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Abstract
Silicon based Mach Zehnder modulators, unlike lithium niobate, suffer from nonlinear pattern-dependent behavior beyond simple intersymbol interference. We experimentally demonstrate a novel predistortion method based on the iterative learning control (ILC) technique to address this issue using quasi-real-time adaptation with hardware-in-the-loop. We compare bit error rate performance to that of linear solutions at several M-QAM modulation levels and baud rates. We demonstrate 256QAM at 20 Gbaud which linear compensation alone cannot achieve. For 40 Gbaud 128QAM, we improve power sensitivity by 4.4 dB. We combine optical compensation with ILC to improve power sensitivity by ∼ 5 dB for 60 Gbaud 32QAM.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The volume of data processed in data centers is rising exponentially. Videos and other files uploaded via social media, must be stored and processed in the cloud. To keep up with the expanding need for bandwidth, communications technology in data centers must be updated at low cost and low energy consumption. Silicon photonic (SiP) technology can provide the required bandwidth, at smaller footprint, lower cost, and higher power efficiency than other technologies. The cost can be especially attractive when SiP subsystems are integrated with electrical circuitry on a single chip [1,2].
The capacity of SiP technology has been demonstrated for various types of modulation. Single carrier demonstrations include 84 Gbaud 16QAM and 70 Gbaud 32QAM with BER below the 20% forward error correction (FEC) threshold [3]. Wavelength-division multiplexing (WDM) demonstrations with a silicon photonic transmitter have focused on low symbol rate (four-channels each at 28 Gb/s) [4]. The receiver design complexity of single carrier and multi-carrier are similar, hence single carrier demonstrations are sufficient for proof of concept [6] at aggressive rates/modulation orders; we also focus on single carrier demonstrations.
Increasing symbol rate (baud rate) and modulation order are the path to increasing capacity, but they can pose technical challenges which limit end-to-end performance. The symbol rate is constrained by the electrical bandwidth limitations of digital-to-analog converters (DAC), RF amplifiers, and electro optical (E/O) converters used with external modulators. Pushing to higher baud rate despite limited bandwidth leads to intersymbol interference (ISI), as the transmitter introduces memory to the channel. The combination of ISI and nonlinearity causes the sampled received signal to cluster (ISI effect) at constellation points that are not aligned on a rectangular grid (nonlinearity); this is known as constellation warping. [8]. These combined effects (ISI and nonlinear) create distortion that cannot be entirely compensated by a linear equalizer [9].
One source of nonlinearity is the Mach Zehnder modulator (MZM) transfer function when driven at high peak-to-peak RF voltage. SiP modulators suffer from an additional source of nonlinearity. Unlike lithium niobate devices, phase shifters in SiP modulators are not linear in applied voltage [10]. This leads to the nonlinear pattern dependent distortion in signals generated by SiP.
To mitigate pattern dependent distortion, a wide range of linear and nonlinear predistortion methods have been reported. In [11], a Volterra series was used to model a predistorter. An indirect learning architecture (ILA) method was used to estimate the coefficients of that model in the presence of transmitter I/Q skew and nonlinear effects. The ILA method first estimates the parameters of a post-distortion module. The post-distortion module is placed in a parallel feedback path with the pre-distortion module. In order to force the pre-distortion and post-distortion modules to behave identically, an adaptation takes place to minimize the error between outputs of pre- and post-distortion modules.
The ILA has also been used with a memory polynomial model (reduced complexity vis-a-vis a Volterra series) [12]. In order to avoid the two step method with ILA estimation of coefficients, a direct learning architecture (DLA) approach was proposed In [13]. The DLA uses the input signal and the calculated error to estimate predistorter coefficients.
Lookup tables (LUT) [5] are another well-known technique to mitigate pattern-dependent distortion in high data rate transmission [6], as well as nonlinear distortion introduced by transmitter components. A LUT with memory depth of L for M-QAM signaling has a predistortion for each of ML unique symbol sequences. Typically, sliding windows of L symbols from the recovered received signal are exploited to build LUT entries [5]. The common problem associated with LUT is the size of the table. At high M, even moderate memory length L can lead to a large table size [7].
Recently an iterative learning control (ILC) method, well known in control theory applications, was proposed to linearize power amplifiers [14, 15]. Unlike other adaptive methods, such as (ILA/DLA), no estimate or identification of a predistorter model is necessary. The ILC approach finds a non-unique input signal which leads to the desired symbol sequence output [16]. While not required, knowledge of the system model can improve ILC convergence speed. In essence, the ILC method iteratively finds the optimum input signal resulting in the desired output, thus linearizing the system.
