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Symbol-interleaved precoding technique for DDO-OFDM systems with the KK receiver

Yi Liu, Ming Chen, Xu Gao, and Yun Cheng

Open Access Open Access

Abstract

Receiver sensitivity of direct-detection optical orthogonal frequency-division multiplexing (DDO-OFDM) systems can be significantly improved by using the Kramers-Kronig (KK) receiver. However, the KK algorithm is sensitive to the high signal-to-average power ratio (PAPR) of the OFDM signal, which may violate the minimum phase condition (MPC) and lead to severe signal distortions during the optical field reconstruction. Channel-independent discrete Fourier transform (DFT) precoding technique can efficiently reduce the PAPR and realize subcarrier signal-to-noise ratio (SNR) equalization, but the PAPR of the precoded signal will increase due to chromatic dispersion. Besides, the relevant researches exhibit that it is hard to accurately reconstruct the complex field from the sampled signals based on KK relation at the low carrier-to-signal power ratio (CSPR) even if the MPC is satisfied. To combat the signal distortions induced by the KK algorithm and improve the bit error rate (BER) performance, we propose and verify DFT/Walsh-Hadamard transform (WHT)-based precoding with a simple symbol-interleaving technique for DDO-OFDM transmission systems. In addition to PAPR reduction, the proposed technique can equalize both subcarrier and inter-symbol SNRs efficiently. The simulated results show that the BER performance can be improved by up to one order of magnitude at a CSPR of 6 dB after up to 1000km single-mode fiber transmission, compared to the conventional DFT precoding technique. It is expected that the proposed technique can be employed to improve the receiver sensitivity of medium-reach DDO-OFDM transmission systems with the KK receiver.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Direct detection optical orthogonal frequency-division multiplexing (DDO-OFDM) or discrete multi-tone (DMT) has been widely considered a promising candidate for short-to-medium reach optical fiber transmission systems, due to its high spectral efficiency (SE), robustness against fiber dispersions, and low cost [1–2]. In general, a strong optical carrier is transmitted along with the information-bearing signal in a DD transmission system, and a single-ended photodiode (PD) with square-law detection is employed to recover the desired signal. However, high optical carrier power to signal power ratio (CSPR) leads to poor receiver sensitivity. One may enhance the receiver sensitivity by lowering the CSPR and eliminating the signal-to-signal beating interference (SSBI) simultaneously. Besides, power fading induced by chromatic dispersion (CD) can be easily avoided by using a single-side-band (SSB) modulation scheme [3], but a more cyclic prefix (CP) overhead is required to resist CD-induced inter-symbol interference (ISI) for long-distance transmission. Therefore, it is highly desirable to explore efficient methods which can digitally eliminate SSBI as well as compensate the CD-induced ISI. In recent years, this topic has attracted extensive attention for DDO-OFDM systems.

In [4–6], a frequency guard band (FGB) with a bandwidth equal to the OFDM signal bandwidth between the optical carrier and signal spectrum is applied to avoid SSBI. Even though this FGB method can provide superior receiver sensitivity at the optimal CSPR of 0 dB, the optical SE is reduced by half. Three SSBI cancellation receivers with two photodiodes are proposed in [7–9], which increase hardware implementation complexity. An iterative SSBI cancellation algorithm at the receiver was proposed and verified in [10]. However, it requires multiple iterations to achieve the best bit error rate (BER) performance, which results in significant digital signal processing (DSP) implementation complexity. Z. Li et al. proposed a simplified SSBI mitigation technique [11] based on pre-distortion and post-compensation. And the CD is digitally pre-compensated in the transmitter. However, electrical dispersion compensation (EDC) in the transmitter may increase the peak-to-average power ratio (PAPR) of the OFDM signal, which will enhance the quantization noise from digital-to-analog converters (DACs) and other nonlinearities. In [12], the Kramers-Kronig (KK) receiver scheme was experimentally demonstrated for DD systems with the receiver-based EDC. In this scheme, the optical field can be recovered from the intensity information converted by the photodiode when the minimum phase condition (MPC) is satisfied. Moreover, a low-complexity KK receiver without digital up-sampling was also studied for DDO-OFDM [13].

