Abstract

In this work, we experimentally investigate the performance improvement in IM/DD systems using constellation switching (CS), which is simple to implement with a reasonably low complexity. By encoding extra bits on the selection of a PAM constellation pattern from a set of constellations, a lower symbol rate can be used to achieve the same system bit rate compared with standard PAM systems based on a single constellation pattern. In our experiments with bandwidth limited components, including a 14 GHz bandwidth digital-to-analog converter (DAC), we demonstrate that the CS signals can improve the receiver sensitivity. In particular, in the 112 Gbit/s PAM4 case, the required receiver power was reduced by 0.8 dB and 1.1 dB using the CS at the HD FEC threshold of BER = 4 × 10−3 in the back-to-back (B2B) and 3 km fiber transmission, respectively. Similarly, in the 84 Gbit/s two-dimension (2D) PAM4 case, the required receiver power was reduced by 1.05 dB and 3.5 dB at the HD FEC threshold of BER = 4 × 10−3 in the back-to-back (B2B) and 5 km fiber transmission, respectively. Moreover, we show by simulations that the improved performance of using the CS signals is also observed over a wide range of transmitter bandwidth, further indicating the merits of using the CS in IM/DD PAM transmissions.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Due to the rapidly increasing demand of bandwidth in applications such as intra-/inter- data center interconnects (DCI) in recent years, the capacity of short-range optical transmission systems has been growing continuously with a data rate up to 400 Gb/s today. Meanwhile, compared with coherent systems based on advanced modulation formats, intensity modulation with direct detection (IM/DD) is more suitable for short-range applications considering cost, footprint and power consumption [1]. 8-lane × 50 Gb/s/λ solutions have been considered for the 400 Gb/s Ethernet (400GbE) transceivers [2]. Recently, the research community moves focus towards techniques that support 100 Gb/s/λ and beyond. Several ≥100 Gb/s single-carrier short-reach IMDD transmission experiments have been reported using various modulation formats, such as pulse amplitude modulation (PAM) [3–6], multi-band carrierless amplitude phase modulation (multi-CAP) [7,8], discrete multi-tone (DMT) and Nyquist subcarrier modulation (SCM) [9–12]. Among these formats, PAM-N has attracted many attentions for short reach DCI since it only requires a low resolution digital-to-analog converter (DAC) which offers reduction of cost and power consumption. In [13], 100 Gb/s PAM-4 and 150 Gb/s PAM-8 signals have been demonstrated over a few kilometers of standard single mode fiber (SSMF) using a low resolution 3-bit DAC. Meanwhile, since the power budget and distance in short reach DCI may vary, multi-dimensional (MD) formats based on multiple time-domain consecutive PAM symbols have been investigated to optimize the capacity for specific link conditions [14–16].

On the other hand, constellation switching (CS) was first proposed to increase the spectral efficiency (SE) of wireless communications [17]. Although CS has also been comprehensively studied in coherent optical systems [18], the gain diminishes rapidly at a soft-decision forward error correction (FEC) threshold and/or in the presence of laser phase noise.

In this work, we experimentally demonstrate and study the use of the CS technique in IM/DD systems. In this scheme, transmitted data is encoded not only in the symbols selected from a constellation pattern but also in the switching between multiple constellation patterns. Two constellation sets are studied: 1) two PAM4 patterns originated from PAM8 through Ungerboeck set partitioning; 2) two 2D-PAM4 patterns [14]. By applying the CS to increase the number of bits per symbol, given the same bit rate the system can operate at lower symbol rate compared with its counterpart without the CS, and thus alleviate the requirement on system bandwidth. Moreover, the added computational complexity for the CS is reasonably low. In our experiments with bandwidth limited components, the utilization of the CS is demonstrated to enhance receiver sensitivity. Note that this work is extended from our previous work presented in [19], and more detailed analysis is conducted here with additional experimental and numerical results. The remainder of the paper is organized as follows: in Section 2, we introduce the implementation of the CS in IM/DD systems; In Section 3, after describing the experiment setup, we report extensive experimental results in comparing the system performance with and without using the CS on PAM4 and 2D-PAM4 modulation formats, respectively. Specifically, in the 112 Gbit/s PAM4 case, the receiver sensitivity can be improved by 0.8 dB and 1.1 dB using the CS at the hard-decision (HD) FEC threshold, i.e. bit error ratio (BER) = 4 × 10−3, in the back-to-back (B2B) and 3 km fiber transmission scenarios, respectively. Similarly, in the 84 Gbit/s 2D-PAM4 case, the receiver sensitivity can be improved by 1.05 dB and 3.5 dB using the CS at the HD FEC threshold in the B2B and 5 km fiber transmission scenarios, respectively. In addition, the impact of transmitter-side bandwidth is also investigated in simulations. The conclusion is drawn in Section 4.

