Behavior Modeling and Characteristic Measurement of Key Impairments in Optical Transceivers

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

Behavior Modeling and Characteristic

Measurement of Key Impairments in Optical Transceivers

陳, 浩

https://doi.org/10.15017/2534464

出版情報：九州大学, 2019, 博士（学術）, 課程博士バージョン：

権利関係：

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Behavior Modeling and Characteristic Measurement of Key Impairments in Optical Transceivers

Graduate School of Information Science and Electrical Engineering Kyushu University

Hao Chen

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Behavior Modeling and Characteristic Measurement of Key Impairments in Optical Transceivers

Dissertation Submitted to

Kyushu University

in partial fulfillment of the requirement for the degree of

Doctor of Philosophy

in

Electronics Science and Technology

by

Hao Chen

Dissertation Supervisor: Professor Kazutoshi Kato

May, 2019

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Abstract

As optical communication system develops to require high speed, low latency, small size, low power consumption and low cost, both direct-detection system and coherent-detection system are widely developed and utilized in recent decades in various application scenarios. To improve the transmission capacity, advanced signal processing technique to mitigate the impairments of optical transceivers is as the potential solution. The basis of the efficient signal processing is the deep understanding, the accurate behavior modeling and the efficient measurement and extraction for the key distortions and impairments in the optical transceiver modules. In this thesis, I focus on the behavior modeling and characteristic measurement of wideband electrical amplitude, in-phase and quadrature-phase (IQ) modulator, digital-to-analog convertor (DAC) and analog-to-digital convertor (ADC) in the coherent and direct-detection optical transceivers.

For the nonlinearity in the wideband electrical amplifier in the optical coherent transceiver, this thesis proposes a gain-isolated nonlinear model based on Volterra series. It improves the estimation accuracy of the conventional Volterra model in estimating the nonlinear waveform as well as the nonlinear BER penalty. The proposed nonlinear model is experimentally validated under various conditions (48 cases in total) including dual polarization Nyquist/NRZ PAM2/4/8 signals corresponding to 4/16/64QAM modulation and 8 different amplifier gain conditions. The estimated nonlinearity penalty in all validated cases have small errors of less than 0.5 dB. Based on this model, nonlinearity penalty of amplifier and Mach-Zehnder modulator in dual polarization coherent optical system are further investigated by simulation.

For the IQ skew in the coherent optical transmitter, this thesis proposes a measurement method based on analyzing the shape of image spectrum when sending single sideband signal. It solves the problem of measuring the transmitter IQ skew by the transmitter itself. Experiments demonstrate that the proposed IQ skew measurement method has as high as sub-picosecond accuracy and is quite robust under various configuration conditions including deviated child and parent Mach-Zehnder modulator bias and test signal magnitudes. Multiple phase average method is proposed and experimentally validated to further reduce the measurement error fluctuations possibly interfered from the DAC imperfections.

For the optical DMT transceiver, the hardware efficient channel probing scheme based on training sequence is proposed. Based on the investigations on the performance dependence of the modulation randomness of training sequence, the proposed training sequence scheme has the repeated short PRBS on the probing subcarrier in order to reduce the hardware complexity and meanwhile the long PRBS on the interfering subcarriers to generate the correct level of inter- subcarrier interference. Experiment has validated that the proposed training sequence scheme has high accuracy considering the probing performance in both channel equalization estimation and subcarrier SNR measurement in the DMT probing stage. With the proposed training sequence scheme, an efficient and accurate DMT probing scheme with realizable and low hardware complexity enables the 100Gbps experiment transmission over 500m standard single-mode fiber.

For the distortions of DAC and ADC in optical transceiver, the measurement and behavior modeling method is proposed. Missing-tone method is proposed to measure the signal-dependent distortions in DAC and ADC. From the measured total noise and distortion spectrum, white noise, colored noise and strong narrowband interference are observed, extracted and separated. Based

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on these distortion results, sophisticated DAC and ADC model are built and the simulation platforms of two promising candidates, DMT and PAM modulation for 100Gbps optical direct- detection systems are established. Finally the tolerance of DAC and ADC distortions in the optical DMT and PAM system is investigated and compared.

This thesis focuses on the nonlinearity investigations in the high performance optical transceivers, which is different from the previous researches on fiber characters in optical communication systems. Nonlinear elements are found to need Volterra series for nonlinear modeling. Accurate and efficient measurement method and novel models are proposed and validated.

Keyword: behavior model; amplifier nonlinearity; Volterra series; in-phase and quadrature phase skew; discrete multi-tone; pulse-amplitude modulation; training sequence design

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List of used abbreviation

ADC analog-to-digital converter

ADSL asymmetrical DSL

AEQ adaptive equalization

AWGN additive white Gaussian noise ASE amplified spontaneous emission BER bit error rate

CPR carrier phase recovery

CDC chromatic dispersion compensation CI-BCH Continuously-Interleaved BCH code CP cyclic prefix and cyclic postfix

CW continuous-wave

DAC digital-to-analog converter DB sequence De Bruijn sequence DML directly modulated laser DMT discrete multi-tone

DP dual polarization

DSB double sideband

DSL digital subscriber lines DSO digital storage oscilloscope DSP digital signal processing

DUT device under test

EATT electrical attenuator

EDFA Erbium-doped Optical Fiber Amplifier FEC forward error correction

FFT fast Fourier transform

FOC frequency offset compensation

GSOP Gram-Schmidt orthogonalization process IFFT inverse fast Fourier transform

IMSR image-to-main signal ratio IQ in-phase and quadrature phase ISI inter-symbol interference LO local oscillator

LSE least square estimation

MA margin adaptive

MMA multi-modulus algorithm

MMSE minimum mean square error

MultiCAP multi-band carrierless amplitude phase modulation

MZ Mach-Zehnder

MZM Mach-Zehnder modulator

NMSE normalized mean square error NRZ non-return-to-zero

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OATT optical attenuator OBPF optical band pass filter

OFDM orthogonal frequency division multiplexing OSA optical spectrum analyzer

OSNR optical signal-to-noise ratio PAM pulse amplitude modulation PC polarization controller PRBS pseudorandom bit sequence QAM quadrature amplitude modulation QPSK quadrature phase shift keying

RA rate adaptive

RF radio frequency

RWD relative waveform difference SNR signal-to-noise ratio

SSB single sideband

TIA trans-impedance amplifier

TS training sequence

VDSL very high data rate DSL VOA variable optical attenuator WDM Wavelength Division Multiplexing

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1. Foreword

1.1. Foreword introduction

Communication techniques have been unceasingly developed in the human history dated from ancient ages, and been split into different directions such as cable communication, wireless communication, satellite communication, optical communication and so on ^[1-3]. From 2000s, the information age came, accompanied with the continuous requirements for the communication systems and techniques. High capacity, long distance, low latency and high reliability communications are urgently required by the system operations of not only the industry, the business and the transportation, but also the consumer electronics, the daily entertainments and the knowledge exchange terminals^[1-3]. Fig 1. 1 shows the trend of global Internet throughput in past years by Cisco statistics ^[4]. Global Internet traffic reached hundreds of terabits per second in 2015 and keeps increasing on. To meet such large amount of communication requirements, optical fiber communication is widely adopted as the proper solution taking the advantages of high capacity and long transmission distance.

