracking ability up to speed of 2.5 Mrad/s, with a large accumulated CD tolerance as large as 2040 ps/nm at a low OSNR penalty. The results also show the proposed scheme has a comparable computational complexity compared with a generally used scheme which includes a CD compensation, combined with polarization demultiplexing algorithm constant modulus algorithm (CMA) and multiple modulus algorithm (MMA).

Fiber Vector Eigenmode Multiplexing Based High Capacity Transmission Over 5-km FMF With Kramers-Kronig Receiver

Vector modes (VMs) with spatially nonhomogeneous polarization patterns, combining polarization and spatial-mode degrees of freedom together, are regarded as the eigenmodes of optical fiber. Such eigenmodes have potential to be used to implement the high capacity spatial division multiplexing (SDM)/mode division multiplexing (MDM) based optical fiber communication systems. Here, we respectively demonstrate 2 VMs of the same order (l = +2, EH11o and EH11e modes) and 2 VMs of different orders (l = 0, HE11o mode and l = +2, EH11e mode) multiplexing transmissions over 5-km few-mode fiber (FMF). The mode crosstalk after FMF transmission has been minimized to prepare high capacity signal transmission without multiple-input-multiple-output (MIMO) digital signal processing (DSP). To increase the transmission capacity and decrease the system cost, the Kramers-Kronig (KK) receiver has been employed and the single-wavelength two sets of eigenmodes multiplexed 360 Gbit/s and 400 Gbit/s signal transmissions over 5-km FMF link are realized, respectively. Then, to further burst the transmission capacity, 2 vector-mode-division-multiplexing channels (HE11o and EH11e modes) in combination of 5 wavelength-division-multiplexing (WDM) channels with a total of 10 parallel channels have been demonstrated. A total data rate of 1.12 Tbit/s with 28 GBaud 16-state quadrature amplitude modulation (QAM) signal has been realized over the 5-km FMF. In all data-carrying experimental demonstrations, the bit-error rate (BER) performance of each VM channel is below the 7% hard decision forward error correction threshold (FEC) at BER of 3.8 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> . The experimental results highlight that the VMDM-based optical transmission technology could find enormous potential in huge-capacity short-reach optical interconnects.

Precoded OVSB-OFDM transmission system using DML with Kramers-Kronig receiver

Different from the use of IQ modulator or a dual-drive Mach-Zehnder modulator, we investigate an optical vestigial sideband orthogonal frequency division multiplexing (OVSB-OFDM) transmission system using a low-cost directly modulated laser. Precoding techniques and Kramers-Kronig (KK) algorithm are employed to enhance the bit error rate (BER) performance. The KK receiver is applied to eliminate the signal-signal beat interference caused by the square-law detection of the photodetector. Discrete Fourier transform (DFT) and orthogonal circulant matrix transform (OCT) precoding techniques can effectively average the signal-to-noise ratio over the data-carrying subcarriers and improve the BER performance. Through numerical simulation, we discuss the relationship between the order and center frequency of the optical filter and the carrier signal power ratio (CSPR). Furthermore, we study the influence of the KK algorithm and precoding techniques on receiver sensitivity under different transmission distances. The results show that both KK algorithm and precoding technologies can improve receiver sensitivity and extend the transmission distance. The performance of DFT precoding is better than that of OCT precoding. When there is no KK algorithm, the transmission distance of OCT precoding or no precoding is only 20 km at the BER of 3.8e-3; while DFT precoding can extend the transmission distance up to 60 km. With the help of the KK algorithm, OCT precoding and DFT precoding can further extend the transmission distance by up to 40 and 60 km, respectively, compared to the DFT precoding and without the KK algorithm case. The DFT precoding with the KK algorithm can improve the receiver sensitivity of more than 7 dB in terms of received optical power, compared to the no precoding and without the KK algorithm case.