2. Implementation of constellation switching in IM/DD systems

In optical communication, transmitted data bits are generally encoded based on a single constellation pattern. However, additional information can be transmitted by employing multiple constellation patterns and encoding extra bits into the switching between them. Multiple constellation patterns can be generated by simply phase rotating the original pattern as demonstrated in coherent systems [18]. However, in PAM systems the phase information is not available. Therefore, we propose to partition a larger constellation into multiple subsets for the same purpose. For example, we can generate two PAM4 patterns from PAM8 through Ungerboeck set partitioning, or two 2D-PAM4 patterns from 2 consecutive PAM4 symbols, as illustrated in Fig. 1(a). To implement the CS in the transmitter-side, the bit stream is first split into different block bit streams, in which the last bit is the parity-bit and it determines the mapping constellation pattern, as shown in Fig. 1(b). With the help of the red marked parity-bit, each consecutive P bits within the block are used to generate the corresponding (P + 1) bits by parity checking. P is the number of bits per symbol. As for the generation of CS 2D-PAM4 and CS PAM4 signals, P are 3 and 2, respectively. Finally, symbol mapping is applied according to the constellation table in Fig. 1(a). As we can see, for each block containing (P·L + 1) input bits, L consecutive symbols from the same constellation are generated. The SE of the CS PAM signals (considering two switching constellation patterns) is given as follows:

SECSPAM=(LP+1)/L=P+1/L
Specifically, the SE of CS 2D-PAM and CS PAM4 signals is (1 + 1/(P·L)) that of 2D-PAM4 and PAM4 signals, respectively.

 figure: Fig. 1

Fig. 1 (a) Constellations of CS PAM4 and CS 2D-PAM4; (b) Generation of the CS PAM4 and CS 2D-PAM4 signals; (c) Demodulation of the CS PAM4 and CS 2D-PAM4 signals; (d) Illustration of constellation pattern identification.

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At the receiver-side, to demodulate the CS signals, the received symbols after equalization are first divided into multiple blocks each containing L symbols, and then applied with constellation pattern identification (CPI) for de-mapping. Afterwards, data bits are decoded from each symbol, and the parity-bit obtained from the CPI is inserted at the end of each block, as shown in Fig. 1(c). As we can see, (P·L + 1) bits are obtained for each block in this case. The operation of the CPI is described as follows. First, hard decision is applied to each block with different constellation patterns, and then the symbol error for each constellation set is evaluated. Specifically, the symbol error is defined as the sum of squared error between hard-decided symbols and original symbols over the symbol block. Finally, the constellation pattern with a smaller symbol error is chosen for the following symbol de-mapping, as shown in Fig. 1(d). The additional complexity to implement CS is reasonably low, which mainly comes from conducting the CPI. Take the CS PAM4 signal as an example, implementing CPI only requires 2L Decisions, (4L-2) Real Adders, 2L Real Multipliers and 1 Comparator for each block. Note that the hard decisions obtained during CPI can also be re-used for the final symbol-to-bit de-mapping.

Generally, the BER of the CS signal depends on the performance of CPI, which is determined by signal to noise ratio (SNR) and the block length L. In Fig. 2, we use Monte Carlo simulations to evaluate the relationship between the probability of false CPI and SNR per symbol under different block length L. Note that the probability of false CPI is equal to the probability of misidentifying the parity-bit. The number of blocks is set to be 219. As we can see, the probability of false CPI for both the CS PAM4 signal and CS 2D-PAM4 signal decreases dramatically as the block length L increases. However, as mentioned earlier, with a larger L, less information bits can be conveyed through the CS. When L further increases, the probability of successful CPI becomes closer to 100%, but both the sensitivity and SE of the CS PAM4/2D-PAM4 signals will converge to PAM4/2D-PAM4 signals, respectively. In practice, we need to optimize L to make a trade-off between the CPI accuracy and SE improvement. Another observation from Fig. 2 is that a larger block length is required for the CS PAM4 signal compared with the CS 2D-PAM4 signal. Given a fixed SNR, the probability of false CPI for the CS 2D-PAM4 signal decreases more rapidly when increasing L, compared with the CS PAM4 signal. Then, we evaluated the BER as a function of SNR per bit, a metric that accounts for different number of bits per symbol of the studied signals, as shown in Fig. 3. As we can see, the CS 2D-PAM4 scheme (L = 4 or L = 6) outperforms the conventional 2D-PAM4 scheme under various SNR per bit conditions. As for the CS PAM4 scheme (L = 6 or L = 8), better BER performance is obtained when SNR per bit is larger than 14 dB compared with the PAM4 scheme.

 figure: Fig. 2

Fig. 2 Probability of false pattern identification versus SNR per symbol under different block length L for (a) the CS 2D-PAM4 signal and (b) the CS PAM4 signal.

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 figure: Fig. 3

Fig. 3 The relationship between BER and SNR per bit for different signals.

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3. Experiment results and discussion

The experimental setup is depicted in Fig. 4. In the transmitter-side DSP, a random symbol sequence is first generated by bit-to-symbol mapping. Four formats including PAM4, CS PAM4, 2D-PAM4 and CS 2D-PAM4 are investigated and compared. The symbol sequence is up-sampled to 2 samples per symbol (sps), and then passed through a 128-tap root-raised cosine (RRC) finite impulse response (FIR) filter with a roll-off factor of 0.1 for pulse shaping. Afterwards, the RRC signal is resampled to match the DAC sampling rate, followed by nonlinear compensation (NLC) to handle arcsine function and clipping, and pre-emphasis to combat the limited bandwidth of the transmitter. Finally, the signal is sent to an 8-bit DAC operating at 70 GSa/s. The 3-dB bandwidth of the DAC is about 14 GHz. The DAC output signal is first amplified using a 50 GHz RF amplifier and then fed into a 28 GHz single-drive Mach-Zehnder Modulator (MZM). The MZM has a 5.5 dB insertion loss and biased at the quadrature point. A C-band laser operating at 1550 nm is employed and its output optical power is 15.5 dBm. The power of the modulated optical signal is 8.3 dBm. After transmission over several kilometers SSMF, a variable optical attenuator (VOA) is used to control the receiving optical signal power. At the receiver-side, a 50 GHz photodetector (PD), which does not have an inline transimpedance amplifier (TIA), is used for optical-to-electrical conversion. The received electrical signal is sampled at 160 GSa/s by a 63 GHz real-time oscilloscope (RTO). As for the receiver-side DSP, the digital waveform is first re-sampled to 2 sps, and then passed through a matched RRC FIR filter. As for the synchronization, training symbols are used [20]. In order to compensate for system impairments, instead of a linear feed-forward equalizer (FFE), a Volterra nonlinear equalizer (VNLE) is adopted [21,22]. The time-domain discrete formulation for the Volterra series expansion up to the third order is expressed as