Fig 1. 1 Global Internet traffic trend ^[4]

Because of the high carrier frequency and large optical available bandwidth compared to the electrical wave, light wave is able to carry more informations and support higher capacity transmission. Breakthroughs of optical fiber communication field occurred in 1970s. Corning Corporation drew the first low loss and commercial available fiber for communication use according to Charles Kuen Kao’s theory ^[5,6]. And Bell Lab fabricated the first semiconductor laser available for the fiber communication ^[6]. After that, optical fiber communication jumped up to a fast-developing stage. The transmission rate and the reachable distance were greatly extended in the following years. For example, WDM (Wavelength Division Multiplexing) techniques proposed by Tingye Li from Bell Lab ^[6], exponentially improved the transmission capacity, which allowed multi-channels to transmit in the single fiber in parallel. Moreover, EDFA (Erbium-doped Optical Fiber Amplifier) proposed by David Payne from University of Southampton ^[7-9], realized the direct

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optical amplification for C-band optical signal, which avoids the optical amplification with complex and low-efficiency conversion of optical to electrical to optical and specially benefits the long haul transmission. In recent years, optical communication system has been successfully developed to own high capacity, long distance and high reliability to serve almost all the application fields of human society. In the traditional telecommunications field, the global back-bone networks and metro networks are implemented mainly based on optical communication ^[10]. Newly growing fields are the data communications, which serves the systems of inter- and intra- Internet data center communications, the on-board communications and the chip-level communications, as shown in Fig 1. 2 ^[10].

Fig 1. 2 Application scenarios of optical communication systems ^[10]

One optical fiber communication system typically includes three parts: optical transmitter, fiber link, and optical receiver. Optical transmitter generates and sends the optical signal that carries the informations. Fiber link is the communication channel and transmits the optical signal through the network. The optical signal is then received by optical receiver and is recovered to the transmitted informations after the signal processing.

Fig 1. 3 Components of an optical transmitter

Block diagram of an optical transmitter is shown in Fig 1. 3. It consists of an optical source with driver circuits, a modulator, and a channel coupler. Commonly optical source is semiconductor lasers ^[1-3], which emits the optical carrier wave. Optical modulator is to modulate electrical signals on to the optical carrier wave. Although an external modulator is sometimes used, it can be dispensed in some cases, since the output of a semiconductor lasers can be directly modulated by varying the injection current. Generally, it is more cost-effective to use directly modulated laser than using the external modulator for the modulation. Channel coupler in optical transmitter is to couple the light into the optical fiber with maximum coupling efficiency ^[1-3].

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Fig 1. 4 Components of an optical receiver

Fig 1. 4 shows the block diagram of an optical receiver. It consists of an optical coupler, a photodetector, and a transimpedance amplifier. Optical coupler focuses the optical signal onto the active region of the photodetector with maximum coupling efficiency. Semiconductor photodiodes are usually used as the photodetector to convert the optical intensity into the electrical current.

The received electrical signal is then amplified by the transimpedance amplifier and output for the following signal processing and demodulation.

The characteristics of the modules in the optical transmitter and receiver determines the transmission performance of the optical communication systems. The data capacity, latency, power consumption and cost of the system are also important factors for some application scenarios.

Therefore, hot researches and efforts are still being made to further improve the communication performance based on analyzing, designing, and processing the module characteristics of optical transceivers ^[11-17].

1.2. Optical transceivers of fiber communication system

1.2.1. Direct-detection optical transceiver

Broadband services brings great attractions and interests to beyond 100Gbps transmission in short reach applications with exceptionally low power and cost restrictions. Direct-detection systems with intensity modulation are considered as a very suitable candidate for this. Its major advantages are low cost, easy handling and installation, high flexibility, and robustness. However, it comes at problems that low cost devices have small bandwidth and bad performance. In order to overcome the limitations in short-reach optical communication systems, advanced modulation formats have been researched for many years ^[18-23]. In 2012, the capacity of 112Gbps was realized by Queen’s University by polarization multiplexing and half-cycle 16QAM Nyquist-subcarrier- modulation using two sets of directly modulated lasers and driver and the receiver side polarization tracking and de-multiplexing ^[20]. Huawei realized the capacity of 102Gbps per fiber lane by multi- band carrierless amplitude phase modulation (MultiCAP) in 2013 ^[21]. Fujitsu realized the capacity of 117Gbps per fiber lane by discrete multi-tone (DMT) using commercial 10G-class devices in 2013

[22]. As a straight forward method to increase the capacity under direct-detection system, multi- level pulse amplitude modulation (PAM) was also considered by University of Cambridge in 2008

[23].

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Fig 1. 5 Block diagram of directly-detection optical system. Tx: transmitter; Rx: receiver; DSP: digital signal process; DAC: digital-to-analog convertor; DML: directly modulated laser; PD: photodetector; TIA:

transimpedance amplifier; ADC: analog-to-digital convertor.

The function block diagram of direct-detection optical system is shown in Fig 1. 5. Transmitted data is encoded and mapped into the constellation stars according to the modulation formats e.g.

DMT or PAM. It is implemented in the transmitter digital signal process (DSP) block, together with the pre-compensation and pre-processing functions. Digital-to-analog convertor (DAC) generates the modulated electrical waveforms according to the digital data. After passing the driver circuit, the electrical signal is converted to optical intensity by directly-modulated laser (DML). These blocks compose the optical transmitter of direct-detection system. The optical signal is then launched into the optical fiber link to transmit. At the optical receiver, the optical intensity is received by the photodetector with the amplification of the following transimpedance amplifier.

Analog-to-digital convertor (ADC) converts the electrical waveforms into digital samples. The information data is finally recovered by the receiver DSP block with processing algorithms and demodulation units. The processing algorithms includes synchronization, channel equalization, distortion compensation and others ^[18-23]. For example to combat the channel fading, in optical DMT system there is usually the frequency domain subcarrier equalization, and in optical PAM system there is usually feed forward equalization or decision feedback equalization.