Integrated Components and Solutions for High-Speed Short-Reach Data Transmission

According to different transmission distances, application scenarios of a data center mainly include intra- and inter-data center optical interconnects. The intra-data center optical interconnect is considered as a few kilometers optical interconnect between servers and racks inside a data center, which accounts for nearly 80% of data traffic of a data center. The other one, inter-data center optical interconnect, is mainly applied in tens of kilometers data transmission among different data centers. Since data exchange in data centers generally occurs between many servers and racks, and a lot of transmitter and receiver components are required, optical interconnects become highly sensitive to component costs. In this paper, we firstly review the development and applications of mainstream transmitter components (e.g., VCSEL, DML, EML, MZM, and monolithic integrated transmitter) and receiver components (e.g., single-end photodetector, Kramers-Kronig receiver, Stokes vector receiver, and monolithic integrated receiver), which have been widely applied in short-reach transmission systems. Then, two types of integrated solutions including simplified detection scheme and transceiver integration scheme are presented in detail. Finally, we summarize and discuss the technological and component options for different transmission distances. We believe that monolithic integrated components, especially transceiver integration, will become a powerful solution for next-generation high-speed short-reach transmission systems.

Transmission of a 112-Gbit/s 16-QAM over a 1440-km SSMF with parallel Kramers-Kronig receivers enabled by an overlap approach and bandwidth compensation.

We investigate the parallelized performance of the conventional Kramers-Kronig (KK) and without the digital up-sampling KK (WDU-KK) receivers in a 112-Gbit/s 16-ary quadrature amplitude modulation (16-QAM) system over a 1440-km standard single-mode fiber (SSMF). A joint overlap approach and bandwidth compensation filter (OLA-BC) architecture is presented to mitigate the edge effect caused by the Hilbert transform and the Gibbs phenomenon induced by the FIR filter, respectively. Moreover, the computational complexity of the OLA-BC based parallelized KK/WDU-KK receiver is also discussed. Parallelized KK/WDU-KK receivers based on the presented OLA-BC architecture can effectively mitigate the edge effect and the Gibbs phenomenon together with more than two orders of magnitude improvement in terms of bit-error-ratio (BER) compared with parallelized KK/WDU-KK receivers without OLA-BC receivers in back-to-back (B2B) case. Finally, we successfully transmit the 16-QAM signals over 960-km SSMF with a BER lower than 7% hard-decision forward error correction (HD-FEC) threshold (3.8 × 10-3) and 1440-km SSMF with a BER lower than 20% soft-decision FEC (SD-FEC) threshold (2 × 10-2).

Analog Low-Latency Kramers-Kronig optical Single-Sideband Receiver

Kramers-Kronig (KK) receivers have been shown to be able to cancel the signal-signal beat interference that occurs upon direct-photodetection in optical single-sideband systems, where the transmitted optical field is proportional to the modulation signal (Field-Modulated Direct-Detection systems). They can support complex modulation formats, and being single-sideband, electronic dispersion compensation, without the need for a coherent receiver. All current research has used digital-signal processing (DSP) to implement the KK algorithm. Radio-frequency KK receivers using analog processing were first introduced in the 1960's, and offer an alternative to DSP, particularly in their approximate form. Analog processing is inherently real-time and offers the lowest latency (processing delay). In this paper, we show that with minor changes, analog processing can also be used with signals from a photodetector. We have built an experimental receiver, using standard parts that have functionality that approximates to the desired algorithm. In particular, we show that a differential pair of bipolar transistors can approximate a square-root function, that the Hilbert transform can be approximated by an 8-tap transmission-line delay, and that a commercial analog multiplier chip can be used for a squaring and subtraction function. The system is demonstrated with 16-QAM at 500 Mbit/s with the photocurrent emulated in software and the KK algorithm performed in hardware. Finally, software band-limits the signal, down-converts it to form a constellation, and calculates symbol error rates. We show that the analog KK reduces the symbol error rate by at least a factor of ten, enabling low carrier-to-signal power ratios to be used with standard forward error correction. Much higher data rates could be supported using custom-designed microwave ASICs, which could be incorporated in packaged receivers. This would enable miniaturized receivers using standard DSP to work with dispersion-limited links. Interestingly, half the benefit could be gained by simply re-biasing the differential stage of a standard photoreceiver.

Enhanced CSPR for a multichannel Kramers-Kronig receiver by self-seeded stimulated Brillouin scattering.

