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In this paper, we propose a real-valued interleaved single-carrier frequency-division multiplexing (I-SC-FDM) scheme for intensity-modulation and direct-detection optical interconnects. By simplifying the encoding structure, the computational complexity can be reduced from Nlog2N complex multiplications to N complex multiplications. At the complementary cumulative distribution function of 10−2, a reduction of 10 dB and 7.5 dB for the peak-to-average power ratio (PAPR) of the I-SC-FDM is achieved than that of orthogonal frequency-division multiplexing modulated with QPSK and 16QAM, respectively, when the subcarrier number is set to 4096. We experimentally demonstrate the I-SC-FDM scheme for optical interconnects with data rates of 12 Gbit/s, 24 Gbit/s and 128 Gbit/s transmitted over 22.5-km, 22.5-km and 2.4-km standard single mode fiber, respectively. The I-SC-FDM scheme shows great potential for cost-sensitive and power-sensitive optical interconnects owing to its low computational complexity and low PAPR.
© 2017 Optical Society of America
1. Introduction
Orthogonal frequency-division multiplexing (OFDM) has been widely researched in optical communications due to its high spectral efficiency, resistance to chromatic dispersion, simplicity of one-tap equalization, and flexibility in modulation format [1–4]. Recently, the real-valued OFDM signal (i.e. discrete multi-tone, DMT) exhibits a great potential in intensity-modulation and direct-detection (IM/DD) optical systems, holding considerable promise for the optical interconnects such as passive optical networks, data centers and system-on-chip interconnects [5–8]. However, OFDM signal shows relatively high peak-to-average power ratio (PAPR) and high computational complexity, which hinder its practical application in the cost-sensitive and power-sensitive optical interconnects [9].
Evolving from OFDM, single-carrier frequency-division multiplexing (SC-FDM) is a key technology for the uplink of long term evolution (LTE) networks, which is also referred to the discrete Fourier transform (DFT)-spread OFDM [10–12]. At the transmitter side, an additional DFT is performed prior to inverse DFT (IDFT) to implement the DFT-spread operation. SC-FDM thus has a smaller value of PAPR compared to that of OFDM as it no longer obeys the Gaussian distribution. Benefiting from this, SC-FDM features a higher tolerance to the nonlinear distortion from the devices. Moreover, SC-FDM is more immune to the high-frequency distortion caused by the limited bandwidth because the distortion can be averaged out over the entire data. In general, SC-FDM can fall into two categories: localized SC-FDM (L-SC-FDM) and the distributed SC-FDM (D-SC-FDM) [11]. In the L-SC-FDM, a set of adjacent subcarriers is used to transmit data and in the D-SC-FDM, the occupied subcarriers are equidistant from each other so that the D-SC-FDM is also referred to interleaved SC-FDM (I-SC-FDM).
For the bandwidth-limited IM/DD optical systems, L-SC-FDM is outstanding compared to conventional OFDM because of its tolerance to high-frequency distortion as discussed above. [13–15]. To achieve high bit rate and spectral efficiency, the subcarrier number needs to be set to 8192 for 140-Gbit/s/λ 128QAM-modulated L-SC-FDM signal [14] and 32-Gbit/s/λ 2048QAM-modulated L-SC-FDM signal [15]. However, such a large subcarrier number results in high PAPR and computational complexity. Compared to L-SC-FDM, I-SC-FDM is able to further reduce PAPR and also simplify the multiplexing structure [11, 16]. Nevertheless, the existing I-SC-FDM signal is complex-valued, which cannot be directly applied to the IM/DD optical systems. For L-SC-FDM, a real-valued signal can be generated by implementing Hermitian symmetry. The outputs of the DFT-spread operation are transmitted on the positive-frequency subcarriers while their Hermitian conjugates are transmitted on the negative-frequency subcarriers [13–15]. In an I-SC-FDM system, however, the outputs of DFT-spread operation should be spread over the entire subcarriers, which constrains the application of Hermitian symmetry for generating the real-valued signal.