1.2.2. Coherent-detection optical transceiver

Coherent-detection optical communication techniques have emerged as the fine solution for the long-haul optical fiber communication systems due to their high receiver sensitivity, high special efficiency and high tolerance to link chromatic dispersion, compared with direct-detection systems. The optical coherent techniques were studied early in 1980s ^[1]. A series of techniques were proposed afterwards to make the transmission capacity of single fiber increase ten times every four years ^[24]. In 2011s, 100Gbps coherent optical communication systems based on advanced modulation formats were realized and commercially used ^[25-26]. However, as the Internet data traffic exponentially increases, transmission capacity of optical communication systems still have severe challenges. The super channel of beyond 1Tbps transmission rate per fiber lane is needed. Therefore many recent researches have focused on increasing the capacity of optical communication system by using high baud rate and high order modulation formats in coherent- detection optical system ^[27-29].

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Fig 1. 6 Coherent detection scheme of optical receiver

The basic idea behind coherent detection consists of combining the received optical signal coherently with a continuous-wave (CW) optical field before the photodetector, as shown in Fig 1.

6. The CW signal is generated locally at the receiver by a narrow line width laser, called local oscillator (LO), a term borrowed from the radio and microwave literature ^[1]. The photodetector converts the combined optical signal and outputs the electrical signal. The optical intensity of the combined signal can be expressed as the following equation:

( ) _s _LO 2 _{s LO} cos(2 _s _LO)

P t



P



P



P P 

 

f 





Eq 1. 1

Here P_s and _s are the intensity and phase of received optical signal, and P_LO and _LO are the intensity and phase of LO signal, _f _ f₀ _ f_LO is the frequency offset between the optical carrier signal and LO signal. And typically, it has P_LO _ P_s.

The main advantages of coherent-detection scheme is evident from Eq 1.1. Denoting the average power of received optical signal by

P

_s , when the LO phase is locked to the received optical signal phase so that  _s= _LO and 

f

 0, the average electrical power of the signal after photodetector, compared to that in direct-detection scheme, is multiplied by a factor of

4 P

_LO

/ P

_s with coherent-detection scheme. Since P_LO is much larger than

P

_s , such power enhancement would greatly improve the receiver sensitivity. On the other hand, because the last term contains signal phase _s obviously, the coherent-detection system is able to transmit information by modulating the phase or frequency of the optical carrier. As a result, compared to the direct-detection system, high order or complex modulation formats with high special efficiency can be adopted in the coherent-detection system to achieve high transmission capacity.

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Fig 1. 7 Block diagram of coherent-detection optical system. Tx: transmitter; Rx: receiver; DSP:

digital signal process; DAC: digital-to-analog convertor; DP: dual polarization; IQ modulator: in-phase and quadrature phase modulator; BPD: balanced photodetector; ADC: analog-to-digital convertor.

The block diagram of dual polarization quadrature-amplitude modulation optical coherent- detection system is shown in Fig 1. 7. Transmitted data is modulated on four lanes in the transmitter DSP because two orthogonal polarizations and both in-phase and quadrature phase (IQ) tributaries of each polarization of light are modulated to increase the spectrum efficiency. Possible pre- distortion and pre-processing functions are also included in the transmitter DSP ^[11]. Each lane signal passes the digital-to-analog converter (DAC) and the electrical amplifier, and then drives the modulator to emit the modulated light into fiber. These blocks compose the optical transmitter of coherent-detection optical system. The modulated optical signal is then launched into the optical fiber link to transmit. At the coherent optical receiver, full information of optical signal including real and imaginary parts on two polarizations are received by dual polarization (DP) 90-degree optical hybrid and balanced photodiodes. After analog-to-digital converter (ADC) sampling and receiver digital signal processing, the information data could be recovered.

1.3. Research objective and significance

For future data center interconnections, 5G mobile network, cloud calculation, and high definition video applications, the required features of future communication systems include high speed, low latency, small size, low power consumption and low cost. Optical communication is as the attractive and potential solution due to its natural advantages. The main challenge is how to further increase the transmission capacity, for either the coherent-detection system or the direct- detection system. Limited by the electronics development status, available photoelectric devices and fabrication conditions, practical optical transceivers always have undesired impairments that cause signal distortions and significantly degrade the signal precision and limit the transmission capacity. It is quite possible to improve the optical transceiver performance by using the advanced signal processing techniques based on deeply understanding, accurately modeling and efficiently

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measuring the key impairments and distortions of the optical transceiver modules ^[27-30].

Currently, there are two common models to describe the module impairments. First one is the physical model ^[30]. One example is the rate equation model of the semiconductor laser originally from Maxwell’s equation together with a quantum-mechanical approach for the induced polarization ^[30]. However, this model is not suitable to analyze the performance degradation of a laser in optical transceiver because it is too complex with too many parameters and too complicated mathematics, and the most important reason is that it is rather indirect and difficulty to connect the model with the transmission performance penalties. And its applicability and extendibility are also the shortcomings. Second common model is the behavior model, which focuses on the module behavior to the input signal and builds the relationship of the module input and output ^[30]. Most modules’ specifications use this kind of model. Linear model with static filtering and static noise is a general behavior model for analysis.

Compared to the physical model, the behavior model is usually much simpler and more extensible. With proper approximation and conditions, accuracy of the behavior model can be acceptably high with low complexity. For the most important, the behavior model is directly related to signal distortions and smoothly evolve to the distortion mitigation method, and thereby improving the performance of optical transceiver.

However, in current stage of high speed optical transmission with complex modulation format signal, the accuracy of the traditional behavior models of optical transceiver is low not considering the impairments from nonlinear effect, electrical crosstalk, narrowband interference, and multi- path effect and so on. These may exist in the key modules in optical transceiver including DAC, driver, laser, modulators, optical front-end, electrical amplifier and ADC. These modules and its impairments are the performance limiting factors for the future optical transceiver. And detailed measurements and accurate behavior models need carefully research.

In order to accurately evaluate and mitigate the transceiver impairments, this paper focuses on the analysis, measurement and behavior modeling of the key modules and impairments in the optical transceiver. This paper try to solve some of the problems by proposing the measurement methods and novel models to characterize the key modules and impairments for the future optical transceiver.

