Low-complexity recombined SLM scheme for PAPR reduction in IM/DD optical OFDM systems

Yating Wu; Congcong He; Qianwu Zhang; Yanzan Sun; Tao Wang

doi:10.1364/OE.26.032237

1. Introduction

Optical orthogonal frequency division multiplexing (O-OFDM) is a promising technology for future software defined elastic optical networks due to its numerous advantages such as high spectral efficiency, strong robustness against chromatic dispersion, and great flexibility in dynamic bandwidth allocation [1,2]. However, as a multi-carrier modulation technique, an inherent defect of OFDM is the high peak-to-average power ratio (PAPR) problem [3]. High PAPR requires large dynamic ranges of the devices such as DACs, ADCs and power amplifiers, and makes the system more susceptible to the nonlinear impairments in the optical fiber transmission. The induced signal distortion causes inter-modulation among the subcarriers and degradation of bit error rate (BER) performance. Therefore, PAPR reduction remains a major challenge for O-OFDM systems.

Various PAPR reduction methods, borrowed from wireless communications, have been applied to O-OFDM systems [3–10]. Among them, clipping is a simple and widely-used technique which reduces PAPR by limiting the peak envelope of the transmitted OFDM signal to a predetermined value [3,4]. However, clipping introduces in-band and out-band radiation, which degrades the performance in terms of both BER and spectral efficiency. As an alternative, an electronic pre-distortion technique of companding transform is utilized in a DD O-OFDM system in [5]. It shows better performance than clipping but with the cost of increased average power.

Compared with the above methods, selective mapping (SLM) is shown to be a distortionless and effective PAPR reduction technique [6–10]. In SLM, multiple candidate signals representing the same information are generated for each OFDM symbol, from which the one with the lowest PAPR is selected to be transmitted. Hence, SLM suffers from high signal processing complexity due to the use of multiple IFFT operations per OFDM symbol. To reduce the computational complexity, various modified SLM schemes have been investigated throughout the years [8–10]. For example, conversion matrices are proposed to replace the computation-intensive IFFT blocks in [8]. Nevertheless, it reduces the computational complexity significantly at the cost of BER degradation. Alsusa and Yang [9] proposed a low-complexity SLM where various representations are created by performing linear combinations among consecutive time-domain OFDM symbols. The disadvantage of this scheme is the incurred transmission latency and additional memory usage. Recently, a SLM technique with low complexity is proposed for PAPR reduction in DD O-OFDM systems based on the fast Hartley transform (FHT) [10]. However, as FHT replaces the FFT kernel of OFDM, hardware adaption may be needed for practical implementation.

In general, the computational complexity of SLM is linearly proportional to the number of the candidate signals, which corresponds to the number of IFFTs required to generate the alternative candidates. On the other hand, the more the candidates are available for selection, the higher the probability of finding a candidate with low PAPR value. Therefore, instead of reducing the number of candidates which affects the PAPR reduction performance, we propose a scheme where more candidate signals are generated with the same number of IFFTs. Or equivalently, the proposed scheme reduces the number of IFFT blocks while achieving PAPR reduction performance similar to that of the conventional SLM scheme. The principal idea is to divide each candidate sequence into two clusters, and the sub-signals from one cluster are added to that of the other cluster to generate new candidate signals. Simulations and a real-time end-to-end IM/DD O-OFDM system are set up to verify the performance experimentally.

The rest of this paper is organized as follows. Section 2 describes the definition of PAPR and the conventional SLM technique applied to IM/DD O-OFDM system. The proposed SLM scheme with complexity analysis is presented in Section 3. In Section 4, the simulation and experimental results based on a real-time IM/DD O-OFDM system are provided. Finally, conclusions are drawn in Section 5.