y(n)=l1=0N11h1(l1)x(nl1)+l1=0N21l2=0l1h2(l1,l2)m=12x(nlm)+l1=0N31l2=0l1l3=0l2h3(l1,l2,l3)m=13x(nlm)+e(n)
where x(n)andy(n)is the n-th symbol of the input and output signals of the VNLE, respectively. hi() denotes the i-th order Volterra kernel and Ni is the corresponding memory length. e(n)is the Volterra error which represents other higher order Volterra kernels. The number of taps of the three kernel orders are N1, N2(N2+1)/2andN3(N3+1)(N3+2)/6, respectively. Training symbols are used to estimate the Volterra kernels using a recursive least square algorithm. The memory length of the first, second and third order kernels is set toN1=101,N2=5andN3=3, respectively. It is well known that not all Volterra kernels have equal importance, and there exist some insignificant kernels. To reduce the complexity of the VNLE, all kernels are first identified and those insignificant kernels are then removed [20]. After the symbol-to-bit de-mapping, the BER is counted. For the CS signal, the CPI is applied before the de-mapping.

 figure: Fig. 4

Fig. 4 Experimental setup.

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First, in Fig. 5 we investigate the impact of the block length L on the performance of the 56 GBaud CS 2D-PAM4 and CS PAM4 signals in a B2B measurement. The received optical power for the CS 2D-PAM4 and CS PAM4 signals is 0 and 4 dBm, respectively. The BERs of the 56 GBaud 2D-PAM4 signals with 0 dBm received optical power and 56 GBaud PAM4 signals with 4 dBm received optical power are also given (dashed lines) for reference. Meanwhile, the performance of both the FFE and VNLE is evaluated. As expected, we observe that the use of the VNLE can enhance the system performance. Specifically, in Fig. 5 the VNLE reduces the BER by a factor of 1.4 from 1.1 × 10−3 to 8 × 10−4 for the 2D-PAM4 signal and by a factor of 1.6 from 6.6 × 10−3 to 4.1 × 10−3 for the PAM4 signal. The VNLE is applied in the following investigations for better performance. As per Fig. 5, the BER of the CS signals decreases as the block length L increases. When the block length is sufficiently large, the BER difference between the CS signal and conventional signal becomes rather small, since the accuracy of CPI is very high in this case. However, a larger block length L means a smaller SE improvement relative to the conventional signal. Considering the trade-off between the BER performance and SE improvement, we choose L = 4 and L = 6 for the generation of the CS 2D-PAM4 signal and CS PAM4 signal for the following evaluations, respectively. In this case, the SE of CS PAM4 is (1 + 1/(2 × 6)) that of PAM4 and the SE of CS 2D-PAM4 is (1 + 1/(3 × 4)) that of 2D-PAM4.

 figure: Fig. 5

Fig. 5 BER versus the block length L for (a) the 56 Gbaud CS 2D-PAM4 signal and (b) 56 Gbaud CS PAM4 signal.

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With the optimized block length, the back-to-back performance is investigated. Figure 6 plots the BER as a function of received optical power for various signals based on 2D-PAM4 and PAM4, respectively. As shown in Fig. 6, compared with the conventional 56 GBaud 2D-PAM4/PAM4 signal, penalties are observed for the 56 GBaud CS signals due to the larger average BER of the parity-bit serving to identify the constellation (the ‘CPI’ bit), especially when the received power is small. However, if we consider the same system bit rate, the symbol rate of the CS signals can be reduced to 51.7 Gbaud and low BERs are achieved under various received power conditions. Specifically, in the 2D-PAM4 case in Fig. 6(a), the receiver sensitivity is improved by 1.05 dB and 1.4 dB at the HD FEC threshold (BER = 4 × 10−3) and KP4 FEC threshold (BER = 2 × 10−4), respectively [2,14]. Similarly, in the PAM4 case in Fig. 6(b), the receiver sensitivity improvement is 0.8 dB at the HD FEC threshold. Meanwhile, the 56 GBaud PAM4 signal suffers from an error floor above the KP4 FEC threshold, mainly due to the limited bandwidth of the transmitter, which contains a 14 GHz DAC. In contrast, no error floor is observed for the 51.7 GBaud CS PAM4 signal and the BER is below the KP4 FEC threshold when the received power is larger than 3 dBm. The constellations of the 56 GBaud 2D-PAM4 signal, 51.7 GBaud CS 2D-PAM4 signal (C1 pattern) and 56 GBaud CS 2D-PAM4 signal (C1 pattern) after equalization are also presented in the insets of Fig. 6(a) from top to bottom given 1 dBm received power. The noise variance is much smaller in the 51.7 GBaud CS 2D-PAM4 constellation. Similarly, the eye diagrams of the 56 GBaud PAM4 signal, 51.7 GBaud CS PAM4 signal (C1 pattern) and 56 GBaud CS PAM4 signal (C1 pattern) after equalization are also given in the insets of Fig. 6(b) from top to bottom given 4 dBm received optical power.

 figure: Fig. 6

Fig. 6 Back-to-back BER versus received power for (a) 2D-PAM4 based signals and (b) PAM4 based signals.