In this paper, we propose a simplified encoding structure to generate the first real-valued I-SC-FDM signal for IM/DD optical interconnects. By employing the proposed encoding structure, the computational complexity can be reduced from Nlog2N complex multiplications to N complex multiplications. At the complementary cumulative distribution function (CCDF) of 10−2, the PAPR of I-SC-FDM signal is 10 dB and 7.5 dB smaller than that of OFDM modulated with QPSK and 16QAM, respectively, when the subcarrier number is set to 4096. I-SC-FDM is well-suited to the cost-sensitive and power-sensitive optical interconnects due to its low computational complexity and low PAPR. The rest of this paper is organized as follows. In Section 2, we demonstrate the principle of the real-valued I-SC-FDM scheme and propose the simplified encoding structure. In Section 3, we compare the PAPR between I-SC-FDM and OFDM signals. The CCDFs of PAPR for I-SC-FDM and OFDM signals are simulated. In Section 4, we experimentally demonstrate the I-SC-FDM scheme for optical interconnects with data rates of 12 Gbit/s, 24 Gbit/s and 128 Gbit/s transmitted over 22.5-km, 22.5-km and 2.4-km standard single mode fiber (SSMF), respectively. Finally, the paper is concluded in Section 5.
2. Principle of real-valued I-SC-FDM scheme
Figure 1 shows the block diagram of IM/DD optical I-SC-FDM system for optical interconnects, in which the orange boxes outline the conventional multiplexing structure for generating the complex-valued I-SC-FDM signal. First, the M-QAM samples are sent into N-point DFT. The output signal of N-point DFT can be given as
where n is from 0 to N − 1. Then, the samples of X are assigned to the odd subcarriers of the 2N-point IDFT, Therefore, the distribution of the occupied subcarriers is interleaved. After 2N-point IDFT, the complex-valued I-SC-FDM signal is generated, which can be expressed as where m is from 0 to 2N − 1. By substituting Eq. (2) for Y (l) in Eq. (3), the complex-valued I-SC-FDM signal can be simplified as which is anti-symmetrical (i.e., y(m) = −y(m + N), 0 ≤ m ≤ N − 1). The complex-valued I-SC-FDM sample can be considered as a repetition of the input M-QAM sample with a systematic phase rotation.
Fig. 1 The block diagram of IM/DD optical I-SC-FDM system for optical interconnects. S/P: series-to-parallel; P/S: parallel-to-series; DAC: digital-to-analog converter; ADC: analog-to-digital converter; IM: intensity modulation; DD: direct detection; C2RT: complex-to-real transform; R2CT: real-to-complex transform; CP: cyclic prefix.
As Fig. 2 shows, we design the simplified encoding structure based on the above properties. The simplified encoding structure includes three parts: the simplified multiplexing structure, cyclic prefix (CP) addition, and complex-to-real transform (C2RT). Our previous work proposed the simplified multiplexing structure to generate the complex-valued I-SC-FDM signal [16] and in this paper, the C2RT is proposed to convert the complex-valued signal into the real-valued signal depending on the anti-symmetrical property.
The simplified multiplexing structure is shown in Fig. 2. Its output sequence is equal to input sequence times the phase sequence (i.e., Em = ejπm/N, 0 ≤ m ≤ N − 1). The simplified multiplexing structure only requires N complex multiplications. In the conventional multiplexing structure, N-order DFT and 2N-order IDFT are required. For N-order DFT or IDFT, the number of complex multiplications is (Nlog2N)/2. Since half of subcarriers are empty, 2N-order IDFT requires almost the same complex multiplications as N-order IDFT. Therefore, the conventional multiplexing structure requires Nlog2N complex multiplications.
Figure 3 presents the number of complex multiplications for both conventional and simplified multiplexing structure, and it can be seen that the required number of complex multiplications for the conventional multiplexing structure is larger than that of the simplified multiplexing structure. The difference between conventional and simplified multiplexing structure rises with the increasing of N. When N is set to 2048 (i.e., the number of subcarrier is set to 4096), the conventional multiplexing structure requires 10 times more complex multiplications than the simplified multiplexing structure. Therefore, the computational complexity is significantly reduced by the simplified multiplexing structure, especially when N is set to a very large number.
Fig. 3 The number of complex multiplications for the conventional and simplified multiplexing structures.