2. System model and conventional scheme

2.1 PAPR in IM/DD O-OFDM systems

In OFDM systems, data is transmitted in the form of blocks. The input data block of N complex symbols is denoted as X = [X(0), X(1), ..., X(N-1)]. For IM/DD O-OFDM systems, the output OFDM signal has to be real-valued to make it suitable for optical transmission. Hence, the input data vector to the IFFT block is required to satisfy the Hermitian symmetry. The time-domain output signal is then obtained by taking IFFT of the input vector

s (n) = \frac{1}{\sqrt{2 N}} \sum_{k = 0}^{2 N - 1} S (k) e^{j π k n / N}, n = 0, 1, 2, ..., 2 N - 1,

where 2N is the total number of subcarriers, S(k) = X(k), S(2N-k) = X *(k) for k = 1,2,...,N-1, and S(0) = X(N) = 0, (·)* denotes the conjugation operation.

When the subcarriers add in phase, a high peak appears. PAPR is defined as the ratio between the maximum peak power and the average power, which can be written as

P A P R (s) = 10 \log_{10} (\frac{\max_{0 \leq n \leq 2 N - 1} {| s (n) |}^{2}}{E [{| s |}^{2}]}) (dB),

where E[·] denotes the expected value operation.

The distribution of PAPR is generally expressed in terms of Complementary Cumulative Distribution Function (CCDF). The CCDF is the probability that the PAPR of an OFDM frame exceeds a certain threshold PAPR₀.

C C D F = 1 - P r {P A P R \leq P A P R_{0}} .

2.2 Conventional SLM (CSLM) scheme

In CSLM, the input symbol sequence is multiplied by a set of unit-norm phase vectors to generate a number of candidate sequences which are the alternative representations of the same information.

As shown in Fig. 1, U candidate sequences {X^u}U–1 u = 0are obtained by component-wise vector multiplication of the input frequency-domain sequence X and the phase sequences {P^u}U–1 u = 0, denoted as X^u = P^{u $\otimes$}X, u = 0,1,...,U-1.

Fig. 1 Block diagram of the conventional SLM scheme applied to IM/DD O-OFDM system.

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As mentioned in Section 2.1, in an IM/DD O-OFDM system, the time domain signal modulating the intensity of the optical carrier must be real-valued. This condition is ensured by imposing Hermitian symmetry (HS) on the input symbol blocks {X^u}U–1 u = 0 prior to the IFFT operation, i.e., S^u = HS{X^u} = [0, X(1), X(2),..., X(N-1), 0, X*(N-1),..., X*(1)].

After the IFFT process, the PAPR is calculated for each of the U different candidate signals. Then the candidate with the lowest PAPR is selected and transmitted.

3. Proposed recombined SLM (RSLM) scheme

3.1 RSLM

To achieve large PAPR reduction in conventional SLM scheme, we have to generate a sufficiently large number of candidate signals, which involves high computational complexity because U IFFTs have to be performed to generate U candidate signals. Therefore, it is desirable if more candidate signals can be generated without increasing the number of IFFTs. In the proposed scheme shown in Fig. 2, 2M² rather than M candidate signals are generated with M IFFT blocks. The processing steps are described in detail below.

Fig. 2 Block diagram of the proposed RSLM scheme.

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Step1: The input data block is multiplied element-wise by M fundamental phase sequences. The fundamental phase sequences {P^m}M–1 m = 0 are the same as that of CSLM, where the phase factors $p_{i}^{m} = \exp (j φ_{i})$ , $φ_{i} \in [0, 2 π]$ for i = 0,1,..., N-1. In this paper, to reduce the computational complexity, the phase rotation angle in phase sequences is constrained to either 0 or π, i.e., $φ_{i} = {0, π}$ , $p_{i}^{m} = {+ 1, - 1}$ . Such phase sequences can be generated from pseudo-random binary sequences or Walsh-Hadamard sets [12].

Step2: Each of the phase-rotated data block X^m is partitioned into two subblocks, i.e., Xm a and Xm b. As shown in Fig. 3, there are three partitioning methods: adjacent, interleaved and random partition. For example, an input data block X = [X(0), X(1), ..., X(7)] is divided into two subblocks based on random partition. Given a random partitioning vector H = [0,1,1,0,0,1,1,0], the first subblock would be X_a = H $\otimes$ = [0, X(1), X(2), 0, 0, X(5), X(6), 0], and the second X_b = $\bar{H} \otimes$ X = [X(0), 0, 0, X(3), X(4), 0, 0, X(7)], where $\bar{H}$ is the complement of H. As the elements of the random partitioning vector H are binary, they are generated by Pseudo Random Binary Sequence (PRBS) in this paper. Whereas for adjacent and interleaved partition, the partitioning vector H would be [1,1,1,1,0,0,0,0] and [1,0,1,0,1,0,1,0], respectively.