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Next, we investigate the BER under different received powers at given transmission distances. The results are shown in Fig. 7(a) for the 2D-PAM4 based signals after 3 km and 5 km transmission, and in Fig. 7(b) for the PAM4 based signals after 3 km transmission. As shown in both figures, because of the chromatic dispersion induced power fading and fiber nonlinearities, both signals suffer from performance penalty depending on the symbol rate and transmission distance. For example, for the 56 GBaud 2D-PAM4 signal with 1 dBm received power, the achieved BER under B2B, 3 km and 5 km transmission is 2.4 × 10−4, 1.7 × 10−3, and 5.2 × 10−2 respectively. For the 51.7 GBaud CS 2D-PAM4 signal with 1 dBm received power, the achieved BER under B2B, 3 km and 5 km transmission is 1.8 × 10−5, 6.1 × 10−5, and 2.1 × 10−2 respectively. On the other hand, a better receiver sensitivity is always achieved under all received power conditions using the 51.7 GBaud CS signal compared with the 56 GBaud PAM signal at the same bit rate. Specifically, in the 2D-PAM4 case shown in Fig. 7(a), a sensitivity improvement of 1.5 dB and 2 dB is obtained at the threshold of BER = 4 × 10−3 and BER = 2 × 10−4 after 3 km transmission, respectively. The improvement is increased to 3.5 dB at the threshold of BER = 4 × 10−3 after 5 km transmission. At the same time, as for the 56 GBaud 2D-PAM4 signal, the KP4 FEC threshold cannot be reached after 5 km transmission. On the contrary, the use of lower complexity lower latency KP4 FEC is still feasible in the 51.7 GBaud CS 2D-PAM4 system with at least 6 dBm received optical power. Similarly, 1.1 dB sensitivity improvement is obtained using the 51.7 GBaud CS PAM4 over the 56 GBaud PAM4 at the HD-FEC threshold after 3 km SSMF transmission, as shown in Fig. 7(b).

 figure: Fig. 7

Fig. 7 BER versus received power after transmission in (a) the 2D-PAM measurement and (b) the PAM4 measurement.

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In the last part of the experiments, we compare the BER performance over transmission distance for these signals. The results are plotted in Fig. 8(a) for the 2D-PAM4 transmission and Fig. 8(b) for the PAM4 transmission. The received optical power after transmission is fixed at 6 dBm for all the signals. In accordance with the results in Fig. 7, the achievable BER for the 51.7 GBaud CS signals is smaller than the 56 GBaud signals. Consequently, the achievable transmission distance can be extended using the 51.7 GBaud CS signal at a given FEC threshold. Specifically, in the 2D-PAM4 case as shown in Fig. 8(a), the achievable distance is increased from about 4.2 km to 5 km at the KP4 FEC threshold, and from about 4.9 km to 5.9 km at the HD FEC threshold. In the PAM4 case as shown in Fig. 8(b), the achievable distance is increased from about 3.9 km to 4.9 km at the HD FEC threshold. In addition, the achievable distance of about 4.1 km can be also obtained by using the CS technique considering the KP4 FEC threshold. Note that in order to further increase the transmission distance in the C-band, we can adopt optical CD pre-compensation and single-sideband (SSB) shaping techniques [23,24], which is out of the scope of this paper.

 figure: Fig. 8

Fig. 8 BER versus transmission distance in (a) the 2D-PAM measurement and (b) the PAM4 measurement.

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Finally, based on simulations we study the impact of transmitter bandwidth on the performance of the systems. At the transmitter-side, the optical power of the MZM output is 6 dBm and a 4th order Bessel low pass filter (LPF) with different 3 dB bandwidth is utilized to emulate the transmitter bandwidth. At the receiver-side, the PD responsibility, dark current and thermal noise power density are set as 0.65 A/W, 10 nA and 2 × 10−24 W/Hz, respectively. Another 4th order Bessel LPF with 25 GHz 3 dB bandwidth is applied to simulate the bandwidth limitation of the receiver. We evaluate the required received optical power to achieve the threshold of BER = 4 × 10−3 and BER = 2 × 10−4 under different transmitter bandwidth in B2B transmission. The results are summarized in Fig. 9(a) for the 2D-PAM4 case and in Fig. 9(b) for the PAM4 case. As we can see, in the 2D-PAM4 case, the 56 GBaud signals have similar performance under various transmitter bandwidth. And the use of 51.7 GBaud CS signal can ensure better performance. In the PAM4 case, when the bandwidth is large such as 24 GHz, the required optical power is a little smaller for the 51.7 GBaud CS PAM4 signal compared with the 56 GBaud PAM4 signal. However, when the bandwidth decreases to 14 GHz which relates to the transmitter bandwidth in our experiment, the 56 GBaud PAM4 signal suffers from a larger performance penalty due to the higher symbol rate. Given the HD and KP4 FEC threshold, 0.5 dB and 0.6 dB improvement is obtained using 51.6 GBaud CS PAM4, respectively.

 figure: Fig. 9

Fig. 9 Simulated relationship between required received optical power and transmitter bandwidth in the (a) 2D-PAM4 system and (b) PAM4 system.