An ILC approach reduces complexity and offers faster and more robust adaption of a wide variety of nonlinear systems with memory. The ILC is typically used for repetitive processes. To extend this technique to a random data sequence, we need a mechanism to produce the predistorted version of the sequence in real-time. Having knowledge of the ideal predistortion for a training sequence, several nonlinear solutions (e.g., polynomial model with memory) can be used to obtain a predistorter module. Our paper focuses on the ILC adaptation. We apply the ILC method to linearize a SiP modulator using advanced modulation formats with coherent detection.
This paper is organized as follows. Section 2 describes the ILC method [14] in greater detail as applied to SiP IQ modulators. Section 3 presents the measurement set-up for SiP IQ modulator hardware-in-the-loop experiments to adaptively find the optimum predistorted driving signal. To the best of our knowledge, this is the first time that ILC-based predistortion is used to linearize an optical system and extends our preliminary results in [18] The measurement results for various baud rates and modulation orders are presented in section 4 and discussed. We offer concluding remarks in section 5.
2. Iterative learning control (ILC)
We exploit the ILC method in the time domain, illustrated in Fig. 1, where X̱k is the transmitted data block and Y̱k is the received data block at the kth iteration in the adaptation. The desired received data block is a fixed, known training sequence Ḏ. The received signal is normalized to have average symbol power equal to that of the desired sequence Ḏ (i.e., to standard IQ coordinates ±1±1i, etc). The ILC iteratively transforms the transmitted data, X̱k so that Y̱k converges to Ḏ. The transmitted data is updated at each iteration using error block, e̱k formed by the difference of Y̱k and Ḏ. The learning controller updates the transmit data X̱k+1 for the next iteration. To calculate the error block, the received data block, Y̱k, must be time synchronized after optical frequency offset compensation and carrier phase recovery with the training sequence Ḏ, as detailed in section 2.
One of the first ILC methods was formulated by Arimoto in control theory [16,17]. He defined the ILC learning controller block (ILC law) as a recursive function called F (·)
Several functions can be used for F (·), depending on the application [16]. In [14] the linear, gain-based and Newton ILC laws were investigated for RF power amplifier linearization. In this work, we employ a modified version of the gain-based law, G-ILC, to compensate pattern dependent distortion introduced by a SiP IQ modulator. Let the input signal at iteration k be given by the n-dimensional column vector X̱k ∈ ℜn×1 where where xk (i) is the predistorted version of the ith symbol in the training sequence Ḏ. The error column vector e̱k is the difference of Y̱k and Ḏ. We define Γk ∈ ℜn×n as a learning gain matrix where the function G is an appropriate gain vector for the transmit signal, X̱k. We choose the sample-by-sample gain function With these definitions, (1) can be expressed as where α is the step size. We adopt a variable step size to enhance the convergence speed, changing once or twice during adaptation depending on the amount of distortion. For the case of the identity matrix, i.e., Γk = In ∈ ℜn×n, we have a linear ILC, which is a special case of G-ILC, given by We tested and compared the choice of Γk = In vs. nonlinear Γk defined in Eqs. (4) and (5). We found convergence speed was greatly enhanced for the nonlinear version of Γk. In the case of higher modulation order, i.e., 256QAM, our choice of G-ILC (nonlinear Γk) has sufficient degrees of freedom to combat nonlinear effects, while exhibiting stable convergence. Linear Γk (Γk = In) could not achieve minimal error in any reasonable time. The predistortion adaptation can be initialized with a linear equalizer such as a minimum mean square error (MMSE) equalizer.3. Experimental set-up
We run the G-ILC adaptation as hardware-in-the-loop in quasi-real-time using offline digital signal processing (DSP). Figure 2 gives the experimental set-up. The system under test illustrated in Fig. 1 would ideally be the SiP modulator alone. However, in our back to back experiment, the system under test becomes the entire transmitter (DAC, driver, modulator) as well as the receiver used to access the optical signal at the modulator output. Therefore, all components in Fig. 2, from the DAC to the real-time oscilloscope (RTO), are essentially part of the system under test. At the transmit side, we use a pseudo random bit sequence (PRBS) of order 19 to generate a M-QAM signal with Gray mapping where M =32, 64, 128 or 256. A 5000-symbol subset of the QAM signal is conserved as the training sequence Ḏ. We repeat the training sequence enough times to fill DAC memory (32 k of 8 bit samples).
Fig. 2 Block diagram of DSP and experimental set-up and feedback loop for the G-ILC predistortion method.