1.4. Chapter partition and content structure

This thesis focuses on the research of the behavior modeling and characteristic measurement techniques of the key impairments and modules in optical transceiver in optical communication systems. In this thesis, I propose the accurate and low-complexity gain-isolated Volterra nonlinear behavior model for the nonlinearity characters of wideband electrical amplifier in coherent optical transceiver when input modulated QAM signals. The in-phase and quadrature-phase (IQ) skew characters in coherent optical transmitter are measured by the proposed measurement method measurement method based on image signal spectrum analysis with high accuracy and robustness in this thesis. Hardware efficient channel response and SNR curve probing schemes for DMT transceiver in optical DMT system are also proposed and validated based on training sequence design. This thesis also measures, models and analyzes the distortions of digital-to-analog convertor (DAC) and analog-to-digital convertor (ADC) in the coherent and direct-detection optical transceivers by the proposed missing-tone method. Experiment demonstrates the accuracy of the

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measured impairments and models based on all the proposed methods. Applications using the measured impairments and models to analyze and evaluate its influence for the transmission system performance are performed, which provides the guides to the system design for the optimal optical transceivers. The chapter partition and content structure are organized as follows:

In chapter 2, for the nonlinearity in the wideband electrical amplifier in the optical coherent transceiver, this thesis proposes a gain-isolated nonlinear model based on Volterra series. It improves the estimation accuracy of the conventional Volterra model in estimating the nonlinear waveform as well as the nonlinear BER penalty. The proposed nonlinear model is experimentally validated under various conditions (48 cases in total) including dual polarization Nyquist/NRZ PAM2/4/8 signals corresponding to 4/16/64QAM modulation and 8 different amplifier gain conditions. The estimated nonlinearity penalty in all validated cases have small errors of less than 0.5 dB. Based on the model, nonlinearity penalty of amplifier and Mach-Zehnder modulator in dual polarization coherent optical system are evaluated by simulation.

In chapter 3, for the IQ skew in the coherent optical transmitter, this thesis proposes a measurement method based on analyzing the shape of image spectrum when sending single sideband comb signal. It solves the problem of measuring the transmitter IQ skew by the transmitter itself. Experiments demonstrate that the proposed IQ skew measurement method has as high as sub-picosecond accuracy and is quite robust under various configuration conditions including deviated child and parent Mach-Zehnder modulator bias and test signal magnitudes.

Multiple phase average method is proposed and experimentally validated to further reduce the measurement error fluctuations possibly interfered from the DAC imperfections.

In Chapter 4, for the optical DMT transceiver, the hardware efficient channel probing scheme based on training sequence is proposed. Based on the investigations on the performance dependence of the randomness of training sequence, the proposed training sequence scheme has the repeated short PRBS on the probing subcarrier in order to reduce the hardware complexity and the long PRBS on the interfering subcarriers to generate the correct level of inter-subcarrier interference. Experiment validates that the proposed training sequence scheme has high accuracy considering the probing performance in channel estimation and SNR measurement in the DMT probing stage. With the proposed training sequence scheme, an efficient and accurate DMT probing scheme has realizable and low hardware complexity enabling the 100Gbps experiment transmission over 500m standard single-mode fiber.

In Chapter 5, for the distortions of DAC and ADC in optical transceiver, the measurement and behavior modeling method is proposed. Missing-tone method is proposed to measure the signal- dependent distortions in DAC and ADC. From the measured total noise spectrum of DAC and ADC, white noise, colored noise and strong narrowband interference are observed, extracted and separated. Based on these distortion results, sophisticated DAC and ADC model are built and the simulation platforms of two promising candidates, DMT and PAM modulation for 100Gbps optical direct-detection systems are established. Tolerance of DAC and ADC distortions in the optical DMT and PAM system is evaluated and analyzed using the simulation platform.

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2. Nonlinear behavior modeling and measurement of wideband electrical amplifier in optical transceiver based on Volterra series

2.1. Foreword of this chapter

For the future high capacity coherent optical transmissions, high baud rate and high order multi-level modulation formats are considered as the potential directions, such as based on 64- or 100-giga-baud 64QAM ^[27-29]. As shown in Fig 1. 7, for such systems, the electrical amplifier with large bandwidth of tens of gigahertz is one important component which amplifies the electrical signal to an optimal level to drive the electro-optical modulator in the transmitter and amplifies the small magnitude received signal with low noise character in the receiver. However, it is well known that the electrical amplifier is always not linear, which generates nonlinear harmonics distortions and causes waveform compression or asymmetries, and then degrades the signal precision and the transmission performance and bit error rate (BER) ^[12-14].

In this chapter, nonlinear characteristic of a commercially-available electrical amplifier with 20 gigahertz (GHz) bandwidth was measured and modelled based on Volterra series. At the beginning, conventional Volterra model was used. However, it was observed to have as large as 1.4 decibel (dB) error in estimating nonlinearity penalty when the input signal was 32-giga-baud (GBaud) NRZ 16QAM modulation. Then, a gain-isolated Volterra model is proposed, which multiplies a correction factor related to signal power onto the nonlinear terms of the Volterra series.

Experiment validated that the proposed model has less than 0.5 dB estimation error, even when the electrical amplifier gain changed over the whole operation range (11.6 dB ~ 18.6 dB) and the input signal types cover PAM2/4/8 corresponding to 4/16/64QAM modulation with Nyquist and NRZ pulse shaping. Using the gain-isolated Volterra model, the performance penalty induced by the electrical amplifier nonlinearity was simulated and compared with the modulation nonlinearity of Mach-Zehnder modulator (MZM), which is another important but much simpler nonlinearity source in coherent optical transmitter.

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2.2. Nonlinearity of electrical amplifier in optical transmitter

Fig 2. 1 Block diagram of QAM optical transmitter. QAM: quadrature amplitude modulation Quadrature amplitude modulation (QAM) optical transmitter, supporting the high order modulation, such as 16 and 64QAM, are widely used in optical coherent-detection systems ^[35-36]. The block diagram of QAM optical transmitter is shown in Fig 2. 1. Electrical amplifier is as the driver to provide proper signal magnitude to the modulator. However, it is well known that the electrical amplifier is nonlinear, which generates nonlinear harmonics distortions and causes waveform compression or asymmetries.

Fig 2. 2 Relationship between input and output with nonlinearity

As shown in Fig 2. 2, the nonlinearity of electrical amplifier causes the amplification gain varying with the input voltage, and the relationship between input voltage and output voltage is no longer linear. Generally small signal gain is taken as the linear gain, and for the large signal case, the gain is compressed and deviated from the linear gain. Fig 2. 2 shows the curve of electrical amplifier input and output relation with nonlinearity. The dash line represents the linear gain case, and the solid line is the nonlinear gain case. When the efficient signal spectrum is within the amplifier pass band and the signal bandwidth is smaller than the amplifier pass band width, the output signal waveform shape would not change. It is considered no memory effect for the input signal. In this case, one input voltage value has only one corresponding output voltage value, as shown in the left figure of Fig 2. 2. When the signal bandwidth is comparable with or smaller than the amplifier pass band width, the current output voltage value depends on not only the current input voltage but also previous input voltages. In this case, one input voltage value could have multi-output voltage values, as shown in the right figure of Fig 2. 2. It is considered that the input signal suffers nonlinearity with memory effect.