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4. Conclusions

In this paper, we report the first experimental demonstration of constellation switching (CS) for direct detection PAM systems. Through selecting a constellation pattern from a set of constellation patterns, additional information can be encoded, which enables the use of a lower symbol rate to achieve the same system bit rate. We propose to partition a larger constellation such as PAM8 to obtain the constellation patterns such as PAM4. It is experimentally demonstrated that the utilization of the CS signals can improve the system performance compared with standard PAM signals given the same bit rate. Specifically, in the 112 Gbit/s PAM4 case, the receiver sensitivity can be improved by 0.8 dB and 1.1 dB using the CS at the HD FEC threshold of BER = 4 × 10−3 in the B2B and 3 km fiber transmission, respectively. Furthermore, the use of lower complexity lower latency KP4 FEC becomes feasible using the CS technique after 3 km transmission. Similarly, in the 84 Gbit/s 2D-PAM4 case, the receiver sensitivity can be improved by 1.05 dB and 3.5 dB using the CS at the HD FEC thresholdin the B2B and 5 km fiber transmission, respectively. In addition, the benefit of the CS is also demonstrated over a wide range of transmitter bandwidth in simulations.

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22. J. Pan and C. H. Cheng, “Nonlinear electrical compensation for the coherent optical OFDM system,” J. Lightwave Technol. 29(2), 215–221 (2011). [CrossRef]  

23. N. Kikuchi, R. Hirai, and T. Fukui, “Quasi Single-Sideband (SSB) IM/DD Nyquist PAM signaling for high spectral efficiency DWDM transmission,” in Proceedings of OFC (Anaheim, California, 2016), paper Th2A.41. [CrossRef]  

24. Q. Zhang, N. Stojanovic, C. Prodaniuc, T. J. Zuo, E. B. Zhou, L. Zhang, F. Karinou, and C. S. Xie, “Demonstration of CD pre-compensated direct detection PAM4 40km transmission in C-band using DDMZM,” in Proceedings of ACP (Wuhan, China, 2016), paper ATh3C.2. [CrossRef]  