As Fig. 2 depicts, after the simplified multiplexing structure, CP addition and complex-to-real transform (C2RT) are employed to generate real-valued I-SC-FDM signal. Due to the anti-symmetrical property, the first half of the complex-valued I-SC-FDM symbol contains all the information and the second half is redundant. Therefore, the real part of the first half symbol is transmitted on its original position and its imaginary part is transmitted on the second-half period. Using this approach, the real-valued symbol is generated without losing any information. In general, the samples from 2N − L to 2N − 1 should be employed as the CP to resist the inter-symbol interference (ISI) where L is CP length. Due to the nonuse of the second half of the symbol, we can also use the opposite number of the samples from N − L to N − 1 as the CP. As a result, the complex-valued I-SC-FDM signal can be converted into the real-valued I-SC-FDM signal by employing the proposed simplified encoding structure with N complex multiplications.
Figure 4 shows the schematic diagram for the real-valued I-SC-FDM symbol. The modulated constellation is QPSK; the subcarrier number is set to 256 (i.e., N = 128) and the CP length is set to 8. The real and imaginary part of the first half of the complex-valued I-SC-FDM symbol can be obtained from
respectively, where m is from 0 to N − 1, and real{.} denotes the extraction of real part and imag{.} denotes the extraction of the imaginary part. As illustrated by Fig. 4, the first half of the real-valued I-SC-FDM symbol is the real part and the second half is the imaginary part. The real-valued I-SC-FDM signal has sine and cosine envelope so that its PAPR should be lower than that of OFDM signal with Gaussian distribution. In the rest of content, the term “I-SC-FDM” denotes “real-valued I-SC-FDM”.3. PAPR of I-SC-FDM signal
The PAPR is defined as the ratio between the maximum peak power and the average power of the discrete signal,
where s is the discrete signal and E{.} denotes the statistical expectation.According to the Eqs. (5) and (6), we can calculate the PAPR of the I-SC-FDM signal modulated with QPSK. When x(m) is QPSK, the values of real{x(m)} and imag{x(m)} are equal to 1 or −1. Therefore, the maximum peak power and average power of the I-SC-FDM signal are equal to 0.5 and 0.25, respectively. The PAPR of the I-SC-FDM signal can be calculated from
thus the QPSK-modulated I-SC-FDM signal has a constant PAPR of 3 dB.We employ the CCDF of the PAPR to evaluate the PAPR performance of the I-SC-FDM and OFDM signals. The CCDF of the PAPR is defined as the probability that the PAPR exceeds a threshold PAPR0,
Figure 5 shows the CCDFs of PAPR for I-SC-FDM, L-SC-FDM and OFDM signals. The simulation shows the PAPR of QPSK-modulated I-SC-FDM is 3 dB, which agrees well with the theoretical analysis in Eq. (8). In L-SC-FDM and OFDM, the PAPR goes up with the increase of the subcarrier number. However, the PAPR of I-SC-FDM signal is almost constant when the subcarrier number increases. For the high-speed and high spectral efficiency optical interconnects, the subcarrier number is usually set to larger than 1024 to obtain reasonably good performance [14,15], and thus it’s obvious that the constant PAPR is an advantage of the I-SC-FDM. Moreover, the I-SC-FDM signal has the lowest PAPR among three signals. At the CCDF of 10−2, the PAPR of I-SC-FDM is 6.6 dB and 5 dB smaller than that of L-SC-FDM and 10 dB and 7.5 dB smaller than that of OFDM for the QPSK and 16QAM, respectively, when the subcarrier number is set to 4096.
4. Experimental setup and results
In order to investigate the performance of I-SC-FDM for optical interconnects, we experimentally demonstrated 12 Gbit/s, 24 Gbit/s, and 128 Gbit/s I-SC-FDM systems. The experiments were built based on the block diagram as depicted in Fig. 6. The 12 Gbit/s and 24 Gbit/s I-SC-FDM systems were demonstrated for access network scenarios, which employed the cost-effective devices. However, for 128 Gbit/s I-SC-FDM system, higher-bandwidth devices were used. The detailed information about the devices will be given in the following sections.
Fig. 6 The block diagram for the experimental setup of I-SC-FDM system. AWG: arbitrary waveform generator; ECL: external cavity laser; EA: electrical amplifier; MZM: Mach-Zehnder modulator; VOA: variable optical attenuator; DC: direct current; PD: photodiode.