To model the amplifier nonlinearity without memory effect, there are several methods. For

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example, the left figure of Fig 2. 2 provides an input-output curve model for the memoryless nonlinearity. Memoryless polynomial model based on measured harmonic distortions of sine wave was used in ^[38]. Generalized memory polynomial model which was proposed for radio frequency (RF) power amplifier was used in digital pre-distortion to linearize the optical transmitter ^[39-40]. Pre- measured look-up table containing the nonlinear input-output relationship was used for nonlinear pre-distortion in ^[41]. Volterra model is widely used in RF power amplifiers where the amplifier bandwidth is much lower than the carrier frequency ^[42-44].

However, for the electrical amplifier in the optical transmitter with high baud rate signal e.g.

32Gbaud, 64Gbaud, 100Gbaud ^[27-29], the pass band width of electrical amplifier in practical use is not enough. Long memory effect and strong inter-symbol interference exists in the transmitted signals, which significantly raises the difficulty of measuring, modeling, evaluating and compensating the nonlinear distortions.

2.3. Volterra nonlinear modeling for commercial electrical amplifier

2.3.1. Methodology for nonlinearity modeling

Fig 2. 3 Experiment setup to measure and model the electrical amplifier nonlinearity. DUT: device under test; EATT: electrical attenuator; DSO: digital storage oscilloscope

Experimental setup for measuring and modelling the nonlinearity of electrical amplifier is shown in Fig 2. 3. The device under test is one typical commercial-available amplifier with 20 GHz bandwidth and adjustable gain from 11.6 dB to 18.6 dB, whose type number is Inphi IN3214SZ.

Noise-like signal with Gaussian distribution is used as the test signal because it covers all the concerned signal levels in time domain and spectrum components in frequency domain, which is believed able to extend the applicable range of the measured nonlinear character and the trained nonlinear model to other input signal types. 64-giga-samples per second (Gsps) sampling rate DAC is used to generate the 262144 samples test signal with 32 GHz bandwidth. An electrical attenuator (EATT) was used before the Analog-to-Digital Converter (ADC) realized by 80Gsps digital storage oscilloscope (DSO). The purpose of this is to keep similar ADC input levels under various conditions so that the ADC’s working status, such as internal gain and noise floor, were the same. Same 10MHz reference clock are applied to DAC and DSO to synchronize the sampling phase of them.

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Fig 2. 4 Receiver signal processing to train nonlinear model. ↑↓N: sampling rate conversion; SYN:

synchronization; Cal. : DAC/ADC filter response calibration; LSE: Least-square-estimation The offline signal processing of nonlinear model training is shown in Fig 2. 4. The captured data was firstly resampled to 64Gsps sampling rate based on sinc-function pulse interpolated sampling rate conversion, and then synchronized with transmitted data by correlation operation.

Because the test signal is sent by DAC in a periodic manner, the captured data could have as long as 150 periods of transmitted length. Thus the noise fluctuations in the received data could be significantly removed by simply 150 times averaging. The filter response of DAC and ADC, which were measured beforehand, could be calibrated by applying transmitted data passing through DAC filter and received data passing through inverse of ADC filter. Then, the input and output waveforms of the amplifier under test were obtained. The final step is to compare the input and output waveforms of amplifier in digital domain and train the nonlinear model coefficients by Least Square Estimation (LSE) method with the specified nonlinear model and parameters. Here the nonlinear model parameters e.g. nonlinear order number and memory length could be determined by measurement, which are discussed in latter section.

When doing modelling or training process, a notable issue is the over-training problem, which limits the trained model accurate only for the test signal or the training set while significantly inaccurate if the input is changed to the validating set with various types.

To avoid the over-training problem for amplifier nonlinearity modeling, open set validation is used in this paper. Specifically, noise-like signal is selected as the test signal in training stage, while the modulated QAM signals i.e. 4/16/64QAM with NRZ and Nyquist pulse shaping are chosen the input signals to validate the trained nonlinear models.

Fig 2. 5 Receiver signal processing to validate nonlinear model. AWGN: additive white Gaussian noise; AEQ: adaptive equalization; BER: bit error rate

Different from the model training stage, in the model validating stage, the modulated QAM signals of 4/16/64QAM with NRZ and Nyquist pulse shaping were as input signals in the experiment.

Experimental setup was exactly same with Fig 2. 3. To validating the accuracy of nonlinear model, waveforms and BERs measured by experiment are taken as the reference. MATLAB-based offline

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signal processing for the model validating are shown in Fig 2. 5. After sampling rate conversion, waveform synchronization and noise removal, amplifier output waveform from measurement could be compared to that from model output. For BER comparison, additional white Gaussian noise was added to both output waveforms and adaptive linear equalization of 31 taps was applied to demodulate and decode the transmission bits and count BER to both model output and measured amplifier output signals.

Fig 2. 6 Comparison indicators of measured and model estimated nonlinear waveforms and nonlinear BER penalties

To evaluate the estimation accuracy of nonlinear model, three indicators for nonlinear waveform comparison and one for the nonlinear BER penalty comparison are defined.

For the waveform comparison, normalized mean square error (NMSE) of the model estimated waveform error is used and defined by

1 2

NLmodel

10 2

1

( ) ( )

NMSE=10log ( )

( )

measure N

y n y n

y n







^{Eq 2. 1}

Here N is the waveform sample length, y_measure is the waveform from the experiment measurement, and y_NLmodel is the estimated waveform by the nonlinear model. The lower the NMSE is, the more accurate the estimated nonlinear waveform of the model is.

Nonlinear correlation coefficients (Rho) and Relative Waveform Difference (RWD) of the model estimated waveform are also defined to compare the nonlinear waveform in different aspects and expressed by

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1

NLmodel

2 2

1 1

NLmodel

[( ( ) ( )) ( ( ) ( ))]

rho=

( ( ) ( )) ( ( ) ( ))

linear measure linear

N

N N

y n y n y n y n

  

  

^{Eq 2. 2}

1 2

NLmodel

10 1 2

[( ( ) ( )) ( ( ) ( ))]

RWD=10log (

( ( ) ( ))

N

measure linear

N

y n y n y n y n

y n y n

  



 

^{Eq 2. 3}

As can be seen, rho is the correlation coefficient between the measured nonlinear waveform and the estimated one by nonlinear model, which describes the similarity degree of the estimated nonlinear waveforms by the nonlinear model referring to the measurement. And RWD is to describe the normalized nonlinear waveform error power of nonlinear model referring to the measurement. Please note that to make the rho and RWD value credible, the accurate linear response or waveform y_linear is needed.