To initialize the adaptation, the first iteration applies a 200-tap MMSE equalizer for the training sequence Ḏ. The MMSE equalizer compensates linear distortion and initiates the G-ILC for faster and more stable convergence. The MMSE predistortion X̱1 is used in the update equation for X̱2 (5).
Before transfer to the DAC, the signal is upsampled to 20 samples per symbol and pulse shaped with a root raised cosine filter with roll-off of 0.01. The shaped pulse is resampled to accommodate the 84 GSa/s Fujitsu DAC (8-bit, 18 GHz) for operation at the desired baud rate; 20, 40 and 60 Gbaud are tested at various times. The resampled symbol sequence is clipped to X̱ before being uploaded to the DAC.
The two DAC outputs, corresponding to the in-phase and quadrature components of M-QAM signal, are passed through tunable RF phase shifters (PS) to synchronize them in time. They are finally amplified with two RF power amplifiers (SHF, 50 GHz, 18 dBm) to achieve a 5 V peak-to-peak swing. We use an external cavity laser (ECL) with output power of 10 dBm to generate a continuous wave carrier at 1530 nm. A high power erbium doped fiber amplifier (EDFA) boosts the laser power to 22 dBm to overcome coupling losses (9 dBm) in working with the SiP modulator chip (8 dBm modulator loss along with 3 dBm on-chip adiabatic 50:50 coupler). We use a ground signal-signal ground (GS-SG) configured RF probe and operate the SiP IQ modulator at the null point. The SiP IQ modulator has a single drive, push-pull configuration that is reverse biased. We measured the 3 dB bandwidth to be approximately 20 GHz and Vπ to be approximately 7.25 V. The operating point and bias mode are controlled through DC voltage sources.
The optical output of the SiP IQ modulator is amplified by a two stage EDFA to overcome coupling losses, and an optical bandpass filters (OBPF) suppress out-of-band amplified spontaneous emission (ASE) noise. A waveshaper (a programmable optical filter) is used as an OBPF at 20 GBaud and 40 GBaud; at 60 GBaud it serves as optical pre-emphasis. We use a variable optical attenuator (VOA) to sweep received optical power. A discrete coherent receiver (CoRx) with 70 GHz bandwidth and a local oscillator (LO) with 18 dBm power are used for reception. Electrical outputs are digitalized by a 160 GSa/s, 60 GHz front end, real-time oscilloscope (RTO) from Keysight.
4. Digital signal processing
While adaptation takes place during repeated transmission, the receiver DSP in Fig. 2 is applied offline, hence we refer to our approach as only “quasi” real-time. We use a fourth order Butterworth low-pass filter (LPF) to suppress noise at the RTO sampling rate. We resample to one sample per symbol. We perform FFT-based frequency offset compensation (FOC) and a blind phase search with 64 test angles for carrier phase recovery (CR).
During adaptation, we suppress additive white Gaussian noise (AWGN) by averaging over ten received copies of identical data blocks at each iteration before calculating the error block. All ten frames being averaged are time synchronized with respect to the transmitted training sequence. Suppression of AWGN achieves more stable convergence.
We tested many scenarios to find the optimum iteration number. Ten iterations are sufficient to reach asymptotic levels of the mean squared error e̱k. As mentioned earlier, we use an adaptive step size to enable smooth and fast convergence. We stepped through several values of α in (4), depending on the distortion level. We found the optimum values for α by trial and error. For example, for 64QAM at 40 Gbaud, we started the G-ILC with α = 0.5, and reduced it to α = 0.1 after three iterations.
The updated predistorted signal after G-ILC, X̱k+1, is passed to the TX DSP block. This updated signal is upsampled to 20 samples per symbol (sps), pulse shaped and resampled.The maximum swing of the pulse-shaped signal varies at each iteration. We therefore implemented a variable clipping ratio given by
where ΔVk+1 indicates the maximum swing of the predistorted pulsed shaped signal at iteration (k +1), i.e., Finally, the clipped, predistorted data is uploaded to the DAC. Note that the MMSE equalizer is only used during the first iteration. The G-ILC updates the predistorted data on a symbol-by-symbol basis. Symbol-by-symbol refers to each element of the block being updated via a separate error calculation. That is, instead of one scalar error ek for the kth block, there is a vector error e̱k with a separate adaptation for each element (or symbol). The complete adaptation (ten iterations of data acquisition, precompensation adaptation, and retransmission) would take on average 15 minutes in the experimental setup.5. Experimental results and discussion
The experiment was run for three baud rates at various modulation levels, each yielding an acceptable bit error rate. At 20 Gbaud we transmitted M-QAM for M = 32, 64, 128 and 256), at 40 Gbaud M = 32, 64 and 128, and at 60 Gbaud we transmitted only 32-QAM. Received constellations are plotted. The bit error rate (BER) was estimated over 12 blocks (5000 symbols per block) of data, without noise averaging. We used a VOA before the CoRx to sweep BER as a function of received optical power. We investigate BER with and without the use of a receiver side 500-tap MMSE filter for post-compensation. The MMSE postcompensation filter is updated at each iteration to help the G-ILC convergence. At 60 Gbaud, we also use an optical pre-emphasis stage to provide robust convergence.