References

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  1. H. Mardoyan, M. A. Mestre, R. Rios-Muller, A. Konczykowska, J. Renaudier, F. Jorge, B. Duval, J. Y. Dupuy, A. Ghazisaeidi, P. Jenneve, M. Achouche, and S. Bigo, “Single carrier 168-Gb/s line-rate PAM direct detection transmission using high-speed selector Power DAC for optical interconnects,” J. Lightwave Technol. 34(7), 1593–1598 (2016).
    [Crossref]
  2. IEEE P802.3bs 400 Gb/s Ethernet Task Force, accessed on Dec. 19, 2016. [Online]. Available: http://www.ieee802.org/3/bs/ .
  3. X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
    [Crossref]
  4. N. Kikuchi and R. Hirai, “Intensity-modulated / direct-detection (IM/DD) Nyquist pulse-amplitude modulation (PAM) signaling for 100-Gbit/s/λ optical short-reach transmission,” in Proceedings of ECOC (Cannes, France, 2014), paper 4.12.
    [Crossref]
  5. M. Zhu, J. Zhang, X. Yi, Y. Song, B. Xu, X. Li, X. Du, and K. Qiu, “Hilbert superposition and modified signal-to-signal beating interference cancellation for single side-band optical NPAM-4 direct-detection system,” Opt. Express 25(11), 12622–12631 (2017).
    [Crossref] [PubMed]
  6. C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.
  7. L. Sun, J. Du, and Z. He, “Multiband three-dimensional carrierless amplitude phase modulation for short reach optical communications,” J. Lightwave Technol. 34(13), 3103–3109 (2016).
    [Crossref]
  8. M. I. Olmedo, T. Zuo, J. B. Jensen, Q. Zhong, X. Xu, S. Popov, and I. T. Monroy, “Multiband carrierless amplitude phase modulation for high capacity optical data links,” J. Lightwave Technol. 32(4), 798–804 (2014).
    [Crossref]
  9. L. Zhang, T. Zuo, Y. Mao, Q. Zhang, E. Zhou, G. N. Liu, and X. Xu, “Beyond 100-Gb/s transmission over 80- km SMF using direct-detection SSB-DMT at C-band,” J. Lightwave Technol. 34(2), 723–729 (2016).
    [Crossref]
  10. Z. Li, M. S. Erkılınç, S. Pachnicke, H. Griesser, R. Bouziane, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Signal-signal beat interference cancellation in spectrally-efficient WDM direct-detection Nyquist-pulse-shaped 16-QAM subcarrier modulation,” Opt. Express 23(18), 23694–23709 (2015).
    [Crossref] [PubMed]
  11. K. Zhong, X. Zhou, T. Gui, L. Tao, Y. Gao, W. Chen, J. Man, L. Zeng, A. P. K. Lau, and C. Lu, “Experimental study of PAM-4, CAP-16, and DMT for 100 Gb/s Short Reach Optical Transmission Systems,” Opt. Express 23(2), 1176–1189 (2015).
    [Crossref] [PubMed]
  12. Y. Gao, J. C. Cartledge, A. S. Kashi, S. Yam, and Y. Matsui, “Direct Modulation of a Laser Using 112-Gb/s 16-QAM Nyquist Subcarrier Modulation,” IEEE Photonics Technol. Lett. 29(1), 35–38 (2017).
    [Crossref]
  13. M. A. Mestre, H. Mardoyan, A. Konczykowska, R. Rios-Müller, J. Renaudier, F. Jorge, and B. Duval, J-Y. Dupuy A. Ghazisaeidi, P. Jennevé, and S. Bigo, “Direct detection transceiver at 150-Gbit/s net data rate using PAM8 for optical interconnects,” in Proceedings of ECOC (Valencia, Spain, 2015), paper 1071.
  14. J. Renaudier, R. Rios-Müller, M. A. Mestre, and H. Mardoyan, “Multi Rate IMDD Transceivers for Optical Interconnects Using Coded Modulation,” in Proceedings of OFC (Los Angeles, California, 2015), paper Tu2J.2.
  15. R. Rios-Müller, J. Renaudier, M. A. Mestre, and H. Mardoyan, “Multi-Dimension Coded PAM4 Signaling for 100Gb/s Short-Reach Transceivers,” in Proceedings of OFC (Los Angeles, California, 2015), paper Th1G.4.
  16. X. Lu, V. S. Lyubopytov, and I. T. Monroy, “24-Dimensional Modulation Formats for 100 Gbit/s IM-DD Transmission Systems Using 850 nm Single-Mode VCSEL,” in Proceedings of ECOC (Gothenburg, Sweden, 2017), paper M.1.D.1.
  17. W. Lu, D. Wang, and Y. Liu, “Transmission efficient modulation based on constellation switching in wireless communication,” in Communications and Networking in China (CHINACOM),20116th International ICST Conference on, 2011), 98–101.
    [Crossref]
  18. S. S. Dash, F. Pythoud, D. Hillerkuss, B. Baeuerle, A. Josten, P. Leuchtmann, and J. Leuthold, “Constellation modulation - an approach to increase spectral efficiency,” Opt. Express 25(14), 16310–16331 (2017).
    [Crossref] [PubMed]
  19. M. Xiang, Q. Zhuge, Z. Xing, K. Zhang, T. M. Hoang, F. Zhang, and D. V. Plant, “Performance enhanced IM/DD 112 Gb/s/λ Transmission Using Constellation Switching PAM4,” in Proceedings of OFC (San Diego, California, 2018), paper W1J.3.
    [Crossref]
  20. M. Xiang, Q. Zhuge, M. Qiu, X. Zhou, F. Zhang, M. Tang, D. Liu, S. Fu, and D. V. Plant, “Modulation format identification aided hitless flexible coherent transceiver,” Opt. Express 24(14), 15642–15655 (2016).
    [Crossref] [PubMed]
  21. F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express 25(7), 7230–7237 (2017).
    [Crossref] [PubMed]
  22. J. Pan and C. H. Cheng, “Nonlinear electrical compensation for the coherent optical OFDM system,” J. Lightwave Technol. 29(2), 215–221 (2011).
    [Crossref]
  23. N. Kikuchi, R. Hirai, and T. Fukui, “Quasi Single-Sideband (SSB) IM/DD Nyquist PAM signaling for high spectral efficiency DWDM transmission,” in Proceedings of OFC (Anaheim, California, 2016), paper Th2A.41.
    [Crossref]
  24. Q. Zhang, N. Stojanovic, C. Prodaniuc, T. J. Zuo, E. B. Zhou, L. Zhang, F. Karinou, and C. S. Xie, “Demonstration of CD pre-compensated direct detection PAM4 40km transmission in C-band using DDMZM,” in Proceedings of ACP (Wuhan, China, 2016), paper ATh3C.2.
    [Crossref]

2017 (5)

2016 (4)

2015 (2)

2014 (1)

2011 (1)

Achouche, M.

Baeuerle, B.

Bayvel, P.

Bigo, S.

Bouziane, R.

Cartledge, J. C.

Y. Gao, J. C. Cartledge, A. S. Kashi, S. Yam, and Y. Matsui, “Direct Modulation of a Laser Using 112-Gb/s 16-QAM Nyquist Subcarrier Modulation,” IEEE Photonics Technol. Lett. 29(1), 35–38 (2017).
[Crossref]

Chen, W.

Cheng, C. H.

Dash, S. S.

Deng, L.

Dippon, T.

C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.

Du, J.

Du, X.

Dupuy, J. Y.

Duval, B.

Erkilinç, M. S.

Fu, S.

Gaiarin, S.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Gao, F.

Gao, Y.

Y. Gao, J. C. Cartledge, A. S. Kashi, S. Yam, and Y. Matsui, “Direct Modulation of a Laser Using 112-Gb/s 16-QAM Nyquist Subcarrier Modulation,” IEEE Photonics Technol. Lett. 29(1), 35–38 (2017).
[Crossref]

K. Zhong, X. Zhou, T. Gui, L. Tao, Y. Gao, W. Chen, J. Man, L. Zeng, A. P. K. Lau, and C. Lu, “Experimental study of PAM-4, CAP-16, and DMT for 100 Gb/s Short Reach Optical Transmission Systems,” Opt. Express 23(2), 1176–1189 (2015).
[Crossref] [PubMed]

Ghazisaeidi, A.