Fig. 3 Example of the three partitioning methods. (a) Adjacent. (b) Interleaved. (c) Random.

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Note that in order for the candidate sequences to be successfully recombined, the subblock partition scheme is the same for all the candidate sequences for the same input X. Meanwhile, in the case of random partition, the partition is changed with each different input data block.

Step3: The subblocks Xm a and Xm b are fed into the modified discrete multitone (DMT) modulation.

In conventional DMT with symmetry constraint in Fig. 4(a), it is a waste of resources to use only the real parts of a complex IFFT operation. In this paper, we employ an efficient DMT implementation [11] as shown in Fig. 4(b). Since both sm a and sm b are real-valued time frames resulting from Xm a and Xm b respectively, they can be computed in parallel by one single IFFT block, i.e.,

Fig. 4 Block diagram of (a) conventional DMT, (b) the efficient modified DMT implementation.

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I F F T {S_{a}^{m} + j \cdot S_{b}^{m}} = I F F T {S_{a}^{m}} + j \cdot I F F T {S_{b}^{m}} = s_{a}^{m} + j \cdot s_{b}^{m} .

Hence, instead of using two IFFTs, a two-channel input–output DMT modulator is created with a single IFFT.

Step4: The $2 M^{2}$ candidate signals are generated by

s^{u} = s_{a}^{m} \pm s_{b}^{n} (m, n = 0, 1, ..., M - 1) .

To show that s^u is a phase-rotated representation of the original data, Eq. (5) can be rewritten as

\begin{matrix} s^{u} = D M T {X_{a}^{m}} \pm D M T {X_{b}^{n}} \\ = D M T {H \otimes X^{m}} \pm D M T {\bar{H} \otimes X^{n}} \\ = D M T {(H \otimes P^{m} \pm \bar{H} \otimes P^{n}) \otimes X} \\ ≜ D M T {{\tilde{P}}^{u} \otimes X} \end{matrix}

where

{\tilde{P}}^{u} ≜ H \otimes P^{m} \pm \bar{H} \otimes P^{n}

. Since the elements of

{\tilde{P}}^{u}

all have unit magnitude,

{\tilde{P}}^{u}

can be equivalently viewed as the effective phase sequences that are used to rotate the phase of the input data sequences.

Step5: Among the generated 2M² candidate signals, the one with the lowest PAPR is selected for transmission. The index of the selected candidate is transmitted to the receiver as side information (SI).

At the receiver, the reverse operation is performed to recover the original data. After the FFT process, the index v of the chosen candidate can be extracted along with the frequency-domain symbol sequence. Then the data sequence is multiplied element-wise with the complex conjugate of the corresponding phase rotation vector ${\tilde{P}}^{v}$ to recover the original data block.

3.2 Correlation Analysis of Candidate Signals

The correlation among the candidate signals has an impact on the PAPR reduction performance [12]. In order to provide more degrees of freedom when selecting the minimum PAPR, it is desirable that the candidate signals are mutually uncorrelated. In this section, we provide the correlation analysis of candidate signals of the proposed scheme.

Assuming that ${S (k)}_{k = 0}^{2 N - 1}$ are independent and identically distributed with E[|S(k)|²] = 1, the correlation between the u-th and v-th OFDM candidate signals is

\begin{matrix} R_{u v} (τ) = E [s^{u} (n) \cdot {s^{v} (n + τ)}^{*}] \\ = \frac{1}{2 N} E [\sum_{k = 0}^{2 N - 1} {\tilde{p}}^{u} (k) S (k) e^{j π k n / N} \sum_{k = 0}^{2 N - 1} {\tilde{p}}^{v *} (k) S^{*} (k) e^{- j π k (n + τ) / N}] \\ = \frac{1}{2 N} \sum_{k = 0}^{2 N - 1} {\tilde{p}}^{u} (k) {\tilde{p}}^{v *} (k) e^{- j π k τ / N} \end{matrix}

where

s^{u} (n)

represents the n-th element of u-th candidate signal,

{\tilde{p}}^{u} (k)

represents the k-th element of u-th equivalent phase sequence, and

0 \leq τ \leq 2 N -1

. As the phase sequence is given by

{\tilde{P}}^{u} = H \otimes P^{m} \pm \bar{H} \otimes P^{n}

in Eq. (6), Eq. (7) indicates that the correlation between candidate signals is highly related to the partitioning vector H.