5.1. Electrical compensation alone
At moderate baud rates, 20 Gbaud and 40 Gbaud, electrical compensation alone was sufficient to achieve convergence in the G-ILC. We present constellation diagrams at 2 dBm received power for various compensation strategies. First we consider no G-ILC, but instead use typical linear compensation techniques. We consider linear MMSE predistortion at the Transmitter side (a), linear postcompensation at the receiver side, in addition to the linear MMSE predistortion (b). Next, we apply G-ILC at the transmitter without any postcompensation (c) and with linear postcompensation (d). The case of 20 Gbaud 256QAM is presented in Fig. 3, and 40 Gbaud 128QAM is presented in Fig. 4. Note that the standard MMSE equalizer for predistortion in (a) is also the initial input signal, X̱1, for G-ILC (c). Coloring is used in the constellations to show density in the constellation, with black to red to yellow showing the transition from lowest to highest density of samples.
Fig. 3 Received constellations at optical power of 2 dBm. (a–d) 20 Gbaud/256QAM: (a) linear predistortion at TX, (b) linear predistortion at TX and linear postcompensation at RX, (c) G-ILC predistortion at TX, and (d) G-ILC predistortion at TX and linear post-compensation at RX. Black-red-yellow is the transition from lowest to highest density of samples.
Fig. 4 Received constellations at optical power of 2 dBm. (a–b) 40 Gbaud/128QAM: (a) linear predistortion at TX and linear postcompensation at RX, and (b) G-ILC predistortion at TX and linear post-compensation at RX. Black-red-yellow is the transition from lowest to highest density of samples.
When comparing (a) to (c) we see the improvement when using nonlinear techniques at the transmitter. The performance improvement from (a) to (b) is matched by the improvement from (c) to (d). This indicates that the linear postcompensation is effective for both linear and nonlinear precompensation. Comparing (b) and (d), clearly the greatest improvement is achieved when combining G-ILC with postcompensation. The constellation using G-ILC suffers from less distortion and points are more tightly packed.
Also, it can be observed in Fig. 4 that increasing baud rate up to 40 Gbaud severely introduces ISI due to the bandwidth constraint of DAC and modulator which limited us to scale down the modulation order to 128QAM. Though, the constellation plots are still highly affected by nonlinear distortion induced by ISI. The pattern dependent distortion is visible in the phase and amplitude distortion in (a). The use of G-ILC in (b) leads to much more uniform constellations.
Mitigating pattern dependent distortion can also be validated through BER curves as a function of received optical power. Figure 5 (a, b) give the BER versus received optical power for 256QAM at 20 Gbaud and 128QAM at 40 Gbaud, respectively. In each plot, the four constellation scenarios are compared.
Fig. 5 BER performance versus optical received power for a) 256QAM at 20 Gbaud, and b) 128QAM at 40 Gbaud which correspond to constellation plots in Figs. 3 and 4, respectively.
The BER for standard linear predistortion (MMSE pre-equalizer) is given by a dashed blue curve with triangle markers. When applying post-compensation we obtain the BER given by a solid blue curve with triangular markers. When using G-ILC in-the-loop we record BER in a dashed black curve with circle markers; results when applying post-compensation is given in a solid black curve with circle markers. Two forward error correction (FEC) threshold levels are provided: target BER =2.4 × 10−2 for FEC with 20% overhead, and target BER =3.8 × 10−3 for FEC with 7% overhead.
The G-ILC predistortion outperforms linear predistortion with or without the use of post-compensation. The MMSE techniques, at transmitter or receiver, can only compensate the linear distortion. The G-ILC not only compensates the linear impairments, but it can also deal with nonlinear distortion. In Fig. 5(a), two solutions are available for the 20% overhead FEC: 1) a combination of MMSE pre- and post-compensation, or 2) G-ILC with post-compensation. When combining G-ILC with post-compensation, we improve power sensitivity by almost 4.4 dB. BER below 20% FEC threshold is barely possible with linear compensation techniques alone. At high baud rate, i.e., at 40 Gbaud for 128QAM, the nonlinear effects are more pronounced. Fig. 5(b) reports that in this case, G-ILC is again a clear winner over linear predistortion. By implementing G-ILC, we are able to fall below the 20% FEC level, which is not achievable with linear predistortion.