Griesser, H.

Gui, T.

He, Z.

Hillerkuss, D.

Iglesias, M.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Jacobsen, G.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Jenneve, P.

Jensen, J. B.

Jorge, F.

Josten, A.

Kakkar, A.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Karinou, F.

C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.

Kashi, A. S.

Y. Gao, J. C. Cartledge, A. S. Kashi, S. Yam, and Y. Matsui, “Direct Modulation of a Laser Using 112-Gb/s 16-QAM Nyquist Subcarrier Modulation,” IEEE Photonics Technol. Lett. 29(1), 35–38 (2017).
[Crossref]

Killey, R. I.

Konczykowska, A.

Lau, A. P. K.

Leuchtmann, P.

Leuthold, J.

Li, X.

Li, Z.

Liu, D.

Liu, G. N.

Llorente, R.

C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.

Lu, C.

Man, J.

Mao, Y.

Mardoyan, H.

Matsui, Y.

Y. Gao, J. C. Cartledge, A. S. Kashi, S. Yam, and Y. Matsui, “Direct Modulation of a Laser Using 112-Gb/s 16-QAM Nyquist Subcarrier Modulation,” IEEE Photonics Technol. Lett. 29(1), 35–38 (2017).
[Crossref]

Mestre, M. A.

Monroy, I. T.

Olmedo, M. I.

Ozolins, O.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Pachnicke, S.

Pan, J.

Pang, X.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Plant, D. V.

Popov, S.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

M. I. Olmedo, T. Zuo, J. B. Jensen, Q. Zhong, X. Xu, S. Popov, and I. T. Monroy, “Multiband carrierless amplitude phase modulation for high capacity optical data links,” J. Lightwave Technol. 32(4), 798–804 (2014).
[Crossref]

Prodaniuc, C.

C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.

Pythoud, F.

Qiang, Z.

C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.

Qiu, K.

Qiu, M.

Renaudier, J.

Rios-Muller, R.

Rodrigo Navarro, J.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Schatz, R.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Song, Y.

Stojanovic, N.

C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.

Sun, L.

Tang, M.

Tao, L.

Thomsen, B. C.

Udalcovs, A.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Westergren, U.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Xiang, M.

Xu, B.

Xu, X.

Yam, S.

Y. Gao, J. C. Cartledge, A. S. Kashi, S. Yam, and Y. Matsui, “Direct Modulation of a Laser Using 112-Gb/s 16-QAM Nyquist Subcarrier Modulation,” IEEE Photonics Technol. Lett. 29(1), 35–38 (2017).
[Crossref]

Yang, Q.

Yi, X.

Zeng, L.

Zhang, F.

Zhang, J.

Zhang, L.

Zhang, Q.

Zhong, K.

Zhong, Q.

Zhou, E.

Zhou, S.

Zhou, X.

Zhu, M.

Zhuge, Q.

Zibar, D.

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Zuo, T.

IEEE Photonics Technol. Lett. (2)

X. Pang, O. Ozolins, S. Gaiarin, A. Kakkar, J. Rodrigo Navarro, M. Iglesias, R. Schatz, A. Udalcovs, U. Westergren, D. Zibar, S. Popov, and G. Jacobsen, “Experimental Study of 1.55-um EML-Based Optical IM/DD PAM-4/8 Short Reach Systems,” IEEE Photonics Technol. Lett. 29(6), 523–526 (2017).
[Crossref]

Y. Gao, J. C. Cartledge, A. S. Kashi, S. Yam, and Y. Matsui, “Direct Modulation of a Laser Using 112-Gb/s 16-QAM Nyquist Subcarrier Modulation,” IEEE Photonics Technol. Lett. 29(1), 35–38 (2017).
[Crossref]

J. Lightwave Technol. (5)

Opt. Express (6)

M. Xiang, Q. Zhuge, M. Qiu, X. Zhou, F. Zhang, M. Tang, D. Liu, S. Fu, and D. V. Plant, “Modulation format identification aided hitless flexible coherent transceiver,” Opt. Express 24(14), 15642–15655 (2016).
[Crossref] [PubMed]

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express 25(7), 7230–7237 (2017).
[Crossref] [PubMed]

Z. Li, M. S. Erkılınç, S. Pachnicke, H. Griesser, R. Bouziane, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Signal-signal beat interference cancellation in spectrally-efficient WDM direct-detection Nyquist-pulse-shaped 16-QAM subcarrier modulation,” Opt. Express 23(18), 23694–23709 (2015).
[Crossref] [PubMed]

K. Zhong, X. Zhou, T. Gui, L. Tao, Y. Gao, W. Chen, J. Man, L. Zeng, A. P. K. Lau, and C. Lu, “Experimental study of PAM-4, CAP-16, and DMT for 100 Gb/s Short Reach Optical Transmission Systems,” Opt. Express 23(2), 1176–1189 (2015).
[Crossref] [PubMed]

M. Zhu, J. Zhang, X. Yi, Y. Song, B. Xu, X. Li, X. Du, and K. Qiu, “Hilbert superposition and modified signal-to-signal beating interference cancellation for single side-band optical NPAM-4 direct-detection system,” Opt. Express 25(11), 12622–12631 (2017).
[Crossref] [PubMed]