4.1. Experimental setup of 12 Gbit/s and 24 Gbit/s I-SC-FDM systems
At the transmitter, the digital I-SC-FDM signal was generated by the simplified encoding structure. The subcarrier number was set to 256. Eight samples were employed as CP. One frame included 128 I-SC-FDM symbols and 6 training symbols. The digital signal was uploaded into 12 GS/s arbitrary waveform generator (AWG, Tektronix AWG7122C) to implement digital-to-analog conversion. The resolution of the AWG was set to 8 bit and the 3-dB bandwidth of AWG is no more than 5 GHz. When QPSK was modulated, the overall link rate was approximately 12 Gbit/s and net bit rate after excluding the 7% forward error correction (FEC) overheads was approximately 10.1 Gbit/s (12 × 2 × 128/272 × 128/134 × 1/(1 + 7%) ≈ 10.1 Gbit/s). When 16QAM is modulated, the overall link rate was approximately 24 Gbit/s and net bit rate after excluding the 7% FEC overheads was approximately 20.2 Gbit/s (12 × 4 × 128/272 × 128/134 × 1/(1+7%) ≈ 20.2 Gbit/s). An external cavity laser (ECL) was used to generate the optical carrier at 1549.7 nm with less than 100 kHz linewidth. A Mach-Zehnder modulator (MZM) with 3-dB bandwidth of 10 GHz was employed to modulate the optical carrier with the generated electrical signal. The peak-to-peak voltage (Vpp) of the electrical signal was within the linear region of the MZM for reducing the nonlinear clipping distortion. The generated optical signal with power of 3 dBm was fed into 22.5-km SSMF. The total loss of 22.5-km SSMF was approximately 4.5 dB.
At the receiver, an variable optical attenuator (VOA) was used to adjust the received optical power (ROP). The received optical signal was converted into an electrical signal by a photo-diode (PD, Agilent 11982A) with a low noise preamplifier and 3-dB bandwidth of 15 GHz. The electrical signal was fed into a 50 GS/s digital phosphor oscilloscope (DPO, Tektronix DPO72004C) to realize analog-to-digital conversion. The 3-dB bandwidth of DPO is about 20 GHz. The generated digital signal was decoded by off-line processing, including resampling, time synchronization, real-to-complex transform (R2CT), CP removal, 256-point DFT, one-tap frequency-domain equalization, 128-point IDFT, and M-QAM demapping.
4.2. Experimental results of 12 Gbit/s and 24 Gbit/s I-SC-FDM systems
Figure 7 depicts the BER versus ROP for 12 Gbit/s QPSK-modulated and 24 Gbit/s 16QAM-modulated I-SC-FDM systems after back-to-back (BTB) and 22.5-km SSMF transmission. After BTB transmission, the required ROP at the 7% FEC limit (i.e., BER = 3.8 × 10−3) was measured to be approximately −19.9 dBm and −12.7 dBm for the QPSK and 16QAM-modulated I-SC-FDM signals, respectively. After 22.5-km SSMF transmission, the required ROP at the 7% FEC limit was measured to be approximately −18.8 dBm and −12 dBm for the QPSK and 16QAM-modulated I-SC-FDM signals, respectively. The optical power budget was 21.8 dB and 15 dB for the 12-Gbit/s and 24-Gbit/s I-SC-FDM systems over 22.5-km SSMF, respectively. Compared to the QPSK-modulated I-SC-FDM, the power penalty for the 16QAM-modulated I-SC-FDM was about 6.8 dB after 22.5-km SSMF transmission. The insets of Fig. 7 show the constellation diagrams of the QPSK and 16QAM at the BER lower than 10−3 after 22.5-km SSMF transmission.
Fig. 7 The BER versus ROP for 12 Gbit/s QPSK-modulated and 24 Gbit/s 16QAM-modulated I-SC-FDM signals after back-to-back (BTB) and 22.5-km SSMF transmission.