However, the KK algorithm is sensitive to the high signal-to-average power ratio (PAPR) of the OFDM signal, which may violate the minimum phase condition (MPC) and lead to severe signal distortions during the optical field reconstruction [14]. The channel-independent discrete Fourier transform (DFT) precoding (or DFT-spread) can efficiently reduce the PAPR and realize subcarrier signal-to-noise ratio (SNR) equalization, which has been widely used in DDO-OFDM for PAPR reduction and unbalanced impairments compensation [15–17]. This precoding technique has also been employed to improve the transmission performance of the DDO-OFDM systems with the KK receiver in [18–19]. Besides, the research work [20] showed that it is hard to accurately reconstruct the complex field from the sampled signals based on KK relation at the low CSPR even if the MPC is satisfied. In [21], the authors proposed an enhanced KK receiver based on a negative-going peak clipping scheme to enable low CSPRs and support low-latency KP4 forward error correction (FEC) with Reed-Solomon code, RS(544, 514). In addition, a modified KK receiver operating at the low CSPR was proposed and experimentally demonstrated [22], but the signal spectrum is broadened due to the additional exponentiation.

At the low CSPRs, some reconstructed OFDM symbols from the KK algorithm may suffer large errors even if the PAPR reduction methods are performed, resulting in degradation in overall BER performance. Inspired by the subcarrier SNR equalization of the precoding technique, we propose a simple symbol-interleaved precoding technique for DDO-OFDM systems with the KK receiver. In addition to PAPR reduction and subcarrier SNR equalization, the proposed technique can also achieve inter-symbol SNR equalization and then improve overall BER performance. The transmission performance of the DFT/Walsh-Hadamard transform (WHT)-based precoding with symbol-interleaving (SI) technique is investigated in short-to-medium reach DDO-OFDM transmission systems via numerical simulations. The rest of this paper is constructed as follows. Section 2 briefly introduces the operation principle of the proposed technique. The simulation setup is described in Section 3. Simulated results and related discussions are presented in Section 4. The last section concludes the paper.

2. Operation principle

In the OFDM transmitter, the transmitted OFDM symbols are usually organized into frames. Assume that each frame consists of N data-carrying OFDM symbols and M subcarriers (SCs) are used to carry data, the precoded symbols for the n-th OFDM symbol can be obtained with the mapped symbols Xn = [x1,n, x2,n, …, xM,n] by

$$\begin{array}{l} X_n^p = P{X_n}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = P{[{{x_{1,{\kern 1pt} n}},{x_{2,{\kern 1pt} n}},x, \cdots ,{x_{M,{\kern 1pt} n}}} ]^T}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{1}{{\sqrt M }}\left[ \begin{array}{l} {p_{1,1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{1,2}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{1,M}}\\ {p_{2,1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{2,2}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{2,M}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \\ {p_{M,1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{M,2}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{M,M}} \end{array} \right]\left[ \begin{array}{l} {x_{1,n}}\\ {x_{2,n}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \\ {x_{M,n}} \end{array} \right], n \in [{1,M} ]\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {[{x_{1,n}^p,x_{2,n}^p, \cdots ,x_{M,n}^p} ]^T} \end{array}$$
where P denotes the M×M precoding matrix. Among various precoding techniques [17], DFT precoding can provide outstanding capabilities of PAPR reduction as well as equalize SC SNR, and has been widely used in optical OFDM transmission systems [16]. Here, the element in the m-th row and the n-th column of the DFT precoding matric can be expressed as
$${p_{m,n}} = exp \left( {\frac{{ - j2\pi ({m - 1} )({n - 1} )}}{M}} \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} m,n \in [{1,M} ]$$
In contrast, the WHT precoding technique, with a Hadamard precoding matrix containing only +1 and -1 as its elements, exhibits low computational complexity. The Hadamard matrix of order M can be constructed using M/2 order Hadamard matrices in the following way
$${H_M} = \left[ \begin{array}{l} {H_{{M / 2}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {H_{{M / 2}}}\\ {H_{{M / 2}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {H_{{M / 2}}} \end{array} \right],{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {H_2} = \left[ \begin{array}{l} 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1\\ 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 1 \end{array} \right]$$
where H2 is the Hadamard matrix of order 2. In general, M should be 1, 2 and multiple of 4.