For the nonlinear BER penalty comparison, BERs from the experiment measurement, the linear model without nonlinearity and the nonlinear model were counted as shown by Fig 2. 6. As in the figure, input QAM signal and loaded noise for the experiment measurement, the linear model without nonlinearity and the nonlinear model are same, and demodulation are also same with same adaptive equalization methods. The only difference for the three is the generation type of the nonlinearity. Nonlinear BER penalty is defined as

( ) ( ) ( )

NLpenalty NL linear

Q dB  Q dB  Q dB

Eq 2. 4

And

( ) 20 log ( 2

10

(2 ))

Q dB _ _ erfcinv BER

_{Eq 2. 5}

Here BER was transferred to Q factor for comparison because Q factors relate to signal-to-noise ratio (SNR) in dB unit, and nonlinear Q factor penalty was calculated by the Q factor in nonlinear case directly subtracting that in linear case in dB unit. And erfcinv(∙) is the inverse of complementary error function. Measured nonlinear Q penalty is from practical amplifier output waveform in experiment, while model estimated Q penalty is from built nonlinear model output waveform.

2.3.2. Nonlinearity modeling with Volterra series

1

2 2

3 3 3

( 1)/2 (1) ( 1)/2

( 1)/2 ( 1)/2

(2) ,

( 1)/2 ( 1)/2

( 1)/2 ( 1)/2 ( 1)/2

(3) , ,

( 1)/2 ( 1)/2 ( 1)/2

( ) ( )

( ) ( ) ( )

+...



 

 

   

  

     

  

   



   



 

  

N

k

k N

N N

k l

k N l N

N N N

k l m

k N l N m N

y n h x n k

h x n k x n l

h x n k x n l x n m

Eq 2. 6

Equation of Volterra series is given by Eq 2.6. Here

x n

( ) and

y n

( ) are input and output

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15

signals of nonlinear system in the discrete time domain, as shown in Fig 2. 6. As can be seen, Volterra series is to model the system transfer function by high order polynomial with full memory terms. Here N₁ is the linear response length, and N₂, N₃ are the truncated memory length of 2^nd, 3^rd order nonlinear terms separately. When transformed to nonlinear compensation apparatus, N1 , N₂ and N₃ are corresponding to the tap number of the nonlinear equalizer.

Correspondingly,

{ h

⁽¹⁾

}

are linear response coefficients, and

{ h

⁽²⁾

}

_,

{ h

⁽³⁾

}

are nonlinear coefficients of each polynomial term which are called Volterra kernels of system and can be trained by Least-Square-Estimation (LSE) method ^[39]. However, for the amplifier in coherent optical system, the signal bandwidth is larger than 32 GHz and sampling period of nonlinear model is as small as 15.625 picosecond (ps) corresponding to the sampling rate of 64Gsps. It is obviously different from the previous Volterra-based nonlinear modeling with input signal of much narrower bandwidth e.g.

2-megahertz (MHz) and large sampling period of 250 nanosecond (ns) ^[42]. The large signal bandwidth and small sampling period would lead to significant inter-symbol interference and long memory for the nonlinearity modeling of the wideband electrical amplifier in the optical transmitter.

Volterra nonlinear model theoretically has high accuracy when nonlinear parameters e.g.

order number and tap number are properly selected. The capability to dealing with memory effect is rather suitable for the electrical amplifier with large signal bandwidth. Another advantage of Volterra model is that it naturally separates linear term and various orders of nonlinear terms, which can analyze the nonlinearity impact isolated with linear effect, and could be easily applied for nonlinear distortion compensation. What’s more, flexible truncation and selection on polynomial terms can efficiently trade off the computational complexity and the model accuracy.

There are three model parameters in Volterra series, i.e. nonlinearity order, nonlinear tap number, and nonlinear coefficients. These parameters could be different case by case related to the specific input signal and device under test. Therefore, extra measurements are needed to determine the values of these parameters.

Fig 2. 7 Electrical amplifier gain value versus gain control voltage

The measured electrical amplifier gain curve versus the gain control voltage is shown in Fig 2.

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7. As can be seen, the largest amplifier gain is 18.6dB when gain control value is 1.98V. As a start, the electrical amplifier with the largest gain setting 18.6dB is chosen as the basic case to build the nonlinear model because it is the most serious case of nonlinearity. The model parameters of the Volterra-based model i.e. nonlinearity order number, nonlinear memory length, needs determinations before training the coefficients were firstly extracted from the measurements. The nonlinearity order number was judged by harmonics measurement. The nonlinear tap number was determined by increasing the values until the estimated waveform error between the nonlinear model and experimental measurement saturated. As discussed previously, the nonlinear coefficients were trained by least square estimation (LSE) method using the wideband but known noise-like signal.

2.3.3. Results of amplifier nonlinear model based on conventional Volterra series

Fig 2. 8 Nonlinearity order number is measured up to the third order by input 1 GHz sine wave In the model training stage, to measure the required nonlinear order number, 1 GHz sine wave with peak-to-peak voltage 0.5 volt was sent into amplifier, the output harmonics power were recorded and shown in Fig 2. 8. In the figure, harmonics powers of 2^nd, 3^rd, 4^th, 5^th nonlinearity order are normalized by fundamental wave power and shown in unit of dBc. When amplifier gain set 18.6 dB compared to 11.6 dB, 4th and 5th order harmonics are almost unchanged, while 2nd and 3rd order increases larger than 7dB. From this, the measured nonlinear order number is selected up to 3rd order.

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17

Fig 2. 9 Tap number (or memory length) of the second and third order nonlinearity are measured by input noise-like signal. a) Coefficients number and NMSE value; b) Waveform error trend. Here ‘11&9’

in horizontal axis means that 11 taps assigned for the second order and 9 taps assigned for the third order nonlinearity, and others are similar

To measure the required tap number (or memory length) of the second and third order nonlinearity, test signal of noise-like signal was input to amplifier with gain setting 18.6dB. Tap number were swept until the model output waveform error denoted by NMSE stopped decreasing, which was shown in Fig 2. 9. From the result, tap number of 11 samples and 9 samples are selected for the 2nd and 3rd order nonlinearity separately while sampling period is 15.625 ps.

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Fig 2. 10 Nonlinear coefficients of Volterra model of electrical amplifier at gain 18.6 dB. a) The second order nonlinear coefficients; b) The third order nonlinear coefficients

With selected nonlinear parameters, nonlinear coefficients of Volterra model were trained by standard LSE method. There are 66 coefficients for 2nd order nonlinearity and 165 coefficients for 3rd order nonlinearity. Parts of these coefficients are shown in Fig 2. 10. Coefficients at time index 0 represents the contribution from current sample, while those at non-zero time indexes represent the nonlinearity memory effect contributed by neighboring samples. Significant memory effect can be observed from these nonlinear coefficients.