In Fig. 5, we examine the correlation among candidate signals with three partitioning methods. As we can see, the correlation with random partition exhibits less peaks and is relatively flat, while adjacent and interleaved partition have large peak values.

Fig. 5 Correlation property of candidate sequences with different partitioning methods (N = 32).

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The variance of correlation (VC) among candidate signals is defined as

V C = (\sum_{0 \leq v < u \leq U - 1} V a r {{| R_{u v} (τ) |}^{2}}_{τ = 0}^{N - 1}) / (\begin{matrix} U \\ 2 \end{matrix}),

where Var[·] denotes the variance operation.

It is shown that lower VC yields better PAPR reduction performance [13]. Hence we measure the VC with three partitioning methods in Table 1, where the length of phase sequences is 64 and 128, and the number of candidates is 8. From the table, we can see the VC value with random partition is the lowest among the three partitions. The reason of this phenomenon is that the interleaved and adjacent partitioning vectors have periodical structures which lead to high peaks of correlation among candidate signals, while the random partition makes the candidate signals more “different”. Hence, we employ random partition for the proposed RSLM.

Table 1. The VC of the candidate signals with different partitioning methods

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3. 3 Computational complexity analysis

In this section, the computational complexity of CSLM and the proposed RSLM are presented in terms of complex multiplication and complex addition. To generate U candidate signals when the number of subcarriers is 2N, U 2N-point IFFT operations are required in CSLM, which involves UNlog₂(2N) complex multiplications and 2UNlog₂(2N) complex additions. While in RSLM scheme, 2U² rather than U candidate signals are generated with U IFFTs. Therefore, in order to generate the same number of candidate signals in RSLM, it only needs $M = ⌈ {(U / 2)}^{1/2} ⌉$ IFFT operations. In addition, extra 2M²N complex additions are needed in RSLM to generate the additional candidate signals as in Eq. (5).

In summary, the RSLM scheme requires MNlog₂(2N) complex multiplications, 2MNlog₂(2N) + 2M²N complex additions, which is considerably lower than CSLM. The computational complexity reduction ratio (CCRR) of the proposed RSLM over CSLM scheme is defined as

C C R R = (1 - \frac{complexity of RSLM}{complexity of CSLM}) \times 100% .

Table 2 presents the number of complex multiplications and additions of the two schemes with different values of U, M and N. It is shown that the proposed scheme can reduce computational complexity significantly. For example, in the case of N = 32 and M = 3, the proposed scheme can reduce the number of complex multiplications by 81.3% and complex additions by 71.9%, while achieving a similar PAPR reduction performance to CSLM with U = 16. Moreover, the proposed scheme becomes even more computationally efficient as the number of carriers and candidate signals increases.

Table 2. Computational complexity of CSLM and proposed RSLM

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4. Simulation and Experimental Results

4.1 Simulation of PAPR

We consider a typical OFDM system with 64 subcarriers and QAM constellation. Binary random phase sequences are employed with element { $\pm$ 1} to facilitate the implementation. The PAPR performance of a system is numerically measured in terms of complementary cumulative density function (CCDF), which is obtained from 10⁵ independent OFDM symbols.

Figure 6 shows the PAPR performance of the proposed RSLM with three different partitioning methods. As low CCDF value for a given threshold shows better PAPR reduction performance, it is shown that the random partition outperforms adjacent and interleaved method. This can be attributed to the fact that the candidates signal are less correlated with random partition. Thus, the simulation result is consistent with the correlation analysis in Section 3.2.

Fig. 6 PAPR reduction performance of RSLM with three partitioning methods. (N = 32, M = 2, 16-QAM).