Figure 8(a) reveals the BER versus ROP for QPSK-modulated I-SC-FDM, OFDM, and clipped OFDM after 22.5-km SSMF transmission. Compared to the electrical I-SC-FDM signal, the electrical OFDM and clipped OFDM signals had the same peak value, which is in the linear region of the devices. The required ROP at the 7% FEC limit was measured to be approximately −12.5 dBm for the QPSK-modulated OFDM signal. Compared to the QPSK-modulated I-SC-FDM, the power penalty for the QPSK-modulated OFDM was approximately 6.3 dB. Under the same peak power, because of the higher PAPR, the electrical OFDM signal had lower average power than the electrical I-SC-FDM signal. After optical modulation, the generated optical OFDM signal had lower OSNR than the generated optical I-SC-FDM signal. Meanwhile, I-SC-FDM is more immunizing to high-frequency distortion as discussed above; therefore, the BER performance of I-SC-FDM signal is much better than that of OFDM.
Fig. 8 (a) The BER versus ROP for QPSK-modulated I-SC-FDM, OFDM, and clipped OFDM after 22.5-km SSMF transmission; (b) The CCDF of PAPR for QPSK-modulated I-SC-FDM, OFDM, and clipped OFDM signals.
The clipping operation can be employed to reduce the PAPR of the OFDM [17], which is simple and effective. The clipping ratio (CR) is defined as the ratio between the square of clipping level and the average power of the unclipped OFDM symbol.
where A is the clipping level and y is the unclipped OFDM symbol.Figure 8(b) shows the CCDF of PAPR for QPSK-modulated I-SC-FDM, OFDM, and clipped OFDM signals. The PAPR of clipped OFDM is lower than that of the OFDM, which is decreasing with the decrement of the CR. When the CR is set to 3 dB, the PAPR of clipped OFDM is about 1.5 dB higher than that of I-SC-FDM at the CCDF of 10−2. As Fig. 8(a) depicts, compared to unclipped OFDM, the improvements of ROP for the clipped OFDM with the CR of 10 dB and 7 dB were about 2 dB and 3 dB, respectively. However, the clipping operation induces the clipping distortion, which increases with the decrease of the CR. When the CR is smaller than 7 dB, the clipping distortion seriously deteriorated the BER performance. Therefore, the BER performance of clipped OFDM was no longer improved when CR is smaller than 7 dB. When QPSK was modulated, I-SC-FDM had at least 3-dB improvement of the ROP compared to clipped OFDM. With the increase of the modulated constellation size, the influence of the clipping distortion on the clipped OFDM signal will be more serious. I-SC-FDM is thus advantageous that the PAPR reduction does not induce the distortion.
4.3. Experimental setup of 128 Gbit/s I-SC-FDM system
For the 128 Gbit/s I-SC-FDM system, the higher-bandwidth devices were required. The digital I-SC-FDM signal was also generated by the simplified encoding structure as shown in Fig. 2. In high-speed systems, the BER performance improves with the increase of the subcarrier number but the improvement is less effective when the subcarrier number is larger than 4096 [14, 15]. Therefore, the subcarrier number of I-SC-FDM was set to 4096. Sixteen samples were employed as CP. In one frame, 10 I-SC-FDM symbols and 1 training symbol were transmitted. The modulated constellation was 16QAM. The generated digital signal was uploaded into Fujitsu digital-to-analog converter (DAC) with 8-bit resolution, 80 GSa/s maximum sampling rate and 16 GHz 3-dB bandwidth. The linear electrical amplifier with 25-GHz 3-dB bandwidth amplified the generated electrical signal. An ECL at 1549.7 nm with less than 100 kHz linewidth was used to generate the optical carrier. The amplified electrical signal was modulated to the optical carrier by an MZM with 3-dB bandwidth of 25 GHz. When the sampling rate of DAC was set to 64 GSa/s, the link rate of I-SC-FDM was 128 Gbit/s and the net data rate after excluding the 7% FEC overheads was 105.4 Gbit/s (64 × 4 × 10/11 × 2000/4128 × 1/(1 + 7%) ≈ 105.4 Gbit/s). The generated optical signal with power of 3 dB was fed into the 2.4-km SSMF. The total loss of 2.4-km SSMF was about 0.48 dB.