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19

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Fig 2. 11 Waveform estimation performance denoted by a) NMSE, b) rho and c) RWD for the conventional Volterra nonlinear model at amplifier gain 18.6dB

In the model validation stage, with the signal processing as shown in Fig 2. 5, nonlinear waveform estimation accuracy of the conventional Volterra nonlinear model could be evaluated.

The indicators are NMSE, rho and RWD. The validation input signals are QAM modulated signals with Nyquist/NRZ QPSK/16QAM PRBS/DB modulation. Here Nyquist/NRZ are pulse shaping shape, QPSK/16QAM are modulation format type, and PRBS/DB are information bit source where ‘PRBS’

is random bit sequence generated by MATLAB function and ‘DB’ is De Bruijn sequence.

Results of nonlinear waveform estimation performance are shown in Fig 2. 11. With the input validation signals, it is obvious that waveform error NMSE of conventional Volterra model improves about 2.5dB compared with the pure linear model with no nonlinearity. The nonlinear waveform correlation coefficient rho is larger than 0.85. With the NMSE and rho results, one can see the nonlinear model is more efficient and accurate to model the nonlinear waveform than the pure linear model. It implies that it may be used to compensate the nonlinearity. However, RWD value shows unexpected bad, which is as large as -3dB to -1dB. By overall consideration of NMSE, rho and RWD, one can see that the shape of the nonlinear waveform has been accurately but the power has not been estimated well. It reminds to improve RWD value and waveform estimation performance by adding correction factor because that rho and RWD are related with each other on the definitions, which is the basis of our next direction to improve the accuracy of conventional Volterra model in the following parts.

For the nonlinear BER penalty estimation, signal processing is also shown in Fig 2. 5. The loaded white noise power is always fixed to make the BER 2.2e-3 and Q factor 9dB under linear channel case. Because adaptive equalizer always equalizes the linear response, the measured Q degradation compared to linear case were considered as nonlinear Q factor penalty as defined by Eq 2.4.

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21

Nonlinear Q penalty between the measured one and the model estimated one is compared to evaluate the accuracy. Nonlinear Q penalty estimation error is the indicator. Here the measured penalty is from the amplifier output waveform in experiment, while the model estimated penalty is from the nonlinear model output waveform.

Fig 2. 12 Nonlinear Q factor penalty estimation by conventional Volterra model at amplifier gain 18.6dB.

NL: nonlinearity

Results of nonlinear Q penalty estimation are shown in Fig 2. 12. Horizontal axis is the validation signal type and the vertical axis is nonlinear penalty of Q factor. Significant nonlinear penalty are observed in the measured results, which shows significant difference between different input modulation types including 32GBaud 2/4/8PAM modulation corresponding to 4/16/64QAM with Nyquist and NRZ pulse shaping. NRZ pulse shows more penalty than Nyquist because it has larger power when peak-to-peak value of optical transmitter is limited. PAM8 is more sensitive to nonlinearity than PAM2 because its amplitude levels is four times of PAM2. Taking the measured results as the reference, conventional Volterra model exhibits significant estimation error that is as large as 1.4 dB for NRZ PAM4 signal and 1.9 dB error for NRZ PAM8 signal. In the following section, an improved nonlinear model is proposed to improve the estimation accuracy of conventional model.

2.3.4. Gain-isolated Volterra nonlinearity modeling method

Fig 2. 13 Proposed nonlinear model with gain isolation and power correction based on Volterra series Structure of proposed nonlinear model with gain isolation and power correction based on Volterra series is shown in Fig 2. 13. On one hand, the amplifier gain is isolated from the Volterra series, which assumes that nonlinear effect is mainly ascribed to the latter stage after variable gain.

This also makes the model available for different gain conditions. On the other hand, an additional

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nonlinear term correction factor determined by signal power after gain is added to the Volterra series. Then, transfer function of the proposed model can be expressed by

( ) ( )

z n x n Gain _{Eq 2. 7}

and

1

2 2

3 3 3

( 1)/2 (1) ( 1)/2

( 1)/2 ( 1)/2

(2) ,

( 1)/2 ( 1)/2

( 1)/2 ( 1)/2 ( 1)/2

(3) , ,

( 1)/2 ( 1)/2 ( 1)/2

( ) ( )

{

( ) ( )

( ) ( ) ( )

+...}







 

 

   

  

     

  

 

  



   



 

  

N

k

k N

N N

k l

k N l N

N N N

k l m

k N l N m N

y n h x n k

P h

x n k x n l

h x n k x n l x n m

Eq 2. 8

Here z n( ) is signal after an ideal gain,

  = P

^ is the nonlinear term correction factor, and P is the average power of z n( ), α and β are free fitting parameters to be determined. It is assumed that the relationship between signal power P and correction factor



is a fractional order power function with two parameters α, β. The designed correction factor could extend the nonlinear model to cover more nonlinear behaviors and characters of higher order polynomial terms and efficiently reduce the estimation error for both distorted waveform and nonlinear power.

Fig 2. 14 Correction factor α and β are determined as 1.16 and -0.28 separately by sending noise-like signal with different signal powers

In the experimental measurement, the correction factor γ could be measured by using the training signal such as noise-like signal with different input powers. The measured results using eight training signal powers are shown in Fig 2. 14. It is found that higher signal power requires smaller value of correction factor. It seems a bit strange because high signal power should have larger nonlinearity and require more correction. The reason is that the nonlinear coefficients for correction are trained in the highest gain or signal power case. By applying power function fitting

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23

with least squares criterion, parameters α and β were fitted as shown in Fig 2. 14 that α equal to 1.16 and β equal to -0.28 and the fitting R-square value is 99%.

2.3.5. Results of gain-isolated Volterra nonlinear model for

amplifier

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Fig 2. 15 Waveform estimation performance denoted by a) NMSE, b) rho and c) RWD for the gain- isolated Volterra nonlinear model with correction factor γ at amplifier gain 18.6dB

When fix the electrical amplifier gain 18.6dB same as previous, estimation performance of proposed gain-isolated Volterra model is evaluated.

Fig 2. 15 shows the waveform estimation results. With the gain-isolated Volterra model with correction factor γ, NMSE of nonlinear model is improved by at least 2dB compared to that with conventional model, rho value keeps unchanged larger than 0.85, and RWD value of nonlinear model is improved to similar level of optimal RWD that determined by rho that is

10 log (2 210 )

  

optimal

RWD rho Eq 2. 9

The optimal RWD is the best RWD value at given rho value on assumption that nonlinear noise power is accurately estimated.