The precoded symbols in a frame can be written as ${X^p} = [X_1^p,X_2^p, \cdots ,X_N^p] = \{{x_{m,n}^p} \}$. To maintain the property of the SC SNR equalization of the precoding technique and realize the inter-symbol SNR equalization, the elements in the m-th row are circularly shifted to the right by m-1. The symbols, $\{{x_{m,n}^{SI}} \}$, after SI is defined as

$$\{{x_{m,n}^{SI}} \}= {[{X_1^{SI},X_2^{SI},X_3^{SI}, \cdots ,X_M^{SI}} ]^T}$$
$$X_m^{SI} = circshift({\{{x_{m,n}^p} \},m - 1} ),n \in [{1,N} ]$$
After channel transmission and KK reception, signal distortions may occur within some OFDM symbol durations, resulting in poor BER performance on these OFDM symbols. Once channel equalization (CEQ) is done, the corresponding symbol de-interleaving (SDI) is performed on equalized symbols $\{{x_{m,n}^{CEQ}} \}$ and its outputs are represented by $\{{x_{m,n}^{SDI}} \}= {[{X_1^{SDI},X_2^{SDI}, \cdots ,X_M^{SDI}} ]^T}$, where the SDI operation for the m-th SC is described by
$$X_m^{SDI} = circshift({\{{x_{m,n}^{CEQ}} \}, - m + 1} ),n \in [{1,N} ]$$
where -m+1 represents the elements in a row that are circularly shifted to the left by m-1. The symbol index is reordered while the SC index is retained after SDI. As shown in Fig. 1(a), we assume that the SNR of the second reconstructed OFDM symbol is seriously degraded caused by the large signal distortions, and the M SC data are dispersed over M data-carrying OFDM symbols by performing the SDI, but with SC index unchanged. Namely, the circular shifting operations for the proposed SI and SDI schemes are only performed on the data-carrying SCs with the same SC index in an OFDM frame. In addition, the degraded QAM symbols are evenly distributed over M OFDM symbols. Subsequently, both SC and inter-symbol SNR equalizations are achieved after realizing the corresponding decoding with the inverse matrix P-1
$$\hat{X}_n^{} = {P^{ - 1}}{[{{y_{1,n}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {y_{2,n}}, \cdots ,{y_{M,n}}} ]^T}$$

 figure: Fig. 1.

Fig. 1. Schematic diagram of (a) the operation principle and (b) two-dimensional SNR equalization of the proposed OFDM symbol-interleaved precoding technique.

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The two-dimensional SNR equalization of the proposed symbol-interleaved precoding technique is schematically plotted in Fig. 1(b). Note that the proposed technique may also be applied for other unbalanced impairments compensation in both time and frequency domains.

3. Simulation setup

The simulation setup, as illustrated in Fig. 2, is established with Matlab procedures to investigate the performance of the proposed technique in the short-to-medium reach DDO-OFDM transmission system with the KK receiver. A pseudo-random binary sequence (PRBS) with a length of 215-1 is generated and repeated 12 times for 16-ary quadrature amplitude modulation (16QAM) mapping. Afterward, the mapped symbols are precoded with DFT/WHT matrices and interleaved according to Eq. (3). In this work, M and N are chosen for 96 and 1000, respectively. Note that only 96 positive-frequency SCs are modulated with 16QAM symbols, direct current (DC) SC is reserved for DC bias, and other high-frequency SCs are filled with zeros for oversampling and SSB signal generation. Both cyclic prefix (CP) and cyclic suffix (CS) with a length of 16 points are added for each 256-point inverse fast Fourier transform (IFFT) output to resist ISI. Eight training symbols (TSs) are inserted in the front of the data-carrying OFDM symbols to realize both timing synchronization and channel estimation [23]. The zero-order-hold DACs with the 8-bit vertical resolution are emulated at the sampling rate of 25 GSa/s. The image spectra of the converted I/Q signals are suppressed by two sixth-order Chebyshev low-pass filters (LPFs), and then drive an optical IQ modulator (IQM) with a half-wave voltage of 4 V. The optical source is a 1552.5 nm continuous-wave laser diode (LD) with the linewidth of 100 kHz. And the CSPR of the optical OFDM signal is adjusted by changing the bias voltage for the IQM [24]. For the electrical-to-optical conversion (EOC), the optical modulation index [25] is set to 1.11 for achieving small modulation nonlinear distortions.