At the receiver, the VOA was used to adjust the ROP. The received optical signal was converted into an electrical signal by the PD (Finisar XPDV2120R) with 3-dB bandwidth of 50 GHz. The converted electrical signal was boosted by an electrical amplifier and then fed into 120 GSa/s oscilloscope (Teledyne LeCroy LabMaster 9 Zi-A Oscilloscopes) with 3-dB bandwidth of 45 GHz. The generated digital signal was decoded by off-line processing, including resampling, time synchronization, R2CT, CP removal, 4096-point DFT, one-tap frequency-domain equalization, 2048-point IDFT, decision feedback equalizer with recursive least square adaptive (DFE-RLS) and M-QAM demapping. DFE-RLS with 7-taps feed-forward filter and 7-taps feedback filter was employed to compensate the distortions in collaboration with the frequency-domain equalization [18, 19].
4.4. Experimental results of 128 Gbit/s I-SC-FDM systems
Figure 9(a) depicts the optical spectrum of the 128 Gbit/s 16QAM-modulated I-SC-FDM signal. The optical spectrum was captured by the optical spectrum analyzer with the resolution of 0.01 nm. Figure 9(b) shows the electrical spectrum of the received 128 Gbit/s 16QAM-modulated I-SC-FDM signal. The sample rate of DAC was set to 64 GSa/s, thus the baseband bandwidth of the 128 Gbit/s I-SC-FDM signal was 32 GHz. The power of high-frequency part near 32-GHz frequency was about 20-dB lower than that of low-frequency part near zero frequency. The high-frequency distortion on the 128 Gbit/s 16QAM-modulated I-SC-FDM signal is very serious.
Fig. 9 (a) The optical spectrum of the 128 Gbit/s 16QAM-modulated I-SC-FDM signal; (b) The electrical spectrum of the received 128 Gbit/s 16QAM-modulated I-SC-FDM signal.
Figure 10 depicts the BER versus ROP for 128 Gbit/s 16QAM-modulated I-SC-FDM signals after BTB and 2.4-km SSMF transmission. The required ROP at the 7% FEC limit was measured to be approximately −2 dBm and −1 dBm for 128 Gbit/s 16QAM-modulated I-SC-FDM signal after BTB and 2.4-km SSMF transmission, respectively. Compared to the BTB transmission, the power penalty for 2.4-km SSMF transmission was approximately 1 dB. The optical power budget was 4 dB for 128 Gbit/s I-SC-FDM system over 2.4-km SSMF. The insets show the constellation diagrams of 16QAM at the BER near 10−3 after 2.4-km SSMF transmission. This experimental result demonstrates the capability of the proposed I-SC-FDM scheme to be applied in the 100 Gbit/s/λ optical interconnects.
Fig. 10 The BER versus ROP for 128 Gbit/s 16QAM-modulated I-SC-FDM signals after BTB and 2:4-km SSMF transmission.
5. Conclusion
In this paper, to the best of our knowledge, we present the first low computational complexity and low PAPR I-SC-FDM signal for IM/DD optical systems. By employing the simplified encoding structure, the computational complexity can be reduced from Nlog2N complex multiplications to N complex multiplications. Therefore, the computational complexity is significantly reduced by the simplified encoding structure, especially when N is set to a very large number. At the CCDF of 10−2, the PAPR of I-SC-FDM signal is 10 dB and 7.5 dB smaller than that of OFDM modulated with QPSK and 16QAM, respectively, when the number of subcarrier is set to 4096. Moreover, the PAPR of I-SC-FDM signal is almost constant when the number of subcarrier increases. We experimentally demonstrate the I-SC-FDM scheme for optical interconnects with data rates of 12 Gbit/s, 24 Gbit/s and 128 Gbit/s transmitted over 22.5-km, 22.5-km and 2.4-km SSMF, respectively. In the future work, we will study the pre-equalization technique for I-SC-FDM to eliminate the high-frequency distortion and compare the performance among conventional OFDM, L-SC-FDM and I-SC-FDM schemes. In conclusion, the I-SC-FDM signal is well-suited to the cost-sensitive and power-sensitive optical interconnects due to its low computational complexity and low PAPR.
Funding
National Natural Science Foundation of China (61427813, 61331010); BUPT Excellent Ph.D. Students Foundation; China Scholarship Council Foundation.
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