Fig 2. 16 Waveform estimation when input 32-giga-baud NRZ PAM4 at amplifier gain 18.6 dB. a) Measured output waveform as reference; b) waveform error by conventional Volterra model and gain-

isolated Volterra model

Results of waveform estimation of the proposed model is shown in Fig 2. 16. The amplifier output waveform measured by experiment is taken as the reference. Input signal is 32-giga-baud NRZ PAM4 signal. Results show that proposed nonlinear model has higher accuracy than

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25

conventional Volterra model in aspect of estimating the nonlinear waveform, which implies that nonlinear compensation scheme based on the proposed nonlinear model would be better than that using conventional Volterra model.

Fig 2. 17 Nonlinear Q-factor penalty estimation by gain-isolated Volterra model at amplifier gain 18.6dB. NL: nonlinearity

For the nonlinear Q penalty estimation, results of proposed gain-isolated Volterra model are shown in Fig 2. 17. As can be seen, penalty estimation error are greatly reduced compared to conventional Volterra model. For all the considered modulation cases, penalty estimation accuracy of gain-isolated Volterra model is seen within 0.3 dB.

Fig 2. 18 Reference Q factors under linear case without nonlinearity for different modulations In the previous section, nonlinear model of amplifier at the highest gain is built and validated to have high accuracy. To evaluate the applicability of the proposed nonlinear model, amplifier gain were set from 11.6 dB to 18.6 dB with 1dB interval. Nonlinear Q penalty estimation evaluation setup are same as in Fig 2. 5. There are eight considered input QAM signals including Nyquist/NRZ PAM2/4/8/16 corresponding to Nyquist/NRZ QPSK/16QAM/64QAM/256QAM modulation in dual polarization coherent optical system. The loaded white noise power for each signal is individually adjusted to make the reference Q factors under linear case without nonlinearity at around 9dB, shown in Fig 2. 18.

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Fig 2. 19 Nonlinear Q-factor estimation error of gain-isolated Volterra model and conventional Volterra model

Indicator to evaluate the nonlinear model accuracy under various amplifier gain settings is the nonlinear Q penalty estimation error. It is defined as the difference between Q-factors by the nonlinear model and by the experiment measurement, are shown in Fig 2. 6.

Results of nonlinear Q penalty estimation error are shown in Fig 2. 19. Due to the impact from the residual nonlinearity, the conventional Volterra model has large penalty estimation errors of maximum 2 dB, especially at the large gain region. However, for all the signal types of Nyquist/NRZ PAM2/4/8 modulations with 8 electrical amplifier gain settings, the proposed gain-isolated Volterra nonlinear model has always less than 0.5 dB penalty estimation errors, which is significantly smaller than the conventional Volterra model. Experiment results confirm that the proposed gain-isolated Volterra model has high accuracy in estimating the nonlinearity-induced performance penalty under various amplifier gain settings, and can credibly emulate the impact of wideband electrical amplifier nonlinearity in the coherent optical transmitter.

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27

2.4. Nonlinearity impact evaluation in optical transmitter in coherent-detection optical system

Fig 2. 20 Dual polarization 16QAM/64QAM coherent system simulation setup with electrical amplifier modeled by proposed gain-isolated Volterra model

Having accurate nonlinear behavior model of the amplifier, performance penalty from electrical amplifier and Mach-Zehnder modulator nonlinearity could be simulated and evaluated entirely. Simulation platform of single wavelength dual polarization (DP) 16QAM/64QAM coherent optical back-to-back transmission system is built as shown in Fig 2. 20. Transmitted signals are DP Nyquist/NRZ 16QAM/64QAM signals of 32-giga-baud. Total transmission bit rate are 256 and 384 gigabits per second for 16QAM and 64QAM respectively.

Fig 2. 21 Nonlinear model for the electrical amplifier in simulation platform. Model parameters are from the measurement

At the transmitter side, both the amplifier and the modulator induce nonlinear distortions.

For the amplifier, the nonlinear characteristics are emulated by the proposed gain-isolated Volterra model measured from experiment, as shown in Fig 2. 21. To switch it off, linear response model measured from the same experiment is used.

For the modulator, the nonlinear input-output relation is modelled as memoryless sine function, which is the theoretical relation derived from Mach-Zehnder structure. Assuming that pi- phase-shift-voltage for direct-current and radio frequency signals are same and unchanged and balanced drive Mach-Zehnder modulator configuration is used, the complete modulator transfer function can be derived and expressed by

Behavior Modeling and Characteristic Measurement of Key Impairments in Optical Transceivers

Behavior Modeling and Characteristic

Measurement of Key Impairments in Optical Transceivers

Behavior Modeling and Characteristic Measurement of Key Impairments in Optical Transceivers

Graduate School of Information Science and Electrical Engineering Kyushu University

Hao Chen

Behavior Modeling and Characteristic Measurement of Key Impairments in Optical Transceivers

Kyushu University

Doctor of Philosophy

Electronics Science and Technology

Hao Chen

Dissertation Supervisor: Professor Kazutoshi Kato

May, 2019

Abstract

List of used abbreviation

Contents

1. Foreword

1.1. Foreword introduction

1.2. Optical transceivers of fiber communication system

1.2.1. Direct-detection optical transceiver

1.2.2. Coherent-detection optical transceiver

P t

P

P

P P 

f 



P

f

4 P

/ P

P

1.3. Research objective and significance

1.4. Chapter partition and content structure

2. Nonlinear behavior modeling and measurement of wideband electrical amplifier in optical transceiver based on Volterra series

2.1. Foreword of this chapter

2.2. Nonlinearity of electrical amplifier in optical transmitter

2.3. Volterra nonlinear modeling for commercial electrical amplifier

2.3.1. Methodology for nonlinearity modeling

y n y n

y n





[( ( ) ( )) ( ( ) ( ))]

rho=

( ( ) ( )) ( ( ) ( ))

y n y n y n y n

y n y n y n y n

  

  

  

[( ( ) ( )) ( ( ) ( ))]

RWD=10log (

( ( ) ( ))

y n y n y n y n

y n y n

  



 

( ) ( ) ( )

Q dB  Q dB  Q dB

( ) 20 log ( 2

(2 ))

Q dB   erfcinv BER

2.3.2. Nonlinearity modeling with Volterra series

( ) ( )

( ) ( )

( ) ( ) ( )

+...

  

   



   



 

  

y n h x n k

h x n k x n l

h x n k x n l x n m

x n

Q dB _ _ erfcinv BER