5.2. Electrical compensation and optical pre-emphasis
When moving to higher baud rates, 60 Gbaud, the previous techniques broke down, even at 32QAM. The limited bandwidth of the transmitter (a 3 dB bandwidth of 20 GHz at −0.75 V bias voltage [3]) created more ISI distortion. The signal corruption was too great for the G-ILC to converge. We suspected the clipping step was too severe for the adaptation. Therefore, we turned to a combination of optical and electrical precompensation used effectively in [3] when using linear compensation.
The waveshaper used as an OBPF in previous experiments was now programmed to adopt a frequency response similar to the MMSE linear compensation. As described in [3], the mix of linear MMSE in optical filtering and DSP can reduce the clipping required and allow the dynamic range of the DAC greater latitude. In our experiments we adopted one mix of optical filter and linear MMSE to initialize adaptation. The optical filter remains fixed from that point on. This filter attenuates lower frequencies more than higher frequencies to compensate the bandwidth limitation of the transmit system, and is provided in Fig. 6.
In Fig. 7 we again present constellations for the two cases of compensation techniques, this time for 60 Gbaud 32QAM, and again at 2 dBm received power. In contrast to Figs. 3 and 4, an optical filter is present in all two cases to relax demands on the digital compensation techniques. We see the same trends identified previously.
Fig. 7 Received 32QAM constellations at optical power of 2 dBm at 60 Gbaud. (a) linear predistortion at TX and linear postcompensation at RX, and (b) G-ILC predistortion at TX and linear post-compensation at RX. Black-red-yellow is transition from lowest to highest density of samples.
BER curves in Fig. 8 also verify the improvement of the performance while using G-ILC at 60 Gbaud. For instance, for a target BER =2.4 ×10−2, G-ILC accompanied by post-compensation can improve the performance by almost 5 dB with respect to the combination pre/post linear compensation. While [3] demonstrated the effectiveness of optical signal processing combined with linear compensation techniques, we show that this approach can also be used to enable compensation of nonlinear distortion through G-ILC.
5.3. G-ILC vs. linear predistortion
In Fig. 9, we represent the summary of BER measurement for M-QAM, M =32, 64, 128 and 256) over different bit rates, assuming received optical power is set at 2 dBm. BER versus bit rates are shown in three categories of 20 Gbaud (at the left side of the plot), 40 Gbaud (in the center of the plot), and 60 Gbaud (at the right side of the plot). And in each category, we show BER measurement for different orders of modulation format.
Fig. 9 BER for different modulation orders, M-QAM (M=32, 64, 128 and 256) at 2 dBm optical received power before CoRx at 20 Gbaud, 40 Gbaud, and 60 Gbaud.
From the category of 20 Gbaud, we already presented BER versus optical received power for 256QAM in Fig. 5 (a). We can observe the same trend for the rest of the modulation orders at this baud rate. 128QAM at 40 Gbaud from the second category was already shown in Fig. 5 (b) along with M =32 and 64 in Fig. 9. Finally, the last category (60 Gbaud) was already discussed in Fig. 8. Thus, we can conclude the effectiveness of G-ILC to compensate nonlinear distortion over wide range of baud rate and modulation orders.
6. Conclusion
We have discussed the implementation of G-ILC predistortion to mitigate pattern dependent distortion in a SiP IQ modulator. We have experimentally shown and verified the effectiveness of G-ILC with respect to linear compensation in quasi-real-time adaptation with hardware-in-the-loop. Back-to-back experiment has been performed for different baud rates (20 Gbaud and 40 Gbaud) and modulation formats of M-QAM (M =32, 64, 128 and 256) to present the feasibility of G-ILC in the presence of high nonlinear distortion. For 20 Gbaud and 40 Gbaud, predistortion was only carried out in electrical domain but at higher baud rate such as 32QAM at 60 Gbaud, we used optical pre-emphasis in addition to G-ILC predistion to overcome severe distortion due to bandwidth constraint. The experimental results prove the robustness of G-ILC predistortion and fast convergence at high baud rates for complex modulation formats.
Funding
Huawei Canada and NSERC (CRDPJ 486716-15).
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