 figure: Fig. 2.

Fig. 2. Block diagram of simulation setup.

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The optical SSB-OFDM signals are transmitted over a fiber link with only chromatic dispersion (CD) [26] considered in our simulations. The dispersion coefficient for the CD is 17 ps/nm/km. On the receiver side, the additive white Gaussian noise (AWGN) is added after the fiber channel to emulate various optical noises and obtain different optical SNRs (OSNRs). The noised optical signals are then directly detected by a single square-law photodiode (PD) and sampled by an 8-bit analog-to-digital converter (ADC) operating at 100 GSa/s for processing with receiver procedures. The receiver DSP flow includes optical field reconstruction with the KK relation, post CD compensation (CDC), downsampling to 25 GSa/s, TSs-aided timing synchronization, CP/CS removal, FFT, TSs-based least-square channel estimation and one-tap equalization, SDI, DFT/WHT decoding and 16QAM demapping. The BER and SNR [17] are calculated for performance evaluation. Note that the KK algorithm [12] is performed with an oversampling factor of 4, and the negative-going peak clipping scheme [21] is applied to enhance the KK algorithm. The corresponding normalized clipping level is set to 0.0156 to mitigate large signal distortions. The optical spectrum after EOC and the electrical spectra after PD and KK algorithm are plotted in Fig. 2(a), 2(b), and 2(c), respectively.

4. Simulated results and discussion

In this section, we will investigate PAPR performance in electrical/optical back-to-back and fiber transmission configurations. Besides, the SNR equalization in both time and frequency domains and the BER performance for short-to-medium reach transmissions are studied to verify the feasibility of the proposed technique.

4.1 PAPR analysis

We calculate the PAPR values over 10,000 OFDM symbols with an oversampling factor of 4, and the corresponding complementary cumulative distribution function (CCDF) curves are plotted in Fig. 3. Here, the PRBS sequence mentioned above is repeated 118 times to generate the OFDM symbols for PAPR analysis. After passing through LPFs, the PAPR of the DFT precoded electrical signal can be significantly reduced by 4 dB at the CCDF of 1e-3, compared to the original one. In contrast, the proposed techniques only provide a 2-dB reduction in PAPR due to the proposed SI operation. The PAPR of the complex envelope of the optical SSB-OFDM signals after EOC with IQM and 480km fiber transmission are also analyzed at the CSPR of 6 dB. Lower PAPRs are observed for various optical signals. This fact is attributed to the optical carrier. We can see clearly that the PAPR is obviously increased by 1 dB at the CCDF of 1e-2 after fiber transmission for the conventional DFT precoded signal, compared to other cases. It indicates that the conventional DFT precoding is more sensitive to CD.

 figure: Fig. 3.

Fig. 3. PAPR performance for various electrical/optical OFDM signals.

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4.2 SNR equalization and BER performance in short-reach transmission

At the OSNR of 24 dB, the SC SNR values are estimated with the recovered 16QAM symbols of four types of OFDM signals after 80km fiber transmission, and the related results are shown in Fig. 4. For the original OFDM case, significant SNR degradation is observed for low-frequency SCs. This is mainly due to the residual SSBI caused by the MPC violation induced by the high PAPR. With the help of the DFT precoding technique, the SNRs on the low-frequency SCs are obviously improved, benefiting from its notable PAPR reduction. Besides, SC SNRs are also well equalized except for several edge SCs. This phenomenon can be also found in the bandwidth-limited system [27] due to ISI. In contrast, the proposed techniques can provide better SC SNR equalization, but with lower SNR performance in comparison to the conventional DFT precoding technique.

 figure: Fig. 4.