S. S. Dash, F. Pythoud, D. Hillerkuss, B. Baeuerle, A. Josten, P. Leuchtmann, and J. Leuthold, “Constellation modulation - an approach to increase spectral efficiency,” Opt. Express 25(14), 16310–16331 (2017).
[Crossref] [PubMed]

Other (11)

M. Xiang, Q. Zhuge, Z. Xing, K. Zhang, T. M. Hoang, F. Zhang, and D. V. Plant, “Performance enhanced IM/DD 112 Gb/s/λ Transmission Using Constellation Switching PAM4,” in Proceedings of OFC (San Diego, California, 2018), paper W1J.3.
[Crossref]

C. Prodaniuc, N. Stojanovic, F. Karinou, Z. Qiang, T. Dippon, and R. Llorente, “PAM-n Solutions for Low-Cost Implementations of 100 Gbps/Lambda Transmissions,” in Proceedings of ECOC (Dusseldorf, Germany, 2016), 842–844.

M. A. Mestre, H. Mardoyan, A. Konczykowska, R. Rios-Müller, J. Renaudier, F. Jorge, and B. Duval, J-Y. Dupuy A. Ghazisaeidi, P. Jennevé, and S. Bigo, “Direct detection transceiver at 150-Gbit/s net data rate using PAM8 for optical interconnects,” in Proceedings of ECOC (Valencia, Spain, 2015), paper 1071.

J. Renaudier, R. Rios-Müller, M. A. Mestre, and H. Mardoyan, “Multi Rate IMDD Transceivers for Optical Interconnects Using Coded Modulation,” in Proceedings of OFC (Los Angeles, California, 2015), paper Tu2J.2.

R. Rios-Müller, J. Renaudier, M. A. Mestre, and H. Mardoyan, “Multi-Dimension Coded PAM4 Signaling for 100Gb/s Short-Reach Transceivers,” in Proceedings of OFC (Los Angeles, California, 2015), paper Th1G.4.

X. Lu, V. S. Lyubopytov, and I. T. Monroy, “24-Dimensional Modulation Formats for 100 Gbit/s IM-DD Transmission Systems Using 850 nm Single-Mode VCSEL,” in Proceedings of ECOC (Gothenburg, Sweden, 2017), paper M.1.D.1.

W. Lu, D. Wang, and Y. Liu, “Transmission efficient modulation based on constellation switching in wireless communication,” in Communications and Networking in China (CHINACOM),20116th International ICST Conference on, 2011), 98–101.
[Crossref]

IEEE P802.3bs 400 Gb/s Ethernet Task Force, accessed on Dec. 19, 2016. [Online]. Available: http://www.ieee802.org/3/bs/ .

N. Kikuchi and R. Hirai, “Intensity-modulated / direct-detection (IM/DD) Nyquist pulse-amplitude modulation (PAM) signaling for 100-Gbit/s/λ optical short-reach transmission,” in Proceedings of ECOC (Cannes, France, 2014), paper 4.12.
[Crossref]

N. Kikuchi, R. Hirai, and T. Fukui, “Quasi Single-Sideband (SSB) IM/DD Nyquist PAM signaling for high spectral efficiency DWDM transmission,” in Proceedings of OFC (Anaheim, California, 2016), paper Th2A.41.
[Crossref]

Q. Zhang, N. Stojanovic, C. Prodaniuc, T. J. Zuo, E. B. Zhou, L. Zhang, F. Karinou, and C. S. Xie, “Demonstration of CD pre-compensated direct detection PAM4 40km transmission in C-band using DDMZM,” in Proceedings of ACP (Wuhan, China, 2016), paper ATh3C.2.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 (a) Constellations of CS PAM4 and CS 2D-PAM4; (b) Generation of the CS PAM4 and CS 2D-PAM4 signals; (c) Demodulation of the CS PAM4 and CS 2D-PAM4 signals; (d) Illustration of constellation pattern identification.
Fig. 2
Fig. 2 Probability of false pattern identification versus SNR per symbol under different block length L for (a) the CS 2D-PAM4 signal and (b) the CS PAM4 signal.
Fig. 3
Fig. 3 The relationship between BER and SNR per bit for different signals.
Fig. 4
Fig. 4 Experimental setup.
Fig. 5
Fig. 5 BER versus the block length L for (a) the 56 Gbaud CS 2D-PAM4 signal and (b) 56 Gbaud CS PAM4 signal.
Fig. 6
Fig. 6 Back-to-back BER versus received power for (a) 2D-PAM4 based signals and (b) PAM4 based signals.
Fig. 7
Fig. 7 BER versus received power after transmission in (a) the 2D-PAM measurement and (b) the PAM4 measurement.
Fig. 8
Fig. 8 BER versus transmission distance in (a) the 2D-PAM measurement and (b) the PAM4 measurement.
Fig. 9
Fig. 9 Simulated relationship between required received optical power and transmitter bandwidth in the (a) 2D-PAM4 system and (b) PAM4 system.

Equations (2)

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S E CSPAM =(LP+1)/L=P+1/L
y( n )= l 1 =0 N 1 1 h 1 ( l 1 )x( n l 1 ) + l 1 =0 N 2 1 l 2 =0 l 1 h 2 ( l 1 , l 2 ) m=1 2 x( n l m ) + l 1 =0 N 3 1 l 2 =0 l 1 l 3 =0 l 2 h 3 ( l 1 , l 2 , l 3 ) m=1 3 x( n l m ) +e( n )

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