We demonstrate carrier-to-signal power ratio (CSPR) enhancement by self-seeded stimulated Brillouin scattering to improve the performance of Kramers-Kronig (KK) detection for multichannel single-sideband (SSB) signals. By virtue of low-CSPR transmission and high-CSPR detection, our proposed scheme effectively advances system performance by reducing propagation-induced distortion while maintaining the minimum phase condition. We experimentally demonstrate the improvement in CSPR and bit error rate of 5×10-Gbaud 16-QAM SSB signals by applying the carrier recovery block after 80-km transmission. Under optimum pump power, the average Q factor improvement of all five channels is 3.0 dB. We also analyze the performances of different channels and the major limiting factor. The results verify that our scheme offers a promising solution to enhance SSB self-coherent KK detection in wavelength-division multiplexing systems.

Optical Single Sideband Signal Reconstruction Based on Time-Domain Iteration

Due to its low cost, simple architecture and robustness to fiber dispersion, single sideband (SSB) transmission with direct detection (DD) system is an attractive solution for 80-km inter data center interconnects (DCIs). However, it will suffer performance penalty caused by the signal-to-signal beating interference (SSBI). Kramers-Kronig (KK) receiver has been extensively investigated for SSBI elimination by reconstructing the SSB signal. The nonlinear operations in KK algorithm require up-sampling to cope with spectral broadening, which results in high complexity for practical application. Optical signal phase retrieval method based on the minimum phase signal has also been investigated for SSB signal recovery, in which the SSB and DC-Value properties are iteratively imposed on the amplitude signal in frequency domain. In this paper, we propose a low complexity iterative algorithm for minimum phase signal recovery without up-sampling in time domain. Finite impulse response (FIR) filter is applied to iteratively generate the SSB signal and update the phase component. Based on the proposed scheme, the transmission of 30 GHz SSB 16-QAM discrete multitone (DMT) signal over 80 km single mode fiber (SMF) is successfully demonstrated with the bit error rate (BER) below the hard-decision forward error correction (HD-FEC) threshold of 3.8 × 10−3. The experimental results show that, the BER performance of KK scheme with up-sampling factor of 2, frequency-domain iterative scheme and our proposed scheme is almost the same. However, compared with the KK scheme, the proposed method can save the numbers of adders and multipliers by the factors of 29 and 7, while the factors are 5.5 and 4 comparing to the frequency-domain iteration scheme.

Direct detection based φOTDR using the Kramers-Kronig receiver.

A Kramers-Kronig (KK) receiver is applied to a phase-sensitive optical time domain reflectometry based on direct detection. An imbalanced Mach-Zehnder interferometer with a 2×2 coupler is used in sensing system to encode the phase information into optical intensity. The directly obtained signal is treated as the in-phase component, and the KK receiver provides the quadrature component by Hilbert transform of the obtained signal, so that the optical phase can be retrieved by IQ demodulation. The working principle is well explained, and the obtained phase variance is theoretically analyzed. The experiment demonstrates the functionality of the sensor and validates the theoretical analysis.

909.5 Tbit/s Dense SDM and WDM Transmission Based on a Single Source Optical Frequency Comb and Kramers-Kronig Detection

Space-division multiplexing (SDM) has become a promising technology for optical communications to seek a sustainable increment in data capacity to keep up with the ever-increasing bandwidth demand. SDM could provide massive parallelization in addition to existing WDM technology for a high capacity without compromising the signal-to-noise ratio requirement of the transmitter. However, the WDM-SDM combination requires arrays of laser sources with associated active wavelength controls. This is not favorable in terms of energy consumption and cost. Single source optical frequency comb (SS-OFC) is an appealing candidate to provide coherent WDM sources, replacing the laser arrays and surpassing the performance by providing the possibility of joint digital signal processing across multiple channels. In this paper, we have generated an SS-OFC based on highly nonlinear fibers (HNLF) for short-reach applications such as optical interconnects, taking advantage of the high pump-to-comb efficiency. The 50-GHz spaced SS-OFC across the whole C band provides 99 comb lines as WDM channel sources. Combining adaptive-rate modulation, a 7.9-km 37-core fiber, and a polarization diversity Kramers-Kronig receiver, we have successfully transmitted 3663 channels (37 SDM × 99 WDM) with reliably error-free performance, achieving 909.5 Tbit/s net rate with an aggregated spectral efficiency of 184.42 bit/s/Hz.