Fig. 4. SC SNR equalization after 80 km transmission.

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The electrical SNR values for each data-carrying OFDM symbol are also estimated and plotted in Fig. 5. Since the high PAPR of the original OFDM signal, large signal distortions frequently occur for numerous OFDM symbols after KK reconstruction. As a result, these OFDM symbols suffer severe SNR degradation. The recovered 16QAM constellation points of the original signal are spread greatly, as depicted in Fig. 5(a). Owing to the PAPR reduction of the DFT precoding, large signal distortions are obviously mitigated. However, the optical field is hard to accurately reconstruct with the KK relation at the low CSPRs [21]. Therefore, we observe that some DFT precoded symbols still suffer signal distortions and obtain degraded SNR performance as shown in Fig. 5(b). The corresponding constellation becomes more distinct but the outer constellation points. In Figs. 5(c) and 5(d), a good inter-symbol SNR equalization is achieved by using the proposed DFT/WHT precoding with SI techniques. And similar SNR equalization capacities in both time and frequency domains are observed for the two proposed techniques. Regarding the DSP implementation complexity in terms of real multiplications and additions as shown in Table 1, WHT precoded one may be a better option.

 figure: Fig. 5.

Fig. 5. Inter-symbol SNR equalization after 80 km transmission.

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Tables Icon

Table 1. Computational complexity

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We also investigate the effect of CSPR on the BER performance and average SNRs over five OFDM frames containing 5,000 data-carrying OFDM symbols, and the relevant curves are plotted in Fig. 6. The BER performance is limited in low and high CSPRs due to large signal reconstruction distortions and system noise, respectively. However, the BER performance can be significantly improved by both the conventional DFT precoding and the proposed techniques at low CSPRs under different OSNRs. This is mainly attributed to the enhanced SNR performance due to its notable PAPR reduction for the conventional DFT precoding technique, as presented in Fig. 6(d). While the inter-symbol SNR equalization is the dominant factor for the BER improvement with the proposed techniques. Therefore, the proposed technique will outperform the conventional DFT when signal reconstruction distortion is the main reason for the BER performance degradation and the electrical SNR is relatively high in short-reach transmission. Note that less than 7 dB CSPR can support low-latency KP4 FEC with the clipping-enhanced KK algorithm [21]. Therefore, we set the CSPR to 6 dB for performance comparison in the following sections.

 figure: Fig. 6.

Fig. 6. The measured BER and SNR performances versus CPSR under different OSNRs.

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4.3 Performance investigation in medium-reach transmission

After 480 km fiber transmission, the estimated SC and symbol SNRs and the corresponding recovered constellations are presented in Fig. 7. The CSPR and OSNR are 6 and 24 dB, respectively. Slightly degraded SC SNR equalization can be observed for the conventional DFT technique caused by CD. In contrast, the proposed techniques can provide a similar SC SNR equalization as the 80 km case does. It is well-known that the laser phase noise will be converted into intensity noise after fiber transmission [30]; therefore, the SNR performance is degraded compared to the short-reach case. And the capacity of inter-symbol SNR equalization is degraded obviously due to the increased PAPR induced by CD. However, the proposed techniques still exhibit superior inter-symbol SNR equalization and obtain more concentrated constellations, as shown in Figs. 7(b)–7(e).

 figure: Fig. 7.

Fig. 7. Two-dimensional SNR equalization after 480km transmission.