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Figure 7 shows the CCDF curves with CSLM, proposed RSLM, and the original OFDM scheme without PAPR reduction. The CCDFs of PAPR are obtained for CSLM scheme with U = 8, 16 and the proposed RSLM scheme with M = 2, 3, where Figs. 7(a) and 7(b) show the probabilities that the PAPR of OFDM sequences exceeds a given PRPR₀ for 16-QAM and 64-QAM, respectively. As illustrated in Fig. 7, compared with the CCDF curve of the original OFDM signals, the CSLM and RSLM schemes reduce the PAPR significantly. RSLM with M = 2, 3 has almost the same PAPR reduction performance as CSLM with U = 8, 16, respectively. The performance gap between them is less than 0.05 dB. In summary, the proposed scheme with M = 2 (3) reduces the complex multiplications and additions by 75% (81.3%) and 66.7% (71.9%) while keeping almost the same PAPR reduction performance as CSLM scheme with U = 8 (16).

Fig. 7 PAPR reduction performance of the CSLM scheme with U = 8, 16 and the proposed RSLM scheme with M = 2, 3 for (a) 16-QAM and (b) 64-QAM.

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4.2 Real-time O-OFDM experimental set up and results

To experimentally verify the proposed RSLM scheme, we set up a real-time end-to-end IM/DD O-OFDM as shown in Fig. 8(a). The key system parameters are summarized in Table 3. Real-time transmitter and receiver are realized using Virtex-6 FPGA from Xilinx with fully pipelined DSP architecture. The total number of subcarriers is 64. Among these subcarriers, 56 subcarriers carry complex data mapped from 16-QAM and 64-QAM, while the rest are used for pilot tones or set to null for guard interval. The time-domain OFDM frame structure is shown in Fig. 8(b), where each frame contains a header with 80 samples, two training sequences (TSs), and 100 data-carrying OFDM symbols. The length of the cyclic prefix (CP) corresponding to 16 samples in each OFDM symbol. The OFDM signal are then quantized to match the resolution of the employed 4 GS/s 12-bit DAC. After passing a low pass filter and an electrical amplifier, the 2.45 Vpp electrical OFDM signal directly modulates a 1330 nm DFB laser with 3 GHz modulation bandwidth. The time-domain waveform and the electrical spectrum are shown in Figs. 8(c) and 8(d). Finally, 8.75 dBm optical power is injected into 25 km single mode fiber (SSMF) without inline optical amplifier.

Fig. 8 A real-time end-to-end IM/DD O-OFDM transmission system. (a) Experimental system setup. (b) Time-domain OFDM frame structure. (c) The waveform of the transmitted OFDM signal. (d) The spectrum of the transmitted OFDM signal. (e) Transmitter. (f) Receiver. SI: side information, LPF: low pass filter, VEA: variable electrical amplifier, VOA: variable optical attenuator, DFB: distributed feedback laser, PIN: positive intrinsic negative.

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Table 3. Transceiver and system configurations

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At the receiver side, the optical signal passes through a variable optical attenuator (VOA) and is then directly detected by a 3 GHz PIN detector to convert into the electrical domain. The electrical OFDM signal is amplified by a variable electrical amplifier (VEA) to ensure that the signal occupies the entire amplitude dynamic range of a 4 GS/s 10-bit ADC. The digital signal sampled by the ADC is directed to the receiver’s FPGA to perform the DSP processes including symbol synchronization by utilizing the frame header and TSs, CP and TS removal, 64-point FFT, channel estimation and equalization, and de-mapping. The transceiver parameters under different configurations are fixed before system performance measurement.

Figure 9 shows the BER performance of CSLM (U = 8) and RSLM (M = 2). The performance of original OFDM system without PAPR reduction is also measured. The constellation diagrams of the OFDM signal with 16-QAM (at a received optical power of −12 dBm and −15dBm) or 64-QAM (at received optical power of −7 dBm and −10 dBm) are shown for reference. As we can see, when BER is below 10⁻³, the dots in the constellation diagrams can be clearly distinguished which indicates that they can be correctly demodulated.

Fig. 9 BER performances versus received optical power, and signal constellations with (a) 16QAM, (b) 64QAM for IM/DD O-OFDM transmission over 25 km SSMF.