Employing a Kramers–Kronig receiver with a directly-modulated laser

The Kramers–Kronig (KK) receiver has been used for dispersion compensation in direct detection systems employing external-modulator-based transmitters. In this paper, we investigate the use of the KK receiver for dispersion compensation in a direct detection system based on a directly-modulated laser. We show the validity of the KK receiver by considering a linear contribution to the optical phase arising from a numerical solution of the laser rate equations as a frequency shift of the optical spectrum. The modulated optical signal can then meet the minimum-phase condition required by the KK algorithm. A system model is presented based on the direct modulation of a 10 Gb/s distributed feedback laser with 16-ary quadrature amplitude modulation (16-QAM) subcarrier modulation. The modulated signal is optically filtered, and a 53.5 Gb/s vestigial sideband signal is transmitted over 65 km of standard single-mode fiber. Dispersion compensation is applied after direct detection with a KK receiver. In addition, the impact of the optical filter parameters and receiver sampling rate on system performance are investigated.

Generalized Kramers–Kronig receiver for coherent terahertz communications

High-speed communication systems rely on spectrally efficient modulation formats that encode information both on the amplitude and on the phase of an electromagnetic carrier. Coherent detection of such signals typically uses rather complex receiver schemes, requiring a continuous-wave (c.w.) local oscillator (LO) as a phase reference and a mixer circuit for spectral down-conversion. In optical communications, the so-called Kramers-Kronig (KK) scheme has been demonstrated to greatly simplify the receiver, reducing the hardware to a single photodiode. In this approach, an LO tone is transmitted along with the signal, and the amplitude and phase of the complex signal envelope are reconstructed from the photocurrent by digital signal processing. This reconstruction exploits the fact that the real and the imaginary part, or, equivalently, the amplitude and the phase of an analytic signal are connected by a KK-type relation. Here, we transfer the KK scheme to high-speed wireless communications at THz carrier frequencies. We use a Schottky-barrier diode (SBD) as a nonlinear element and generalize the theory of KK processing to account for the non-quadratic characteristics of this device. Using 16-state quadrature amplitude modulation (16QAM), we transmit a net data rate of 115 Gbit/s at a carrier frequency of 0.3 THz over a distance of 110 m.

DLI-Based DP-QPSK Reception Scheme for Short-Range Optical Communication

The unacceptably high cost of digital coherent receivers for short-range optical communication has in recent years fueled the study of alternative transmission schemes that are simultaneously highly performing and cost-effective. However, the proposed solutions (e.g., Kramers–Kronig receiver) usually require the cooperation of a transmitting end, which is unachievable in the context of non-cooperative communication. In this work, we mainly studied a dual-polarization quadrature phase-shift-keying (DP-QPSK) non-coherent reception scheme based on a delay-line interferometer (DLI) and measures to improve the reception performance. A data recovery algorithm was proposed for DLI-based DP-QPSK demodulation. The simulation results demonstrated that the algorithm could accurately recover the raw data from the transmitter. Moreover, decreasing the differential delay of the DLI could effectively increase the chromatic dispersion tolerance of the receiver and the optimal delay scaled inversely with the chromatic dispersion. It was found that a DLI-based DP-QPSK reception scheme is a better choice for short-range, non-cooperative communication with less severe transmission impairments.

Minimum Phase Conditions in Kramers-Kronig Optical Receivers

Theoretically, for continuous signals, Kramers-Kronig (KK) receivers should be able to precisely reconstruct a band-limited signal's phase from its envelope if the minimum phase (MP) condition is satisfied. However, simulations have shown that even if the MP condition is satisfied, for sampled signals the reconstructed signal can contain large errors. In this article, we explore the effectiveness of DSP-based KK receivers close to the MP condition using single-tone and multi-tones. For a single-tone signal, we use a precise mathematical analysis to demonstrate large distortions in the reconstructed signal even when the MP condition is clearly satisfied and illustrate this on the complex-plane. We then study multi-tone signals spectrally, and show that there are strong intermodulation distortion tones when MP is satisfied that either directly fall in-band, or could be aliased back in-band during down-sampling. Satisfying MP does, however, eliminate signal-signal beat interference, as expected with KK processing. High up-sampling ratios reduce the spectral density of these tones, which explains the improved performance seen in experimental systems. We then explore the error-vector magnitude close to the boundary of MP for various carrier-to-signal power ratios and up-sampling ratios, to illustrate that there is a soft transition around the MP condition. Overall, this work shows that the MP condition is not sufficient to guarantee error-free performance in noiseless conditions.