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To further investigate the transmission performance of the proposed technique in medium-reach transmission systems, the BER performance as a function of transmission distance (up to 1000 km) under 20, 22 and 24 dB OSNRs are measured and presented in Fig. 8. The CSPR is set to 6 dB. The BER fluctuations over transmission distance can be observed. The possible reason is the PAPR variations induced by different accumulated dispersions [14], which leads to different signal distortions after the KK algorithm. Both the conventional DFT and the proposed techniques can significantly improve the BER performance after up to 160 km of fiber transmissions. And better BER performance at the OSNR of 20 dB in these short-reach transmissions can be achieved with the conventional DFT precoding technique as depicted in Fig. 8(a). However, the BER performance improvement of the transmission system with the proposed DFT/WHT + SI schemes becomes more obvious as the OSNR increases. At high OSNRs, the proposed schemes outperform the traditional DFT scheme in both short and medium-reach transmission cases. The BER performance can be improved by up to one order of magnitude after up to 1000 km fiber transmission by employing the proposed technique. Therefore, the proposed techniques may be more suitable for the medium-reach DDO-OFDM system with the KK receiver.

 figure: Fig. 8.

Fig. 8. The measured BER performance in different transmission distances.

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5. Conclusion

In this work, we proposed a simple symbol-interleaved precoding technique for short-to-medium reach DDO-OFDM with the KK receiver. The transmission performance is verified via numerical simulations. The results exhibited that the proposed technique can reduce the PAPR and achieve both SC and inter-symbol SNR equalizations. At a low CSPR of 6 dB, the proposed technique showed better two-dimensional SNR equalization and then strong robustness against the signal reconstruction distortion induced by the KK relation. Moreover, by using the proposed technique, the BER performance can be improved by up to one order of magnitude for medium-reach (up to 1000 km) fiber transmission, compared to the conventional DFT precoded and original ones. It is expected that the proposed technique can be employed to improve the receiver sensitivity of medium-reach DDO-OFDM transmission systems with the KK receiver and enable low-latency FECs.

Funding

National Natural Science Foundation of China (61805079); Natural Science Foundation of Hunan Province (2020JJ4433); Scientific Research Foundation of Hunan Provincial Education Department (20B330, 21A0562); Construct Program of the Key Discipline in Hunan Province.

Disclosures

The authors declare no conflicts of interest

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. Schematic diagram of (a) the operation principle and (b) two-dimensional SNR equalization of the proposed OFDM symbol-interleaved precoding technique.

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Fig. 2.
Fig. 2. Block diagram of simulation setup.

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Fig. 3.
Fig. 3. PAPR performance for various electrical/optical OFDM signals.

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Fig. 4.
Fig. 4. SC SNR equalization after 80 km transmission.

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Fig. 5.
Fig. 5. Inter-symbol SNR equalization after 80 km transmission.

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Fig. 6.
Fig. 6. The measured BER and SNR performances versus CPSR under different OSNRs.

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Fig. 7.
Fig. 7. Two-dimensional SNR equalization after 480km transmission.

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Fig. 8.
Fig. 8. The measured BER performance in different transmission distances.

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Tables (1)

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Table 1. Computational complexity

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Equations (7)

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X n p = P X n = P [ x 1 , n , x 2 , n , x , ⋯ , x M , n ] T = 1 M [ p 1 , 1 p 1 , 2 ⋯ p 1 , M p 2 , 1 p 2 , 2 ⋯ p 2 , M ⋮ ⋮ ⋮ p M , 1 p M , 2 ⋯ p M , M ] [ x 1 , n x 2 , n ⋮ x M , n ] , n ∈ [ 1 , M ] = [ x 1 , n p , x 2 , n p , ⋯ , x M , n p ] T
p m , n = e x p ( − j 2 π ( m − 1 ) ( n − 1 ) M ) , m , n ∈ [ 1 , M ]
H M = [ H M / 2 H M / 2 H M / 2 − H M / 2 ] , H 2 = [ 1 1 1 − 1 ]
{ x m , n S I } = [ X 1 S I , X 2 S I , X 3 S I , ⋯ , X M S I ] T
X m S I = c i r c s h i f t ( { x m , n p } , m − 1 ) , n ∈ [ 1 , N ]
X m S D I = c i r c s h i f t ( { x m , n C E Q } , − m + 1 ) , n ∈ [ 1 , N ]
X ^ n = P − 1 [ y 1 , n , y 2 , n , ⋯ , y M , n ] T
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