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Both RSLM and CSLM signal have better receiver sensitivity than the original signal. For example, in Fig. 9(a), the sensitivity of received optical power at a BER of 10⁻³ is about −14.1 dBm, −15.8 dBm, and −15.7 dBm for the original, CSLM and RSLM schemes after 25-km SSMF transmission in the IM/DD O-OFDM system, respectively. Notes that the BER performances of the CSLM and proposed RSLM scheme are very close, while RSLM reduces the complexity significantly as previously shown in Table 2. Compared to the original signal, the receiver sensitivity of the RSLM scheme is increased by 1.8 dB at a BER of 10⁻⁴. It is shown that the RSLM scheme can alleviate influence of nonlinearity impairments by restraining the PAPR of the OFDM signal, make the constellations more focused than the original signal, and improve the receiver sensitivity as well.

5. Conclusions

We have proposed and experimentally demonstrated a real-time IM/DD O-OFDM system with a RSLM scheme to reduce the high PAPR of OFDM signals. The proposed RSLM scheme reduces the computational complexity significantly while keeping similar PAPR reduction performance compared with conventional SLM scheme. Experimental results have shown that in the case of 64 subcarriers with 16-QAM and 8 candidate signals, the PAPR reduction of RSLM is 4.23 dB at a CCDF of 10⁻⁴ and the received sensitivity is improved by 1.8 dB at a BER of 10⁻⁴ after transmission over 25km SSMF. Meanwhile, the RSLM scheme reduces the complex multiplications by 75% and additions by 66.7% compared with the comparable conventional SLM scheme.

Funding

National Natural Science Foundation of China (NSFC) (61601279, 61420106011, 61601277,61635006, 61671011, 61501289); Shanghai Science and Technology Development Funds (17010500400, 16511104100, 16YF1403900).

References

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	N = 32	N = 64
Adjacent partition	0.0024	0.0011
Interleaved partition	0.0039	0.0019
Random partition	0.0015	0.0006

	CSLM	RSLM	CCRR	CSLM	RSLM	CCRR
	U = 8	M = 2	-	U = 16	M = 3	-
	N = 32
Complex Multiplications	1536	384	75%	3072	576	81.3%
Complex Additions	3072	1024	66.7%	6144	1728	71.9%
	N = 64
Complex Multiplications	3584	896	75%	7168	1344	81.3%
Complex Additions	7168	2304	67.9%	14336	3840	73.2%

Parameter	Value
FFT/IFFT points	64
Data-carrying subcarriers	56
Number of candidate signal	8
Modulation format	16/64-QAM
ADC/DAC resolution	10/12-bit
ADC & DAC sampling rate	4GS/s
Cyclic prefix	16 samples
Transmitter output power	+ 8.75 dBm
DFB wavelength	1330 nm
DFB modulation bandwidth	3 GHz
PIN detector bandwidth	3 GHz

	N = 32	N = 64
Adjacent partition	0.0024	0.0011
Interleaved partition	0.0039	0.0019
Random partition	0.0015	0.0006

	CSLM	RSLM	CCRR	CSLM	RSLM	CCRR
	U = 8	M = 2	-	U = 16	M = 3	-
	N = 32
Complex Multiplications	1536	384	75%	3072	576	81.3%
Complex Additions	3072	1024	66.7%	6144	1728	71.9%
	N = 64
Complex Multiplications	3584	896	75%	7168	1344	81.3%
Complex Additions	7168	2304	67.9%	14336	3840	73.2%

Low-complexity recombined SLM scheme for PAPR reduction in IM/DD optical OFDM systems

Abstract

1. Introduction

2. System model and conventional scheme

2.1 PAPR in IM/DD O-OFDM systems

2.2 Conventional SLM (CSLM) scheme

3. Proposed recombined SLM (RSLM) scheme

3.1 RSLM

3.2 Correlation Analysis of Candidate Signals

3. 3 Computational complexity analysis

4. Simulation and Experimental Results

4.1 Simulation of PAPR

4.2 Real-time O-OFDM experimental set up and results

5. Conclusions

Funding

References

Cited By

Figures (9)

Tables (3)

Equations (9)

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